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abstract: 'Quantum spectroscopy was performed using the frequency-entangled broadband photon pairs generated by spontaneous parametric down-conversion. An absorptive sample was placed in front of the idler photon detector, and the frequency of signal photons was resolved by a diffraction grating. The absorption spectrum of the sample was measured by counting the coincidences, and the result is in agreement with the one measured by a conventional spectrophotometer with a classical light source.'
author:
- Atsushi Yabushita
- Takayoshi Kobayashi
title: Spectroscopy by frequency entangled photon pairs
---
Introduction
============
The polarization entanglement of spatially separated photon pairs, generated by spontaneous parametric down-conversion (SPDC), has been used in a variety of quantum experiments to demonstrate quantum teleportation, entanglement-based quantum cryptography, Bell-inequality violations, and others [@quantum-review].
These SPDC photon pairs are also entangled in their wave vectors. The entanglement in wave-vector space is used in various experiments, such as quantum imaging [@quantum-imaging-lithography; @quantum-imaging], photonic de Broglie wavelength measurement [@deBroglie; @deBroglie2; @deBroglie3], quantum interference [@quantum-interference2; @quantum-interference3], and quantum lithography [@quantum-lithography; @quantum-lithography2; @quantum-lithography3]. It was claimed and experimentally verified that they have higher resolution than the classical limit, which will be used to various applications.
On the other hand, in the case of quantum imaging, the information about the shape of a spatial filter is transferred by entangled photon pairs, therefore it is useful for secure two-dimensional information transfer. Compared with the case using the polarization entanglement, it can send much more information by the entanglement of the wave-vector space taking an advantage of the continuity of the entangled parameter. A protocol for quantum key distribution was proposed based on a system whose dimension is higher than 2 in Ref.[@QKD-highdimension].
In this paper, a frequency entanglement was used to measure the spectroscopic property of a sample, which can also be used for nonlocal pulse shaping [@nonlocal-pulseshaping]. The state was maximally entangled by using a continuous-wave (cw) pump [@coincident-frequencies-theory; @coincident-frequencies-theory2; @coincident-frequencies-theory3]. Focusing the pump on a SPDC crystal by an objective lens, the bandwidth of SPDC was extended. It enabled to measure the absorption spectrum of a sample in broadband. Various types of spectroscopic measurements were performed utilizing classical light source including sophisticatedly constructed extremely short pulses [@classical-spectroscopy]. However, in the following situations, spectroscopy utilizing the frequency entanglement can be a powerful way to measure the spectroscopic properties of the sample.
One of the situations is the case when it is difficult to analyze frequency of photons transmitted through an absorptive sample. For example, to measure the spectroscopic property of a sample in vacuum ultraviolet (VUV) range, a spectrometer must be settled in a vacuum chamber in case of the conventional VUV spectroscopy, and the spectrometer must be aligned and controlled under the vacuum condition. However, in case of spectroscopy using the frequency-entangled photon pair consists of a VUV photon and a longer wavelength photon (like a visible photon), the VUV photon transmitted thorough the sample is to be detected by the photodetector without capability of energy resolution. Instead the visible photon is to be resolved by a spectrometer in the atmospheric pressure which is easily handled. This is one of the useful features of this method. The method is also useful when the sample is in space and the photons transmitted through the sample are not easy to be resolved by their energy.
Another situation is the case when the spectroscopic property of a fragile sample is to be measured in infrared range. The power of the light source must be very low not to damage the sample, and yet an infrared photodetector is usually so noisy that a signal-to-noise ratio is very low in the measurement using the classical light source. However, when the frequency-entangled photon pairs are used, the coincidence counts are measured, then one of the photons of each photon pair works as a timing gate for the measurement of another photon of the pair which is to be resolved by its energy, and the signal-to-noise ratio is expected to be enhanced substantially. This is another advantage of this method.
The SPDC photon pairs are emitted conically from the point where the pump light is focused on the beta-barium borate (BBO) crystal. Using a parabolic mirror, all SPDC photon pairs in the light cones are collimated without achromatic aberration, and travel to a beam splitter without expanding their beam diameters keeping their polarization entanglement. It is not aimed to use the polarization in this experiment, so signal and idler photons are separated from each other by a polarizing beam splitter which destroys the polarization entanglement. When a nonpolarizing beam splitter is used in place of the polarizing beam splitter, the photon pairs detected at the photon counters have polarization entanglement, which can be used for quantum cryptography [@quantum-cryptography; @B92]. Therefore using the polarization entanglement and the frequency entanglement simultaneously, it will become possible to implement wavelength-division multiplexing (WDM), quantum cryptography, or WDM quantum key distribution, which can send much more information compared with the case of polarization-only entanglement.
Experiment
==========
![Experimental setup. Details are given in the text.\[figures-setup\]](setup.eps)
The schematic drawing of our experimental setup is shown in Fig. \[figures-setup\]. Frequency-nondegenerate photon pairs are generated by SPDC in a 1-mm-thick type-II BBO ($\beta$-BaB$_2$O$_4$) crystal pumped by the second harmonic light (1.5 mW) of a cw Ti:sapphire laser operated at 859.4 nm. To generate photon pairs by bandwidth extended SPDC, the pump light is focused on the BBO crystal by a microscope objective lens of 8-mm focal length (L1). Generated SPDC photons diversing from the focal waist are collimated by an off-axis parabolic mirror (M) of 25.4-mm focal length. A prism pair (P1,P2) is used to eliminate the remainder of the pump light, which can be a noise source in the experiments. When the light beam passes through the prism pairs, the beam height was lowered by a mirror (M3), and only the SPDC pairs are picked out by another mirror (M2). The signal and idler photons were separated from each other by a polarizing beam splitter (PBS). Vertically polarized photons (signal) are reflected by the PBS, and diffracted by a grating (G) (1400 grooves/mm). Horizontally polarized photons (idler) pass through the PBS and a partially absorptive sample.
A 2.5-mm-thick plate of Nd$^{3+}$-doped glass was used as a sample. The main absorption transitions in Nd$^{3+}$ around the visible spectral range are from $^4I_{9/2}$ to $^2G_{7/2}(^2G_{5/2})$, $^4F_{7/2}$, $^4F_{5/2}$, and $^4F_{3/2}$, of which peak wavelengths are located at about 580, 750, 810, and 870 nm, respectively. The spectrum of signal light is centered at 840 nm, and it overlaps with the absorption peaks at 810 and 870 nm (see Fig. \[data-Nd-absorbance\]), of which transitions are studied in this paper.
![Absorption spectrum of Nd$^{3+}$-doped glass.\[data-Nd-absorbance\]](absorbance.eps)
Both signal and idler photons are collimated into fibers by lenses (L2,L3) and then detected by a set of two single-photon counting modules (SPCM: EG&G SPCM-AQR-14). Delaying the electric signal from the SPCM detecting signal photons by a nanosecond delay unit (ORTEC 425A), the coincidences are counted by a time-to-amplitude converter/single-channel analyzer (ORTEC 567) followed by a computer-controlled multichannel pulse-height analyzer (MCA:ORTEC TRUMP-PCI-2k).
Theory
======
In this paper, we discuss the application of the frequency-entanglement of SPDC photon pairs to absorption spectroscopy. The state of generated SPDC photon pairs can be written as [@coincident-frequencies-theory; @coincident-frequencies-theory2; @coincident-frequencies-theory3] $$\begin{aligned}
|\psi\rangle \propto \int d\omega_s \int d\omega_i \tilde{E}(\omega_s+\omega_i) \Phi_L(\omega_s,\omega_i) |\omega_s\rangle_s |\omega_i\rangle_i,
\label{eq-state}\end{aligned}$$ where $|\omega\rangle_j \equiv a^{\dagger}_j(\omega) |0\rangle$ is a single-photon state whose frequency is $\omega$. $a^{\dagger}_j(\omega)$ is the photon creation operator for frequency $\omega$ photons. Here $j=p,s,$ and $i$ indicate the pump, signal, and idler waves, respectively. $\tilde{E}(\omega)$ is the Fourier transform of the classical field of pump laser, and $$\begin{aligned}
\Phi_L(\omega_s,\omega_i) \propto \mathrm{sinc} \left[\Delta k(\omega_s,\omega_i)L/2\right] \equiv \frac{\sin(\Delta k(\omega_s,\omega_i)L/2)}{\Delta k(\omega_s,\omega_i)L/2}
\label{eq-phasematch}\end{aligned}$$ is the phase-matching function [@shg-phasematch] with the phase mismatch parameter $\Delta k$ expressed as $$\begin{aligned}
\Delta k(\omega_s,\omega_i) = k_s(\omega_s)+k_i(\omega_i)-k_p(\omega_s+\omega_i).\end{aligned}$$ As easily seen from Eq. (\[eq-phasematch\]), thinner crystals are preferable for the generation of broadband SPDC photon pairs. Generally, in our quantum experiments using SPDC photon pairs, thick crystals as thick as 5 mm are usually used to obtain high SPDC conversion efficiency. However in this paper broadband SPDC photon pairs are indispensable for its spectroscopic measurement, thin crystal of 1-mm thickness is used.
The wavelength resolution of this system is calculated as 4 nm using $d_g\phi_f/2F$, where $d_g$, $\phi_f$, and $F$ are a gap of grating grooves($1/1400$ mm), a fiber diameter ($125~\mu$m), and a focal length of lens L2 (11 mm), respectively. It is much narrower than the full width of the parametric fluorescence spectrum.
Since a cw laser is being used as a pump, the pump bandwidth is also much narrower than the width of the parametric fluorescence spectrum. It forces the sum of signal and idler frequency to be equal to the pump frequency as $\omega_p = \omega_s + \omega_i$ , and it entangles a SPDC photon pair in its frequency as a signal photon at frequency $\omega_p/2-\omega$ is generated with an idler photon at frequency $\omega_p/2+\omega$.
The wavelength resolution of the system and the pump bandwidth are so narrow that, using some approximations, the absorption spectrum of the sample can be easily calculated as [@quantum-spectrum-filter; @quantum-spectrum-filter2] $$\begin{aligned}
A(\omega^{'})=-\log_{10}\frac{R_c(\omega_p-\omega^{'})}{R_{c,\mathrm{sample}}(\omega_p-\omega^{'})},\end{aligned}$$ where $R_{c,\mathrm{sample}}(\omega)$ is a coincidence counting rate when the sample was placed in the idler path, and $R_c(\omega)$ is the one without the sample.
Results and Discussion
======================
Figure \[data-coincidence-sample\] shows coincidence counts accumulated for 30 s without a sample when the grating angle is set to diffract the center wavelength of signal light, and the coincidence counts have a peak at the delay time of 18 ns determined by the delay unit.
![Coincidence counts accumulated for 30 seconds when the grating angle is set to diffract center wavelength of signal light\[data-coincidence-sample\]](coincidence-sample.eps)
Coincidence counts averaged from 5 to 45 ns except between 14 and 22 ns delay is used for the calculation of a background noise. In this paper, coincidence sum counts is obtained by the integration of the coincidence counts during the delay from 14 to 22 ns and by subtracting the background noise.
Rotating the grating around the vertical axis crossing the incident point of the idler beam, coincidence counts are accumulated for 120 s at each step without a sample. The dashed curve in Fig. \[data-ndglass\] shows the wavelength dependence of the sum of coincidence counts at each step (coincidence spectrum). The center wavelengths of the signal and idler are centered at 883 and 840 nm, respectively, and the full width at half maximums (FWHMs) are 63 and 69 nm, respectively.
![Coincidence spectrum with the Nd$^{3+}$-doped glass sample, accumulated scanning a grating angle (solid line) and one without a sample (dashed line).\[data-ndglass\]](ndglass.eps)
Then, rotating the grating, coincidence spectrum was measured with the Nd$^{3+}$-doped glass sample in the idler light path, accumulating for 120 s at each step (see Fig. \[data-ndglass\]). The wavelength calibration is performed using a reflection, zeroth-order diffraction, from the grating. Beside the center wavelength of the SPDC light, the glass sample has two absorption peaks at about 810 and 870 nm (see Fig. \[data-Nd-absorbance\]), so the idler photons are absorbed and the coincidence counts are reduced in the spectral ranges.
![The absorption spectrum calculated by dividing the coincidence spectrum without a sample by one with the sample. The solid line shows the absorption spectrum measured by a UV-visible-near-infrared spectrophotometer.\[data-compareAbs\]](compareAbs2.eps)
By dividing the coincidence spectrum without a sample by one with the sample, the absorption spectrum of the sample is calculated, and compared with one measured by a UV-visible-near-infrared spectrophotometer in Fig. \[data-compareAbs\]. The absorption spectrum determined from the ratio of the coincidence counts fits fairly well with the one measured by the conventional spectrophotometer except in the spectral range from 860 to 880 nm, where the sample absorbance is large and the amount of transmitted photons is so small that the error tends to be substantially large.
Conclusion
==========
In conclusion, the absorption spectrum of the Nd$^{3+}$-doped glass plate was measured using an frequency-entangled two-photon state generated by spontaneous parametric down-conversion. This method is performed without resolving the frequency of idler photons which transmit through the sample. It is greatly advantageous in the case when the transmitted photons are not easy to be resolved by their energy, like in vacuum ultraviolet range or in space. The method using the frequency entangled photon pairs has an advantage over the one using classical method, when the spectroscopic property of a fragile sample is analyzed in a spectral range where any low-noise photodetector is not available, like in infrared range. Not to damage fragile samples, the pump light power must be very low, but the photodetector for a infrared range is usually so noisy that it is difficult to measure the characteristics under such situation. However, using the frequency-entangled photon pairs, one of the photon of each pair can be used as a timing gate for the other photon of the pair resolved by its energy. It increases the signal-to-noise ratio, so the measurement becomes much easier than the case using the classical spectroscopy apparatus.
A type-II BBO crystal is used for production of SPDC photon pairs entangled in polarization. All photon pairs in the SPDC light cones are collimated by a parabolic mirror, and travel to a beam splitter. In this experiment, a polarizing beam splitter was used to separate signal and idler photons from each other effectively. However it destroys the polarization entanglement. Using a nonpolarizing beam splitter in place of the polarizing beam splitter, the SPDC photon pairs detected at photon counters have the polarization entanglement besides the frequency entanglement, and it will enable WDM cryptography to be implemented which has an ability to send much more information than the case not using the frequency entanglement.
We thank Dr. Haibo Wang and Tomoyuki Horikiri for their valuable discussion.
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---
abstract: 'We review hard diffractive results and prospects at the Tevatron with an emphasis on factorization breaking in diffractive processes. Upper limits on the exclusive di-jet and $\chi_c^0$ production cross sections at CDF and the status of the 0 Forward Proton Detectors are discussed.'
author:
- Krisztian Peters
bibliography:
- 'k\_peters.bib'
title: 'Hard Diffractive Results and Prospects at the Tevatron[^1]'
---
[ address=[Department of Physics & Astronomy, University of Manchester,\
Manchester M13 9PL, UK\
]{} ]{}
Diffraction at the Tevatron
===========================
Diffractive events are mediated by the exchange of color singlets with vacuum quantum numbers and have clear experimental signatures. These are on one hand rapidity gaps: the absence of particles in some regions of rapidity in contrast to non-diffractive events where gaps are filled by additional soft parton interactions which yields an exponential suppression of rapidity gaps. On the other hand tagged protons or anti-protons: $p$ or $\bar p$ scattered at small angle and measured in Roman Pots far away from the interaction point. Depending on these rapidity gaps and tags, the main diffractive event topologies at the Tevatron are: single diffraction (SD), double diffraction (DD) or double Pomeron exchange (DPE). Single diffraction is characterized by a leading proton or anti-proton which escapes the collision intact and the presence of a further particle or a di-jet resulting from a hard scattering separated by a rapidity gap. Double diffraction is characterized by a gap in the central region and dissociated protons and anti-protons. In DPE there is a gap on both, the proton and anti-proton side, with a central produced di-jet or other particles. Proton and anti-proton remains intact.
In RUN II the CDF Forward Proton Detectors (FPDs) include Roman Pot detectors approximately 57m from the interaction point. These consist of three stations and each station comprises one scintillation fiber detector and one trigger counter. Beam Shower Counters, a set of scintillation counters around the beam pipe, are used to reject non-diffractive (ND) background at the trigger level. This makes it possible to collect diffractive data at high luminosities. The energy flow in the event in the very forward direction is measured by Miniplug Calorimeters. These consist of alternating layers of lead plates and liquid scintillator readout. It has a towerless geometry without dead regions due to the lack of internal mechanical boundaries. The 0 FPDs are described in the following.
Factorization in diffraction
============================
(a)![Ratio of diffractive to non-diffractive di-jet event rates as a function of $x_{B_j}$ at CDF for different $\xi$ ranges and compared to Run I (a) and for different $Q^2$ values.[]{data-label="fig:SD"}](plots/rpj5_sf_23_bless_2.eps "fig:"){width=".5\textwidth"} (b)![Ratio of diffractive to non-diffractive di-jet event rates as a function of $x_{B_j}$ at CDF for different $\xi$ ranges and compared to Run I (a) and for different $Q^2$ values.[]{data-label="fig:SD"}](plots/rpj5_sf_23_q2_bless.eps "fig:"){width=".5\textwidth"}
One of the central issues of diffraction is whether hard diffractive processes obey QCD factorization. As Collins proved [@Collins:1997sr] for the general class of diffractive DIS processes, the cross section can be described as the convolution of a hard scattering matrix element (process dependent) and parton density functions (process independent). The question arises if this factorization theorem is more general, are parton densities really universal in diffractive exchange? Can we use them for different collider processes and energies? To answer this question is fundamental for the understanding of diffraction and it is also important to extrapolate Tevatron results to the LHC. The general strategy to prove factorization is to extract parton density functions (PDFs) and compare predictions to measurements of other processes and experiments.
Before the extraction of PDFs, diffractive fractions already yield some insight. Both, CDF and 0 , measured for different processes the fractions of events with one gap to all events. It was found that all ratios are at the order of 1% at the Tevatron which would support a more general factorization theorem. However the ratio of 1% yields an uniform gap suppression w.r.t. HERA where the diffractive rates are approximately 10 times higher. This discrepancy indicates already the breakdown of QCD factorization.
CDF measured in Run I the diffractive structure function of the anti-proton from SD di-jets and the result was compared with expectations from diffractive DIS measurements of H1 [@Affolder:2000vb]. These events can be described in terms of a Pomeron emitted from the anti-proton and scattering with a parton from the proton. The diffractive structure function was obtained by measuring the ratio of the diffractive and non-diffractive cross section. The product of this ratio with the known non-diffractive structure function gives the result on the diffractive structure function. Although the shapes of the structure functions from HERA and Tevatron have similar shapes, there is a normalization discrepancy of a factor of 10. This result again confirms the breakdown of QCD factorization between the Tevatron and HERA as already expected from the result on the diffractive fractions. One possibility to explain these observations is that, since there are more spectator partons in $p\bar p$ collisions w.r.t. $\gamma^* p$ collisions, the rate of gap destructions due to soft partonic interactions is larger at the Tevatron. One attempt to describe this rate is made with the introduction of the concept of the “gap survival probability” $|S|^2$ [@Bjorken:1992er; @Gotsman:1993vd]. The observation that the shapes of the two structure functions from HERA and the Tevatron are similar supports this concept. The gap survival probability factor corrects the normalization discrepancy.
In Run II the diffractive di-jet sample was collected with a dedicated trigger which selects events with at least one calorimeter tower above the 5 GeV $E_{\rm T}$ threshold and a threefold Roman Pot Spectrometer coincidence. Calorimeter information is used to determine the momentum loss of the anti-proton, $$\xi_{\bar p}=\frac{\sum E_T e^{-\eta}}{\sqrt s} \, .$$ SD and background regions are selected according to the measured $\xi_{\bar p}$ values. A large number of events are at $\xi_{\bar
p}\sim 1$ which are due to the overlap of at least one ND contribution. A plateau is observed in the $\xi_{\bar p}$ distribution which results from a distribution proportional to $1/\xi_{\bar p}$ as expected for diffractive production.
(a)![Di-jet mass fraction for different rapidity gap selections (a) and di-muon plus photon invariant mass in the exclusive sample compared to Monte Carlo predictions (b) at CDF.[]{data-label="fig:DPE"}](plots/dpe_mjj_gerp_st5_18_syst_bless.eps "fig:"){width=".55\textwidth"} (b)![Di-jet mass fraction for different rapidity gap selections (a) and di-muon plus photon invariant mass in the exclusive sample compared to Monte Carlo predictions (b) at CDF.[]{data-label="fig:DPE"}](plots/mchic.eps "fig:"){width=".45\textwidth"}
In Fig. \[fig:SD\] the ratio of SD to ND event rates is plotted versus Bjorken $x_{B_j}$. In Fig. \[fig:SD\](a) this ratio is integrated over three different $\xi$ regions and compared to the Run I result. There is no $\xi$ dependence observable between 0.03 and 0.1. Furthermore the slope and normalization agrees and thus confirms the Run I result. In Fig. \[fig:SD\](b) the same ratio is plotted for three different $Q^2$ values, where $Q^2$ is the mean jet transverse energy $\langle (E^1_{\rm T} + E^2_{\rm
T})/2\rangle^2$. There is no appreciable $Q^2$ dependence in the observable region of $100 < Q^2 < 1600$ GeV$^2$. Thus both structure functions seem to have a similar $Q^2$ evolution which does not disfavor any of the two mechanisms of hard diffraction, namely the existence of a hard Pomeron (exchange of a colorless object) or a soft color rearrangement in the final state.
Using the di-jet sample CDF extracted also DPE di-jet production events. These events are characterized by a leading anti-proton, two jets in the central pseudorapidity region and a large rapidity gap on the outgoing proton side. If factorization holds the ratio of DPE to SD, $R^{DPE}_{SD}(x_p, \xi_p)$ has to be equal to the ratio of SD to ND, $R^{SD}_{ND}(x_{\bar p}, \xi_{\bar p})$ (in LO QCD) for a fixed $x_{B_j}$ and $\xi$ value. Although the collected events have different $\xi$ ranges for the proton and anti-proton, the weighted average of $R^{SD}_{ND}(x_{\bar p}, \xi_{\bar p})$ is flat in $\xi$ and the ratio was extrapolated to $\xi = 0.02$. At this $\xi$ values the above mentioned ratios differ by a factor of 5 [@Affolder:2000hd]. The deviation from unity yields again a breakdown of factorization. Since the formation of a second gap is less suppressed this result is coherent with the concept of the gap survival probability. The number of spectator partons does not changed with the formation of a second gap, i.e. one does not have to pay the price for the gap two times.
In the same manner as was previously done by the SD/ND ratio, the diffractive structure function can be extracted from the DPE/SD ratio. The obtained result now approximately agrees with expectations from H1 leading again to the above mentioned conclusions.
Search for exclusive events
===========================
Since the CDF Roman Pots have been installed at the end of Run I, CDF collected in Run II two orders of magnitude more di-jet data which made a study of exclusive di-jet production in DPE possible. The strategy was to obtain an inclusive DPE di-jet event sample and look for exclusive signature using the di-jet mass fraction $R_{jj}=M_{jj}/M_X$ where the di-jet mass is divided by the mass of the rest of the system excluding the proton and anti-proton. In Fig. \[fig:DPE\](a) the obtained number of events is plotted versus the di-jet mass fraction where no gap, a narrow gap and a wide gap was required on the proton side. The result is a smoothly falling spectrum all the way down to $R_{jj}=1$ and a similar event yield at high mass fraction regardless of the gap requirements. From this it follows that no significant excess due to exclusive di-jets is seen at high $R_{jj}$. An upper limit on the exclusive di-jet production cross section is calculated based on events with $R_{jj} > 0.8$. For example requiring a minimum jet transverse energy of 10 GeV, this upper limit is $$E_{\rm T}^{\rm min}=10 ~{\rm GeV}:~~\sigma (R_{jj}>0.8)<1.1\pm 0.1(stat)\pm 0.5 (sys)~~ {\rm nb} \, .$$
There are also other production channels available in DPE, we mention here the exclusive $\chi_c^0$ production. This process is of particular interest, since the $\chi_c^0$ quantum numbers are very similar to the ones of the Higgs boson, thus it can be used to test and normalize the predictions for exclusive Higgs production at the LHC. CDF did an analysis in Run II where the $\chi_c^0$ further decays in a $J/\psi +\gamma$. 93 pb$^{-1}$ of di-muon triggered data was used and events have been selected in the $J/\psi$ mass window. A large rapidity gap on both the proton and anti-proton side was required. 10 events have been found which are exclusive $\chi_c^0 (\to
J/\psi +\gamma)$ candidates. In Fig. \[fig:DPE\](b) the di-muon plus photon invariant mass of the 10 events is compared with a sample of generated $\chi_c^0$ events passed through a detector simulation. The invariant mass is consistent with that of the $\chi_c^0$, although the mean mass is higher and the distribution broader in the data than in the simulation. This may be due to the simulation or there may be contributions from cosmic events, higher mass $\chi_c$ mesons or $J/\psi + \pi^0$ events. Since it is very difficult to fully understand the background one may calculate an “upper limit” on exclusive $\chi_c^0$ production assuming that the observed 10 events are all $J/\psi +\gamma$ events. This upper limit is: $$\sigma = 49 \pm 18 (stat) \pm 39(sys)~~ {\rm pb} \, .$$
0 Forward Proton Detectors
===========================
![The Forward Proton Detector at 0 . Quadrupole Pots are named P or A when placed on the proton or the anti-proton side, respectively. Dipole Pots are named D.[]{data-label="fig:fpd"}](plots/fpdlayout.eps){width="100.00000%"}
The 0 Forward Proton Spectrometers have been installed and recently commissioned. These are 9 momentum spectrometers with 2 scintillating fiber detectors each, as schematically shown in Fig. \[fig:fpd\]. There is a dipole spectrometer located on the scattered anti-proton side behind the dipole magnets approximately 58 m away from the interaction point. They have in the range of $|t|\approx
0 - 1$ GeV$^2$ and $\xi\approx 0.03 - 0.07$ a very good acceptance. Eight quadrupole spectrometers are on both side of the main detector approximately 23 and 31 m away from the interaction point, behind the quadrupole magnets and the separators. They have a very good acceptance in the region of: $|t|\approx 0.8 - 3.0$ GeV$^2$ and $\xi\approx 10^{-3} - 0.05$. The position detectors housed inside Roman Pots operate millimeters away from the beam, however outside of the ultra high vacuum. They enable the reconstruction of the high energy protons and anti-protons directly, thus providing a first time possibility of tagging both the protons and anti-protons and measuring their $\xi$ and $t$ dependence at the Tevatron.
The 0 scintillating fiber detectors have 6 layers of scintillating fiber channels and one trigger scintillator layer. The fibers are oriented within $\pm 45^o$ to reconstruct hits and obtain redundancy. Furthermore every second channel is offset by 2/3 fiber for a finer hit resolution. All 18 detectors regularly brought close to the beamline and diffractive samples being collected.
In a small dedicated test run of the FPDs in a stand alone mode the slope of the elastic cross section of proton anti-proton scattering was measured. In Fig. \[fig:elastic\] the result (not normalized) is plotted and compared with theory predictions [@Block:1989gz]. 0 access a new kinematic domain with these measurements and the result of the slope agrees well with the model of Bock [*et al.*]{} [@Block:1989gz]. This measurement is being redone using the fully integrated FPDs. The alignment and detector understanding of the Roman Pot detectors is in progress and more physics results are expected soon.
![The slope of the elastic $p\bar p$ cross section measured at 0 and compared to predictions of [@Block:1989gz] (solid line).[]{data-label="fig:elastic"}](plots/dsigdt.eps){width=".6\textwidth"}
[^1]: Presented at the XXXV International Symposium on Multiparticle Dynamics 2005, August 9-15, 2005, Kromeriz, Czech Republic
|
---
abstract: 'In this paper we investigate the usage of adversarial perturbations for the purpose of privacy from human perception and model (machine) based detection. We employ adversarial perturbations for obfuscating certain variables in raw data while preserving the rest. Current adversarial perturbation methods are used for data poisoning with *minimal* perturbations of the raw data such that the machine learning model’s performance is adversely impacted while the human vision cannot perceive the difference in the poisoned dataset due to minimal nature of perturbations. We instead apply relatively *maximal* perturbations of raw data to conditionally damage model’s classification of one attribute while preserving the model performance over another attribute. In addition, the maximal nature of perturbation helps adversely impact human perception in classifying hidden attribute apart from impacting model performance. We validate our result qualitatively by showing the obfuscated dataset and quantitatively by showing the inability of models trained on clean data to predict the hidden attribute from the perturbed dataset while being able to predict the rest of attributes.'
author:
-
bibliography:
- 'expert\_matcher.bib'
title: 'Maximal adversarial perturbations for obfuscation: Hiding certain attributes while preserving rest'
---
Introduction
============
In this paper we investigate the usage of adversarial perturbations for privacy from both human and model (machine) based classification of hidden attributes from the perturbed data. With the advent of distributed learning, several methods such as federated learning and split learning have become prominent for distributed deep learning. At the same time, privacy preserving machine learning is a very active area of research. Adversarial approaches of minimally perturbing the data to fool the model performance while fooling human perception in detecting such a perturbation have become popular. We investigate whether an advesarial perturbation can be used to achieve the following goals:
1. Damage the model performance for predicting a chosen sensitive attribute while keeping the perfomance of predicting another attribute intact.
2. Obfuscate with maximal perturbation to make it difficult for human to detect the hidden attribute.
Such perturbations are necessitated in sectors like finance [@bateni2018fair; @srinivasan2019generating; @chen2018interpretable; @chen2018fair], healthcare [@vepakomma2018split; @chang2018distributed], government, retail [@zhao2017men; @yao2017beyond] and hiring [@kay2015unequal; @Harrison2018Bias] due to privacy, fairness, ethical and regulatory issues.
We validate our results qualitatively by presenting the perturbed datasets and quantitatively by showing the inability of models trained on clean data to predict the hidden attribute from the perturbed dataset while being able to predict the rest of attributes. We consider these to be useful intermediate experiments and results towards the goal of using adversarial methods for generating perturbations such that when a model is trained from the perturbed data for predicting the hidden attribute, the model performance is under control. We also would like to mention that these approaches can motivate a theoretical study of privacy guarantees of adversarial approaches under worst-case settings.
Contributions and method
------------------------
[r]{}[0.5]{}\[Appp\]
{width="40.00000%"}
Method - Setting up technical aspects of problem and what we do to achieve it Add equations for adversarial perturbation
In view of hiding certain latent information from data, we employ adversarial perturbation based methods for adding noise in the data. Majority of the past research in adversarial attacks has focused on a small value of $\epsilom$-ball of perturbation because their goal is to alter the model’s outcome without producing any imperceptible change for humans. However, the goal of this work is to obfuscate the data to hide certain latent information from the both model as well as humans, therefore, we conduct our experiments over a broad spectrum of the epsilon values which encompass the output range which fool machine as well as human. In this study we consider only images but our technique is generic in design and applicable wherever adversarial perturbations can be performed successfully [@advnlp1; @advnlp2; @advrl1; @advgan1]. To generate adversarial perturbation, we use VGG-16 [@vgg16] model with pretrained layers on ImageNet [@imagenet]. We employ an architecture as shown in Figure \[Appp\] where a fork is created after three blocks of the VGG-16 architecture where each block consists of two convolution layers and one pooling layer. The hidden attribute fork consists of few layers of DNN for local computation.
We employ adversarial perturbation methods with a larger $\epsilon$-ball of perturbations to generate images that are pretty hard for humans to classify with respect to the hidden attribute. Typically adversarial perturbations have only been used with small $\epsilon$-ball perturbations to the best of our knowledge in rest of the works. However, the goal of this work is to obfuscate the data to hide certain latent information from the both model as well as humans, therefore, we conduct our experiments over a broad spectrum of the epsilon values which encompass the output range which fool machine as well as human. In this study we consider only images but our technique is generic in design and applicable wherever adversarial perturbations can be performed successfully. To generate adversarial perturbation, we use VGG-16 [@simonyan2014very] model with pretrained layers on ImageNet [@imagenet]. We employ an architecture as shown in Figure \[Appp\] where a fork is created after three blocks of the VGG-16 architecture where each block consists of two convolution layers and one pooling layer. The hidden attribute fork consists of few layers of DNN for local computation. The rest of the network after the red fork is used to predict the label attribute that is supposed to be preserved. We then train the network. Upon training, we then employ adversarial methods of fast gradient sign method (FGSM) and projected gradient based perturbation (PG), to perturb the layer preceding the hidden attribute fork (shown by grey arrow). We although choose a higher $\epsilon$-ball of possible perturbations in order to generate the perturbation of this layer with respect to loss function corresponding to only the hidden label attribute. We show detailed results with regards to the quality of our results in the experiments section. In addition we weight the two loss functions with $\alpha_1,\alpha_2$.
Related work
------------
In Figure \[Appp2\] we share the landscape of deep learning based approaches for hiding certain attributes in data. We categorize it broadly into 4 approaches that include a.) perturbation of raw data which includes our approach, b.) overlaying mask on raw data to hide certain parts of image, [@relbalanceDist], c.) modifying the output of intermediate (encoded) representations [@lample2017fader; @vepakomma2018split; @vepakomma2019reducing], d.) transforming the entire (or partial) natural image into another natural image [@du2014garp; @relchen2019distributed; @relwu2018privacy]. We share 6 example methods within these categories. Our approach belongs to the category of ‘adversarial attack based perturbations’. As a sub note, all the above approaches can be further categorized into sub approaches that fool humans and/or machines while our intermediate work focuses on both.
\[Appp2\]
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Results
=======
We detail our experimental setup and results in this section. A condensed version of the architecture in our setup is shown in figure \[Appp\]. We perform experiments where VGG16 is the initially trained model that is adversarially perturbed with large $\epsilon$-ball of perturbations. We then predict over the perturbed data with a clean model trained on unperturbed data. Three clean models were trained with architectures of VGG16 and VGG19. The two methods of adversarial perturbations used with large choice of $\epsilon$-ball were fast gradient sign method (FGSM), [@goodfellow2014explaining] and projected gradient adversarial method (PGD), [@athalye2017synthesizing]. The dataset used was UTKFace which is a large-scale face dataset with long age span. The dataset consists of over 20,000 face images with annotations of age, gender, and ethnicity. The images cover large variation in pose, facial expression, illumination, occlusion, resolution, etc. This dataset could be used on a variety of tasks, e.g., face detection, age estimation, age progression/regression and landmark localization. In Table \[tab1\], we show results of the different methods of PGD abd FGSM employed in the large $\epsilon$-ballsetting with different weights $\alpha_1,\alpha_2$ for the weighted loss and the accuracy of predicting on perturbeddata generated by our approach using clean models of VGG16 and VGG19 trained on unperturbed data. We note that $82.92\%$ of race predictions by the clean model after our perturbation belong to the majority race class. This shows that our method is successfully able to increase the no-information rate in our predictions of the hidden label attribute of race while preserving gender accuracy. $39.2\%$ of ground truth of race belong to the same class as well. Therefore we reach the required level of obfuscation.
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\[tab1\]
$\alpha_1$ $\alpha_2$ Epsilon Race Accuracy Gender Accuracy Method Clean Model
------------ ------------ --------- --------------- ----------------- -------- -------------
1 1 0.2 0% 72.9% PGD VGG16
1 1 0.2 30% 54% PGD VGG19
1 1.00E-05 0.3 40% 65% FGSM VGG16
1 1.00E-05 0.4 40% 58% FGSM VGG16
1 1.00E-05 0.5 40% 54% FGSM VGG16
: We show results of the different methods of PGD and FGSM employed in the large $\epsilon$-ball setting with different weights $\alpha_1,\alpha_2$ for the weighted loss and the accuracy of predicting on perturbed data generated by our approach using clean models of VGG16 and VGG19 trained on unperturbed data. The baseline performance on clean data prior to perturbation is $87\%$ for race accuracy and $97\%$ for gender accuracy. We note that $82.92\%$ of race predictions by the clean model after our perturbation belong to the majority race class. This shows that our method is successfully able to increase the no-information rate in our predictions of the hidden label attribute of race while preserving gender accuracy. $39.2\%$ of ground truth of race belong to the same class as well. Therefore we reach the required obfuscation target.
Conclusion and future work
==========================
We investigate large $\epsilon$-ball perturbations obtained via adversarial methods for obfuscating one label attribute while preserving rest. We show better performance in fooling human with projected gradient descent based approach and better utility in preserving accuracy of public label attributes with fast gradient sign method based prediction. For future work, we aim to enhance this approach with information theoretic and other statistical dependency minimizing loss functions like distance correlation, Hilbert Schmidt Independence Criterion and Kernel Target Alignment. We note that $82.92\%$ of race predictions by the clean model after our perturbation belong to the majority race class. This shows that our method is successfully able to increase the no-information rate in our predictions of the hidden label attribute of race while preserving gender accuracy. $39.2\%$ of ground truth of race belong to the same class as well. Therefore we reach the required baseline. Therefore the other goal would be to raise the performance of predicting the public label attribute as we reach the required obfuscation performance on hidden label attribute.
|
---
abstract: 'A finite element method simulation of a carbon fibre reinforced polymer block is used to analyse the nonlinearities arising from a contacting delamination gap inside the material. The ultrasonic signal is amplified and nonlinearities are analysed by delayed Time Reversal – Nonlinear Elastic Wave Spectroscopy signal processing method. This signal processing method allows to focus the wave energy onto the receiving transducer and to modify the focused wave shape, allowing to use several different methods, including pulse inversion, for detecting the nonlinear signature of the damage. It is found that the small crack with contacting acoustic nonlinearity produces a noticeable nonlinear signature when using pulse inversion signal processing, and even higher signature with delayed time reversal, without requiring any baseline information from an undamaged medium.'
author:
- |
Martin Lints$^{1),2)}$, Andrus Salupere$^{1)}$, Serge Dos Santos$^{2)}$\
$^{1)}$ Tallinn University of Technology, Department of Cybernetics,\
Akadeemia tee 21, 12618 Tallinn, Estonia. martin.lints@cens.ioc.ee\
$^{2)}$ INSA Centre Val de Loire, Blois Campus;\
COMUE “Léonard de Vinci”, U930 “Imagerie et Cerveau” Inserm,\
3 rue de la Chocolaterie, CS23410, 41034 Blois, France.
title: Simulation of detecting contact nonlinearity in carbon fibre polymer using ultrasonic nonlinear delayed time reversal
---
Introduction
============
In the past, the use of carbon fibre reinforced polymer (CFRP) has been limited to non-structural parts of high-tech aeronautical products. In recent times, due to the effort of weight reduction and product lifetime enhancement, the application areas of CFRP have widened to the load-bearing parts of the aeronautical, automotive and civil engineering products. Due to the increased demands on the strength of the CFRP products and possible complex failure mechanisms, the Non-Destructive Testing (NDT) methods of CFRP have been an important applied and academic problem.
The complex failure mechanisms of CFRP include microcracking and delamination. Microcracking can occur at lower loads or due to aging and can be difficult to examine using ultrasonic NDT. With increased loading, the damage can evolve to delaminations, a very fine cracking between the layers of the CFRP. These damages are difficult to detect using ultrasonic methods due to their small thicknesses. The damage can exhibit itself as a contact acoustical nonlinearity [@solodov2002can]. A statistical distribution of microcracks or delamination damage in the material could also be described by hysteresis in a continuum material model [@AleshinAbeeleHyst2004; @solodov2001PhysRev; @Solodov_nonlinearultrasonic]. This can also be applicable for other materials than CFRP, for example biological tissues [@DosSantosPrevorovsky2011; @riviere2012time]. Nonlinear ultrasonic methods have been in development for detecting and localizing fatigue and micro-crack damage by their nonlinear effects [@guo2012detection; @kober2014theoretical]. The detection of harmonic overtones is one of the simplest measures of nonlinearities [@blanloeuil2016closed]. Many nonlinear analysis methods not requiring filtering have been developed, for example scaling subtraction method [@bruno2009analysis; @scalerandi2008nonlinear] or pulse inversion (PI) with its generalizations [@ESAMDosSantos2008; @ciampa2012nonlinear], and applications of time reversal using scattering as new sources.
This paper proposes a delayed TR-NEWS signal processing method [@Lints2016] for detecting the nonlinear signature of a single small crack in CFRP as contact acoustical nonlinearity. In the Finite Element Method (FEM) simulation, the CFRP is modelled as anisotropic, layered medium. The ultrasonic signal is focused by TR-NEWS to the region of the material with the defect. The nonlinear signature of the crack is analysed by PI and compared with the delayed TR-NEWS method, which allows to create arbitrary wave envelope at the focusing region of TR-NEWS. It is used here to create an interaction of waves near the damage. The signature of the damage appears as the nonlinear effect of the wave interaction on the contacting crack. This signal processing requires only one transmitting and one receiving transducer. The effectiveness of the delayed TR-NEWS method has been shown in the previous work by physical experiments and simulations in undamaged and linear materials [@Lints2016]. In this paper, the FEM simulation model is advanced further by including absorbing boundary conditions and the contacting crack defect in the material.
Mathematical and simulation model
=================================
This section describes the simulation which is based on a physical experiment, and describes the differences and similarities between the simulation and the experiment. It shows some important points about the mathematical model, the delayed TR-NEWS signal processing and the FEM simulation. Detailed information about the derivation of mathematical and FEM model is available at [@MLRR2017].
Mathematical model {#sec:matmodel}
------------------
The test object is a CFRP block consisting of 144 layers (Fig. \[fig:cfrptextconf\]). It is composed of fabric woven from yarns of fibre and impregnated with epoxy. The cross-section of the yarns have elliptical shape (Fig. \[fig:cfrpcloseup\]) and the material has inclusions of pure epoxy, so a wave propagating through the material will encounter yarns (fibres with epoxy) and areas of pure epoxy.
The simulation is in time domain, since the TR-NEWS procedure relies on transient echoes and complex wave motion for the wave energy focusing process. Due to the heavy computational cost of time domain simulation, a simple laminate model is used where: i) the material consists of homogeneous layers, ii) each layer has its own elasticity properties, and iii) dispersion arises due to the periodical discontinuity of the material properties. It consists of CFRP layers with 90$^\circ$/0$^\circ$ weave, 45$^\circ$/45$^\circ$ weave and epoxy layer. The thicknesses of the layers are given by random variable functions which reflect the actual structure of the material. The random variable distribution, describing the CFRP structure, is measured from a close-up image of the CFRP test object [@MLRR2017]. This links the distribution of the microstructure inside the actual material with the thicknesses of the layers in the laminate model. It should enable a more realistic simulated material having dispersion effects due to discontinuities.
![The layered structure of the CFRP with the fabric yarns in tight packing and epoxy in the voids[]{data-label="fig:cfrpcloseup"}](./shapeBMT.jpg){width="45mm"}
The three different kind of layers have the following mechanical properties: i) isotropic pure epoxy: $E = 3.7$ GPa, $\nu = 0.4$, $\rho = 1200$ kg/m$^3$; ii) transversely isotropic composite with 0/90$^\circ$ weave: $E_1 = E_2 = 70$ GPa, $G_{12} = 5$ GPa, $\nu_{12} = 0.1$, $\rho = 1600$ kg/m$^3$; and iii) transversely isotropic composite with 45$^\circ$/45$^\circ$ weave: $E_1 = E_2 = 20$ GPa, $G_{12} = 30$ GPa, $\nu_{12} = 0.74$, $\rho = 1600$ kg/m$^3$. For the simulation, a laminate model was constructed using 50 pairs of epoxy and carbon fibre layers, where carbon fibre weave direction alternated between each pair.
![(Colour online) The laminate material model with layers of stochastic thicknesses and absorbing boundary conditions on bottom and left boundaries and four fixed degrees of freedom[]{data-label="fig:laminatefig"}](./sim_layout_absorb.jpg){width="45mm"}
The boundary conditions of the model (Fig. \[fig:laminatefig\]) include Lysmer-Kuhlemeyer absorbing boundary conditions [@nielsen2006absorbing] so the wave energy would pass through the simulation region. Four degrees of freedom (DOFs) are fixed, the rest are free. The simulation model includes a contacting delamination defect in the material near the receiving transducer (Fig. \[fig:simsetup\]). The transmitting shear wave transducer can send maximum 50 kPa pulse at 70$^\circ$ degree angle.
![Schematic (not to scale) of the simulation geometry, location of crack, transmitter and receiver points without the layers[]{data-label="fig:simsetup"}](./simsetup.eps){width="55mm"}
TR-NEWS signal processing {#sec:TRNEWS}
-------------------------
This subsection repeats the signal processing method that is applied for this problem and has been published in the previous work [@Lints2016]. It is included for a self-contained discourse in this paper.
In the physical experiments, on which the simulation is based on, the CFRP block (Fig. \[fig:cfrptextconf\]) was studied using TR-NEWS NDT equipment and signal processing methods [@Lints2016]. The 2D FEM simulations reflect it as closely as possible in terms of transducer placement, frequencies and signal processing.
The roles of the transducers are not changed during the experiment: the focusing of the ultrasonic wave relies on the TR-NEWS signal processing. This is a two-pass method where the receiving and transmitting transducers do not change their roles. In this sense the “Time Reversal” describes the signal processing method which accounts for internal reflections of the material as virtual transducers, used for focusing the wave in the second pass of the wave transmission. The placement of the transducers is not important from the signal processing standpoint: in NDT investigation they could be placed arbitrarily and they do not have to be in line with each other, but the configuration must remain fixed during the complete TR-NEWS procedure.
Figure \[virttr\] outlines the TR-NEWS signal processing steps. The simulation uses the same signal processing steps as are usually applied to physical experiments. Firstly the chirp-coded excitation $c(t)$ is transmitted through the medium. $$\label{eq:chirp}
c(t) = A \cdot \sin \left( \psi (t) \right),$$ where $\psi(t)$ is linearly changing instantaneous phase. In this work, a linear sweep from 0 to 2 MHz was used. Then the chirp-coded coda response $y(t)$ with a time duration $T$ is recorded at the receiver $$y(t,T) = h(t) \ast c(t) = \int_{\mathbb{R}} h(t - t',T) c(t') dt',$$ where $h(t-t',T)$ is the impulse response of the medium. The $y(t,T)$ is the direct response from the receiving transducer when the chirp excitation $c(t)$ is transmitted through medium. Next the correlation $\Gamma(t)$ between the received response $y(t,T)$ and chirp-coded excitation $c(t)$ is computed during some time period $\Delta t$ $$\label{gammaeq}
\Gamma (t) = \int_{\Delta t} y(t - t',T) c(t') dt' \simeq h(t) \ast c(t) \ast c(T-t,T),$$ where the $h(t) \ast c(t) \ast c(T-t,T)$ is pseudo-impulse response which is proportional to the impulse response $h(t)$ if using linear chirp excitation for $c(t)$ because $ \Gamma_c (t) = c(t) \ast c(T-t) = \delta(t - T) $. Therefore the actual correlation $\Gamma (t) \sim h(t)$ contains information about the wave propagation paths in complex media.
![(Colour online) Schematic process of TR-NEWS with the virtual transducer concept. (1) The initial broadband excitation $T_x(t)$ propagates in a medium. (2) Additional echoes coming from interfaces and scatterers in its response $R_x$ could be associated to a virtual source $T_x^{(2)}$. (3) Applying reciprocity and TR process to $R_x$. (4) The time reversed new excitation $T_x = R_x(-t)$ produces a new response $R_x$ (the TR-NEWS coda $y_{TR}(t)$) with a spatio-temporal focusing at $z = 0; y=0; t=t_f$ and symmetric side lobes with respect to the focusing.[]{data-label="virttr"}](./TR-NEWS-virt-tr_vector.eps){width="75mm"}
Time reversing the correlation $\Gamma(t)$ from the previous step results in $\Gamma(-t)$ used as a new input signal. Re-propagating $\Gamma(-t)$ in the same configuration and direction as the initial chirp yields $$\label{resultequation}
y_{TR}(t, T) = \Gamma (T-t) \ast h(t) \sim \delta(t - T),$$ where $y_{TR} \sim \delta(t - T)$ is now the focused signal under receiving transducer where the focusing takes place at time $T$. This is because $\Gamma (t)$ contains information about the internal reflections of the complex media, and transmitting its time reversed version $\Gamma (T-t)$ will eliminate these reflection delays by the time signal reaches the receiver, resulting in the focused signal $y_{TR}$ (Eq. ). The test configuration must remain constant during all of these steps, otherwise the focusing is lost. The steps of this focusing process in a physical experiment are shown in Fig. \[steps4\].
![Bi-layered aluminium experimental chirp-coded TR-NEWS signal processing steps: (1) chirp excitation, (2) output recorded at Rx, (3) cross-correlation between input and output, (4) focusing resulting from taking time-reversed cross-correlation as new input [@Lints2016].[]{data-label="steps4"}](./steps4.eps){width="75mm"}
PI is an established method for detecting nonlinearities [@ESAMDosSantos2008]. The procedure used here involves conducting TR-NEWS measurements with positive and negative sign for $A$ in Eq. and comparing the focused signals. Differences could indicate the presence of nonlinearities.
Delayed TR-NEWS signal processing considers a single $y_{TR}$ focusing wave as a new basis which can be used to build arbitrary wave shapes at the focusing. This is done by time-delaying and superimposing $n$ time-reversed correlation $\Gamma(T-t)$ signals (Fig. \[steps6\] left column) $$\label{dTReq}
\Gamma_s(T-t) = \sum_{i=0}^n a_i \Gamma (T- t + \tau_i) = \sum_{i=0}^n a_i \Gamma (T-t + i \Delta \tau),$$ where $a_i$ is the $i$-th amplitude coefficient and $\tau_i$ the $i$-th time delay. In case of uniform time delay the $\Delta \tau$ is the time delay between samples. Upon propagating this $\Gamma_s(t-T)$ through the media according to the last step of TR-NEWS, a delayed and scaled shape of signal at the focusing point can be created. The delayed TR-NEWS signal processing optimization can be used for amplitude modulation, signal improvement and sidelobe reduction [@Lints2016].
![Delayed TR-NEWS signal processing steps in bi-layered aluminium, starting from the cross-correlation step (left column) and prediction of linear superposition of waves (right column): (1) cross-correlation(Eq. ), (2) delayed and scaled cross-correlation, (3) linear superposition of two cross-correlations which becomes the new excitation, (4) focusing (Eq. ), (5) delayed and scaled focusing, (6) linear superposition of the two focusing peaks.[]{data-label="steps6"}](./steps6.eps){width="75mm"}
It is possible to predict what the delayed TR-NEWS focusing output would be in a linear material (Fig. \[steps6\] right column): $$\begin{gathered}
\label{dTRpred}
y_{dTR}(t) = \left[ \sum_i a_i \Gamma_c \left( T - t + \tau_i \right) \right] \ast h(t) \xlongequal{\text{linearity}} \\= \sum_i a_i \Gamma_c \left( T - t + \tau_i \right) \ast h(t) = \sum_i a_i y_{TR} (t - \tau_i).\end{gathered}$$ The purpose of the prediction is twofold. Firstly it can be used to figure out optimal delay and amplitude parameters $a_i$ and $\tau_i$ beforehand for the delayed TR-NEWS experiment, using the original focusing peak $y_{TR}$. Secondly it could be possible to analyse the differences between the measured delayed TR-NEWS result and its prediction, which acts as a baseline for comparison. The difference could indicate the magnitude of nonlinearity, because the prediction relies on the applicability of linear superposition and is found to be accurate in experiments with linear material [@Lints2016].
The FEM simulation model
------------------------
The simulation program considers 2D wave propagation in a solid material with linear elasticity. The nonlinearity comes from an internal defect, a crack in the computational region (Fig. \[fig:simsetup\]) which can come into contact with itself. This contacting nonlinearity has asymmetric stiffness and is therefore nonclassically nonlinear. Since the CFRP is a complex material, then in this work it is modelled as a laminate with anisotropic layers arranged in a periodic manner, described in Section \[sec:matmodel\]. Because the physical experiment was conducted on the corner of a large CFRP block, the simulation is also in a semi-infinite quarter-space. The region has two free surfaces for reflection and two absorbing boundaries for the wave energy to escape.
The constitutive equation of the material itself is linear (although anisotropic). The linear plane strain elastodynamics problem is solved $$\label{eq:linelast}
\rho \ddot{u}_i - \sigma_{ij,j} = b_i,$$ where $\rho$ is material density, $u_i$ is displacement component, $\sigma_{ij}$ is stress component and, $b_i$ is body force component [@reddy]. Einstein summation convention is used and comma in index denotes spatial derivative. The constitutive equation in the variational formulation is $$\begin{gathered}
\label{eq:varform}
0 = \int_{\Omega} \left( \sigma_{ij} \delta \varepsilon_{ij} + \rho \ddot{u}_i \delta u_i \right) dx dy - \\
\int_{\Omega} b_i \delta u_i dx dy - \int_{\Gamma} t_i \delta u_i ds
$$ where $\varepsilon_{ij}$ is strain and $t_i$ is traction component on boundary. In our case the region $\Omega$ is a 2D space and boundary $\Gamma$ surrounding it is a 1D line. The body forces are zero in this simulation. Strain is assumed to be small.
The matrix formulation of the finite element model with damping is $$\label{eq:dynsystem1}
M\ddot{\Delta} + C\dot{\Delta} +K \Delta = F,$$ where $M$ is mass matrix, $K$ is stiffness matrix, $F$ is external forcing and $\Delta$ is displacement vector [@MLRR2017]. The damping matrix $C$ is used to apply the Lysmer-Kuhlemeyer absorbing boundary conditions [@nielsen2006absorbing] as a diagonal matrix, allowing to take advantage of the explicit solution scheme.
The element matrices are $$\label{eq:consistentmass}
M_e = \int_\Omega \rho \Psi^T \Psi dx dy,$$ $$K_e = \int_\Omega B^T C_e B dx dy,$$ $$F = \int_\Gamma \Psi^T f ds,$$ where $C_e$ is here the constitutive matrix for the plane strain elasticity.
Linear triangular three-node elements (T3), also known as constant strain triangles [@reddy], were chosen for this problem for the following reasons. Firstly because the epoxy layers in the laminate model can be very small, therefore small elements are needed anyway, with T3 being computationally cheapest. Secondly, linear elements are well suited for nonlinear problems: since the strain is constant throughout the element, the computation of nonlinear constitutive relations would also be simple. In this simulation, the material itself is linear but future work might include nonlinearity or hysteresis.
The T3 element lumped mass matrix [@zienkiewicz1] is $$M_e = \frac{\rho A_e}{3} I_6,$$ where $I_6$ is $6 \times 6$ identity matrix and $A_e$ is the area of the element. The element stiffness matrix is $$K_e = A_e B_e^T C_e B_e,$$ where matrix $B$ is $$\label{eq:bmatrixT3}
B = \frac{1}{2A_e}
\begin{bmatrix}
\beta_1 & 0 & \beta_2 & 0 & \beta_3 & 0 \\
0 & \gamma_1 & 0 & \gamma_2 & 0 & \gamma_3 \\
\gamma_1 & \beta_1 & \gamma_2 & \beta_2 & \gamma_3 & \beta_3 \\
\end{bmatrix},$$ and with $x_i$ and $y_i$ being the node coordinates [@reddy], then $$\begin{array}{llcrr}
\beta_1 &= y_2 - y_3, & \quad & \gamma_1 &= x_3 - x_2, \\
\beta_2 &= y_3 - y_1, & \quad & \gamma_2 &= x_1 - x_3, \\
\beta_3 &= y_1 - y_2, & \quad & \gamma_3 &= x_2 - x_1. \\
\end{array}$$
The external distributed force is simply divided into relevant nodes. The Lysmer-Kuhlemeyer absorbing boundary conditions are applied as viscous stresses on the boundaries, which means that they can be applied on DOF basis, making the damping matrix $C$ diagonal. The viscous stresses on the boundary DOFs are $$\begin{aligned}
c_{ii} &= \int_\Gamma a \rho V_p ds, \quad \text{normal motion DOF},\\
c_{ii} &= \int_\Gamma b \rho V_s ds, \quad \text{shear motion DOF},\end{aligned}$$ where $\Gamma$ is the boundary portion of the element [@nielsen2006absorbing]. In this work the scaling parameters are $a=1$ and $b=1$. The wave velocities used for these boundary conditions are $V_p = 2972$ m/s and $V_s = 1956$ m/s [@advancedNDT].
Equation is solved for each timestep $\Delta t = 5\cdot 10^{-10}$ s by explicit central difference scheme $$\begin{gathered}
\label{eq:lumpedsol}
\left( \frac{M}{\Delta t^2} + \frac{C}{2 \Delta t} \right) u_{n+1} = F_n -\\ \left( K - \frac{2 M}{\Delta t^2} \right) u_n - \left( \frac{M}{\Delta t^2} - \frac{C}{2 \Delta t} \right) u_{n-1}.\end{gathered}$$ This scheme is solved by dividing the equation by the term in the first parentheses, which is simple if $M$ and $C$ are diagonal matrices. Each simulation considers a 60 $\mu$s time window.
### Contact gap treatment
There is a single source of nonlinearity in this simulation: the contacting crack fully inside the material (Fig. \[fig:simsetup\]). If the material is at rest, then the crack is small and straight. In this work, there is neither a preload nor an initial gap in the contacting crack. This simple material defect results in a localised nonclassical nonlinearity, which can be analysed by various signal processing methods.
It is known that frictional contact problems can be sensitive to timestep length and loading path [@MijarArora2004]. In this work, it is assumed that the small timestep length and relatively small forces involved keep the error small. Therefore an explicit solution method scheme is utilized, similarly to [@SchutteEtAl2010]. A more precise solution could be expected from an implicit scheme, but that is left for the future. Further refinements could include thermoelastic contribution to the constitutive equation at the frictional contact gap [@solodov2001PhysRev].
The node-to-node contact model is used [@zienkiewicz2] with Coulomb friction. The defect is horizontal, simplifying the calculation of normal gap between the nodes. If the position of a node on a slave (lower) surface is $(n^s_x, n^s_y)$ and on master (higher) $(n^m_x, n^m_y)$, then the normal contact gap is $g_N = n^s_y - n^m_y$ and the tangential gap (offset) is $g_T=n^s_x - n^m_x$. In case of normal penetration of one surface into another, then $g_N > 0$. If there is no penetration, then $g_N \leq 0$. The coefficient of friction is $\mu = 0.6$, and the solution aims to satisfy the Kuhn-Tucker conditions on the crack surface: $$\begin{cases}
g_N \leq 0,\\
\lambda_N = \sigma \cdot n \leq 0,\\
g_N \cdot \lambda_N = 0,
\end{cases}$$ where $\lambda_N$ is the normal force on crack, $\sigma$ is stress and $n$ is the normal vector of the surface. The penalty plus Lagrange multiplier method is used for normal contact and the penalty method for friction [@MijarArora2004p1].
The contact logic for the node pairs can be summarized by following steps.
- The initial contact forces are zeroed: normal $\lambda_N = 0$ and tangential $\lambda_T = 0$.
- System in Eq. is solved.
- Vector gap functions are found: $g_N = n^s_y - n^m_y$ and $g_T = n^s_x - n^m_x$.
- Normal forcing is updated $\lambda_N = \lambda_N + g_N b$ where $b$ is some big penalty value and $\lambda_N \geq 0$.
- Logic diverges to 3 paths:
No force is applied
: in case of no contact.
Only normal force
: is applied if preceding step had no contact or had contact with tangential slip.
Normal and tangential forces
: are applied if previous iteration had non-slip contact.
- The normal contact condition is verified by setting the penetration value $g_P = g_N$ where $g_N \geq 0$. Then the $L^2$-norm of penetration is evaluated $\langle g_P | g_P \rangle < \varepsilon$ where $\varepsilon$ is the limiting value for the error due to contact penetration. If the condition is not fulfilled, the iteration is repeated, otherwise new timestep is taken.
A more thorough explanation of this contact gap logic is available at [@MLRR2017].
Results
=======
The signal analysis of the time domain simulation results of the damaged and undamaged medium are compared, describing some analysis measures which could allow to detect the presence of damage as nonlinearity. The simulation follows ultrasonic TR-NEWS NDT procedures where the transducer data is available as time-series, measured at some specific location. The signals are low-pass filtered to keep only the ultrasonic component. Here five measurement points are analysed at various distances from the crack damage and transmitting transducer (Fig. \[fig:simsetup\]). A video of the displacement fields for TR-NEWS focusing to point 3 in cracked medium is available at [@CFTRfoc3video].
TR-NEWS with pulse inversion
----------------------------
Figure \[fig:noncr\] shows the undamaged CFRP TR-NEWS focusing for the receiver positions 1 to 5 (Fig. \[fig:simsetup\]). It is an ordinary TR-NEWS focusing where at the middle of the signal (30 $\mu$s) is the focusing, surrounded by the sidelobes. There are two aspects to note about this is figure. Firstly, the sidelobes shift toward the main focusing and comparatively decrease in amplitude as the receiving transducer position shifts toward the transmitting transducer (from position 1 to position 5), indicating lower noise as the signal gets stronger. Secondly, the sidelobes are symmetrical in respect to the main lobe. This does not happen in nonlinear (damaged) material. The PI results are identical, indicating no nonlinearity, and are not shown here.
![(Colour online) Unnormalized TR-NEWS focusing of undamaged CFRP simulation[]{data-label="fig:noncr"}](./noncrfocusBW.eps){width="75mm"}
Figure \[fig:cr5plot\] shows the TR-NEWS results of the cracked CFRP test object simulation for the receiving transducer positions 1 to 5 (Fig. \[fig:simsetup\]). Here the PI signal processing is also applied and it shows the nonlinearity as difference between results from initial chirp signals with positive and negative sign. This nonlinear, damaged case exhibits nonlinearity particularly strongly in receiving position 3 (near the middle of the crack). Also, the sidelobes are unsymmetrical in respect to the main lobe.
![(Colour online) Normalized TR-NEWS focusing of damaged CFRP simulation with PI applied to detect nonlinearities as difference between negative and positive excitations[]{data-label="fig:cr5plot"}](./cracked5plotBW.eps){width="75mm"}
Figure \[fig:nl\] shows the envelopes of the PI measure of nonlinearity across the measuring points. The nonlinearity magnitude depends on the measuring point location in respect to the crack: point 3 near the middle of the crack shows strongest nonlinearity, points 2 and 4 show less, and points 1 and 5 show the least.
![(Colour online) Envelopes of the PI nonlinearity measures from all of the measuring points[]{data-label="fig:nl"}](./nonlinearitiesBW.eps){width="75mm"}
Figure \[fig:cr\] shows the unnormalized focusing signal for the damaged medium, which can be compared with corresponding undamaged result in Fig. \[fig:noncr\]. The focused signals have some interesting properties:
1. The highest signal amplitude comes from the receiver position closest to the crack midpoint (pos. 3), not from the position closest to the transmitter (pos. 5).
2. Comparing the amplitudes of the positions 2 and 4, at far and near side of the crack end respective to transmitter: the farther position has larger focusing amplitude than the nearer position. Since the simulation region has two absorbing boundaries, the wave propagation is mostly in one direction, therefore the defect between pos. 2 and 4 must be capturing the wave energy and the TR-NEWS signal processing is using that energy as a new “virtual transducer” for the pos. 2 focusing. This could be further analysed in future works from the correlation signals which generate these focused signals.
3. Amplitudes from the measurement positions 1 and 5 are “right way” around: the nearer measurement point has larger focusing amplitude than the farther.
![(Colour online) Unnormalized TR-NEWS focusing of damaged CFRP simulation[]{data-label="fig:cr"}](./crfocusBW.eps){width="75mm"}
Figure \[fig:virtsource\] shows a snapshot of the simulation $u_2$ displacement at a time moment $t=32.6$ $\mu$s, just after the focusing. The defect in material is acting as a source of new excitation after TR-NEWS focusing. Wave energy is captured between the damage and outside wall of the material and emitted as a wave.
![(Colour online) Displacement $u_2$ field at time $t=32.6$ $\mu$s with a wave emission coming from the damaged region. Video available at [@CFTRfoc3video][]{data-label="fig:virtsource"}](./virtual_source1.png){width="75mm"}
Delayed TR-NEWS analysis
------------------------
Section \[sec:TRNEWS\] describes the delayed TR-NEWS signal processing method which allows to create arbitrary envelope wave at the focusing (Eq. ), instead of the simple peak of the TR-NEWS. Equation shows that in linear material, the outcome of the delayed TR-NEWS process can be predicted. Since this method with prediction works very well in physical NDT measurements of linear materials [@Lints2016], it is now tested in simulation with the nonlinearity, supposing that the difference between the simulation result and the linear prediction (Eq. ) is due to nonlinear interaction of waves in the presence of nonlinearities or damage. Figure \[fig:dTR\] shows the comparison between the linear superposition prediction and the simulation result of a simple delayed TR-NEWS process where two focusing peaks are at superposition with 1 $\mu$s time delay. The difference between the prediction and the simulation is large and obvious, indicating the presence of nonlinearity. This measure of nonlinearity seems to be stronger than the measure calculated from PI (Fig. \[fig:cr5plot\]), making it a good candidate for further investigation.
![(Colour online) Delayed TR-NEWS signal processing with one delay of amplitude $a_i=1$ and delay value $\tau=1$ $\mu$s (Eq. ): comparison between the linear prediction and the nonlinear simulation outcome[]{data-label="fig:dTR"}](./dTR1msBW.eps){width="75mm"}
The delayed TR-NEWS signal processing could also be used for activating the contacting gap as the energy pocket. This could be done by creating a new focusing wave envelope which would have the resonant frequency of the defect, permitting higher amplitude waves near the damaged region, which would enhance the extraction of the nonlinear signature. This study is left for the future.
Conclusion
==========
This paper investigated nonlinear NDT by using a simple FEM simulation model for a crack nonlinearity in CFRP. In the laminate model, the damage is a simple horizontal contacting crack near the receiving transducer. The signal processing uses TR-NEWS method for focusing the available wave energy near the receiving transducer. The magnitude of nonlinearity due to the damage is measured firstly with PI, secondly with the proposed delayed TR-NEWS signal processing procedure. While PI indicates the presence of the nonlinearity, the simple delayed TR-NEWS procedure shows it even more strongly and is promising for future investigations and further development due to its signal processing flexibility.
Since the delayed TR-NEWS procedure allows to generate a wave at the focusing with arbitrary envelope, it could be used in the future to excite the crack damage by its resonance frequencies, using the damage as an energy pocket. Other perspectives include a more detailed simulation model for the CFRP in order to take more of its microstructure geometry into account to have stronger focusing. Additionally, the damage could be modelled either by a collection of various cracks at various angles or by hysteresis. Moreover, heating from the frictional forces at the damage could be considered for a more precise simulation model.
Acknowledgement {#acknowledgement .unnumbered}
---------------
This research has been conducted within the [*co-tutelle*]{} PhD studies of Martin Lints, between the Tallinn University of Technology, Department of Cybernetics in Estonia and the Institut National des Sciences Appliquées Centre Val de Loire at Blois, France. The research is supported by Estonian Research Council (project IUT33-24).
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abstract: 'We investigate the influence of ram pressure on the star-formation rate and the distribution of gas and stellar matter in interacting model galaxies in clusters. To simulate the baryonic and non-baryonic components of interacting disc galaxies moving through a hot, thin medium we use a combined N-body/hydrodynamic code GADGET2 with a description for star formation based on density thresholds. Two identical model spiral galaxies on a collision trajectory with three different configurations were investigated in detail. In the first configuration the galaxies collide without the presence of an ambient medium, in the second configurations the ram pressure acts face on on the interacting galaxies and in the third configuration the ram pressure acts edge on. The ambient medium is thin ($10^{-28}$ g/cm$^3$), hot (3 keV $\approx 3.6\times10^7$K) and has a relative velocity of 1000 km/s, to mimic an average low ram pressure in the outskirts of galaxy clusters. The interaction velocities are comparable to galaxy interactions in groups, falling along filaments into galaxy clusters. The global star formation rate of the interacting system is enhanced in the presence of ram pressure by a factor of three in comparison to the same interaction without the presence of an ambient medium. The tidal tails and the gaseous bridge of the interacting system are almost completely destroyed by the ram pressure. The amount of gas in the wake of the interacting system is $\sim50$% of the total gas of the colliding galaxies after 500 Myr the galaxies start to feel the ram pressure. Nearly $\sim10-15\%$ in mass of all newly formed stars are formed in the wake of the interacting system at distances larger than 20 kpc behind the stellar discs. As the tidal tails and the gaseous bridge between the interacting system feel the ram pressure, knots of cold gas ($T<1\times10^5$K) start to form. These irregular structures contain several $10^6$ M$_{\sun}$ of cold gas and newly formed stars and, as the ram pressure acts on them, they move far away (several 100 kpc) from the stellar discs. They can be classified as ’stripped baryonic dwarf’ galaxies. These ’stripped baryonic dwarfs’ are strongly affected by turbulence, e.g. Kelvin-Helmholtz instabilities, which are not resolvable within the presented SPH simulations. Heat conduction, which is not included, would affect these small structures as well. Therefore we give some estimate on the lifetime of these objects.'
author:
- |
W. Kapferer$^{1}$[^1], T. Kronberger$^{1}$, C. Ferrari$^{1}$, T. Riser$^{2}$ and S. Schindler$^{1}$\
$^{1}$Institute for Astro- and Particle Physics, University of Innsbruck, Technikerstr. 25, A-6020 Innsbruck, Austria\
$^{2}$Institute for Mathematics, University of Innsbruck, Technikerstr. 13, A-6020 Innsbruck, Austria
date: '- -'
title: 'On the influence of ram-pressure stripping on interacting galaxies in clusters'
---
\[firstpage\]
galaxies: interactions - intergalactic medium - galaxies: stellar content - galaxies: structure - methods: numerical - hydrodynamics
Introduction
============
Multiwavelength observations have shown that the star formation rate in interacting galaxies is enhanced in comparison to isolated galaxies (Bushouse 1987, Sulentic 1976, Stocke 1978, Solomon & Sage, 1988, Combes et al. 1994, Kewley et al. 2006, and references therein). To understand the physical processes involved in interacting galaxies numerical simulations are an ideal tool. Since the first publications in this field (Pfleiderer 1963, Toomre & Toomre 1972) it is evident that interactions are the sources of tidal tails, bridges and other signs of massive perturbations in and around the discs of galaxies. First calculations including star formation and gas depletion in interacting systems (e.g. Noguchi & Ishibashi 1986, Noguchi 1991, Olson & Kwan 1990 a,b, Mihos 1992, Mihos & Hernquist 1996) indicated that galaxy mergers are able to increase the total star formation of the system up to an order of magnitude and that these events of strong starbursts are able to deplete the cold gas reservoir of the system significantly. Many numerical investigations put special emphasis on modelling observed interacting systems, like NGC7252 (Mihos et al. 1998) or on the dependence of the star formation rates on interaction parameters like spatial alignment and minimum separation (di Matteo et al. 2007, Kapferer et al. 2005). Bournaud et al. (2005) investigated the remnants of galaxy mergers with different mass ratios; Mayer et al. (2006) showed via numerical simulations that dwarf galaxies lose their gas by ram pressure and tidal stripping during passages through a disc. Cox et al. (2004) investigated galaxy mergers with special emphasis on the heating process of gas due to shocks. The formation of dwarf galaxies in the debris of interacting galaxies was investigated by Duc et al. (2004). The accretion onto supermassive black holes in merging galaxies and the resulting suppression of star formation and the morphology of the elliptical remnants were investigated by Springel et al. (2005).\
From observations of the Virgo cluster of galaxies it is reported that galaxies moving through a galaxy cluster suffer mass loss gas by the ram pressure of the intra-cluster medium (ICM) onto the inter-stellar medium (ISM) (Gunn & Gott 1972, Cayatte et al. 1990, Kenney et al. 2004, Vollmer et al. 2004). To study the effects of ram-pressure stripping in detail several numerical approaches were carried out, from very simple descriptions for the gas phase, like sticky particles (e.g. Vollmer et al. 2001), to advanced Eulerian grid techniques (e.g. Roediger et al. 2006) or Lagrangian descriptions of hydrodynamics (e.g. Schulz & Struck 2001, J[á]{}chym et al. 2007), so called smoothed particle hydrodynamics (SPH). All these methods were applied to single model disc galaxies interacting with a time-dependent or a constant pressure from a hot gas phase outside the disc. Recently the investigation of the dependence of ram pressure on the star formation of a single model galaxy moving through an ambient medium was done by Kronberger et al. (2007).\
On larger scales the effect of ram-pressure stripping on the chemical enrichment was studied in cosmological galaxy-cluster simulations. To model the physics involved below the resolution of such simulations an analytical approach for the mass loss by ram-pressure stripping was introduced, based on the Gunn & Gott criterion (Schindler et al. 2005, Domainko et al. 2006). In combination with the mass losses by galactic winds and starbursts, ram-pressure stripping is able to enrich the ICM to observed levels (Kapferer et al. 2007).\
In this paper we concentrate on the distribution of baryonic, stellar and gaseous matter, in interacting disc galaxies moving through an ambient hot medium. By applying a density threshold technique to model star formation (Hernquist & Springel 2003) we present the impact of ram-pressure stripping on the star formation rate and the positions of star formation in comparison to galaxy interactions without the presence of ram pressure.
The simulation setup
====================
The initial model for the model galaxies
----------------------------------------
The model galaxies were created with an initial disc galaxy generator developed by Volker Springel. Details and analysis can be found in Springel et al. (2004). The total mass and the virial radius of the galaxies’ halo are given by $$M_{200}=\frac{v^{3}_{200}}{10\,G\,H(z)} \qquad\mbox{and}\qquad
r_{200}=\frac{v_{200}}{10\,H(z)}$$ with $H(z)$ the Hubble constant at redshift z and $G$ the gravitational constant. We constructed a model galaxy with a disc-scale length of 3.3 kpc and a circular velocity of the halo of 160 km/s. The gas fraction in the disc is initially 25% of the total disc mass. The mass resolution of the different components of the galaxies (gas, stellar, dark matter) can be found in section 2.4. In Fig. \[gal\_fo\] the gas distribution of the model galaxy after 2 Gyr of evolution without ram pressure and interaction is shown.
The star formation and feedback model
-------------------------------------
We applied the so called hybrid method for star formation and feedback introduced by Springel & Hernquist (2003). The basic assumption of this model is the conversion of cold clouds into stars on a characteristic timescale $t_*$ and the release of a certain mass fraction $\beta$ due to supernovae (SNe). The relation can be expressed as $$\frac{d\rho_{*}}{dt}=\frac{\rho_{c}}{t_{*}}-\beta\frac{\rho_{c}}{t_{*}}=(1-\beta)\frac{\rho_{c}}{t_{*}},$$ where $\rho_*$ and $\rho_c$ are the stellar and cold gas phase densities, respectively. The factor $\beta$ gives the mass fraction of stars above $8 M_{\sun}$, which is in our simulations 0.1, adopting a Salpeter initial mass function with a slope of -1.35 in the mass interval $0.1$ to $40$ M$_{\sun}$. Each SN explosion heats the surrounding gas in the bubble leading to an evaporation of cold gas. The minimum temperature the gas can reach due to radiative cooling is $10^{4}$K. From observations and analytical models it is evident, that a certain fraction of matter can escape the galaxies potential due to thermal and or cosmic-ray driven winds due to SN explosions (Breitschwerdt et al. 1991). This decreases the mass and energy budget of a galaxy, especially in the case of starbursts. To take this into account, we applied the same method as Springel & Hernquist (2003) and scale the mass outflow in such a way, that the mass outflow is proportional to the star formation rate of the underlying system, with a proportionally factor of two. Following the mass budget for the hot and cold gas due to star formation, mass feedback, cloud evaporation, growth of clouds due to radiative cooling and mass outflow by a galactic wind lead to a selfconsistent description of star formation in disc galaxies. More details on the model can be found in Springel & Hernquist (2003).
The merger and the ram pressure
-------------------------------
To study galaxy-galaxy interactions in a ICM wind we place the two model galaxies on Keplerian orbits (see Duc et al. 2000 for details) with a maximum separation of 200 kpc, a minimum separation of 1 kpc and oriented edge on, after they have evolved for two Gyrs without any interaction. The relative velocity of the encounter is $\sim$ 300 km/s, which makes the studied merger comparable to galaxy mergers within groups, falling along filaments into galaxy clusters. After 2 Gyr of evolution the spiral structure of the disc galaxies is evolved.
![The gas distribution in the disc of the model galaxy after 2 Gyr of evolution without ram pressure and interaction seen face on.[]{data-label="gal_fo"}](single_fo.eps){width="\columnwidth"}
In addition to the interaction trajectory we gave the whole system a directed velocity of 1000km/s through a hot ($3.6 \times 10^7$ K) and thin ($10^{-28}$ g/cm$^3$) medium. This is done in two different configurations. In the first configuration the galaxies fly face on through the ICM and in the second configuration edge on while they interact with each other, see Fig. \[configurations\]. To compare the effects of the ram pressure we calculate as a third configuration of the same galaxy interaction without an ambient medium. The galaxies collide always edge on. The simulation of the ram pressure on the interacting galaxy pair was carried out for one Gyr, so that the system has a first encounter but has not merged yet.\
To study the influence of ram pressure by an ambient medium on the distribution of the tidal tails, the gaseous bridges and the star formation rates, we let the interaction take place in a homogenous density and a constant temperature distribution of the ambient medium. Therefore the effects of ram-pressure stripping are accessible. The influence of a varying ICM temperature and density distributions will be investigated in an upcoming work.
![Interaction geometry and the two configurations. The galaxies’ positions and velocities are chosen such that the galaxies are point masses moving on Keplerian orbits. See Duc et al. (2000) for further descriptions. In configuration A the galaxies move with 1000 km/s face on through the hot ambient medium, whereas configuration B represents the edge on flight through the medium. Between the configurations is an angle of 90$^{\circ}$. Configuration C (not imprinted) is the same encounter without an ambient medium acting on it.[]{data-label="configurations"}](configurations.eps){width="\columnwidth"}
The resolution
--------------
The influence of the resolution on the global star formation rate of interacting galaxies was investigated in Kapferer et al. (2005). It was found that for interacting disc galaxies the global star formation does not vary for mass resolutions below $10^{6}$ M$_{\sun}$ for the stellar and gas particles in the disc. To resolve star forming regions in the bridges and tidal tails a higher resolution was adopted in the present simulations. The total mass of the model galaxy is $1.09 \times 10^{12}$ M$_{\sun}$, the disc mass is $2.7 \times 10^{10}$ M$_{\sun}$ with a 25% gas content. We assigned $7 \times 10^{5}$ particles to each model galaxy. In Table \[run properties\] the particle numbers and corresponding mass resolutions for a single model galaxy and the ICM are listed. The ambient medium was sampled by $1 \times 10^{6}$ gas particles in a box with 1 Mpc on a side. To study the influence of a higher resolved ambient medium we performed a simulation with 10 times more gas particles. The star formation rate and the distribution of the gaseous matter did not change significantly. The same trend for the star formation rate is present in the high resolution and the low resolution simulation, see Fig. \[sfr\_hr\_normal\]. The mean smoothing length for ISM gas particles in our simulations is 0.978 kpc and the mean smoothing length for ICM particles is 1.17 kpc. The gravitational softening for gas particles is 0.03 kpc, for halo particles 0.02 kpc and for collisionless disc particles 0.05 kpc.
[c | c c c c c]{} & number of & mass resolution & total mass& particles & \[M$_{\odot}$/particle\] & \[M$_{\odot}$\] DM halo & $3\times10^5$& $3.5\times10^6$ & $1.05\times10^{12}$gaseous disc & $2\times10^5$ & $3.4\times10^4$ & $6.8\times10^9$stellar disc & $2\times10^5$ & $1\times10^5$ & $2\times10^{10}$ICM & $1\times10^6$ & $1.46\times10^6$ & $1.5\times10^{12}$ICM hr$^1$ & $1\times10^7$ & $1.46\times10^5$ & $1.5\times10^{12}$$^1$ high resolution
\[run properties\]
![The star formation rates for two interacting galaxies moving edge on through the ambient medium as a function of time. The same trend in the normal resolution and high resolution simulation is present.[]{data-label="sfr_hr_normal"}](sfr_hr_normal.eps){width="\columnwidth"}
Kelvin-Helmholtz instabilities and thermal conductivity
-------------------------------------------------------
Kelvin-Helmholtz instabilities develop in the case of a velocity shear within a fluid or when there is a velocity difference along the interface between two fluids. To resolve and treat this kind of turbulent behaviour in hydrodynamics is a challenge. A comprehensive analysis of the treatment of Kelvin-Helmholtz instabilities and other turbulent phenomena, such as Rayleigh-Taylor instabilities, was carried out with two commonly used techniques, namely Eulerian grid-based and smoothed-particle hydrodynamics (SPH) by Agertz et al. (2007). They conclude in their comparison that grid codes are able to resolve and treat dynamical instabilities and mixing, while SPH codes are not. According to Agertz et al. (2007) the reason for this is that SPH has severe problems in the case of strong density gradients. SPH particles representing low density regions near to high-density regions are decoupled by erroneous pressure force calculations due to the asymmetric density within the symmetric smoothing kernel. This leads to a decoupling in the different phases of the fluid, leading to an artificial suppression in the growth of turbulent structures, such as Kelvin-Helmholtz instabilities. In addition Agertz et al. (2007) conclude that for timescales below the typical dynamical timescales of the turbulent structure, SPH and Eulerian grid-based schemes agree.\
In McCarthy et al. (2007) the stripping of a hot gaseous matter around galaxies in groups and clusters was investigated by applying the SPH scheme. They give an estimate for the Kelvin-Helmholtz timescale (Eq. (4) McCarthy et al. 2007), based on the work of Mori & Burkert (2000) (eq.22). It is important to note that these estimates are in principle only applicable for spherical distributions and a particular relation between the galaxy core radius and mass. Dropping the relation between the galaxy core radius and mass and assuming that asphericity only introduces a modest change, this leads to timescales in the range of Gyrs for the main galaxies. As we stop our simulation after 1 Gyr, we conclude that although Kelvin-Helmholtz instabilities are not well resolved in SPH simulations our results regarding the main galaxies are not strongly affected by it. For the small clumps the situation is different, here the Kelvin-Helmholtz instabilities are in the range of several hundred Myrs, assuming a relative velocity of 1000 km/s and the low density of the ICM. Therefore the gas clumps can be stripped completely. Many of these clumps form from gas which originates from gas stripped into the slip stream of the galaxies. In this situation the structures are not affected by the ram pressure as strongly as in other regions. When entering the ICM wind the clumps will be deaccelerated to several hundred km/s relative velocities. This leads to longer Kelvin-Helmholtz timescales. Another issue not addressed in the simulation is heat conduction, which should in principle heat the gaseous clumps and therefore prevent them from becoming denser, therefore altering the star formation efficiency. The evaporation timescale due to thermal conductivity of a gas clump with an n$_H$ column density of 50/cm$^3$ and a size of 1 kpc (assuming spherical geometry) in an ambient $3.6\times10^7$K gas with an electron number density of $10^{-1}$ is in the range of 500 Myrs, according to Nipoti & Binney (2007). In the case of the ICM electron number densities, which are two orders of magnitudes smaller, these evaporation times would be even larger. Note that the simulations presented in this work are a simplified step towards the understanding of the complex physics involved in environmental effects in the evolution of galaxies. The multi-phase multi-scale gas physics, together with magnetic fields involved in star formation, require complex and fully consistent theoretical models which are not achieved yet.
Results
=======
The distribution of the different components
--------------------------------------------
The most striking difference between galaxy-galaxy interactions, suffering or not from ram pressure is the evolution of the distribution of the gaseous tidal tails and the gas bridge between the interacting galaxies. In Fig. \[dry\_merger\] the gas distribution for the interacting model galaxies is shown at the apocentre of the interaction. In the upper panel the galaxies interact without the ram pressure acting on them, whereas in the lower panel the effect of the ram pressure is shown. As the galaxies move face on through the ICM the gas feels the pressure of the ambient medium, resulting in an increasing offset of the stripped gas, especially at the tidal tails and the bridge. The gaseous bridge between the galaxies and parts of the tidal tails are compressed and fragmented into massive ($\sim 10^6$ M$_{\sun}$) gas clumps, which cool due to radiation and form new stars.
![The gas density of the interacting galaxies seen face on. In the lower panel the interaction takes place in an ambient medium with a constant ram pressure acting face on at the system, whereas in the upper panel no ambient gas is present.[]{data-label="dry_merger"}](dry_merger.eps){width="\columnwidth"}
In Fig. \[stars\] the stellar density of the interacting galaxies seen face on is shown. In the lower panel the interaction takes place in an ambient medium with a constant ram pressure acting face on at the system, whereas in the upper panel no ambient gas is present. The new formed stars are coloured blue, whereas the old stellar population is coloured yellow.
![The stellar density of the interacting galaxies seen face on. In the lower panel the interaction takes place in an ambient medium with a constant ram pressure acting face on at the system, whereas in the upper panel no ambient gas is present. The new formed stars are coloured blue, whereas the old stellar population is coloured yellow.[]{data-label="stars"}](stars.eps){width="6cm"}
In Fig. \[mass\_wake\_fo\] the fraction of stripped gas to the total amount of gas in the wake (distance larger than 20 kpc from the stellar disc) is shown for an interacting system moving face on through the ambient medium as a function of time. After 500 Myr nearly 50% of the gas in the interacting system is located 20 kpc behind the stellar discs of the system. In this gaseous wake approximately 10% of all new stars are formed after 500 Myr.\
![The amount of gas in the wake (distance to the stellar disc larger than 20 kpc) as fraction of the total amount of gas. In this model the galaxies are moving face on through the ambient medium.[]{data-label="mass_wake_fo"}](mass_wake_fo.eps){width="\columnwidth"}
The evolution of the wake is presented in Fig. \[gas\_icm\_fo\_time\]. The distribution of the gas in the interacting galaxies is shown at different timesteps. The left column shows the face on and the right column the edge-on view. The interval of time between each row is 250 Myr. Panel (a) and (e) show the situation at the first encounter. The ram pressure already distorts the outer gas layers. The effect of the ram pressure on the tidal tails and bridges starts after the galaxies have had the first encounter. In panels (b) and (f) the compression of the gas is already distinct. As the gas gets more compressed it cools and becomes denser. Parts of the cooled gas fall back onto the discs leading to episodes of star formation in the disc. In panels (c) and (g) the tidal tails and the gaseous bridge are nearly completely destroyed. Only dense knots are visible around the discs at distances larger than 50 kpc. The knots are star forming regions with several $10^5$ M$_{\sun}$ of newly formed stars. The last timestep shows that the dense knots of gas become more and more separated from the interacting system. While these knots form stars, more and more gas is transformed into stellar matter, leading to small irregular structures with several $10^6$ M$_{\sun}$ of baryonic matter.\
The evolution of the ratio of heated gas (T$>10^7$K) in the wake originating from the interacting system to the total gas of the system is given in Fig. \[hot\_mass\_wake\_fo\]. After 0.7 Gyr $\sim$5-6% of the stripped material in the interacting system is heated to temperatures above T$>10^7$K. This gas will show its signatures in X-ray observations of the system.
In the case of the galaxies moving edge on through the ambient medium the distribution of matter is very similar. The gaseous bridge and the tidal tails are destroyed by the ram pressure. Dense knots of gas form, which cool in the same way as in the face on situation. The remaining gaseous discs are truncated very similarly as in the face on passage.\
We have additionally investigated the robustness of our results with respect to the resolution. In Fig. \[hr\_lr\_density\] the distribution of gas of the interacting galaxies after one Gyr of evolution in the high resolution ICM simulation (a) and normal resolution ICM simulation (b) are shown. In the highly resolved ICM simulation the knots of gas, which form from stripped matter in the gaseous bridges and the tidal tails are present as in the normal resolved ICM simulation. The calculated amount of gas in the wake, originating from the interacting galaxies, defined as gas which lacks 20 kpc behind the discs, is 48% in the case of the highly resolved ICM simulation and 43% in the normal resolved simulation.
![The distribution of gas in the interacting galaxies after one Gyr of evolution in the high resolution ICM simulation (a) and normal resolution ICM simulation (b), seen edge on.[]{data-label="hr_lr_density"}](hr_lr_density.eps){width="\columnwidth"}
As the ram-pressure is the dominating mechanism, galactic outflows do not alter the mass distribution significantly. The amount of gaseous and stellar matter in the wake of the interacting galaxies is not changed.
The star formation rate and the regions of star formation
---------------------------------------------------------
To study the effects of constant ram pressure on the star formation we did as a first step simulations with a single disc galaxy moving through a hot medium with 1000 km/s relative velocity. The results of these investigations are discussed in detail in Kronberger et al. (2008). In the case of the interacting galaxies in an ambient medium, newly formed stars can be found up to a hundred kpc behind the plane of the disc. In Fig. \[stars\_wake\] the evolution of the ratio of newly formed stars in the wake to the total amount of newly formed stars in the case of the face-on interaction is shown. The definition for newly formed stars in the wake is a distance larger than 20 kpc to the interacting discs. Almost 20 % of all newly formed stars are located at a distance of more than 20 kpc to the interacting discs after a simulation time of roughly one Gyr.
![The evolution of the ratio of newly formed stars in the wake to the total amount of newly formed stars in the case of the face on interaction. The definition for the stars in the wake is a distance larger than 20 kpc to the interacting discs.[]{data-label="stars_wake"}](stars_wake.eps){width="\columnwidth"}
The evolution of the total star formation rate for the interacting systems is shown in Fig. \[sfrs\]. In the case of the interaction without an ambient medium the first encounter increases the star formation rate by a factor of $\sim3$. After the first encounter the global star formation rate decreases to slightly smaller rates than before. This can be explained by the formation of the tidal tails and the bridge, which decrease the gas content of the discs.\
In the case of an external ram pressure the situation is very different. After the first encounter the star-formation enhancement does not decrease. This different behaviour is present in both configurations, face on and edge on acting ram pressure. The reason for the enhancement can be found in the compression and deformation of the tidal tails and the gaseous bridges, as well as the compression of the discs due to the external pressure of the ICM on it. By comparing the star formation rates with a simulation in the ICM with no directed wind reveals, that most of the enhancement, belongs to the pressure of the ICM onto the galaxies. As the bridges and the tails are affected in the same way in the case of face on and edge on ram pressure, the total star formation rate does not vary significantly between the two configurations. The influence of the outflow of a galactic wind on the star formation rate is minor. As the outflow is kept in the range of twice the star formation rate, which is in agreement with observations (Martin, 1999), it is a negligible mass loss of the interacting system in comparison to the amount of stripped matter. The cool, dense knots and the distorted gas in the discs form new stars.
![The star formation rate of the the merging system as a function of time for four different configurations. The black solid line shows the star formation rate for the interacting system without ambient medium, the dashed blue line gives the face on case of the ram pressure, the dotted red line the edge-on ram-pressure case, corresponding to configurations A and B in Fig. 1. The dotted green line gives the star formation rate for an interaction taking place in the ambient medium without a constant wind.[]{data-label="sfrs"}](sfrs.eps){width="\columnwidth"}
In Fig. \[zoo\_fo\] the star-forming regions in the simulation of face on ram pressure are shown. The upper panel shows an edge-on view of the interacting system with the temperature colour-coded gas distribution. Additionally the newly formed stars are highlighted as white points. The lower panels give two star forming regions (a) and (b) in a larger view. Only the cool gas (T$<10^5$ K) and newly formed stars are shown. The total gas mass, originating from the interacting system in insert (a) is $3.5\times10^6$ M$_{\sun}$, the total stellar mass present in the volume shown by insert (a) is $3.5\times10^6$ M$_{\sun}$. The total gas mass, stripped from the interacting system and presented in insert (b) is $2.3\times10^6$ M$_{\sun}$ and the total stellar mass present in insert (b) is $5.9\times10^5$ M$_{\sun}$.
Comparison to observations
==========================
As the merger investigated in this work is more likely happening in groups, located in the infall regions of galaxy cluster, a direct comparison to observations of mergers within galaxy clusters is restricted. Nevertheless some observations, especially those in the outskirts of galaxy clusters, can be compared to study the influence of a low ram pressure. Since the first HI radio atlas of the Virgo cluster by Cayatte et al. (1990), it was found that ram-pressure stripping affects the star formation of cluster galaxies. Up to now, X-ray observations revealed the existence of tails or wakes of gas, probably associated to ram-pressure stripping, in about ten elliptical cluster galaxies or group galaxies (Kim et al. 2007, Jaltema et al. 2008 and Schindler & Diaferio 2008 and references therein). Only in recent years optical, radio and X-ray observations have started to accumulate evidence of gas tails stripped from late-type cluster galaxies at hundred kpc scales (e.g. UGC 6697, CGCG 97-073/97-079 in A1367: Gavazzi et al. 2001a,b, Sun & Vikhlinin 2005 – C153 in A2125: Wang et al. 2004 – ESO 137-001 in A3627: Sun et al. 2006, 2007 – NGC 4525, NGC 4388 and NGC 4532/DDO 137 in Virgo: Yoshida et al. 2002, Oosterloo & van Gorkom 2005, Haynes et al. 2007, Koopmann 2007 – D100 in Coma: Yagi et al. 2007). Most of these galaxies show signatures of enhanced star formation. The fact that some of them are interacting objects is well proven (e.g. CGCG 97-073/97-079 in A1367: Gavazzi et al. 2001b, NGC 4532/DDO 137 in Virgo, Koopmann 2007), still debated in other cases (e.g. UGC 6697 in A1367, Gavazzi et al. 2001a, NGC 4254 in Virgo, Vollmer et al. 2005). Concerning this point, it is worth to be stressed here that ram-pressure deeply alters the “expected”[^2] morphology of interacting galaxies, as clearly shown in Fig. \[dry\_merger\]. This point is extremely important to properly identify the different physical mechanisms acting on galaxies when interpreting observational results.\
Since stripped tails are diffuse sources that require high sensitivity observations with a sufficiently large field of view, only for very few of them we have detailed information coming from multi-wavelength data. Our simulations indicate that we should be able to detect in X-rays heated gas (T $> 10^7$ K) in the wake of interacting systems. At least three extended X-ray tails of late-type galaxies have been observed in clusters (UGC 6697 in A1367: Sun & Vikhlinin 2005 – ESO 137-001 in A3627: Sun et al. 2006 – C153 in A2125: Wang et al. 2004). The gas temperature measured in two of these sources is in the expected range (T $\sim
0.5 - 1.2 \times 10^7$ K, Wang et al. 2004, Sun & Vikhlinin 2005). In ESO 137-001 the X-ray tail coincides positionally with a detected H$\alpha$ tail and 29 HII regions have been detected in the wake of gas (Sun et al. 2007). Similarly to the results of our simulations, the HII regions closest to the galactic disc form a bow-like front with the axis nearly in the same direction as the tail. Additionally, Sun et al. (2006) detected three possible ultra-luminous X-ray sources (ULXs) related to active star formation in the tail. In UGC 6697 hints of correlation between X-ray and H$\alpha$ emission have also been shown (even if on smaller scales). Additionally, at least two X-ray point sources have been detected in the tail of this galaxy, which may be associated with star clusters. The detailed analysis of Sun & Vikhlinin (2005) also demonstrates that ram pressure alone cannot explain the peculiarities of UGC 6697 (i.e. its complex velocity field, the presence of a warp in the South-East region of the stellar disk). In agreement with our numerical results, they thus suggest that, together with ram pressure stripping, tidal effects related to galaxy interactions could play a role in determining the observed properties of UGC 6697 and its X-ray tail.\
On the other hand, it has also been proven that tidal effects alone cannot explain the observational properties of some interacting galaxies. Two irregular objects (CGCG 97-073 and 97-079) located in the North-West region of the galaxy cluster A1367 show extended ($\sim$75 kpc) tails of H$\alpha$ and synchrotron emission (Gavazzi et al. 1995, 2001b). Observations indicate that the tails host $\sim$40% of the original gas of the two galaxies. Gavazzi et al. (2001b) concluded that both the high SFR of these irregular galaxies and their head-tail morphology can be explained by ram-pressure effects. However, due to the low ICM density in the peripheral cluster region where the two galaxies are located, they claimed the need of an additional physical mechanism (i.e. the interaction between the two objects), able to lose their potential well, thus making ram-pressure more effective in stripping and/or compressing the ISM.\
The fraction of stripped gas in the wake of a given galaxy is significantly higher (and comparable to the observed value) in the case of interacting systems than in isolated ones (see Kronberger et al. 2008). The combination of a galaxy encounter and of moderate ram-pressure was already suggested to be responsible for the perturbed atomic gas distribution observed in interacting galaxies (NGC 4654/NGC 4639 and NGC 4254 in the Virgo cluster, Vollmer 2003, Vollmer et al. 2005). Haynes et al. (2007) and Duc et al. (2007) have recently suggested that more extended ($\geq$ 100 kpc) HI tails can be created by galaxy harassment and high-speed galaxy collisions. Our simulations show that wakes of gas and newly formed stars of several hundred of kpc are originated in interacting galaxies subject to ram pressure stripping. In addition, our results can explain: a) the origin of the blue colour and of the ionized gas detected in several gas wakes of interacting/disturbed galaxies (e.g. Gavazzi et al. 2001a, Yoshida et al. 2002, Sun & Vikhlinin 2005, Sun et al. 2007), and b) the existence of ULXs and isolated intra-cluster HII regions, observed at several tens of kpc from the closest galaxy and related to stars that have formed within the intracluster volume (e.g. Gerhard et al. 2002, Sun et al. 2007).\
Finally, our results can explain the origin of the region with the highest density of star formation activity ever observed in a local cluster (A1367, Cortese et al. 2006). Two giant galaxies and several dwarfs/extragalactic HII regions, together with an extended ($\geq$150 kpc) wake of ionized gas, were observed in a compact group infalling towards the cluster centre. Both ram-pressure stripping and galaxy-galaxy interactions are extremely efficient in this case. Due to their lower velocity dispersion compared to clusters, galaxy groups are actually the natural site for tidal interactions. This group is furthermore not isolated, but moving with a high velocity ($\sim$1700 km/s) in the ICM of A1367 (Cortese et al. 2004). Note that the simulation presented in this work is not representative for high velocity encounters/mergers in the central regions of galaxy clusters. Here additional effects, like galaxy harassment, play an important role. On the other hand the ram pressure increases with the ICM density and the square of the relative velocity of the galaxy with respect to the ICM. Therefore the effect of ram pressure stripping increases in the denser regions of galaxy clusters. The relative strength and the interplay of ram pressure in high velocity encounters will be investigated in an upcoming paper.
Discussion and conclusions
==========================
In order to investigate the influence of the ram pressure on interacting galaxies, we compared the results of a simulation of an interaction with and without a constant wind. We focus on the evolution of the star formation rate of the interacting system and on the distribution of the different baryonic components (i.e. gas and stars). The results can be summarised as follows:
- The star formation rate of the interaction is enhanced in the presence of an ambient hot (3keV) and rare medium ($10^{-28}
$g/cm$^3$) together with ram pressure by a factor of three in comparison to the same interaction without the ambient medium.
- The morphology of the interacting system is strongly distorted. The tidal tails and the bridge between the interacting system are completely destroyed by the ram pressure. The resulting gas and stellar mass distributions of the two galaxies would not be characterised by observers as interacting system after the first close encounter.
- The amount of gas in the wake of the interacting system is 50% of the total gas mass of the interacting system after 500 Myr of ram pressure acting on it. Approximately 10-15% of the gas is heated up to temperatures above $1\times10^7$K, which would be observable in X-rays.
- After 500 Myr of ram pressure $\sim10\%$ of all newly formed stars are formed in the wake of the interacting system at distances larger than 20 kpc behind the stellar disc in the case of the face on ram pressure. The same behaviour can be observed in the case of the edge on ram pressure.
- As the tidal tails and the gaseous bridge between the interacting system feel the ram pressure, knots of cold gas ($T<10^5$K) start to form and these irregular structures start to form new stars. These knots are found to contain several $10^6$ M$_{\sun}$ of cold gas and newly formed stars. They can be classified as ’stripped baryonic dwarf’ galaxies. The lifetime of these ’stripped baryonic dwarfs’ is limited by turbulence and heat conduction. If the objects are in the slipstream of the disc galaxies, they can survive for a several hundred Myrs up to a Gyr. As the ram pressure in the gaseous wake is decreasing due to the decreasing relative velocities, the timescales for Kelvin-Helmholtz instabilities is increasing as well, therefore extending the lifetime of these objects. Thermal conduction is affecting ’stripped baryonic dwarfs’ as well, but the low densities of the surrounding ICM seems to result in evaporation timescales at least larger than 500 Myrs.
Concluding we found that interacting galaxies affected by a moderate ram pressure show a completely different behaviour compared to the case of no ram pressure acting on them. The simulations presented in this work are idealised in order to be able to distinguish between different effects. As galaxies move through a real cluster, the ram pressure changes and complex interactions with the cluster potential and other galaxies are present. These factors will be investigated in upcoming papers.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank the anonymous referee for fruitful comments which helped to improve the quality of the paper. The authors thank Volker Springel for providing them with GADGET2 and his initial-conditions generator. The authors acknowledge the Austrian Science Foundation (FWF) through grants P18523-N16 and P19300-N16. Thomas Kronberger is a recipient of a DOC fellowship of the Austrian Academy of Sciences. The authors further acknowledge the UniInfrastrukturprogramm des BMWF Forschungsprojekt Konsortium Hochleistungsrechnen, the ESO Mobilitätsstipendien des BMWF (Austria), and the Tiroler Wissenschaftsfonds (Gefördert aus Mitteln des vom Land Tirol eingerichteten Wissenschaftsfonds).
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[^1]: E-mail: wolfgang.e.kapferer@uibk.ac.at
[^2]: Based on the results of previous models of galaxy interactions, which do not take into account the ambient medium effects (e.g. Kapferer et al. 2005).
|
---
abstract: |
The Kramers problem for quantum Bose-gases with specular-diffuse boundary conditions of the kinetic theory is considered. On an example of Kramers problem the new generalized method of a source of the decision of the boundary problems from the kinetic theory is developed. The method allows to receive the decision with any degree of accuracy. At the basis of a method lays the idea of representation of a boundary condition on distribution function in the form of a source in the kinetic equation. By means of integrals Fourier the kinetic equation with a source is reduced to the integral equation of Fredholm type of the second kind. The decision is received in the form of Neumann’s series.
[**Key words:**]{} quantum Bose–gas, constant collision frequency, the Kramers problem, reflection–diffusion boundary conditions, the Neumann series.
PACS numbers: 51. Physics of gases, 51.10.+y Kinetic and transport theory of gases.
---
**The Kramers problem for quantum bose-gases with constant collision frequency and specular–diffusive boundary conditions**
**E. A. Bedrikova[^1] and A. V. Latyshev[^2]**
[*Faculty of Physics and Mathematics,\
Moscow State Regional University, 105005,\
Moscow, Radio str., 10–A*]{}
Введение. О точных решениях граничных задач кинетической теории
===============================================================
Задача Крамерса является одной из важнейших задач кинетической теории газов. Эта задача имеет большое практическое значение. Решение этой задачи изложено в таких монографиях, как [@Ferziger] и [@Cerc73].
История проблемы
----------------
Более полувека тому назад К. М. Кейз в своей знаменитой работе [@Case60] заложил основы аналитического решения граничных задач теории переноса. Идея этого метода состояла в следующем: найти общее решение неоднородного характеристического уравнения, отвечающего уравнению переноса, в классе обобщенных функций в виде суммы двух обобщенных функций – главного значения интеграла и слагаемого, пропорционального дельта–функции Дирака.
Первое из этих слагаемых является частным решением неоднородного характеристического уравнения, а второе является общим решением соответствующего однородного уравнения, отвечающего неоднородному характеристическому. Коэффициентом пропорциональности в этом выражении стоит так называемая дисперсионная функция. Нули дисперсионной функции связаны взаимно однозначно с частными решениями исходного уравнения переноса.
К характеристическому уравнению мы приходим после экспоненциального разделения переменных в уравнении переноса. С помощью спектрального параметра мы разделяем пространственную и скоростную переменные в уравнении переноса.
Общее решение характеристического уравнения содержит в качестве частного решения сингулярное ядро Коши, знаменатель которого есть разность скоростной и спектральной переменной.
Именно ядро Коши позволяет использовать всю мощь методов теории функций комплексного переменного – в частности, теории краевых задач Римана—Гильберта.
Итак, построение собственных функций характеристического уравнения приводит к понятию дисперсионного уравнения, корни которого находятся во взаимно однозначном соответствии с частными (дискретными) решениями исходного уравнения переноса.
Общее решение граничных задач для уравнения переноса ищется в виде линейной комбинации дискретных решений с произвольными коэффициентами (эти коэффициенты называются дискретными коэффициентами) и интеграла по спектральному параметру от собственной функции непрерывного спектра, умноженных на неизвестную функцию (коэффициент непрерывного спектра). Некоторые дискретные коэффициенты задаются и считаются известными. Дискретные коэффициенты отвечают дискретному спектру, или, в некоторых случаях, отвечают спектру, присоединенному к непрерывному.
Подстановка общего решения в граничные условия приводит к интегральному сингулярному уравнению с ядром Коши. Решение этого уравнения позволяет построить решение исходной граничной задачи для уравнения переноса.
Действуя именно таким способом, К. Черчиньяни в 1962 г. в работе [@Cerc62] построил точное решение задачи Крамерса об изотермическом скольжении. Эта задача является важной содержательной задачей кинетической теории.
Работы [@Case60; @Cerc62] заложили основы аналитических методов для получения точных решений модельных кинетических уравнений.
Затем в работах [@5]–[@8] Черчиньяни и его соавторы получили ряд значительных результатов для кинетической теории газов. Эти результаты получили дальнейшее обобщение в наших последующих работах.
Обобщение этого метода на векторный случай (системы кинетических уравнений) наталкивается на значительные трудности (см., например, [@Siewert74]). С такими трудностями столкнулись авторы работ [@Siewert74; @Cerc77; @Aoki1], в которых делались попытки решить задачу о температурном скачке (задача Смолуховского).
Преодолеть эти трудности удалось в работе [@lat90], в которой впервые дано решение задачи Смолуховского. Затем эта задача была обобщена на случай слабого испарения [@ly92c]–[@ly94], на молекулярные газы [@ly93] и [@ly98], на безмассовые Бозе–газы [@ly97], на скачок температуры в металле (случай вырожденной плазмы) [@ly03] и [@ly05], и на другие проблемы [@ly02] и [@ly01].
Затем в работах [@LYPMTF] и [@ly92b] было дано решение задачи об умеренно сильном испарении (конденсации). Одномерная задача о сильном испарении была поставлена в работе [@Arthur] и была сделана попытка получить ее точное решение.
Задача о температурном скачке для БГК–уравнения с частотой столкновений, пропорциональной модулю скорости молекул, была решена методом Винера—Хопфа в работе [@Cassell]. Затем в более общей постановке с учетом слабого испарения (конденсации) эта задача была решена методом Кейза в нашей работе [@ly96].
Задача Крамерса в дальнейшем была обобщена на случай бинарных газов [@ly91b]–[@40], была решена с использованием высших моделей уравнения Больцмана [@ly97j]–[@ly04s], была обобщена на случай зеркально–диффузных граничных условий [@lyfd04]–[@41].
Нестационарные задачи для кинетических уравнений рассматривались в наших работах [@ly92a] и [@ly98a].
Различные проблемы теории скин–эффекта рассматривались в работах [@lyjvm99]–[@44].
В последнее десятилетие задачи Крамерса и Смолуховского были сформулированы и аналитически решены для квантовых Ферми–газов в работах [@Lat2001TMF] и [@lyt03].
Вопросам теории плазмы посвящены наши работы [@ly01jv]–[@53].
В наших работах [@54]–[@64] были развиты приближенные методы решения граничных задач кинетической теории с зеркально–диффузными граничными условиями.
В настоящей работе применяется новый эффективный метод решения граничных задач с зеркально–диффузными граничными условиями, развитый в работе [@LatYushk2012].
Настоящая работа является продолжением работ [@LatIVUS02] и [@Bedrikova]. В [@LatIVUS02] была решена полупространственная задача Крамерса для квантовых Бозе–газов: найдена скорость изотермического скольжения вдоль плоской поверхности и построена функция распределения летящих к стенке молекул непосредственно у стенки.
В работе [@Bedrikova] для задачи Крамерса построено распределение массовой скорости Бозе–газа в полупространстве и находится ее значение непосредственно у стенки. Для этого выводится формула факторизации дисперсионной функции. Проводится сравнение коэффициентов массовой скорости Бозе– и Ферми–газов.
В настоящей работе решается полупространственная задача Крамерса для квантовых Бозе–газов с зеркально–диффузными граничными условиями. Для квантовых Ферми–газов такая задача была решена в работе [@Ivanisenko].
Обобщенный метод источника
--------------------------
В основе предлагаемого метода лежит идея включить граничное условие на функцию распределения в виде источника в кинетическое уравнение. Так что предлагаемый метод можно называть обобщенным методом источника.
Суть предлагаемого метода состоит в следующем. Сначала формулируется в полупространcтве $x>0$ классическая задача Крамерса об изотермическом скольжении с зеркально–диффузными граничными условиями. Затем функция распределения продолжается в сопряженное полупространство $x<0$ четным образом по пространственной и по скоростной переменным. В полупространстве $x<0$ также формулируется задача Крамерса.
После того как получено линеаризованное кинетическое уравнение разобьем искомую функцию (которую также будем называть функцией распределения) на два слагаемых: чепмен—энскоговскую функцию распределения $h_{as}(x,\mu)$ и вторую часть функции распределения $h_c(x,\mu)$, отвечающей непрерывному спектру: $$h(x,\mu)=h_{as}(x,\mu)+h_c(x,\mu),$$ ($as \equiv asymptotic, c\equiv
continuous$).
В силу того, что чепмен—энскоговская функция распределения есть линейная комбинация дискретных решений исходного уравнения, функция $h_c(x,\mu)$ также является решением кинетического уравнения. Функция $h_c(x,\mu)$ обращается в нуль вдали от стенки. На стенке эта функция удовлетворяет зеркально–диффузному граничному условию.
Далее мы преобразуем кинетическое уравнение для функции $h_c(x,\mu)$, включив в это уравнение в виде члена типа источника, лежащего в плоскости $x=0$, граничное условие на стенке для функции $h_c(x,\mu)$. Подчеркнем, что функция $h_c(x,\mu)$ удовлетворяет полученному кинетическому уравнению в обеих сопряженных полупространствах $x<0$ и $x>0$.
Это кинетическое уравнение мы решаем во втором и четвертом квадрантах фазовой плоскости $(x,\mu)$ как линейное дифференциальное уравнение первого порядка, считая известным массовую скорость газа $U_c(x)$. Из полученных решений находим граничные значения неизвестной функции $h^{\pm}(x,\mu)$ при $x=\pm 0$, входящие в кинетическое уравнение.
Теперь мы разлагаем в интегралы Фурье неизвестную функцию $h_c(x,\mu)$, неизвестную массовую скорость $U_c(x)$ и дельта–функцию Дирака. Граничные значения неизвестной функции $h_c^{\pm}(0,\mu)$ при этом выражаются одним и тем же интегралом на спектральную плотность $E(k)$ массовой скорости.
Подстановка интегралов Фурье в кинетическое уравнение и выражение массовой скорости приводит к характеристической системе уравнений. Если исключить из этой системы спектральную плотность $\Phi(k,\mu)$ функции $h_c(x,\mu)$, мы получим интегральное уравнение Фредгольма второго рода.
Считая градиент массовой скорости заданным, разложим неизвестную скорость скольжения, а также спектральные плотности массовой скорости и функции распределения в ряды по степеням коэффициента диффузности $q$ (это ряды Неймана). На этом пути мы получаем счетную систему зацепленных уравнений на коэффициенты рядов для спектральных плотностей. При этом все уравнения на коэффициенты спектральной плотности для массовой скорости имеют особенность (полюс второго порядка в нуле). Исключая эти особенности последовательно, мы построим все члены ряда для скорости скольжения, а также ряды для спектральных плотностей массовой скорости и функции распределения.
Изотермическое скольжение вдоль плоской поверхности
---------------------------------------------------
Изложим физику скольжения газа вдоль плоской поверхности.
Пусть газ занимает полупространство $x>0$ над твердой плоской неподвижной стенкой. Возьмем декартову систему координат с осью $x$, перпендикулярной стенке, и с плоскостью ($y,z$), совпадающей со стенкой, так что начало координат лежит на стенке.
Предположим, что вдали от стенки и вдоль оси $y$ задан градиент массовой скорости газа, величина которого равна $g_v$: $$g_v=\left( \dfrac{d u_y(x)}{d x}\right)_{x= +\infty}.$$
Задание градиента массовой скорости газа вызывает течение газа вдоль стенки. Рассмотрим это течение в отсутствии тангенциального градиента давления и при постоянной температуре. В этих условиях массовая скорость газа будет иметь только одну тангенциальную составляющую $u_y(x)$, которая вдали от стенки будет меняться по линейному закону. Отклонение от линейного распределения будет происходить вблизи стенки в слое, часто называемом слоем Кнудсена, толщина которого имеет порядок длины свободного пробега $l$. Вне слоя Кнудсена течение газа описывается уравнениями Навье—Стокса. Явление движения газа вдоль поверхности, вызываемое градиентом массовой скорости, заданным вдали от стенки, называется изотермическим скольжением газа.
Для решения уравнений Навье—Стокса требуется поставить граничные условия на стенке. В качестве такого граничного условия принимается экстраполированное значение гидродинамической скорости на поверхности – величина $u_{sl}$.
Отметим, что реальный профиль скорости в слое Кнудсена отличен от гидродинамического. Для получения величины $u_{sl}$ требуется решить уравнение Больцмана в слое Кнудсена. При малых градиентах скорости имеем: $$u_{sl}=K_{v}l G_v, \qquad
G_v=\left( \dfrac{du_y(x)}{dx}\right)_{x=+ \infty}.$$
Задача нахождения скорости изотермического скольжения $u_{sl}$ называется задачей Крамерса (см., например, [@Ferziger]. Определение величины $u_{sl}$ позволяет, как увидим ниже, полностью построить функцию распределения газовых молекул в данной задаче, найти профиль распределения в полупространстве массовой скорости газа, а также найти значение массовой скорости газа на границе полупространства.
Настоящая работа посвящена изучению влияния квантовых эффектов на кинетические явления в разреженных Бозе–газах. Рассмотрение ведется на примере классической задачи об изотермическом скольжении газа (задача Крамерса) вдоль плоской поверхности [@Ferziger] и [@Cerc73]. Рассматриваются как диффузные граничные условия, так и зеркально–диффузные граничные условия Максвелла.
Граничные условия, описывающие взаимодействие молекул газа с поверхностью конденсированной фазы, приблекают внимание исследователей в течение длительного времени. Эта проблема по-прежнему остается открытой, в частности, для реальных поверхностей. В конкретных задачах используются главным образом модельные граничные условия. Одно из таких условий — это зеркально–диффузные граничные условия Максвелла. Все параметры отраженных молекул в задачах скольжения определяются при этом одной величиной — коэффициентом зеркальности, который часто отождествляют с коэффициентом аккомодации тангенциального импульса молекул.
При наличии вдали от поверхности градиента тангенциальной к поверхности компоненты скорости газа возникает скольжение газа вдоль поверхности. Такое скольжение называется изотермическим [@Ferziger] и [@Cerc73]. Задача Крамерса (см. [@Ferziger]–[@8]) состоит в нахождении скорости изотермического скольжения газа.
Пусть газ занимает полупространство $x>0$ над плоской твердой стенкой и движется вдоль оси $y$ со средней (массовой) скоростью $u_y(x)$ . Вдали от поверхности на расстоянии много большем средней длины свободного пробега частиц газа имеется градиент массовой (средней) скорости газа $$g_v=\Big(\dfrac{du_y(x)}{dx}\Big)_{x\to +\infty},$$ т.е. профиль массовой скорости вдали от стенки можно представить в виде $$u_y(x)=u_{sl}+g_vx,\;\qquad x\to +\infty.$$
Наличие градиента массовой скорости вызывает скольжение газа вдоль поверхности, называемое изотермическим. Величина $u_{sl}$ называется скоростью изотермического скольжения ($sl\equiv sliding\equiv$ скольжение).
При малых градиентах $g_v$ скорость изотермического скольжения пропорциональна величине градиента: $$u_{sl}=C_mlg_v.
\eqno{(1.1)}$$ Здесь $C_m$ – коэффициент изотермического скольжения, $l$ – средняя длина свободного пробега частиц.
Величина $C_m$ определяется кинетическими процессами вблизи поверхности. Для ее определения необходимо решить кинетическое уравнение в так называемом слое Кнудсена, т.е. в слое газа, примыкающего к поверхности, толщиной порядка длины свободного пробега $l$.
В качестве кинетического уравнения рассмотрим обобщение на квантовый случай БГК–уравнения (Бхатнагар, Гросс, Крук) $$\dfrac{\partial f}{\partial t}+({\bf v}\nabla f)=\nu (f_{eq}-f).
\eqno{(1.2)}$$
Здесь $f$ – функция распределения молекул по скоростям, [**v**]{} – скорость молекул, $\nu$ – эффективная частота столкновений молекул, $f_{eq}$ – локально равновесная функция распределения, $$f_{eq}=n\left(\dfrac{m}{2\pi kT}\right)^{3/2}\exp \left[-
\dfrac{m({\bf v}-{\bf u})^2}{2kT}\right].$$
Величины $n,\;T,\;{\bf u}$ зависят, вообще говоря, от координаты ${\bf r}$ и определяются как $$n=\int f d^3v,
\eqno{(1.3)}$$ $${\bf u}=\dfrac{1}{n_{eq}}\int {\bf v}f d^3v,
\eqno{(1.4)}$$ $$T=\dfrac{2}{3kn}\int \dfrac{m}{2}({\bf v}-{\bf u})^2 f d^3v
\eqno{(1.5)}$$
Числовая плотность (концентрация) $n$ квантового газа и его температура $T$ в задаче Крамерса считаются постоянными.
Кинетическое уравнение для квантовых Бозе–газов с постоянной частотой столкновений
==================================================================================
Вывод уравнения
---------------
Рассмотрим обобщение кинетического уравнения (1.2) на случай квантового Бозе–газа. Функцию $f_{eq}$ в (1.2) теперь будем понимать как локально–равновесную функцию Бозе $$f_{eq}=\dfrac{1}{-1+\exp\left(\dfrac{{{\mathcal{E}}}_*-\mu}{kT}\right)},
\qquad
{{\mathcal{E}}}_*=\dfrac{m}{2}({\bf v}-{\bf u})^2.$$ Здесь $\mu$ – химический потенциал молекул \[8\], $-\infty<\mu\leqslant 0$.
Вместо соотношений (1.3)–(1.5) теперь имеем следующие соотношения, вытекающие из законов сохранения числа частиц, импульса и энергии: $$\int f_{eq}R_i d\Omega=\int f R_i d\Omega.
\eqno{(2.1)}$$ Здесь $$d\Omega=\dfrac{(2s+1)m^3}{(2\pi \hbar)^3}d^3v,$$ $s$ – спин ферми–частицы, $
R_1=1,\;R_2=v_x,\;R_3=v_y,\;R_4=v_z,\;R_5={{\mathcal{E}}}_*.
$
Рассмотрим теперь применение кинетического уравнения (1.2) к задаче о вычислении скорости изотермического скольжения квантового Бозе–газа. При этом ограничимся рассмотрением малых градиентов [ $g_v$]{}, что позволяет линеаризовать задачу. В этом случае температура и концентрация газа постоянны. Из соотношений (2.1) следует, что величина $\mathbf{u}$ совпадает с массовой скоростью газа (1.4). Кроме того, течение газа предположим стационарным.
Линеаризуем задачу относительно равновесной функции распределения Бозе—Эйнштейна (бозеана) $f_B$ $$f_B=\dfrac{1}{-1+\exp\left(\dfrac{{{\mathcal{E}}}-\mu}{kT}\right)}, \qquad
{{\mathcal{E}}}=\dfrac{mv^2}{2}.$$
Начнем с линеаризации локально равновесной функции распределения $f_{eq}$. Ее линеаризуем относительно бозеана $f_B$ по массовой скорости $\mathbf{u}$: $$f_{eq}=f_{eq}\Big|_{\mathbf{u}=0}+\dfrac{\partial f_{eq}}{\partial
\mathbf{u}}\Big|_{\mathbf{u}=0}\cdot\mathbf{u},$$ что приводит к выражению $$f_{eq}=f_B(v)+g_F(v)\dfrac{mv_y}{kT}u_y,
\eqno{(2.2)}$$ в котором $f_B(v)$ – абсюлютный бозеан (см. рис. 1), $$f_B(v)=\dfrac{1}{-1+\exp\Big(\dfrac{mv^2}{2kT}-\dfrac{\mu}{kT}\Big)}$$ и $$g_B(v)=\dfrac{\exp\Big(\dfrac{mv^2}{2kT}-\dfrac{\mu}{kT}\Big)}
{\Big[-1+\exp\Big(\dfrac{mv^2}{2kT}-\dfrac{\mu}{kT}\Big)\Big]^2}.$$
Функция $g_F(v)$ называется функцией Эйнштейна (см. рис. 2).
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Введем безразмерную скорость $\mathbf{C}=\sqrt{\beta}\mathbf{v},
\beta=\dfrac{m}{2kT}$, и безразмерный (приведенный) химический потенциал $\alpha=\dfrac{\mu}{kT}$. В этих переменных выражение (2.2) записывается как $$f_{eq}=f_B(C)+2g_B(C)C_yU_y(x),
\eqno{(2.3)}$$ при этом в (2.3) $U_y(x)=\sqrt{\beta}u_y(x)$ – безразмерная массовая скорость, $$f_B(C)=\dfrac{1}{-1+\exp(C^2-\alpha)}, \qquad
g_B(C)=\dfrac{\exp(C^2-\alpha)}{\big[-1+\exp(C^2-\alpha)\big]^2},$$ (графики этих функций см. на рис. 1 и 2).
Согласно (2.3) функцию распределения будем искать в виде $$f=f(x,\mathbf{C})=f_B(C)+g_B(C)C_yh(x,C_x).
\eqno{(2.4)}$$
Подставляя (2.3) и (2.4) в уравнение (1.2), приходим к уравнению $$C_x\dfrac{\partial h}{\partial x}=\nu
\sqrt{\beta}(2U_y(x)-h(x,C_x)).$$
Далее удобно ввести безразмерную координату $x_1=x\nu\sqrt{\beta}$. Получим следующее уравнение $$C_x\dfrac{\partial h}{\partial x_1}=2U_y(x_1)-h_1(x_1,C_x).
\eqno{(2.5)}$$
Размерный градиент $g_v=\Big(\dfrac{du_y(x)}{dx}\Big)_{x\to +\infty}$ при переходе к безразмерной координате $x_1$ преобразуется следующим образом: $$G_v=\Big(\dfrac{dU_y(x_1)}{dx_1}\Big)_{x_1\to +\infty}=\sqrt{\beta}
\Big(\dfrac{du_y}{dx}\Big)_{x\to+\infty}\cdot\dfrac{dx}{dx_1},$$ откуда $$G_v=\dfrac{g_v}{\nu}.$$
Массовая скорость газа может быть найдена из закона сохранения импульса (2.1), который в безразмерных параметрах имеет вид $$\int C_y(f_{eq}-f)d\Omega=0,$$ или, $$\int C_y^2g_B(C)\Big[2U_y(x_1)-h(x_1,C_x)\Big]d^3C=0,$$ откуда получаем $$U_y(x_1)=\dfrac{\displaystyle\int C_y^2g_B(C)h(x_1,C_x)d^3C}
{\displaystyle 2\int C_y^2g_B(C)d^3C}.
\eqno{(2.6)}$$
Заметим, что после линеаризации массовой скорости (1.4), т.е. после подстановки в (1.4) выражения (2.4), приходим к следующему выражению $$U_y(x_1)=\dfrac{\displaystyle\int C_y^2g_B(C)h(x_1,C_x)d^3C}
{\displaystyle \int f_B(C)d^3C}.
\eqno{(2.7)}$$
Знаменатели в (2.6) и (2.7) имеют различные выражения. Покажем, что эти выражения совпадают.
Перейдем к сферическим координатам $$C_x=C\cos \theta,\qquad C_y=C\sin \theta\cos\varphi,\qquad
C_z=C\sin \theta \sin \varphi,\qquad$$ $$d^3C=C^2\sin\theta d\varphi dC.$$
Теперь получаем, что знаменатель из (2.7) равен $$\int f_B(C)d^3C=4\pi \int \dfrac{C^2dC}{-1+\exp(C^2-\alpha)}=
4\pi f_2^B(\alpha),$$ где $f_2^B(\alpha)$ – второй (второго порядка) полупространственный момент абсолютного бозеана $$f_2^F(\alpha)=\int\limits_{0}^{\infty}f_B(C)C^2dC.$$
После интегрирования по частям получаем $
f_2^B(\alpha)=-\dfrac{1}{2}l_0^B(\alpha),
$ где $$l_0^B(\alpha)=\int\limits_{0}^{\infty}\ln(1-\exp(\alpha-C^2))dC=
\dfrac{1}{2}\int\limits_{-\infty}^{\infty}\ln(1-\exp(\alpha-C^2))dC.$$
Знаменатель из (2.6) равен $$2\int C_y^2 g_B(C)d^3C=\dfrac{8\pi}{3}g_4^B(\alpha),$$ где $$g_4^B(\alpha)=\dfrac{3}{2}f_2^B(\alpha).$$ Следовательно, знаменатель из (2.6) равен: $$2\int C_y^2
g_B(C)d^3C=\dfrac{8\pi}{3}\cdot\dfrac{3}{2}f_2^B(\alpha),$$ что и означает совпадение формул (2.6) и (2.7).
В числителе (2.6) удобнее перейти к цилиндрическим координатам, полагая $C^2=C_x^2+C_{\bot}^2$, $C_y=C_{\bot}\sin\varphi$, $d^3C=C_{\bot}dC_{\bot}dC_xd\varphi$. Далее получаем: $$\int C_y^2g_B(C)h(x_1,C_x)d^3C=\int\limits_{-\infty}^{\infty}
h(x_1,C_x)dC_x\int\limits_{0}^{\infty}C_{\bot}^3g_B(C)dC_{\bot}
\int\limits_{0}^{2\pi}\cos^2\varphi d\varphi=$$ $$=\pi\int\limits_{-\infty}^{\infty}h(x_1,C_x)dC_x\int\limits_{0}^{\infty}
\dfrac{\exp(C_x^2+C_{\bot}^2-\alpha)C_{\bot}^3dC_{\bot}}
{[-1+\exp(C_x^2+C_{\bot}^2-\alpha)]^2}.$$
Вычисляя внутренний интеграл по частям, имеем: $$\int
C_y^2g_B(C)h(x_1,C_x)d^3C=-\dfrac{\pi}{2}\int\limits_{-\infty}^{\infty}
\ln(1-\exp(\alpha-C_x^2))h(x_1,C_x)dC_x.$$
Следовательно, массовая скорость вычисляется по формуле $$U_y(x_1)=\dfrac{1}{4l_0^B(\alpha)}\int\limits_{-\infty}^{\infty}
\ln(1-\exp(\alpha-C_x^2))h(x_1,C_x)dC_x.
\eqno{(2.8)}$$
Введем функцию $$K_B(\mu,\alpha)=\dfrac{\ln(1-\exp(\alpha-\mu^2))}{2l_0^B(\alpha)}=
\dfrac{\ln(1-\exp(\alpha-\mu^2))}{\int\limits_{-\infty}^{\infty}
\ln(1-\exp(\alpha-\tau^2))d\tau},
\eqno{(2.9)}$$ где $\mu=C_x$.
Эта функция обладает свойством $$\int\limits_{-\infty}^{\infty}K_B(\mu,\alpha)d\mu \equiv 1, \qquad
\forall \alpha\in(-\infty,+\infty).$$
Семейство функций $K_B(\mu,\alpha)=\dfrac{\ln(1-e^{\alpha-\mu^2})}{2l_0^B(\alpha)}$ называется ядром кинетического уравнения (см. рис. 3).
{width="16cm" height="10cm"}
Массовая скорость согласно (2.8) и (2.9) равна $$U_y(x_1)=\dfrac{1}{2}\int\limits_{-\infty}^{\infty}K_B(\mu',\alpha)
h(x_1,\mu')d\mu'.
\eqno{(2.10)}$$ Таким образом, согласно (2.10) уравнение (2.5) представим в стандартном для теории переноса виде: $$\mu\dfrac{\partial h}{\partial x_1}+h(x_1,\mu)=
{\int\limits}_{-\infty}^\infty K_B(\mu',\alpha)h(x_1,\mu')\,d\mu',
\eqno{(2.11)}$$ или, в явном виде $$\mu\dfrac{\partial h}{\partial x_1}+h(x_1,\mu)=
\dfrac{1}{2l_0^B(\alpha)}{\int\limits}_{-\infty}^\infty
\ln(1-\exp(\alpha-{\mu'}^2))h(x_1,\mu')\,d\mu'.
\eqno{(2.12)}$$
Предельный случай уравнения
---------------------------
Рассмотрим предельный случая уравнения (2.12) при $\alpha\to
-\infty$. В этом случае $$\lim_{\alpha \to -\infty}K_B(\mu,\alpha)=\lim_{\alpha \to -\infty}
\dfrac{\ln(1+\exp(\alpha-\mu^2))}
{{\int\limits}_{-\infty}^\infty \ln(1+\exp(\alpha-u^2))\,du}=$$ $$=\dfrac{\exp(-\mu^2)}{{\int\limits}_{-\infty}^\infty \exp(-u^2)\,du}=
\dfrac{\exp(-\mu^2)}{\sqrt{\pi}}$$ и мы получаем уравнение $$\mu\dfrac{\partial h}{\partial x}+h(x_1,\mu)=
\dfrac{1}{\sqrt{\pi}}{\int\limits}_{-\infty}^\infty\exp(-{\mu'}^2)h(x_1,\mu')\,d\mu',$$ которое является БГК–уравнением для одноатомных газов с постоянной частотой столкновений молекул.
Постановка задачи Крамерса
--------------------------
Задание градиента массовой скорости (2.2) означает, что вдали от стенки распределение массовой скорости в полупространстве имеет линейный рост $$u_y(x)=u_{sl}(\alpha)+g_vx, \qquad x\to +\infty,$$ где $u_{sl}(\alpha)$ – неизвестная скорость скольжения.
Умножая это равенство на $\sqrt{\beta}$ и учитывая связь размерного и безразмерного градиентов $g_v=\nu G_v$, для безразмерной массовой скорости получаем $$U_y(x_1)=U_{sl}(\alpha)+G_vx_1,\qquad x_1\to +\infty.
\eqno{(2.13)}$$
Зеркально–диффузное отражение Бозе–частиц от поверхности означает, что последние отражаются от стенки, имея бозеевское распределение, т.е. $$f(x=0,\mathbf{v})=qf_B(v)+(1-q)f(x=0,-v_x,v_y,v_z),\qquad v_x>0,
\eqno{(2.14)}$$ где $0\leqslant q \leqslant 1$, $q$ – коэффициент диффузности, $f_B(v)$ – абсолютный бозеан.
В уравнении (2.14) параметр $q$ (коэффициент диффузности) – часть молекул, рассеивающихся границей диффузно, $1-q$ – часть молекул, рассеивающихся зеркально.
Учитывая, что функцию распределения мы ищем в виде (2.4), из условия (2.14) получаем граничное условие на стенке на функцию $h(x_1,\mu)$: $$h(0,\mu)=(1-q)h(0,-\mu),\qquad \mu>0.
\eqno{(2.15)}$$
Вторым граничным условием является граничное условие “вдали от стенки”. Этим условием является соотношение (2.13). Преобразуем это условие на функцию $h(x_1,\mu)$. Условие (2.13) означает, что вдали от стенки массовая скорость переходит в свое асимптотическое распределение $$U_y^{as}(x_1)=U_{sl}(\alpha)+G_vx_1.$$ Выражение для массовой скорости (2.8) означает, что вдали от стенки функция $h(x_1,\mu)$ переходит в свое асимптотическое распределение $$h_{as}(x_1,\mu)=2U_{sl}(\alpha)+2G_v(x_1-\mu),$$ называемое распределением Чепмена—Энскога (см., например, [@Ferziger], [@Cerc73], [@Cerc62]).
Таким образом, вторым граничным условием является условие: $$h(x_1,\mu)=2U_{sl}(\alpha)+2G_v(x_1-\mu),\qquad x\to+\infty.
\eqno{(2.16)}$$
Теперь задача Крамерса при условии полного диффузного отражения Бозе–частиц от стенки сформулирована полностью и состоит в решении уравнения (2.12) с граничными условиями (2.15) и (2.16). При этом требуется определить безразмерную скорость скольжения $U_{sl}(\alpha)$, величина градиента $G_v$ считается заданной.
Включение граничных условий в кинетическое уравнение
====================================================
Продолжим функцию распределения на сопряженное полупространство симметричным образом: $$f(t,x,\mathbf{v})=f(t,-x, -v_x,v_y,v_z).
\eqno{(3.1)}$$
Продолжение согласно (3.1) на полупространство $x<0$ позволяет включить граничные условия в уравнения задачи.
Такое продолжение функции распределения на полупространство $x<0$ позволяет фактически рассматривать две задачи, одна из которых определена в “положительном” полупространстве $x>0$, вторая – в отрицательном “полупространстве” $x<0$.
Сформулируем зеркально–диффузные граничные условия для функции распределения соответственно для “положительного” и для “отрицательного” полупространств: $$f(t,+0, \mathbf{v})=qf_0(v)+(1-q)f(t,+0,-v_x,v_y, v_z), \quad v_x>0,
\eqno{(3.2)}$$ $$f(t,-0, \mathbf{v})=qf_0(v)+(1-q)f(t,-0, -v_x,v_y, v_z),
\quad v_x<0.
\eqno{(3.3)}$$ где $q$ – коэффициент диффузности, $0 \leqslant q \leqslant 1$.
В уравнениях (3.2) и (3.3) параметр $q$ (коэффициент диффузности) – часть молекул, рассеивающихся границей диффузно, $1-q$ – часть молекул, рассеивающихся зеркально, т.е. уходящие от стенки молекулы имеют максвелловское распределение по скоростям.
Далее безразмерную координату $x_1$ снова будем обозначать через $x$.
Согласно (2.4) и (3.1) мы имеем: $$h(x,\mu)=h(-x,-\mu), \qquad \mu>0.$$
На функцию $h(x,\mu)$ в “положительном” и “отрицательном” полупространствах получаем одно и то же уравнение уравнение: $$\mu\dfrac{\partial h}{\partial x}+h(x,\mu)=
\int\limits_{-\infty}^{\infty}K_B(t,\alpha)h(x,t)\,dt,
\eqno{(3.4)}$$ и соответственно следующие граничные условия: $$h(+0,\mu)=(1-q)h(+0,-\mu)=(1-q)h(-0,\mu), \quad \mu>0,$$ $$h(-0,\mu)=(1-q)h(-0,-\mu)=(1-q)h(+0,\mu), \quad \mu<0.$$
Правая часть уравнения (3.4) есть удвоенная массовая скорость газа: $$U(x)=\int\limits_{-\infty}^{\infty}K_B(t,\alpha)h(x,t)dt.$$
Представим функцию $h(x,\mu)$ в виде: $$h(x,\mu)=h^{\pm}_{as}(x,\mu)+h_c(x,\mu),
\quad \text{если}\quad
\pm x>0,$$ где асимптотическая часть функции распределения (так называемая чепмен—энскоговская функция распределения) $$h_{as}^{\pm}(x,\mu)=2U_{sl}(q,\alpha)\pm 2G_v(x-\mu),
\quad \text{если}\quad
\pm x>0,
\eqno{(3.5)}$$ также является решением кинетического уравнения (3.4).
Здесь $U_{sl}(q,\alpha)$ – есть искомая скорость изотермического скольжения (безразмерная).
Следовательно, функция $h_c(x,\mu)$ также удовлетворяет уравнению (3.4): $$\mu\dfrac{\partial h_c}{\partial x}+h_c(x,\mu)=
\int\limits_{-\infty}^{\infty}K_B(t,\alpha)h_c(x,t)dt.$$
Так как вдали от стенки ($x\to \pm \infty$) функция распределения $h(x,\mu)$ переходит в чепмен—энскоговскую $h_{as}^{\pm}(x,\mu)$, то для функции $h_c(x,\mu)$, отвечающей непрерывному спектру, получаем следующее граничное условие: $h_c(\pm \infty,\mu)=0.$
Отсюда для массовой скорости газа получаем: $$U_c(\pm \infty)=0.
\eqno{(3.6)}$$
Отметим, что в равенстве (3.5) знак градиента в “отрицательном” полупространстве меняется на противоположный. Поэтому условие (3.6) выполняется автоматически для функций $h_{as}^{\pm}(x,\mu)$.
Тогда граничные условия переходят в следующие: $$h_c(+0,\mu)=$$$$=-h_{as}^+(+0,\mu)+(1-q)h_{as}^+(+0,-\mu)+$$$$+
(1-q)h_c(+0,-\mu),
\quad \mu>0,$$ $$h_c(-0,\mu)=$$$$=-h_{as}^-(-0,\mu)+(1-q)h_{as}^-(-0,-\mu)$$$$+
(1-q)h_c(-0,-\mu), \quad \mu<0.$$
Обозначим $$h_0^{\pm}(\mu)=-2qU_{sl}(q,\alpha)+(2-q)2G_v|\mu|.$$
И перепишем предыдущие граничные условия в виде: $$h_c(+0,\mu)=h_0^+(\mu)+(1-q)h_c(+0,-\mu), \quad \mu>0,$$ $$h_c(-0,\mu)=h_0^-(\mu)+(1-q)h_c(-0,-\mu), \quad \mu<0,$$ где $$h_0^{\pm}(\mu)=-h_{as}^{\pm}(0,\mu)+(1-q)h_{as}^{\pm}(0,-\mu)=$$$$=-2qU_{sl}(q,\alpha)+(2-q)2G_v|\mu|.$$
Учитывая симметричное продолжение функции распределения, имеем $$h_c(-0,-\mu)=h_c(+0,+\mu),\qquad
h_c(+0,-\mu)=h_c(-0,+\mu).$$ Следовательно, граничные условия перепишутся в виде: $$h_c(+0,\mu)=h_0^+(\mu)+(1-q)h_c(-0,\mu), \quad \mu>0,
\eqno{(3.7)}$$ $$h_c(-0,\mu)=h_0^-(\mu)+(1-q)h_c(+0,\mu), \quad \mu<0.
\eqno{(3.8)}$$
Включим граничные условия (3.7) и (3.8) в кинетическое уравнение следующим образом: $$\mu \dfrac{\partial h_c}{\partial x}+h_c(x,\mu)=2U_c(x)+
|\mu|\Big[h_0^{\pm}(\mu)-q h_c(\pm 0,\mu)\Big]
\delta(x),
\eqno{(3.9)}$$ где $U_c(x)$ – часть массовой скорости, отвечающая непрерывному спектру, $$2U_c(x)=\int\limits_{-\infty}^{\infty}K_B(t,\alpha)h_c(x,t)\,dt.
\eqno{(3.10)}$$
Уравнение (3.9) содержит два уравнения. В “положительном” полупространстве, т.е. при $x>0$ в правой части уравнения (3.9) следует взять верхний знак “плюс”, а в “нижнем” полупространстве, т.е. при $x<0$ в правой части того же уравнения следует взять знак “минус”.
В самом деле, пусть, например, $\mu>0$. Проинтегрируем обе части уравнения (3.9) по $x$ от $-\varepsilon$ до $+\varepsilon$. В результате получаем равенство: $$h_c(+\varepsilon,\mu)-h_c(-\varepsilon,\mu)=h_0^+(\mu)-
qh_c(-\varepsilon,\mu),$$ откуда переходя к пределу при $\varepsilon\to 0$ в точности получаем граничное условие (3.7).
На основании определения массовой скорости (3.10) заключаем, что для нее выполняется условие (3.6): $U_c(+\infty)=0.$ Следовательно, в полупространстве $x>0$ профиль массовой скорости газа вычисляется по формуле: $$U(x)=U_{as}(x)+\dfrac{1}{2}\int\limits_{-\infty}^{\infty}
K_B(t,\alpha)h_c(x,t)dt,
\eqno{(3.11)}$$ а вдали от стенки имеет следующее линейное распределение: $$U_{as}(x)=U_{sl}(q,\alpha)+G_vx, \qquad x\to +\infty.
\eqno{(3.12)}$$
Кинетическое уравнение во втором и четвертом квадрантах фазового пространства
=============================================================================
Решая уравнение (3.9) при $x>0,\,\mu<0$, считая заданным массовую скорость $U(x)$, получаем, удовлетворяя граничным условиям (3.8), следующее решение: $$h_c^+(x,\mu)=-\dfrac{1}{\mu}\exp(-\dfrac{x}{\mu})
\int\limits_{x}^{+\infty} \exp(+\dfrac{t}{\mu})2U_c(t)\,dt.
\eqno{(4.1)}$$
Аналогично при $x<0,\,\mu>0$ находим: $$h_c^-(x,\mu)=-\dfrac{1}{\mu}\exp(-\dfrac{x}{\mu})
\int\limits_{x}^{-\infty} \exp(+\dfrac{t}{\mu})2U_c(t)\,dt.
\eqno{(4.2)}$$
Теперь уравнения (3.9) и (3.10) можно переписать, заменив второй член в квадратной скобке из (3.9) согласно (4.1) и (4.2), в виде: $$\mu\dfrac{\partial h_c}{\partial x}+h_c(x,\mu)=2U_c(x)+
|\mu|\Big[h_0^{\pm}(\mu)-qh_c^{\pm}(0,\mu)\Big]\delta(x),
\eqno{(4.3)}$$ $$2U_c(x)=\int\limits_{-\infty}^{\infty}K_B(t,\alpha)h_c(x,t)dt.
\eqno{(4.4)}$$
В равенствах (4.3) граничные значения $h_c^{\pm}(0,\mu)$ выражаются через составляющую массовой скорости, отвечающей непрерывному спектру: $$h_{c}^{\pm}(0,\mu)=-\dfrac{1}{\mu}e^{-x/\mu} \int\limits_{0}^{\pm
\infty}e^{t/\mu}2U_c(t)dt=h_c(\pm 0,\mu).$$
Решение уравнений (4.4) и (4.3) ищем в виде интегралов Фурье: $$2U_c(x)=\dfrac{1}{2\pi}\int\limits_{-\infty}^{\infty}
e^{ikx}E(k)\,dk,\qquad
\delta(x)=\dfrac{1}{2\pi}\int\limits_{-\infty}^{\infty}
e^{ikx}\,dk,
\eqno{(4.5)}$$ $$h_c(x,\mu)=\dfrac{1}{2\pi}\int\limits_{-\infty}^{\infty}
e^{ikx}\Phi(k,\mu)\,dk.
\eqno{(4.6)}$$
При этом функция распределения $h_c^+(x,\mu)$ выражается через спектральную плотность $E(k)$ массовой скорости следующим образом: $$h_c^+(x,\mu)=-\dfrac{1}{\mu}\exp(-\dfrac{x}{\mu})
\int\limits_{x}^{+\infty} \exp(+\dfrac{t}{\mu})dt
\dfrac{1}{2\pi}
\int\limits_{-\infty}^{+\infty}e^{ikt}E(k)\,dk=$$ $$=\dfrac{1}{2\pi}\int\limits_{-\infty}^{\infty}\dfrac{e^{ikx}
E(k)}{1+ik\mu}dk.
$$
Аналогично,
$$h_c^-(x,\mu)=\dfrac{1}{2\pi}
\int\limits_{-\infty}^{\infty}\dfrac{e^{ikx}
E(k)}{1+ik\mu}dk.
$$
Таким образом, $$h_c^{\pm}(x,\mu)=\dfrac{1}{2\pi}
\int\limits_{-\infty}^{\infty}\dfrac{e^{ikx}
E(k)}{1+ik\mu}dk.
$$
Используя четность функции $E(k)$ далее получаем: $$h_c^{\pm}(0,\mu)=\dfrac{1}{2\pi}
\int\limits_{-\infty}^{\infty}\dfrac{E(k)}{1+ik\mu}dk=
\dfrac{1}{2\pi}\int\limits_{-\infty}^{\infty}
\dfrac{E(k)\,dk}{1+k^2\mu^2}=
\dfrac{1}{\pi}\int\limits_{0}^{\infty}\dfrac{E(k)\,dk}{1+
k^2\mu^2}.
\eqno{(4.7)}$$
Теперь с помощью равенства (4.7) уравнение (4.3) можно переписать в виде: $$\mu\dfrac{\partial h_c}{\partial x}+h_c(x,\mu)=2U_c(x)+
|\mu|\Bigg[h_0^{\pm}(\mu)-\dfrac{q}{\pi}\int\limits_{0}^{\infty}
\dfrac{E(k)\,dk}{1+k^2\mu^2}\Bigg]\delta(x),
\eqno{(4.3')}$$
Характеристическая система
==========================
Теперь подставим интегралы Фурье (4.6) и (4.5), а также равенство (4.7) в уравнения (4.3) и (4.4). Получаем характеристическую систему уравнений: $$\Phi(k,\mu)(1+ik\mu)=$$$$=E(k)+|\mu|\Bigg[-2qU_{sl}(q,\alpha)+2(2-q)G_v|\mu|
-\dfrac{q}{\pi}
\int\limits_{0}^{\infty}\dfrac{E(k_1)dk_1}{1+k_1^2\mu^2}\Bigg],
\eqno{(5.1)}$$ $$E(k)=\int\limits_{-\infty}^{\infty}K(t,\alpha)\Phi(k,t)dt.
\eqno{(5.2)}$$
Из уравнения (5.1) получаем: $$\Phi(k,\mu)=\dfrac{E(k)}{1+ik\mu}+$$$$+
\dfrac{|\mu|}{1+ik\mu}\Bigg[-2qU_{sl}(q,\alpha)+2(2-q)G_v|\mu|-\dfrac{q}{\pi}
\int\limits_{0}^{\infty}\dfrac{E(k_1)dk_1}{1+k_1^2\mu^2}\Bigg],
\eqno{(5.3)}$$
Подставим выражение для функции $\Phi(k,\mu)$, определенное равенством (5.3), в (5.2). Получаем, что: $$E(k)L(k)=-2qU_{sl}(q,\alpha)T_1(k)+2(2-q)G_v T_2(k)-$$ $$-\dfrac{q}{\pi}\int\limits_{0}^{\infty}
E(k_1)dk_1\int\limits_{-\infty}^{\infty}
\dfrac{K_B(t,\alpha)|t|dt}{(1+ikt)(1+k_1^2t^2)}.
\eqno{(5.4)}$$ Здесь $$T_n(k)=2\int\limits_{0}^{\infty}
\dfrac{K_B(t,\alpha)t^n\,dt}{1+k^2t^2},\quad n=0,1,2,3\cdots,$$ причем для четных $n$ $$T_n(k)=2\int\limits_{0}^{\infty}
\dfrac{K_B(t,\alpha)t^n\,dt}{1+k^2t^2}=
\int\limits_{-\infty}^{\infty}
\dfrac{K_B(t,\alpha)t^n\,dt}{1+k^2t^2},\quad n=0,2,4,\cdots,$$ кроме того, $$L(k)=1-\int\limits_{-\infty}^{\infty}
\dfrac{K_B(t,\alpha)dt}{1+ikt}.$$ Нетрудно видеть, что $$L(k)=1-
\int\limits_{-\infty}^{\infty}\dfrac{K_B(t,\alpha)dt}{1+k^2t^2}=$$$$=
1-2\int\limits_{0}^{\infty}\dfrac{K_B(t,\alpha)dt}
{1+k^2t^2}=2k^2 \int\limits_{0}^{\infty}
\dfrac{K_B(t,\alpha)t^2\;dt}{1+k^2t^2}=k^2 \int\limits_{-\infty}^{\infty}
\dfrac{K_B(t,\alpha)t^2\;dt}{1+k^2t^2},$$ или, кратко, $$L(k)=k^2 T_2(k).$$
Кроме того, внутренний интеграл в (5.4) преобразуем и обозначим следующим образом: $$\int\limits_{-\infty}^{\infty}
\dfrac{K_B(t,\alpha)|t|dt}{(1+ikt)(1+k_1^2t^2)}
=2\int\limits_{0}^{\infty}\dfrac{K_B(t,\alpha)t\,dt}
{(1+k^2t^2)(1+k_1^2t^2)}=J(k,k_1).$$
Заметим, что $$J(k,0)=T_1(k), \qquad J(0,k_1)=T_1(k_1).$$
Перепишем теперь уравнение (5.4) с помощью предыдущего равенства в следующем виде: $$E(k)L(k)=-2qU_{sl}(q,\alpha)T_1(k)+2(2-q)G_v T_2(k)-$$$$-
\dfrac{q}{\pi}\int\limits_{0}^{\infty} J(k,k_1)E(k_1)\,dk_1.
\eqno{(5.5)}$$
Уравнение (5.5) есть интегральное уравнение Фредгольма второго рода.
Ряд Неймана
===========
Считая градиент массовой скорости в уравнении (5.5) заданным, разложим решения характеристической системы (5.3) и (5.5) в ряд по степеням коэффициента диффузности $q$: $$E(k)=G_v2(2-q)\Big[E_0(k)+q\,E_1(k)+q^2\,E_2(k)+\cdots\big],
\eqno{(6.1)}$$ $$\Phi(k,\mu)=G_v2(2-q)\Big[
\Phi_0(k,\mu)+q\Phi_1(k,\mu)+q^2\Phi_2(k,\mu)+\cdots\Big].
\eqno{(6.2)}$$
Скорость скольжения $U_{sl}(q,\alpha)$ при этом будем искать в виде $$U_{sl}(q,\alpha)=G_v\dfrac{2-q}{q}
\Big[U_0+U_1q+U_2q^2+\cdots+U_nq^n+\cdots\Big].
\eqno{(6.3)}$$
Подставим ряды (6.1)–(6.3) в уравнения (5.3) и (5.5). Получаем следующую систему уравнений: $$(1+ik\mu)[\Phi_0(k,\mu)+\Phi_1(k,\mu)q+\Phi_2(k,\mu)q^2+\cdots]=$$ $$=[E_0(k)+E_1(k)q+E_2(k)q^2+\cdots]-(U_0+U_1q+U_2q^2+\cdots)|\mu|+$$ $$+\mu^2-|\mu|\dfrac{q}{\pi}\int\limits_{0}^{\infty}
\dfrac{E_0(k_1)+E_1(k_1)q+E_2(k_1)q^2+\cdots}{1+k_1^2\mu^2}dk_1,$$ $$[E_0(k)+E_1(k)q+E_2(k)q^2+\cdots]L(k)=-[U_0+U_1q+U_2q^2+\cdots]T_1(k)+
T_2(k)-$$ $$-\dfrac{q}{\pi}\int\limits_{0}^{\infty}J(k,k_1)
[E_0(k_1)+E_1(k_1)q+E_2(k_1)q^2+\cdots]dk_1.$$
Последние интегральные уравнения распадаются на эквивалентную бесконечную систему уравнений. В нулевом приближении получаем следующую систему уравнений: $$E_0(k)L(k)=T_2(k)-U_0T_1(k),
\eqno{(6.4)}$$ $$\Phi_0(k,\mu)(1+ik\mu)=E_0(k)+\mu^2-U_0|\mu|,
\eqno{(6.5)}$$
В первом приближении: $$E_1(k)L(k)=-U_1T_1(k)-
\dfrac{1}{\pi}\int\limits_{0}^{\infty}
J(k,k_1)E_0(k_1)dk_1,
\eqno{(6.6)}$$ $$\Phi_1(k,\mu)(1+ik\mu)=E_1(k)-U_1|\mu|-\dfrac{|\mu|}{\pi}
\int\limits_{0}^{\infty}\dfrac{E_0(k_1)dk_1}{1+k_1^2\mu^2}.
\eqno{(6.7)}$$
Во втором приближении: $$E_2(k)L(k)=-U_2T_1(k)-\dfrac{1}{\pi}
\int\limits_{0}^{\infty}J(k,k_2)E_1(k_2)\,dk_2,
\eqno{(6.8)}$$ $$\Phi_2(k,\mu)(1+ik\mu)=E_2(k)-U_2|\mu|-
\dfrac{|\mu|}{\pi}\int\limits_{0}^{\infty}\dfrac{E_1(k_2)dk_2}
{1+k_2^2\mu^2}.
\eqno{(6.9)}$$
В $n$–м приближении получаем: $$E_n(k)L(k)=-U_nT_1(k)-\dfrac{1}{\pi}
\int\limits_{0}^{\infty}J(k,k_n)E_{n-1}(k_n)dk_n,
\eqno{(6.10)}$$ $$\Phi_n(k,\mu)(1+ik\mu)=E_n(k)-U_n|\mu|-\hspace{5cm}$$ $$\hspace{4.5cm}
- \dfrac{|\mu|}{\pi}
\int\limits_{0}^{\infty}\dfrac{E_{n-1}(k_n)dk_n}{1+k_n^2\mu^2},
\quad n=1,2,3,\cdots.
\eqno{(6.11)}$$
Нулевое приближение
-------------------
Из формулы (6.4) для нулевого приближения находим: $$E_0(k)=\dfrac{T_2(k)-U_0T_1(k)}{L(k)}.
\eqno{(6.12)}$$
Нулевое приближение массовой скорости на основании (6.12) равно: $$U_c^{(0)}(x)=G_v\dfrac{2-q}{2\pi}\int\limits_{-\infty}^{\infty}
e^{ikx}E_0(k)\,dk=$$$$=G_v\dfrac{2-q}{2\pi}
\int\limits_{-\infty}^{\infty}
e^{ikx}\dfrac{-U_0T_1(k)+T_2(k)}{L(k)}dk.
\eqno{(6.13)}$$
Согласно (6.13) наложим на нулевое приближение массовой скорости требование: $U_c(+\infty)=0$. Это условие приводит к тому, что подынтегральное выражение из интеграла Фурье (6.13) в точке $k=0$ конечно. Следовательно, мы должны устранить полюс второго порядка в точке $k=0$ у функции $E_0(k)$.
Замечая, что $$T_2(0)=\int\limits_{-\infty}^{\infty}t^2K_B(t,\alpha)dt=
\dfrac{1}{2l_0^B(\alpha)}\int\limits_{-\infty}^{\infty}
t^2\ln(1-e^{\alpha-t^2})dt=\dfrac{l_2^B(\alpha)}{l_0^B(\alpha)}.$$ $$T_1(0)=2\int\limits_{0}^{\infty}tK_B(t,\alpha)dt=
\dfrac{1}{l_0^B(\alpha)}\int\limits_{0}^{\infty}t\ln(1-e^{\alpha-t^2})dt=
\dfrac{l_1^B(\alpha)}{l_0^B(\alpha)},$$ находим нулевое приближение $U_0$: $$U_0=\dfrac{T_2(0)}{T_1(0)}=\dfrac{\int\limits_{-\infty}^{\infty}
t^2\ln(1-e^{\alpha-t^2})dt}
{2\int\limits_{0}^{\infty}t\ln(1-e^{\alpha-t^2})dt}=
\dfrac{l_2^B(\alpha)}{l_1^B(\alpha)}.
$$
Заметим, что $$U_0(-\infty)=\dfrac{\sqrt{\pi}}{2}=0.8862, \qquad
U_0(0)=0.7227.$$
Найдем числитель выражения (6.12): $$T_2(k)-U_0T_1(k)=T_2(k)-\dfrac{T_2(0)}{T_1(0)}T_1(k)=$$$$=
\dfrac{1}{T_1(0)}\Big[T_1(0)T_2(k)-T_2(0)T_1(k)\Big].$$
Замечая, что $$\dfrac{1}{1+k^2t^2}=1-\dfrac{k^2t^2}{1+k^2t^2},$$ получаем $$T_2(k)=T_2(0)-k^2T_4(k),\qquad T_1(k)=T_1(0)-k^2T_3(k).$$
Откуда $$T_1(0)T_2(k)-T_2(0)T_1(k)=k^2\Big[T_2(0)T_3(k)-T_1(0)T_4(k)\Big].$$
Следовательно, мы получаем: $$T_2(k)-U_0T_1(k)=\dfrac{k^2}{L(k)T_1(0)}\Big[T_2(0)T_3(k)-T_1(0)T_4(k)\Big],$$ или, учитывая, что $L(k)=k^2T_2(k)$, запишем предыдущее равенство короче, $$E_0(k)=\dfrac{\varphi_0(k)}{T_2(k)},$$ где $$\varphi_0(k)=\dfrac{T_2(0)T_3(k)-T_1(0)T_4(k)}{T_1(0)}.$$
Согласно (6.5) находим: $$\Phi_0(k,\mu)= \dfrac{E_0(k)+\mu^2-U_0|\mu|}
{1+ik\mu},
$$ и, следовательно, $$h_c^{(0)}(x,\mu)=\dfrac{1}{2\pi}\int\limits_{-\infty}^{\infty}
\Big[E_0(k)+\mu^2-U_0|\mu|\Big]
\dfrac{e^{ikx}dk}{1+ik\mu}.$$
Первое приближение
------------------
Перейдем к первому приближению. В первом приближении из уравнения (6.6) находим: $$E_1(k)=-\dfrac{1}{L(k)}\Big[U_1T_1(k)+\dfrac{1}{\pi}
\int\limits_{0}^{\infty}
\dfrac{J(k,k_1)}{T_2(k_1)}\varphi_0(k_1)dk_1\Big].
\eqno{(6.14)}$$
Первая поправка к массовой скорости имеет вид $$U_c^{(1)}(x)=G_v\dfrac{2-q}{2\pi}\int\limits_{-\infty}^{\infty}
e^{ikx}E_1(k)\,dk.
$$
Требование $U_c(+\infty)=0$ приводит к требованию конечности подынтегрального выражения в предыдущем интеграле Фурье. Устраняя полюс второго порядка в точке $k=0$, находим: $$U_1=-
\dfrac{1}{\pi T_1(0)}\int\limits_{0}^{\infty}
J(0,k_1)\dfrac{\varphi_0(k_1)}{T_2(k_1)}dk_1=$$ $$=-\dfrac{1}{\pi T_1(0)}
\int\limits_{0}^{\infty}\dfrac{T_1(k_1)}{T_2(k_1)}\varphi_0(k_1)\,dk_1=
-\dfrac{1}{\pi T_1(0)}
\int\limits_{0}^{\infty}T_1(k_1)E_0(k_1)\,dk_1.
\eqno{(6.15)}$$
Нетрудно проверить, что $$U_1(-\infty)\approx 0.1405,\qquad U_1(0)=0.1775.$$
Преобразуем с помощью (6.15) выражение в квадратной скобке из выражения (6.14): $$U_1T_1(k)+\dfrac{1}{\pi}
\int\limits_{0}^{\infty}J(k,k_1)\dfrac{\varphi_0(k_1)}{T_2(k_1)}dk_1=$$
$$=\dfrac{1}{\pi}\int\limits_{0}^{\infty}
J(k,k_1)\dfrac{\varphi_0(k_1)}{T_2(k_1)}dk_1-
\dfrac{T_1(k)}{\pi T_1(0)}\int\limits_{0}^{\infty}
T_1(k_1)\dfrac{\varphi_0(k_1)}{T_2(k_1)}dk_1=$$ $$=\dfrac{1}{\pi}\int\limits_{0}^{\infty}
\Big[J(k,k_1)-\dfrac{T_1(k)T_1(k_1)}{T_1(0)}\Big]E_0(k_1)dk_1.
\eqno{(4.16)}$$
Заметим, что $J(0,k_1)=T_1(k_1)$. Найдем выражение $$J(k,k_1)-\dfrac{T_1(k)T_1(k_1)}{T_1(0)}.$$ Рассмотрим разложение на элементарные дроби: $$\dfrac{1}{(1+k^2t^2)(1+k_1^2t^2)}=
\dfrac{[(1+k_1^2t^2)-k_1^2t^2][(1+k^2t^2)-k^2t^2]}
{(1+k^2t^2)(1+k_1^2t^2)}=$$ $$=1-\dfrac{k_1^2t^2}{1+k_1^2t^2}-\dfrac{k^2t^2}{1+k^2t^2}+
\dfrac{k^2k_1^2\,t^4}{(1+k^2t^2)(1+k_1^2t^2)}$$
С помощью этого разложения преобразуем интеграл $$J(k,k_1)=2\int\limits_{0}^{\infty}\dfrac{K_B(t,\alpha)t\,dt}
{(1+k^2t^2)(1+k_1^2t^2)}.$$
Получаем следующее представление этого интеграла: $$J(k,k_1)=T_1(0)-k_1^22\int\limits_{0}^{\infty}\dfrac{K_B(t,\alpha)t^3dt}
{1+k_1^2t^2}-k^2 2\int\limits_{0}^{\infty}
\dfrac{K_B(t,\alpha)t^3dt}{1+k^2t^2}+$$$$+k^2k_1^22
\int\limits_{0}^{\infty}\dfrac{K_B(t,\alpha)t^5dt}{(1+k^2t^2)(1+k_1^2t^2)},$$ или $$J(k,k_1)=T_1(0)-k^2T_3(k)-k_1^2T_3(k_1)+k^2k_1^2J_5(k,k_1),$$ где $$J_n(k,k_1)=2\int\limits_{0}^{\infty}
\dfrac{K_B(t,\alpha)t^ndt}{(1+k^2t^2)(1+k_1^2t^2)}, \qquad n=3,5.$$
Теперь ясно, что $$J(k,k_1)-\dfrac{T_1(k)T_1(k_1)}{T_1(0)}=
k^2k_1^2\Big[J_5(k,k_1)-\dfrac{T_3(k)T_3(k_1)}{T_1(0)}\Big].$$ Представим это выражение в виде $$J(k,k_1)-\dfrac{T_1(k)T_1(k_1)}{T_1(0)}=k^2 S(k,k_1),$$ где $$S(k,k_1)=k_1^2\Big[J_5(k,k_1)- \dfrac{T_3(k)T_3(k_1)}{T_1(0)}\Big].$$
Вернемся к выражению (6.14). С помощью (6.16) теперь получаем: $$E_1(k_1)=-\dfrac{1}{\pi T_2(k_1)}
\int\limits_{0}^{\infty}
\dfrac{S(k_1,k_2)}{T_2(k_2)}\varphi_0(k_2)\,dk_2,
\eqno{(6.17)}$$ или, кратко, $$E_1(k_1)=\dfrac{\varphi_1(k_1)}{T_2(k_1)},$$ где $$\varphi_1(k_1)=-\dfrac{1}{\pi}\int\limits_{0}^{\infty}
\dfrac{S(k_1,k_2)}{T_2(k_2)}\varphi_0(k_2)\,dk_2,$$ или $$\varphi_1(k_1)=-\dfrac{1}{\pi}\int\limits_{0}^{\infty}
S(k_1,k_2)E_0(k_2)\,dk_2.$$
Теперь подставляя (6.17) в (6.7) находим первое приближение спектральной плотности функции распределения: $$\Phi_1(k,\mu)=\dfrac{1}{1+ik\mu}\Big[E_1(k)
-U_1|\mu|-\dfrac{|\mu|}{\pi}
\int\limits_{0}^{\infty}\dfrac{E_0(k_1)\,dk_1}
{1+k_1^2\mu^2}\Big].$$
Второе приближение
------------------
Перейдем ко второму приближению задачи – уравнения (6.8) и (6.9). Из уравнения (6.8) находим: $$E_2(k)=-\dfrac{1}{L(k)}\Big[U_2T_1(k)+\dfrac{1}{\pi}
\int\limits_{0}^{\infty}J(k,k_1)E_1(k_1)\,dk_1\Big].
\eqno{(6.18)}$$
Вторая поправка к массовой скорости имеет вид: $$U_c^{(2)}(x)=G_v\dfrac{2-q}{2\pi}\int\limits_{-\infty}^{\infty}
e^{ikx}E_2(k)\,dk.$$
Условие $U_c(+\infty)=0$ приводит к требованию ограниченности функции $E_2(k)$ в точке $k=0$. Устраняя полюс второго порядка в точке $k=0$ в правой части равенства для $E_2(k)$, находим: $$U_2=-\dfrac{1}{\pi T_1(0)}\int\limits_{0}^{\infty}J(0,k_1)E_1(k_1)dk_1=$$$$=
-\dfrac{1}{\pi T_1(0)}\int\limits_{0}^{\infty}T_1(k_1)E_1(k_1)dk_1.
\eqno{(6.19)}$$
Нетрудно проверить, что $$U_2(-\infty)\approx -0.0116,\qquad U_2(0)=-0.0214.$$
Формулу (6.19) преобразуем к следующему виду: $$U_2=\dfrac{1}{\pi^2T_1(0)}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}
\dfrac{T_1(k_1)S(k_1,k_2)}{T_2(k_1)T_2(k_2)}\varphi_0(k_2)\,
dk_1dk_2.$$
Преобразуем выражение (6.18) с помощью равенства (6.19). Имеем: $$E_2(k)=-\dfrac{1}{L(k)}
\int\limits_{0}^{\infty}\Big[J(k,k_1)-\dfrac{T_1(k)T_1(k_1)}{T_1(0)}\Big]
E_1(k_1)\,dk_1.$$
Выше было показано, что $$J(k,k_1)-\dfrac{T_1(k)T_1(k_1)}{T_1(0)}= k^2 S(k,k_1).$$
Следовательно, предыдущее равенство дает: $$E_2(k)=-\dfrac{1}{\pi T_2(k)}
\int\limits_{0}^{\infty}S(k,k_1)E_1(k_1)\,dk_1=$$$$=
\dfrac{1}{\pi^2T_2(k)}\int\limits_{0}^{\infty}
\int\limits_{0}^{\infty} \dfrac{S(k,k_1)S(k_1,k_2)}{T_2(k_1)T_2(k_2)}
\varphi_0(k_2)dk_1dk_2.$$
Перепишем это равенство в виде: $$E_2(k)=\dfrac{\varphi_2(k)}{T_2(k)},$$ где $$\varphi_2(k)=-\dfrac{1}{\pi}\int\limits_{0}^{\infty}
S(k,k_1)E_1(k_1)dk_1=$$ $$=\dfrac{1}{\pi^2}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}
\dfrac{S(k,k_1)S(k_1,k_2)}{T_2(k_1)T_2(k_2)}
\varphi_0(k_2)dk_1dk_2.$$
Для второго приближения спектральной плотности функции распределения из уравнения (6.9) получаем: $$\Phi_2(k,\mu)=\dfrac{1}{1+ik\mu}\Bigg[E_2(k)-U_2|\mu|-
-\dfrac{|\mu|}{\pi}
\int\limits_{0}^{\infty}\dfrac{E_1(k_1)dk_1}
{1+k_1^2\mu^2}\Bigg].$$
Высшие приближения
------------------
В третьем приближении получаем: $$E_3(k)=-\dfrac{1}{L(k)}\Big[U_3T_1(k)+\dfrac{1}{\pi}
\int\limits_{0}^{\infty}J(k,k_1)E_2(k_1)dk_1\Big].$$ Как и ранее, устраняя полюс второго порядка в точке $k=0$, получаем: $$U_3=-\dfrac{1}{\pi T_1(0)}\int\limits_{0}^{\infty}J(0,k_1)E_2(k_1)dk_1=
-\dfrac{1}{\pi T_1(0)}\int\limits_{0}^{\infty}T_1(k_1)E_2(k_1)dk_1,$$ или $$U_3=-\dfrac{1}{\pi T_1(0)}\int\limits_{0}^{\infty}
\dfrac{T_1(k_1)}{T_2(k_1)}\varphi_2(k_1)\,dk_1.$$
Заметим, что $$U_3(-\infty)=0.0008,\qquad U_3(0)=0.0018.$$
Кроме того, в третьем приближении мы получаем: $$E_3(k)=-\dfrac{1}{\pi L(k)}\int\limits_{0}^{\infty}\Big[J(k,k_1)-
\dfrac{T_1(k)T_1(k_1)}{T_1(0)}\Big]E_2(k_1)dk_1,=$$ $$=-\dfrac{1}{\pi
T_2(k)}\int\limits_{0}^{\infty}S(k,k_1)E_2(k_1)dk_1,$$ или $$E_3(k)=\dfrac{\varphi_3(k)}{T_2(k)},$$ где $$\varphi_3(k)=-\dfrac{1}{\pi}\int\limits_{0}^{\infty}
S(k,k_1)E_2(k_1)dk_1=$$ $$=-\dfrac{1}{\pi^3}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}
\int\limits_{0}^{\infty}\dfrac{S(k,k_1)S(k_1,k_2)S(k_2,k_3)}
{T_2(k_1)T_2(k_2)T_2(k_3)}\varphi_0(k_3)dk_1dk_2dk_3,$$ и $$U_3=-\dfrac{1}{\pi^3}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}
\int\limits_{0}^{\infty}\dfrac{T_1(k_1)S(k_1,k_2)S(k_2,k_3)}
{T_1(0)T_2(k_1)T_2(k_2)T_3(k_3)}\varphi_0(k_3)dk_1dk_2dk_3.$$
Проводя аналогичные рассуждения, для $n$–го приближения согласно (4.10) и (4.11) получаем: $$U_n=-\dfrac{1}{\pi T_1(0)}\int\limits_{0}^{\infty}T_1(k)E_{n-1}(k)\,dk,
\qquad n=1,2,\cdots$$ $$E_n(k)=-\dfrac{1}{\pi T_2(k)}\int\limits_{0}^{\infty}
S(k,k_1)E_{n-1}(k_1)dk_1, \qquad n=1,2,\cdots,$$ или $$E_n(k)=\dfrac{\varphi_n(k)}{T_2(k)}, \qquad n=0,1,2, \cdots,$$ где $$\varphi_{n}(k)=-\dfrac{1}{\pi}
\int\limits_{0}^{\infty}S(k,k_1)E_{n-1}(k_1)dk_1, \qquad
n=1,2,\cdots,$$ $$\Phi_n(k,\mu)=\dfrac{1}{1+ik\mu}\Bigg[E_n(k)-U_n|\mu|-
\dfrac{|\mu|}{\pi}
\int\limits_{0}^{\infty}\dfrac{E_{n-1}(k_1)dk_1}
{1+k_1^2\mu^2}\Bigg].$$
Выпишем $n$–ые приближения $V_n$, $E_n(k)$ и $\varphi_n(k)$, выраженные через нулевое приближение спектральной плотности массовой скорости $E_0(k)=\varphi_0(k)/T_2(k)$. Имеем: $$U_n=\dfrac{(-1)^n}{\pi^n}\int\limits_{0}^{\infty}\cdots
\int\limits_{0}^{\infty}\dfrac{T_1(k_1)S(k_1,k_2)\cdots
S(k_{n-1},k_n)}{T_1(0)T_2(k_1)\cdots T_2(k_n)}\times$$ $$\times\varphi_0(k_n)\,dk_1
\cdots dk_n,$$ $$E_n(k)=\dfrac{(-1)^n}{\pi^n T_2(k)}\int\limits_{0}^{\infty}
\cdots \int\limits_{0}^{\infty}\dfrac{S(k,k_1)S(k_1,k_2)\cdots
S(k_{n-1},k_n)}{T_2(k_1)\cdots T_2(k_n)} \times$$$$\times\varphi_0(k_n)dk_1
\cdots dk_n,\qquad
n=1,2,3,\cdots,$$ $$\varphi_n(k)=\dfrac{(-1)^n}{\pi^{n}}\int\limits_{0}^{\infty}\cdots
\int\limits_{0}^{\infty}\dfrac{S(k,k_1)S(k_1,k_2)\cdots S(k_{n-1},k_n)}
{T_2(k_1)\cdots T_2(k_n)}\times$$ $$\times \varphi_0(k_n)dk_1\cdots dk_n,\qquad n=1,2,\cdots.$$
Сравнение с точным решением. Скорость скольжения
================================================
Сравним нулевое, первое и второе приближения при $q=1$ с точным решением. Ограничимся случаем квантовых Бозе–газов, близких к классическим (т.е. при $\alpha\to -\infty$), и случаем диффузного отражения молекул газа от поверхности.
Точное значение скорости скольжения в случае диффузного рассеяния для квантовых Бозе–газов с постоянной частотой столкновений таково: $$U_{sl}(\alpha,q=1)=V_1(\alpha)G_v.$$
Здесь $$V_1(\alpha)=-\dfrac{1}{\pi}\int\limits_{0}^{\infty}\zeta(\tau,\alpha)d\tau,$$ где $$\zeta(\tau,\alpha)=\theta(\tau,\alpha)-\pi,$$ $$\theta(\tau,\alpha)=\arg \lambda^+(\mu,\alpha)=
\arcctg\dfrac{\lambda(\tau,\alpha)}{\pi \tau K(\tau,\alpha)},$$ $$\lambda(z,\alpha)=1+z\int\limits_{-\infty}^{\infty}\dfrac{K(t,\alpha)dt}
{t-z},$$ или, в явном виде, $$\theta(\tau,\alpha)=\arcctg\Bigg[\dfrac{1}{\pi}
\int\limits_{-\infty}^{\infty}
\dfrac{x\ln(1-e^{\alpha-x^2})}{\tau
\ln(1-e^{\alpha-\tau^2})}\dfrac{dx}{x-\tau}\Bigg].$$
Следовательно, точное значение безразмерной скорости скольжения в случае диффузного рассеяния для квантовых Бозе–газов, близким к классическим газам (т.е. $\alpha\to -\infty$) таково: $$U_{sl}(\alpha=-\infty,q=1)=1.0162G_v.$$
Безразмерная скорость скольжения во втором приближении равна: $$U_{sl}^{(2)}(\alpha,q=1)=G_v \dfrac{2-q}{q}\Big[U_0(\alpha)+U_1(\alpha)q+
U_2(\alpha)q^2\Big].$$
Составим относительную ошибку приближения $$O_n(\alpha)=\dfrac{V_1(\alpha)-U_{sl}^{(n)}(\alpha,q=1)}{V_1(\alpha)}.$$ где $$U_{sl}^{(n)}(\alpha,q)=\sum\limits_{k=0}^{k=n}U_k(\alpha)q^k.$$
Результаты численных расчетов приведем в виде таблиц. В таблицах 1–3 приведем значения коэффициентов $U_0(\alpha), U_1(\alpha),
U_2(\alpha)$ при различных значениях безразмерного химического потенциала $\alpha$ и значения соответствующей относительной ошибки нулевого, первого и второго приближений безразмерной скорости скольжения.
**Таблица 1.**
-------------- --------------- -------------------------
Химпотенциал Коэффициент Относительная ошибка
$\alpha$ $U_0(\alpha)$ в нулевом приближении,%
0 0.7227 18.01
-1 0.8580 13.33
-2 0.8769 12.96
-3 0.8829 12.85
-4 0.8850 12.81
-5 0.8858 12.80
-6 0.8861 12.79
-7 0.8862 12.79
-8 0.8862 12.79
-------------- --------------- -------------------------
**Таблица 2.**
-------------- --------------- ------------------------
Химпотенциал Коэффициент Относительная ошибка
$\alpha$ $U_0(\alpha)$ в первом приближении,%
0 0.1775 -2.12
-1 0.1431 -1.12
-2 0.1413 -1.06
-3 0.1408 -1.05
-4 0.1406 -1.04
-5 0.1406 -1.04
-6 0.1405 -1.04
-7 0.1405 -1.04
-8 0.1405 -1.04
-------------- --------------- ------------------------
**Таблица 3.**
-------------- --------------- -------------------------
Химпотенциал Коэффициент Относительная ошибка
$\alpha$ $U_0(\alpha)$ во втором приближении,%
0 -0.0214 0.30
-1 -0.0121 0.11
-2 -0.0117 0.10
-3 -0.0116 0.10
-4 -0.0116 0.10
-5 -0.0116 0.10
-6 -0.0116 0.10
-7 -0.0116 0.10
-8 -0.0116 0.10
-------------- --------------- -------------------------
Приведенное сравнение последовательных приближений с точным результатом свидетельствует о высокой эффективности предлагаемого метода.
Профиль скорости газа в полупространстве и ее значение у стенки
===============================================================
Массовую скорость, отвечающую непрерывному спектру, разложим по степеням коэффициента диффузности: $$U_c(x)=U_c^{(0)}(x)+qU_c^{(1)}(x)+q^2U_c^{(2)}(x)+\cdots.
\eqno{(8.1)}$$
Тогда профиль массовой скорости в полупространстве можно строить по формуле: $$U(x)=U_{sl}(q,\alpha)+G_vx+U_c(x),
\eqno{(8.2)}$$ где $U_c(x)$ определяется предыдущим равенством (8.1).
Коэффициенты ряда (8.1) вычислим согласно выведенным выше формулам: $$U_c^{(n)}(x)=G_v\dfrac{2-q}{2\pi}\int\limits_{-\infty}^{\infty}
e^{ikx}E_n(k)dk, \qquad n=0,1,2,\cdots .$$
Согласно (8.2) вычислим скорость газа непосредственно у стенки: $$U(0)=U_{sl}(q,\alpha)+U_c^{(0)}(0)+qU_c^{(1)}(0)+q^2U_c^{(2)}(0)+\cdots.
\eqno{(8.3)}$$
В случае чисто диффузного отражения молекул от стенки ($q=1$) согласно (8.3) мы имеем $$U(0)=U_{sl}(1,\alpha)+U_c^{(0)}(0)+U_c^{(1)}(0)+U_c^{(2)}(0)+\cdots.$$
Отсюда в нулевом приближении получаем: $$U^{(0)}=U_{sl}(1,\alpha)+U_c^{(0)}(0).$$
Отсюда видно, что $$U^{(0)}\Big|_{\alpha=-\infty}=
U_{sl}(1,-\infty)+U_c^{(0)}(0)\Big|_{\alpha=-\infty}=0.6747G_v.$$
В первом приближении получаем: $$U^{(1)}(0)=U_{sl}(1,\alpha)+U_c^{(0)}(0)+U_c^{(1)}(0).$$
Отсюда видно, что $$U^{(1)}(0)\Big|_{\alpha=-\infty}=U_{sl}(1,-\infty)+
U_c^{(0)}(0)\Big|_{\alpha=-\infty}+$$$$+U_c^{(1)}(0)\Big|_{\alpha=-\infty}
=0.7103G_v.$$
Во втором приближении получаем: $$U^{(2)}(0)=U_{sl}(1,\alpha)+U_c^{(0)}(0)+U_c^{(1)}(0)+U_c^{(2)}(0).$$
Отсюда видно, что $$U^{(2)}(0)\Big|_{\alpha=-\infty}=U_{sl}(1, -\infty)+
U_c^{(0)}(0)\Big|_{\alpha=-\infty}+$$ $$+U_c^{(1)}(0)\Big|_{\alpha=-\infty}+
U_c^{(2)}(0)\Big|_{\alpha=-\infty}=0.7068G_v.$$
Сравним эти результаты с точным значение скорости у стенки [@Lat2001TMF]: $$U(0,\alpha)=\sqrt{\dfrac{l_2^B(\alpha)}{l_0^B(\alpha)}}G_v.$$
Из этой формулы вытекает, что $$U(0)\Big|_{\alpha=-\infty}=\dfrac{1}{\sqrt{2}}G_v=0.7071G_v.$$ Введем относительную ошибку $$O_n=\dfrac{U(0)-U^{(n)}(0)}{U(0)}\cdot 100\% , \qquad
n=0,1,2,\cdots.$$
В нулевом приближении относительная ошибка равна $4.6\%$, в первом приближении равна $-0.45\%$, во втором приближении равна: $0.044\%$.
****
Приведение формул к размерному виду
===================================
Формулу (6.3) для безразмерной скорости скольжения приведем к размерному виду. Для этого понадобится коэффициент вязкости квантового Бозе–газа.
По определению коэффициент кинематической вязкости равен:
$$\eta=-\dfrac{P_{xy}}{\Big(\dfrac{du_y}{dx}\Big)_\infty},$$ где $u_y(x)$ – размерная массовая скорость, откуда
$$\eta=-\dfrac{m}{g_v}\int fv_xv_y\,d\Omega,
\eqno{(9.1)}$$ где $g_v$ – размерный градиент массовой скорости. Учитывая, что $x_1=\nu \sqrt{\beta}x$, где $x$ – размерная координата, имеем: $$g_v=\Big(\dfrac{du_y(x)}{dx}\Big)_\infty=
\dfrac{\nu d(\sqrt{\beta}u_y(x))}{d(\nu \sqrt{\beta}x)}=
\nu\dfrac{dU_y(x_1)}{dx_1}=\nu G_v.$$ Здесь $G_v$ – безразмерный градиент, $U_y$ – безразмерная массовая скорость в направлении оси $y$.
Перейдем в (9.1) к интегрированию по безразмерным компонентам скорости: $$\eta=-\dfrac{m^4(2s+1)}{\nu G_v(2\pi\hbar)^3(\sqrt{\beta})^5}
\int fC_xC_y\,d^3C=$$
$$=
-\dfrac{m^4(2s+1)}{\nu G_v(2\pi\hbar)^3(\sqrt{\beta})^5}
\int C_x\,C_y^2\,g_B(C)\,h(x,C_x)\,d^3C.$$
Подставляя вместо $h(x,C_x)$ асимптотическую функцию $h_{as}(x,C_x)$, находим, что
$$\eta=\dfrac{2(2s+1)m^4}{\nu (2\pi\hbar)^3(\sqrt{\beta})^5}
\int C_x^2\,C_y^2\,g_B(C)\,d^3C.$$
Вычисляя интеграл в этом выражении, находим коэффициент вязкости: $$\eta=-\dfrac{2(2s+1)m^4\pi l_2^B(\alpha)}{\nu (2\pi\hbar)^3
(\sqrt{\beta})^5},$$ где $$l_2^B(\alpha)=\int\limits_{0}^{\infty}
x^2\ln(1-\exp(\alpha-x^2))\,dx.$$
Выразим коэффициент вязкости через числовую плотность. Нетрудно видеть, что $$N=\int f\,d\Omega=-\dfrac{2\pi (2s+1)m^3l_0^B(\alpha)}
{(2\pi\hbar)^3 (\sqrt{\beta})^3},$$ $$l_0^B(\alpha)=\int\limits_{0}^{\infty}\ln(1-\exp(\alpha-x^2))\,dx.$$
Следовательно, коэффициент вязкости можно представить в виде: $$\eta=\dfrac{N\,m\,l_2(\alpha)}{\nu\,\beta\,l_0^B(\alpha)}=
\dfrac{\rho}{\nu\,\beta}\cdot\dfrac{l_2^B(\alpha)}{l_0^B(\alpha)}.
\eqno{(9.3)}$$
Выражение для размерной скорости с учетом равенства (6.3), в котором все коэффициенты ряда найдены, перепишем в виде: $$\sqrt{\beta}u_{sl}(\alpha,q)=C(\alpha,q)\dfrac{g_v}{\nu},$$ откуда размерная скорость скольжения равна: $$u_{sl}(\alpha,q)=\dfrac{C(\alpha,q)}{\nu \,\sqrt{\beta}\,l}\,l\,g_v.
\eqno{(9.4)}$$
Здесь $$C(q,\alpha)=\dfrac{2-q}{q}\Big[U_0+U_1q+U_2q^2\cdots\Big].
\eqno{(9.5)}$$
Длину свободного пробега $l$ в (9.4) выразим через вязкость $\eta$ согласно Черчиньяни [@Cerc62]–[@8]: $l=\eta \rho^{-1}\sqrt{\pi\beta}$. Подставляя выражение (9.5) в (9.4), получаем искомую размерную скорость скольжения: $$u_{sl}(\alpha,q)=K_v^B(\alpha,q)\,l\,g_v,
$$ где $$K_v^B(\alpha,q)=\dfrac{C(\alpha,q)\,l_0^B(\alpha)}
{\sqrt{\pi}\,l_2^F(\alpha)}$$ есть коэффициент изотермического скольжения.
![Зависимость коэффициента изотермического скольжения от коэффициента диффузности. []{data-label="rateIII"}](Koeff.eps){width="16.0cm" height="10cm"}
![Зависимость коэффициента изотермического скольжения от коэффициента диффузности. []{data-label="rateIII"}](Koefq.eps){width="16.0cm" height="10cm"}
![Зависимость коэффициента $V_1(\alpha)$ от приведенного химического потенциала. []{data-label="rateIII"}](v1alpha.eps){width="16.0cm" height="10cm"}
![Зависимость коэффициента изотермического скольжения и коэффициента $V_1(\alpha)$ от приведенного химического потенциала. []{data-label="rateIII"}](v1kv.eps){width="16.0cm" height="10cm"}
![Зависимость относительного коэффициента вязкости $\dfrac{\eta(\alpha)}{\eta(-\infty)}$ от приведенного химического потенциала. []{data-label="rateIII"}](visco.eps){width="16.0cm" height="10cm"}
![Зависимость коэффициента изотермического скольжения $K_v(\alpha)$ (кривая [*1*]{}) и коэффициента скорости Бозе–газа $K_v(0,\alpha)$ непосредственно у стенки (кривая [*2*]{}) от приведенного химического потенциала. []{data-label="rateIII"}](Kramers2.eps){width="16.0cm" height="10cm"}
![Зависимость коэффициента скорости $C(0,\alpha)$ Бозе–газа непосредственно у стенки от приведенного химического потенциала. []{data-label="rateIII"}](Kramers3.eps){width="16.0cm" height="10cm"}
Заключение
==========
В настоящей работе с помощью развитого недавно [@LatYushk2012] нового метода решена полупространственная граничная задача кинетической теории — задача Крамерса об изотермическом скольжении квантового Бозе–газа с постоянной частотой столкновений молекул и с зеркально–диффузными граничными условиями. В основе метода лежит идея продолжить функцию распределения в сопряженное полупространство $x<0$ и включить в кинетическое уравнение граничное условие в виде члена типа источника на функцию распределения, отвечающую непрерывному спектру. С помощью преобразования Фурье кинетическое уравнение сводим к характеристическому интегральному уравнению Фредгольма второго рода, которое решаем методом последовательных приближений. Для этого разлагаем в ряды по степеням коэффициента диффузности скорость скольжения газа, его функцию распределения и массовую скорость, отвечающие непрерывному спектру. Подставляя эти разложения в характеристическое уравнение и приравнивая коэффициенты при одинаковых степенях коэффициента диффузности, получаем счетную систему зацепленных уравнений, из которых находим все коэффициенты искомых разложений.
Мы находим так называемую скорость скольжения газа вдоль поверхности, функцию распределения и распределение массовой скорости в полупространстве. Скорость скольжения — это фиктивная скорость газа, которая получается, если профиль асимптотического распределения массовой скорости, вычисленную вдали от стенки на основе асимптотического распределения Чепмена—Энскога, пролонгировать до границы полупространства.
Предлагаемый метод обладает высокой эффективностью. Так, сравнение с точным решением показывает, что в третьем приближении ошибка не превосходит $0.1\%$.
Изложенный в работе метод был успешно применен [@53]–[@64] в решении ряда таких сложных задач кинетической теории, которые не допускают аналитического решения.
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[^1]: $bedrikova@mail.ru$
[^2]: $avlatyshev@mail.ru$
|
---
abstract: 'This is the first paper in a general program to automate skein theoretic arguments. In this paper, we study skein theoretic invariants of planar trivalent graphs. Equivalently, we classify trivalent categories, which are nondegenerate pivotal tensor categories over $\bbC$ generated by a symmetric self-dual simple object $X$ and a rotationally invariant morphism $1 \rightarrow X \tensor X \tensor X$. Our main result is that the only trivalent categories with $\dim \Hom(1 \to X^{\otimes n})$ bounded by $1,0,1,1,4,11,40$ for $0 \leq n \leq 6$ are quantum $SO(3)$, quantum $G_2$, a one-parameter family of free products of certain Temperley-Lieb categories (which we call ABA categories), and the $H3$ Haagerup fusion category. We also prove similar results where the map $1 \rightarrow X^{\otimes 3}$ is not rotationally invariant, and we give a complete classification of nondegenerate braided trivalent categories with dimensions of invariant spaces bounded by the sequence $1,0,1,1,4$. Our main techniques are a new approach to finding skein relations which can be easily automated using Gröbner bases, and evaluation algorithms which use the discharging method developed in the proof of the $4$-color theorem.'
author:
- Scott Morrison
- Emily Peters
- Noah Snyder
bibliography:
- '../../bibliography/bibliography.bib'
title: Categories generated by a trivalent vertex
---
Introduction
============
This is the first paper in a general program to automate skein theoretic arguments in quantum algebra and quantum topology. In this paper, we study skein theoretic invariants of planar trivalent graphs following Kuperberg [@MR1265145]. However, the general approach will work much more broadly and in later papers we will consider other situations like those in [@MR2132671; @MR1733737; @MR1972635; @MR2783128; @1410.2876]. One might think of this program as attempting for skein theoretic arguments what [@MR2914056] did for principal graph arguments.
Before getting into the particulars of this paper, we will recall the basic notions of skein theory, which we illustrate using its most famous example. The Jones polynomial invariant [@MR0766964] of framed links can be computed by applying the Kauffman bracket skein relations $$\begin{aligned}
\scalebox{1.4}{$\bigcirc$} & = -A^2 -A^{-2} \notag \\
\intertext{and}
\overcrossing & = A \identity + A^{-1} {\begin{tikzpicture}[baseline=0cm]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}.\end{aligned}$$
These relations not only assign a Laurent polynomial to framed links, they also assign to each tangle a linear combination of noncrossing tangles. Unlike for ordinary tangles, the number of noncrossing tangles with $n$ boundary points is finite (and indeed given by Catalan numbers). Our goal is to find and prove theorems modelled on the following well-known result.
\[thm:bracket\] Suppose a skein theoretic invariant of framed links has the property that the span of $4$-boundary point tangles modulo the relations is $2$-dimensional. This invariant must be a specialization of the Jones polynomial.
The proof of this statement is straightforward. Since the span of $4$-boundary point tangles is only $2$-dimensional, there must be some linear relation between the three diagrams which occur in the bracket relation. Some quick calculations show that only the specific Kauffman bracket relation is compatible with the second and third Reidemeister moves. Finally it is clear that the bracket relations are enough to determine the invariant, since applying the crossing relation will turn any framed link into a linear combination of unlinked, unknotted, circles.
A number of similar results have been proved following this same outline. In each case, assumptions on the dimensions of spaces of diagrams guarantee relations of a certain form must hold, and an involved calculation determines the coefficients of these relations (possibly in terms of some parameters). Subsequently one finds an evaluation algorithm using these relations, demonstrating that they suffice to determine the invariant. Finally, one may also want to know that this invariant actually exists!
Our program has a two-fold goal. First, we are interested in generalizing this approach beyond invariants of links. We take the view that links are ‘merely’ a certain class of planar graphs (with vertices modeled on the under-crossing and the over-crossing), subject to some local relations. We would like to be able to prove theorems about arbitrary such classes. Second, to the extent possible we want to automate the technique — to find very general methods to derive such theorems, and where possible to implement these methods in code. This will enable us to rapidly explore many different skein theoretic settings, and moreover to explore much further out into the space of examples than is possible with by-hand calculations. (This paper restricts itself to the case of planar trivalent graphs — but we go much further than previous investigations of trivalent skein theories.)
In particular, the first step in the outline above — finding relations — is highly amenable to computer calculation, as explained in this paper (see [@F4E6] for another approach to automating skein theory). By contrast, the second step — giving an evaluation algorithm — remains an art. In this paper, our evaluation algorithms all come from the discharging technique developed in the proof of the $4$-color theorem. Indeed, we hope that in the future it may be possible to systematically discover evaluation algorithms based on discharging, adapted to the available relations in a given skein theoretic setting.
In this paper we concentrate on skein theoretic invariants of planar trivalent graphs. By the diagrammatic calculus for tensor categories, this paper can also be thought of as providing classifications of nondegenerate pivotal tensor categories over $\bbC$ generated by a symmetrically self-dual simple object $X$ and a rotationally invariant morphism $1 \rightarrow X \tensor X \tensor X$, where the dimensions of the first few invariant spaces $\Hom(1 \to X^{\otimes n})$ are small. We call such a tensor category a *trivalent category*. We note that the arguments in the paper are elementary and skein theoretic, so no knowledge of tensor categories is needed except for the existence proofs.
We now summarize this paper’s results in a table. To use this table, compute the dimensions of the spaces of diagrams with $n$ boundary points (equivalently the invariant spaces $\Hom(1 \to X^{\otimes n})$ for your tensor category) for the first few values of $n$. If an initial segment of this sequence of dimensions appears in the first column of some row of this table, then the second column explicitly identifies the category for you. If, on the other hand, you only have upper bounds for the dimensions of invariant spaces, then your category appears in the corresponding row, or some previous row.
dimension bounds new examples reference $\quad$
------------------- ---------------------------------------------- -------------------
1,0,1,1,2,… $SO(3)_{\zeta_5}$ Theorem \[thm:4\]
1,0,1,1,3,… $SO(3)_q$ or $OSp(1|2)$ —$\|$—
1,0,1,1,4,8,… $ABA \subset TL_{\sqrt{d t^{-1}}} \ast TL_t$ Theorem \[thm:5\]
1,0,1,1,4,9,… $(G_2)_{\zeta_{20}}$ —$\|$—
1,0,1,1,4,10,… $(G_2)_{q}$ —$\|$—
1,0,1,1,4,11,37,… $H3$ Theorem \[thm:6\]
1,0,1,1,4,11,40,… nothing more —$\|$—
In this table, $\ast$ denotes the free product, and $H3$ denotes the fusion category Morita equivalent to the Haagerup fusion category constructed in [@MR2909758]. It is fascinating to see the Haagerup subfactor once again appearing as the first surprising example in a classification of ‘small’ categories.
The same classification is shown in Figure \[fig:treeoflife\].
Kuperberg proved in [@MR1265145] that if the dimensions are exactly $1,0,1,1,4,10$ and in addition the diagrams with no internal faces give bases for these spaces, then the category must be $(G_2)_{q}$. In order to apply Kuperberg’s result one needs to do a calculation to verify linear independence (cf. [@MR2783128 Lemma 3.9]). Our classification is more satisfying, as it does not include this linear independence assumption. To get an even more satisfying result, one would need to drop the condition that the trivalent vertex generates all morphisms. Dropping the generating assumption would introduce some additional examples (e.g. from subfactors, or from quantum subgroups of $(G_2)_q$).
It is worth noting that for these results up to and including the row $1,0,1,1,4,10$, we can also give proofs that do not use a computer, following Kuperberg. However, these by-hand calculations are not enlightening, and we prefer to give computer-assisted arguments uniformly in all cases, because they are easier to follow and more reliable.[^1] By contrast, the $1,0,1,1,4,11$ results would be quite difficult, and probably impossible, to check by hand.
We also prove classification results when the map $1 \rightarrow X
\tensor X \tensor X$ has a nontrivial rotational eigenvalue. This case turns out to be much easier than the rotationally invariant case (likely easy enough to check by hand in a tedious week), and there are correspondingly many fewer possibilities. If the dimensions are below $1,0,1,1,4,11,40,\ldots$ then the category must be a twisted version of $\mathrm{Rep}(S_3)$ or a twisted version of the Haagerup fusion category, or possibly one other new tensor category. This new candidate category is interesting as it can not come from subfactors or quantum groups. Finding such exotic tensor categories is one of our main motivations for this project.
As a corollary in the spirit of the results in [@MR2783128], we see that if $X$ is a simple object in a pivotal category and $X^{\otimes 2} \iso 1 \oplus X \oplus A \oplus B$ for some simple objects $A$ and $B$, and moreover $\dim \Hom(X \to (A \oplus B)^{\otimes 2}) \leq 3$, then $\dim \Hom(1 \to X^{\otimes 5}) \leq 10$ and so the category generated by the morphism $X^{\otimes 2} \to X$ must be either a twisted $\mathrm{Rep}(S_3)$ category, or an $SO(3)_q$, $ABA$, or $(G_2)_q$ category.
These results also allow a complete classification of braided trivalent categories with $\dim \Hom(1 \to X^{\otimes 4}) \leq 4$. given in Section \[sec:braided\]. A quick argument shows that the braiding guarantees that $\dim \Hom(1 \to X^
{\otimes 5}) \leq 10$. By the table above, any braided trivalent category with $\dim \Hom(1 \to X^{\otimes 4}) \leq 4$ must be $OSp(1|2)$, $SO(3)_q$ or $(G_2)_q$ (the ABA categories are not braided). We also classify all braidings on these categories. Note that these results on braided trivalent categories only use the $1,0,1,1,4,10$ classifications, and so can be checked by hand in a reasonable amount of time.
Source code
-----------
This article relies on a number of computer calculations. In the interests of verifiability, the source code for all these calculations are bundled with the [arXiv]{} source of this article. After downloading the source, you’ll find a [code/]{} subdirectory containing a number of Mathematica notebooks. These notebooks are referenced at the necessary points through the text. As described above, equivalent calculations could also be performed by hand except for the computer calculations in Section \[sec:six\] and parts of Section \[sec:rotationalev\].
Acknowledgements
----------------
Scott Morrison was supported by an Australian Research Council Discovery Early Career Researcher Award DE120100232, and Discovery Projects DP140100732 and DP160103479. Emily Peters was supported by the NSF grant DMS-1501116. Noah Snyder was supported by the NSF grant DMS-1454767. All three authors were supported by DOD-DARPA grant HR0011-12-1-0009. Scott Morrison would like to thank the Erwin Schrödinger Institute and its 2014 programme on “Modern Trends in Topological Quantum Field Theory” for their hospitality. We would like to thank Greg Kuperberg for a blog comment [@how-to-almost-prove-the-4-color-theorem] suggesting applying the discharging method to skein theory, Victor Ostrik for explaining his construction of the twisted Haagerup categories, and David Roe and Dylan Thurston for helpful suggestions.
Trivalent categories
====================
In this section, we introduce the notion of a trivalent category, as a pivotal category which is ‘generated by a trivalent vertex’. In particular, every morphism in such a category is a linear combination of trivalent graphs (possibly with boundary) embedded in the plane, and indeed any such trivalent graph is allowed as a morphism.
In Appendix \[sec:local\] we motivate trivalent categories for a wider audience — particularly graph theorists — by explaining an equivalence between trivalent categories and certain skein theoretic invariants of planar trivalent graphs. While this equivalence is not essential for understanding the paper, reading the appendix may be useful for readers unfamiliar with diagrammatic methods in category theory.
This category theoretic language will be useful, because the examples we will encounter along the way come from fields of mathematics where category theory is very convenient. Nevertheless this point of view is not needed in the proofs of the main theorems. The key to translating category theoretic statements below into graph theoretic statements is to remember that $\Hom_{\mathcal C}(1\rightarrow X^{\otimes n})$ is the vector space of formal linear combination of planar trivalent graphs with $n$ boundary points modulo the relevant skein relations for the category $\cC$.
Recall that a (strict) pivotal category is a rigid monoidal category such that $x^{**} = x$ for all $x$. In this paper, all of our categories will be $\bbC$-linear.
Pivotal categories axiomatize the nicest possible theory of duals, and correspondingly have a diagrammatic calculus allowing arbitrary planar isotopies. As usual, we have string diagrams representing morphisms, with oriented strings labelled by objects of the category, and vertices (or ‘coupons’) labelled by morphisms of the category. The strings may have critical points, which we interpret as the evaluation and coevaluation maps provided by the rigid structure. Arbitrary planar isotopies of a diagram preserve the represented morphism; it is critical that $2\pi$ rotations of the vertices are allowed, and this corresponds exactly to strict pivotality.
Given a pivotal category $\cC$ and a chosen object $X$, we use the notation $\cC_k = \Inv_\cC(X^{\otimes k}) = \Hom_\cC (1 \to X^{\otimes k})$ for the ‘invariant spaces’ of $X$. (If you know about planar algebras [@math.QA/9909027], recall these vector spaces form an unshaded planar algebra.) We say a category $\cC$ is *evaluable* if $\dim \Hom(1 \to 1) = 1$, and in fact $\Hom(1 \to 1)$ may be identified with the ground field by sending the empty diagram to $1$. The category $\cC$ is *nondegenerate* if for every morphism $x: a \to b$, there is another morphism $x': b \to a$ so $\operatorname{tr}(x x') \neq 0 \in \Hom(1 \to 1)$.
A *trivalent* category $(\cC, X, \tau)$ is a nondegenerate evaluable pivotal category over $\bbC$ with an object $X$ with $\dim \cC_1 = 0$, $\dim{\cC_2} = 1$, and $\dim{\cC_3} = 1$, with a rotationally invariant morphism $\tau \in \cC_3$ called ‘the trivalent vertex’, such that the category is generated (as a pivotal category) by $\tau$.
We’ll often simply refer to $\cC$ itself as a trivalent category.
The rotational invariance of $\tau$ allows us to drop the “coupon” attached to $\tau$ in string diagrams and treat it as an undecorated trivalent vertex.
We want one more simplification to our diagrammatic calculus: in the present situation it turns out that we can always ignore the orientations on strings, because the object $X$ is automatically symmetrically self-dual. Said another way, the 2-valent vertex corresponding to the self-duality $X \iso X^*$ is rotationally symmetric.
The object $X$ is symmetrically self-dual.
Suppose that $\alpha: X \rightarrow X^*$ is any self-duality and that $\psi: X \rightarrow X \otimes X$ is an inclusion. Because $\cC$ is nondegenerate and $\dim \cC_3 = 1$, given any two non-zero maps $\beta: X \to X \otimes X$ and $\gamma: X \otimes X \to X^*$, $\operatorname{tr}(\alpha^{-1} \circ \gamma \circ \beta) \neq 0$, so $\alpha^{-1} \circ \gamma \circ \beta$ and $\gamma \circ \beta$ are nonzero too. Taking $\gamma = \psi^* \circ (\alpha \otimes \alpha^*)$ and $\beta = \psi$, we see that the map $$\psi^* \circ (\alpha \otimes \alpha^*) \circ \psi =
\begin{tikzpicture}[baseline=-2, scale=1.5];
\node[rectangle, draw] (A) at (1,0) {$\psi$};
\node[rectangle, draw] (B) at (-1,0) {\rotatebox{180}{$\psi$}};
\node[rectangle, draw] (C) at (0,-.5) {$\alpha$};
\node[rectangle, draw] (D) at (0,.5) {\rotatebox{180}{$\alpha$}};
\begin{scope}[thick, decoration={
markings,
mark=at position 0.5 with {\arrow{<}}}
]
\draw[postaction={decorate}] (D.0) to [in=180, out=0] (A.160);
\draw[postaction={decorate}] (C.0) to [in=180, out=0] (A.200);
\draw[postaction={decorate}] (D.180) to [in=0,out=180] (B.20);
\draw[postaction={decorate}] (C.180) to [in=0, out=180] (B.-20);
\draw[postaction={decorate}] (A.0)-- ++(0:5mm);
\draw[postaction={decorate}] (B.180)-- ++(180:5mm);
\end{scope}
\end{tikzpicture}$$ is nonzero and manifestly rotationally invariant.
Combining the symmetric self-duality of $X$ with the rotational invariance of $\tau$, we can interpret any unoriented planar trivalent graph with $n$ boundary points as an element of $\cC_n$.
To any trivalent category we assign several important parameters as follows. Since $\dim \cC_0=1$, any diagram with a loop in it is a multiple $d$ of the same diagram missing that loop. The loop value $d$ must be nonzero because it is the pairing of the unique-up-to-scalar element of $\cC_2$ with itself, and $\cC$ is nondegenerate. In addition, we must have a relation $$\label{eq:bigon}
{
\begin{tikzpicture}[baseline=0cm]
\foreach \x in {1, ..., 2}
\draw (360*\x/2+90:.8cm)--(360*\x/2+90:.5cm);
\foreach \x in {1, ..., 2}
\draw (360*\x/2+90:.5cm) .. controls +(360*\x/2+120+90:.3cm) and +(360*\x/2+360/2-120+90:.3cm) .. (360*\x/2+360/2+90:.5cm);
\end{tikzpicture}
}
= b \cdot
\begin{tikzpicture}[baseline=.3cm]
\draw (0,0)--(0,1);
\end{tikzpicture}$$ for some parameter $b$ which is again nonzero, since the theta graph must be nonzero. Because one can rescale the trivalent vertex by a constant, without loss of generality we can assume $b=1$. Finally, we see that $$\label{eq:triangle}
{
\begin{tikzpicture}[baseline=0cm]
\foreach \x in {1, ..., 3}
\draw (360*\x/3+90:.8cm)--(360*\x/3+90:.5cm);
\foreach \x in {1, ..., 3}
\draw (360*\x/3+90:.5cm) .. controls +(360*\x/3+120+90:.3cm) and +(360*\x/3+360/3-120+90:.3cm) .. (360*\x/3+360/3+90:.5cm);
\end{tikzpicture}
}
= t \cdot
\begin{tikzpicture}[baseline=.1cm,scale=0.75]
\draw (0,0) -- (0,1);
\draw (0,0) -- (0.7,-0.5);
\draw (0,0) -- (-0.7,-0.5);
\end{tikzpicture},$$ although in this case the parameter $t$ can be zero. These parameters $d$ and $t$ will be the key parameters in this paper.
We now give a simple example of a trivalent category, the ‘chromatic category’. Further examples appear throughout this paper, in particular $SO(3)_q$ (which is essentially the same as the chromatic category) in the proof of Proposition \[prop:4:realization\], $ABA_{(d,t)}$ in the proof of Proposition \[prop:5:realization:ABA\], $(G_2)_q$ in Definition \[def:G2\], and the H3 category of [@MR2909758] which is given a trivalent presentation in Section \[sec:six\].
The chromatic category has as objects finite subsets of an interval, and a morphism between two such sets is a linear combination of trivalent graphs embedded in a strip between these intervals, subject to the following local relations: $$\begin{aligned}
\tikz[baseline=0]{\draw (0,0) circle (4mm);} & = n - 1 \\
{
\begin{tikzpicture}[baseline=0cm]
\foreach \x in {1, ..., 2}
\draw (360*\x/2+90:.8cm)--(360*\x/2+90:.5cm);
\foreach \x in {1, ..., 2}
\draw (360*\x/2+90:.5cm) .. controls +(360*\x/2+120+90:.3cm) and +(360*\x/2+360/2-120+90:.3cm) .. (360*\x/2+360/2+90:.5cm);
\end{tikzpicture}
}
& = (n-2) \cdot
\begin{tikzpicture}[baseline=.3cm]
\draw (0,0)--(0,1);
\end{tikzpicture} \\
{
\begin{tikzpicture}[baseline=0cm]
\foreach \x in {1, ..., 3}
\draw (360*\x/3+90:.8cm)--(360*\x/3+90:.5cm);
\foreach \x in {1, ..., 3}
\draw (360*\x/3+90:.5cm) .. controls +(360*\x/3+120+90:.3cm) and +(360*\x/3+360/3-120+90:.3cm) .. (360*\x/3+360/3+90:.5cm);
\end{tikzpicture}
}
& = (n-3) \cdot
\begin{tikzpicture}[baseline=.1cm,scale=0.75]
\draw (0,0) -- (0,1);
\draw (0,0) -- (0.7,-0.5);
\draw (0,0) -- (-0.7,-0.5);
\end{tikzpicture} \\
{ \begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}}+ {\begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}& = { \begin{tikzpicture}[baseline=0cm]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}
}\; + {\begin{tikzpicture}[baseline=0cm]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\end{aligned}$$ (Here the object $X$ is just a singleton on an interval, and the morphism $\tau$ is just the trivalent vertex.) One can verify that these relations suffice to evaluate any closed trivalent graph: the ‘$I=H$’ relation ensures that we can reduce the size of a chosen face without increasing the total number of vertices in the graph, and once there is a small enough face, one of the other relations lets us reduce the total number of vertices. Indeed, in the case that the parameter $n$ is an integer, one can see that this evaluation is exactly the normalized chromatic number of the graph — that is, the number of ways to color the faces of the planar trivalent graph with $n$ colors such that adjacent faces do not share a color, with the ‘outer face’ of the planar diagram always having a fixed color. The proof of this fact is that in each relation, the total number of colorings is the same on either side of the relation.
Later we will see that this category is actually equivalent to the category we call $SO(3)_q$ below, namely the category of representations of $U_q(\mathfrak{sl}(2))$ consisting of representations whose highest weight is a root, with the equivalence sending the object $X$ to the irreducible $3$-dimensional representation and sending the trivalent vertex to some multiple of the quantum determinant. The parameters match up according to the formula $n=
q^2+2+q^{-2}$. (This is of course a well known equivalence in quantum topology.)
Similarly, the $G_2$ spider defined in [@MR1403861] is an example of a trivalent category, and it is equivalent to the category of representations of $U_q(\mathfrak{g}_2)$ with $X$ the $7$-dimensional representation and $\tau$ the quantum deformation of the defining invariant antisymmetric trilinear form.
There is a well-known theorem about nondegeneracy and negligibles which we will need.
\[prop:general:uniqueness\] An evaluable pivotal category has a unique maximal ideal, the negligible ideal. (cf. [@MR2979509 Proposition 3.5])
\[cor:reductions=>uniqueness\] Given a collection of linear relations amongst planar trivalent graphs, such that any closed diagram can be reduced to a multiple of the empty diagram by those relations, there is a unique nondegenerate trivalent category satisfying those relations.
Any trivalent category is automatically spherical. To see this note that since $X$ is simple we need only check that the dimension of $X$ and the dimension of $X^*$ agree, but since $X$ is self-dual this is obvious.
Small graphs
============
We will write $D(n,k)$ for the collection of trivalent graphs with $n$ boundary points and at most $k$ internal faces having four or more edges. For a fixed trivalent category, we write $M(n,k)$ for the matrix of bilinear inner products, i.e. $$\langle X, Y \rangle =
\begin{tikzpicture}[baseline=0]
\draw (-0.2,0) arc (180:0:1.2);
\draw (0,0) arc (180:0:1);
\draw (0.2,0) arc (180:0:0.8);
\node[draw,fill=white,circle,inner sep=8pt] (X) at (0,0) {$X$};
\node[draw,fill=white,circle,inner sep=8pt] (Y) at (2,0) {$Y$};
\end{tikzpicture},$$ of the elements of $D(n,k)$, and $\Delta(n,k)$ is the determinant of $M(n,k)$. Similarly, we will write $D^\square(n,k)$ for the collection of trivalent graphs with $n$ boundary points and at most $k$ internal faces having five or more edges, and analogously define $M^\square(n,k)$ and $\Delta^\square(n,k)$.
\[thm:zeroalltheway\] For either $\mu=\emptyset$ or $\mu = \square$, if there is a linear relation amongst diagrams in $D^\mu(n,k)$, then $\Delta^\mu(n',k')=0$ for all $n' \geq n$ and $k' \geq k$.
Take the diagrams appearing in the relation and glue a fixed tree (with $n'-n$ leaves) to a fixed boundary point of each of them. There is then a non-trivial relation amongst the resulting diagrams, and hence $\Delta^\mu(n',k')=0$ also.
If $\Delta^\mu(n,k)=0$ and the diagrams in $D^\mu(n,k)$ span $\cC_n$, then $\Delta^\mu(n',k') = 0$ for all $n' \geq n$, $k' \geq k$.
In all our examples, we will have enough conditions on the trivalent category that it will be possible to evaluate each of the $\Delta^\mu(n,k)$ that we consider as a rational function in $d$, $b$, and $t$. We will always normalize to set $b=1$, but it is worth noting that you can recover the rational function up to an overall power of $b$, from the $b=1$ specialization as follows. Notice that rescaling the trivalent vertex by $\lambda$ rescales the values of all closed planar trivalent graphs. We can put a grading on closed planar trivalent graphs according to the number of trivalent vertices. Our parameters $d = \bigcirc, b = \tikz[scale=0.25,baseline=-2]{\draw (0,0) circle (1); \draw (1,0) -- (-1,0);} / d$, and $t = \tikz[scale=0.4,baseline=-2]{\draw (90:1) -- (210:1) -- (330:1) -- (90:1) -- (0,0) -- (210:1) (0,0) -- (330:1);}/bd$ have gradings $0$, $+2$, and $+2$. It is not difficult to see, by looking at the diagrams involved in the calculation, that $\Delta^\mu(n,k)$ is homogenous with respect to this grading. So to recover the rational function from its $b=1$ factorization up to an overall power of $b$, we simply multiply each monomial by a power of $b$ to make it homogenous.
Diagrams with four boundary points {#sec:four}
==================================
Recall that for any trivalent category $\cC$ we get two numbers $(d,t)$ from the loop and the triangle, and furthermore $d \neq 0$. In this section we prove
[@1202.4396 Theorem 3.4] \[thm:4\] A trivalent category $\cC$ with $\dim \cC_4 \leq 3$ has $P_{SO(3)} = d+t-dt-2 = 0$ and must be either an $SO(3)_q$ category for $d = q^2+1+q^{-2}$ if $(d,t) \neq (-1,3/2)$, or $OSp(1|2)$ if $(d,t) = (-1,3/2)$.
(Throughout this paper we will use $P$’s with various subscripts to denote important polynomials in $d$ and $t$ whose vanishing set corresponds to some existing trivalent category. So, for example, $P_{SO(3)}$ is the polynomial which vanishes when $d$ and $t$ have the values that they have for quantum $SO(3)$. By contrast, we will use $Q_{i,j}$ to denote polynomials whose exact form is not important to the reader. Here $i$ and $j$ are the degrees of the polynomial in $d$ and $t$ respectively. The smaller polynomials are listed in the appendix, and all appear in computer readable form in the [polynomials/]{} directory of the [arXiv]{} source of this article.)
This Theorem follows from three propositions.
\[prop:4:nonexistence\] For any $(d,t)$ not satisfying $P_{SO(3)} = 0$ there are no trivalent categories with $\dim \Span D(4,0) \leq 3$.
\[prop:4:uniqueness\] For every pair $(d,t)$ satisfying $P_{SO(3)} = 0$ there is at most one trivalent category with $\dim \cC_4 \leq 4$.
\[prop:4:realization\] The $SO(3)_q$ categories are trivalent categories with $\dim \cC_4 \leq
3$, and realize every pair $(d,t)$ satisfying $P_{SO(3)} = 0$, except $(-1,3/2)$. The remaining point $(-1,3/2)$ is realized by $OSp(1|2)$.
Although versions of these propositions were already proved in [@1202.4396][^2], we give a slightly different argument which is easier to automate and thus scale to the needs of the later sections. For the first proposition, we use the dimension assumption to see that $\Delta(4,0)$ and $\Delta(4,1)$ vanish. A short calculation shows that $P_{SO(3)}$ must vanish. (Later, this sort of calculation will be handled by Gröbner bases, but for now it’s easy enough to do by hand.) For the second proposition, we fix $(d, t)$ satisfying $P_{SO(3)} = 0$. The proof divides into three phases. We use the degeneracy implied by $\dim \cC_4 \leq 3$ and an easy graph theoretic fact to show that $D(n,0)$ spans $\cC_n$ for all $n$. Specializing to $n = 4$, this shows that the kernel of $M(4,0)$ gives a relation in the category. The fact that this relation suffices to evaluate all closed diagrams shows that there is at most one trivalent category at this value of $(d,t)$. Finally, the third proposition is a straightforward statement about a well-understood family of categories.
The proof of Theorem \[thm:4\] only needs the weakening of Proposition \[prop:4:uniqueness\] that covers the cases where $\dim \cC_4 \leq 3$. We will need the full strength later in the paper.
Proof of Proposition \[prop:4:nonexistence\] (Non-existence) {#proof-of-proposition-prop4nonexistence-non-existence .unnumbered}
------------------------------------------------------------
The diagrams $D(4,1)$ are $${\begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\, , \, {\begin{tikzpicture}[baseline=0cm]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\, , \, { \begin{tikzpicture}[baseline=0cm]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}
}\, , \, { \begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}}\, , \, {
\begin{tikzpicture}[baseline=0cm]
\foreach \x in {1, ..., 4}
\draw (360*\x/4+45:.8cm)--(360*\x/4+45:.5cm);
\foreach \x in {1, ..., 4}
\draw (360*\x/4+45:.5cm) .. controls +(360*\x/4+120+45:.3cm) and +(360*\x/4+360/4-120+45:.3cm) .. (360*\x/4+360/4+45:.5cm);
\end{tikzpicture}
},$$ and they have matrix of inner products
$$\begin{aligned}
\left(
\mathfig{0.3}{inner-product-matrix}
\right) & =
\begin{pmatrix}
d^2 & d & d & 0 & d \\
d & d^2 & 0 & d & d \\
d & 0 & d & td & t^2d \\
0 & d & td & d & t^2d \\
d & d & t^2d & t^2d & {{
\begin{tikzpicture}[scale=0.8,baseline=-0.1cm]
\fill[white] (0,0) circle (0.5cm);
\foreach \x in {1, ..., 4} {
\draw (360*\x/4-360/4+45:.3cm) -- (360*\x/4+45:.3cm)--(360*\x/4+45:.5cm) -- (360*\x/4+360/4+45:.5cm);
}
\end{tikzpicture}
}}\end{pmatrix}.\end{aligned}$$
Recall $\Delta(4,1)$ is the determinant of this matrix, and $\Delta(4,0)$ is the determinant of the minor leaving off the last row and column.
In a trivalent category, $$\begin{aligned}
\Delta(4, 0) = d^4 (d+t-dt-2)(d+t+dt).\end{aligned}$$
In a trivalent category, $$\begin{aligned}
\Delta(4, 1) = - d^4(d+t-dt-2)(2d+2dt-4dt^2+2dt^4+2d^2t^4 - {{
\begin{tikzpicture}[scale=0.8,baseline=-0.1cm]
\fill[white] (0,0) circle (0.5cm);
\foreach \x in {1, ..., 4} {
\draw (360*\x/4-360/4+45:.3cm) -- (360*\x/4+45:.3cm)--(360*\x/4+45:.5cm) -- (360*\x/4+360/4+45:.5cm);
}
\end{tikzpicture}
}}(d+t+dt))\end{aligned}$$
A trivalent category with $\dim \Span D(4,0) \leq 3$ must have $\Delta(4,0) =
\Delta(4,1) = 0$. Since $d \neq 0$, we must have either $P_{SO(3)} = 0$, or $(d+t+dt) = 0$ and $2d+2dt-4dt^2+2dt^4+2d^2t^4 = 0$. In the latter case, solving gives $(d,t) =
\left( \frac{1 \pm \sqrt{5}}{2}, \frac{1 \mp \sqrt{5}}{2}\right)$ which also satisfies $P_{SO(3)} = 0$.
In this paper we work over $\bbC$ throughout, but it is also interesting to ask these questions in other characteristics. The argument above works with no modifications outside of characteristic $2$. In characteristic $2$, the argument breaks down since when $(d+t+dt) = 0$, we have that $\Delta(4,1)$ is automatically zero. However, in characteristic $2$, a closer look shows that $\Delta(4,0) = d^4 P_{SO(3)}^2$, so the conclusion holds anyway. We will ignore non-zero characteristic in the rest of the paper.
Proof of Proposition \[prop:4:uniqueness\] (Uniqueness) {#proof-of-proposition-prop4uniqueness-uniqueness .unnumbered}
-------------------------------------------------------
We fix a value of $(d,t)$ satisfying $P_{SO(3)} = 0$.
\[lem:I=H=>spanning\] If there is a relation of the form $$\label{eq:I=H}
{ \begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}}\; = \alpha \; { \begin{tikzpicture}[baseline=0cm]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}
}\; + \beta \; {\begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\; + \gamma \; {\begin{tikzpicture}[baseline=0cm]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}},$$ then any trivalent graph in $\cC_n$ can be reduced to $\operatorname{span} D(n,0)$.
Applying this relation to the largest face gives a sum of terms with either fewer faces or the same number of faces but with a smaller largest face. By induction, we can write any diagram as a sum of terms with no internal faces.
In a trivalent category with $\dim \Span D(4,0) \leq 3$, the diagrams $D(n,0)$ span $\cC_n$ for all $n$.
If the two diagrams and are linearly dependent, we obtain a relation as in Equation by adding an “H” along the top boundary of both pictures. Otherwise, there must be a relation amongst the four diagrams in $D(4,0)$, with the coefficient of at least one of and being nonzero. Rescaling and rotating gives a relation as in Equation .
\[lem:4:spanning\] In a trivalent category with $\dim \cC_4 \leq 4$, the diagrams $D(4,0)$ span $\cC_4$.
If $\dim \Span D(4,0) \leq 3$, then the previous Lemma applies. Otherwise, $\dim \Span D(4,0) = 4$ and the conclusion is immediate.
Note that on $P_{SO(3)} = 0$, $d \neq 1$.
\[lem:recognition:SO3\] In a trivalent category where $D(4,0)$ spans $\cC_4$ and $P_{SO(3)} = 0$, there is a relation of the form $${ \begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}}\; - \; { \begin{tikzpicture}[baseline=0cm]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}
}\; + \frac{1}{d-1}\; {\begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\; - \frac{1}{d-1}\; {\begin{tikzpicture}[baseline=0cm]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}= 0.$$
Since $D(4,0)$ spans, any element of $\cC_4$ with inner product $0$ with all elements of $D(4,0)$ must vanish. Computing the kernel of $M(4,0)$ as given above, we find this relation.
To prove the proposition, we observe that by Lemma \[lem:4:spanning\], $\cC_4 = \Span D(4,0)$, so by Lemma \[lem:recognition:SO3\] there is a relation of the form in Equation . Finally, Lemma \[lem:I=H=>spanning\] shows that this relation suffices to evaluate all closed diagrams as a multiple of the empty diagram, and Corollary \[cor:reductions=>uniqueness\] shows that there is a unique trivalent category at this value of $(d, t)$.
Proof of Proposition \[prop:4:realization\] (Realization) {#proof-of-proposition-prop4realization-realization .unnumbered}
---------------------------------------------------------
The curve $P_{SO(3)} = 0$ is rational and can be parameterized by $d= \delta^2 -1$ and $t= \frac{\delta^2 -3}{\delta^2-2}$ where $\delta \neq \pm \sqrt{2}$. It is more usual to change variables to $q+q^{-1} = \delta$ where $q$ is not a primitive $8$th root of unity. Under this parameterization $q$ and $q^{-1}$ are sent to the same point.
So long as $\delta \neq 0$, that is $(d,t) \neq (-1,3/2)$, we have the following realization. Let $\mathrm{TL}_\delta$ be the Temperley-Lieb category, which consists of linear combinations of planar tangles with the circle equal to $\delta$. This is not a trivalent category, since there is no trivalent vertex. However, we can interpret trivalent graphs in $\mathrm{TL}_\delta$ as follows. Take a trivalent graph and replace each strand with a pair of strands attached to the second Jones-Wenzl projection $$f^{(2)} = {\begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\;
-\frac{1}{\delta} \; {\begin{tikzpicture}[baseline=0cm]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}},$$ and replace each trivalent vertex by $$\sqrt{\frac{\delta}{\delta^2-2}} \; \begin{tikzpicture}[baseline]
\foreach \n in {0, 120, 240} {
\draw (30+\n:0.15) -- ($(30+\n:0.15) + (90+\n:1.33)$) (30+\n:0.15) -- ($(30+\n:0.15)+(-30+\n:1.33)$);
}
\foreach \n in {0, 120, 240} {
\node[draw, rectangle, fill=white,rotate=\n] at (90+\n:0.8) {$f^{(2)}$};
}
\end{tikzpicture}.$$
This trivalent category is called $SO(3)_q$ where $\delta = q+q^{-1}$. The trivalent vertex is normalized so that $b=1$, and quick calculation shows that $d= \delta^2 -1 = q^2+1+q^{-2}$ and $t= \frac{\delta^2 -3}{\delta^2-2} = \frac{q^2-1+q^{-2}}{q^2+q^{-2}}$.
The remaining point $(-1,3/2)$ is realized in a somewhat different way. The Lie supergroup $OSp(1|2)$ has a standard $(1|2)$-dimensional representation which we denote $X$. This representation is simple, and using highest weight theory, $X \otimes X \cong 1 \oplus X \oplus Y$ for some simple object $Y$ distinct from $1$ and $X$. Thus $X$ is self-dual and the map $X \otimes X \rightarrow X$ gives a map $X^{\otimes 3} \rightarrow 1$. A direct calculation shows that this map factors through the symmetric cube of $X$, and thus gives a trivalent vertex. We normalize this vertex so that the value of the bigon is $1$. This gives a trivalent category which we denote $OSp(1|2)$ which has $\dim \cC_4 = 3$ and $d=-1$. Thus, it must realize the remaining point $(-1, 3/2)$.
We will abuse notation somewhat and use $SO(3)_{\pm i}$ to refer to $OSp(1|2)$. This can be justified by giving an appropriately modified definition of $SO(3)_q$. Specifically the categories of representations of $U_{\pm i q}(\mathfrak{su}(2))$ and $U_q(\mathfrak{osp}(1|2))$ are closely related but non-isomorphic due the appearance of certain signs. However, if you restrict attention to the representations of even highest weight the categories become equivalent because these signs all vanish [@MR1230843; @MR3010460].
This strange point $(-1,3/2)$ can also be realized by the nondegenerate quotient of $(G_2)_{\pm i}$. This is a straightforward calculation, but we will delay it until Remark \[rem:G2-i\] since we will discuss $(G_2)_q$ in much more detail in the next section.
When $d$ is generic on $P_{SO(3)} = 0$, the dimension of $\cC_n$ is given by the Motzkin sums $1,0,1,1, 3,6,15,36,\ldots$ (A005043 in [@EIS]). More specifically, by computing the radical of the inner product, all of these categories have $\dim \cC_4 = 3$ unless $d$ satisfies $d^2=d+1$. In this last case (where $d$ is the golden ratio or its Galois conjugate) instead we have $\dim \cC_4=2$ and the category satisfies the additional skein relation $${ \begin{tikzpicture}[baseline=0cm]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}
}\; = \; {\begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\; - \frac{1}{d}\; {\begin{tikzpicture}[baseline=0cm]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}.$$ These two examples are often called the golden categories or the Fibonacci categories. For these categories the dimensions are given by Fibonacci numbers $1,0,1,1,2,3,5,
\ldots$.
One can make sense of $SO(3)_q$ at an $8$th root of unity, by not including the $\sqrt{\frac{\delta}{\delta^2-2}}$ factor in the trivalent vertex. With this normalization, $b=0$ and so the category is degenerate. Its nondegenerate quotient has no trivalent vertices.
Cubic categories
----------------
A *cubic category* is a trivalent category $\cC$ with $\dim{\cC_4} = 4$.
\[prop:cubic\] For any cubic category,
(a) the diagrams $D(4,0)$ form a basis of $\cC_4$,
(b) $d+t+dt \neq 0$ and $P_{SO(3)} = d+t-dt-2 \neq 0$,
(c) ${{
\begin{tikzpicture}[scale=0.8,baseline=-0.1cm]
\fill[white] (0,0) circle (0.5cm);
\foreach \x in {1, ..., 4} {
\draw (360*\x/4-360/4+45:.3cm) -- (360*\x/4+45:.3cm)--(360*\x/4+45:.5cm) -- (360*\x/4+360/4+45:.5cm);
}
\end{tikzpicture}
}}= \frac{2d+2dt-4dt^2+2dt^4+2d^2t^4 }{d+t+dt}$, and
(d) the square satisfies the following relation $$\begin{gathered}
\label{eq:square}
{
\begin{tikzpicture}[baseline=0cm]
\foreach \x in {1, ..., 4}
\draw (360*\x/4+45:.8cm)--(360*\x/4+45:.5cm);
\foreach \x in {1, ..., 4}
\draw (360*\x/4+45:.5cm) .. controls +(360*\x/4+120+45:.3cm) and +(360*\x/4+360/4-120+45:.3cm) .. (360*\x/4+360/4+45:.5cm);
\end{tikzpicture}
} = \frac{d t^2+t^2-1}{d t+d+t}
\left( \;
{ \begin{tikzpicture}[baseline=0cm]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}
}\; + \;
{ \begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}}\; \right)
\; \\+ \; \frac{-t^2+t+1}{d t+d+t}
\left( \;
{\begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\; + \;
{\begin{tikzpicture}[baseline=0cm]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\; \right)\end{gathered}$$
By Lemma \[lem:4:spanning\] if $D(4,0)$ is dependent then $\dim \cC_4 \leq 3$. Hence, $D(4,0)$ must be independent, and so a basis of $\cC_4$. Hence, $\Delta(4,0) \neq 0$, which implies $d+t+dt \neq 0$ and $d+t-dt-2 \neq 0$. On the other hand, we must have $\Delta(4,1) = 0$, giving (c). Finally, equation (\[eq:square\]) is in the radical of the inner product on $D(4,1)$.
We now identify the minimal idempotents in $\cC_4$ for any cubic category (subject to a certain quantity being invertible).
\[thm:idempotents-and-traces\] Suppose $\cC$ is a cubic category with parameters $d$ and $t$. If $$\xi=\sqrt{d^2 t^4+2 d \left(t^4-2 t^3-t^2+4
t+2\right)+\left(t^2-2 t-1\right)^2}$$ is nonzero, then the four minimal idempotents in $\cC_4$ (with respect the multiplication via vertical stacking) are $$\begin{aligned}
\iota & = \frac{1}{d} {\begin{tikzpicture}[baseline=0cm]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\\
x & = { \begin{tikzpicture}[baseline=0cm]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}
}\\
y_\pm & =
\frac{-(d+1) t^2 \pm\xi +1}{\pm2 \xi } {\begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}+ \frac{d \left(t^2-2 t-2\right) \mp \xi +t^2-2 t-1}{\pm 2 d \xi } {\begin{tikzpicture}[baseline=0cm]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\\
& \quad - \frac{d (t+2) t \pm \xi +t^2+1}{\pm 2 \xi } { \begin{tikzpicture}[baseline=0cm]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}
}+ \frac{d t+d+t}{\pm \xi } { \begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}}\end{aligned}$$ with dimensions $$\begin{aligned}
\tr \iota & = 1 \\
\tr x & = d \\
\tr {y_\pm} & = -\frac{d^3 t^2+d^2 \left(\mp\xi +2 t^2+2 t-1\right)+d
(\pm\xi +2 t+3)\pm\xi -t^2+2 t+1}{\pm 2 \xi }\end{aligned}$$ We note that $\tr{y_\pm}$ never vanishes since $d \neq 0$ and $P_{SO(3)} \neq 0$.
This is a direct calculation using Equation , performed in the Mathematica notebook [code/idempotents.nb]{} available with the [arXiv]{} source of this article. That $\iota$ and $x$ are minimal idempotents is clear. We then solve the quadratic equations $y_\pm \iota = y_\pm x = 0$, $y_+ + y_- = 1 - \iota - x$, and $y_\pm^2 = y_\pm \neq 0$.
When $\xi$ vanishes there is no basis of projections and so the category is not semisimple. This does not happen in any of our examples. (Note that for $G_2$ at $q=\pm i$ we have that $\xi$ vanishes, but at this special value $G_2$ is no longer cubic. See Remark \[rem:badpoints\].)
In any cubic category, if $n+2k < 12$, all entries of $M^\square(n,k)$ can be written as rational functions in $d$ and $t$.
There are fewer than twelve faces in each inner product appearing in $M^\square(n,k)$, because the number of faces is bounded by $n+2k$. Any polyhedron with fewer than twelve faces has at least one face which is a square or smaller. The relations for simplifying bigons, triangles and squares (Equations , , and ) then suffice to rewrite this polyhedron as a linear combination of polyhedra with strictly fewer faces; repeating the argument completely evaluates the original polyhedron as a function of $d$ and $t$.
In fact, the denominators in these rational functions are always powers of $Q_{1,1} = dt + d + t$, the denominator appearing in Equation .
We unapologetically state the values of these determinants as facts, even though for larger $n$ and $k$ they are the result of quite intensive calculations (computing the inner products is already time consuming, and subsequently the determinant is even harder). Of course, a computer is doing these calculations (see [code/ComputingInnerProducts.nb]{}).
A careful reader will note that the matrices $M^\square(8,0)$, $M^\square(8,1)$, $M^\square(9,0)$, $M^\square(9,1)$, $M^\square(10,0)$, and $M^\square(11,0)$ are covered by the above lemma, but we do not compute their determinants in what follows. They seem quite difficult without very considerable computational resources. In any case, our analysis of small skein theories meets another hurdle first; we can’t even compute the intersections of $\Delta^\square(7,0)$ and $\Delta^\square(7,1)$, or of $\Delta^\square(7,1)$ and $\Delta^\square(7,2)$.
Diagrams with five boundary points
==================================
In this section we study diagrams with five boundary points. Note that $$\begin{aligned}
D(5,0) & = \left\{
\mathfig{0.1}{tree1}, \mathfig{0.1}{tree2}, \mathfig{0.1}{tree3}, \mathfig{0.1}{tree4}, \mathfig{0.1}{tree5}, \right. \\
& \qquad \left. \mathfig{0.1}{forest1}, \mathfig{0.1}{forest2}, \mathfig{0.1}{forest3}, \mathfig{0.1}{forest4}, \mathfig{0.1}{forest5}\
\right\}\end{aligned}$$ is a set of 10 diagrams, and $$D^\square(5,1) \setminus D(5,0) = \left\{ {
\begin{tikzpicture}[baseline=0cm]
\foreach \x in {1, ..., 5}
\draw (360*\x/5+90:.8cm)--(360*\x/5+90:.5cm);
\foreach \x in {1, ..., 5}
\draw (360*\x/5+90:.5cm) .. controls +(360*\x/5+120+90:.3cm) and +(360*\x/5+360/5-120+90:.3cm) .. (360*\x/5+360/5+90:.5cm);
\end{tikzpicture}
} \right\}$$ has just one element.
\[thm:5\] If $\cC$ is a cubic category (that is $\dim \cC_4 = 4$) and $\dim \cC_5 \leq 10$, then either
1. $(d,t)$ satisfy $P_{ABA}= t^2-t-1 =0$ and $\cC$ is one of the ABA categories described below, or
2. $(d,t)$ satisfy $$P_{G_2} = d^2 t^5+2 d t^5-4 d t^4-d t^3+6 d t^2+4 d t+d+t^5-4 t^4+t^3+7 t^2-2=0$$ and $\cC$ is $(G_2)_q$ with $$\begin{aligned}
d & =q^{10} + q^8 + q^2 + 1 + q^{-2} + q^{-8} + q^{-10}, \\
\intertext{and}
t & = - \frac{q^2 - 1 + q^{-2}}{q^4 + q^{-4}}.\end{aligned}$$
Theorem \[thm:5\] follows from four propositions.
\[prop:5:nonexistence\] For any $(d,t)$ not satisfying $P_{ABA}=0$ or $P_{G_2}=0$ there are no trivalent categories with $\dim \cC_4 = 4$ and $\dim \Span D^\square(5,1) \leq 10$.
\[prop:5:uniqueness\] For every pair $(d,t)$ satisfying $P_{ABA}=0$ or $P_{G_2}=0$ there is at most one trivalent category with $\dim \cC_4 = 4$ and $\dim \cC_5 \leq 11$.
\[prop:5:realization:ABA\] The trivalent categories $ABA_{(d,t)}$, defined below via a free product construction, satisfy $\dim \cC_4 = 4$ and $\dim \cC_5 \leq 10$, and realize every pair $(d,t)$ satisfying $P_{ABA}=0$ and $P_{SO(3)} \neq 0$.
\[prop:5:realization:G2\] The $(G_2)_q$ categories are trivalent categories with $\dim \cC_4 = 4$ and $\dim \cC_5 \leq 10$, and realize every pair $(d,t)$ satisfying $P_{G_2}=0$ and $P_{SO(3)} \neq 0$.
(In fact, in this section we only need the weakening of Proposition \[prop:5:uniqueness\] that covers the cases where $\dim \cC_5 \leq 10$. We will need the full strength in the next section.)
\[rem:badpoints\] By non-degeneracy $d \neq 0$. By Proposition \[prop:cubic\], any cubic category has $P_{SO(3)} \neq 0$ and $d+t +d t \neq 0$. In this remark we catalog what happens at the remaining special points where one of these polynomials does vanish.
The points on the intersection of the $P_{ABA}$ curve and $P_{SO(3)}(d+t+d t) =0 $ are $(d,t) = (\tau, \bar{\tau})$ and $(\bar{\tau}, \tau)$. These two points do correspond to trivalent, but non-cubic, categories: the golden categories with $\dim \cC_2 = 2$.
The intersection points of the $P_{G_2}$ curve and $P_{SO(3)}(d+t+d t) =0 $ are:
1. $(d,t)=(-1,3/2)$, corresponding to $q$ a primitive $4$th root of unity.
2. $(d,t)=(-2,-2)$, corresponding to $q$ a primitive $3$rd or $6$th root of unity.
3. $(d,t)=(2,0)$, corresponding to $q$ a primitive $12$th root of unity.
4. The two points $(d,t) = (\tau, \bar{\tau})$ and $(d,t)=(\bar{\tau},\tau)$, corresponding to $q$ a primitive $30$th root of unity.
The first of these cases is the special trivalent non-cubic category $(G_2)_{q=\pm i} = OSp(1|2)$. For the second, the bigon for $G_2$ at that value is $0$, so the non-degenerate quotient is not trivalent. The third case corresponds to the trivalent non-cubic category $SO(3)_{\zeta_{12}}$. The last case corresponds to the golden categories with $\dim \cC_2 = 2$.
As always we assume $d \neq 0$, corresponding to $q$ is not a primitive $7$th, $14$th, or $24$th root of unity, as otherwise the non-degenerate quotient is trivial.
If $\cC$ and $\cD$ are two pivotal categories, then their tensor product $\cC \boxtimes \cD$ has morphisms consisting of a red planar diagram and a blue planar diagram superimposed on each other, where the blue parts live in $\cC$ and the red parts live in $\cD$ and the red diagrams are allowed to cross the blue diagrams in a symmetric way. Clearly the dimensions of the box spaces multiply, as $$\Hom_{\cC \boxtimes \cD}(1 \to (X \boxtimes Y)^{\otimes n}) \iso \Hom_{\cC}(1 \to X^{\otimes n}) \otimes \Hom_{\cD}(1 \to Y^{\otimes n}),$$ and the invariants of closed diagrams also just multiply. Let $\cG_\tau$ denote the Golden category with loop value $\tau$ (and triangle value $\bar{\tau}$). The tensor products $\cG_\tau \boxtimes \cG_\tau$, $\cG_\tau \boxtimes \cG_{\bar{\tau}}$, and $\cG_{\bar{\tau}} \boxtimes \cG_{\bar{\tau}}$ give pivotal categories with a distinguished trivalent vertex with $n$-boundary point spaces of dimension $(1,0,1,1,4,9, \ldots)$. It is natural to wonder how they fit into our classification. The category $\cG_\tau \boxtimes \cG_{\bar{\tau}}$ has $(d,t)=(-1,-1)$ which lies on the $G_2$ curve and corresponds to $q$ a primitive $20$th root. The categories $\cG_\tau \boxtimes \cG_\tau$ and $\cG_{\bar{\tau}} \boxtimes \cG_{\bar{\tau}}$ have $(d,t)$ being $(\tau^2, \bar{\tau}^2)$ and $(\bar{\tau}^2, \tau^2)$. These points lie on the $\mathrm{SO}(3)$ curve corresponding to $q$ being a primitive $20$th root of unity (half of which give $d=\tau^2$ and half of which give $d=\bar{\tau}^2$). In particular, these categories are not generated by the trivalent vertex. Instead, they correspond to the even part of the $D_6$ subfactor and its Galois conjugate. These categories contain a trivalent subcategory $\mathrm{SO}(3)_{\zeta_{20}}$ and a single extra generating $4$-box satisfying a version of the relations from [@MR2559686]. The relationship between $D_6$ and the tensor products of golden categories is explained by level-rank duality between $\mathrm{SO}(3)_4$ and $\mathrm{SO}(4)_3$ together with the coincidence of Lie algebras between $\mathfrak{so}(4)$ and $\mathfrak{sl}(2) \oplus \mathfrak{sl}(2)$, as explained in [@MR2783128 Theorem 4.1].
We follow the same outline as in the previous section.
Proof of Proposition \[prop:5:nonexistence\] (Non-existence) {#proof-of-proposition-prop5nonexistence-non-existence .unnumbered}
------------------------------------------------------------
In any trivalent category, $$\begin{aligned}
\Delta(5, 0) = d^{10} P_{ABA}^2 P_{SO(3)}^4 Q_{1,2}.\end{aligned}$$
In any trivalent category, $$\Delta^\square(5,1) = d^{10} P_{ABA}^2 P_{SO(3)}^4 \left(-5 d t\left( d t^5+2 t^5 -2t^4 -2 t^3 +2 t^2 + t\right)+ Q_{1,2} {{
\begin{tikzpicture}[scale=0.8,baseline=-0.1cm]
\fill[white] (0,0) circle (0.5cm);
\foreach \x in {1, ..., 5} {
\draw (360*\x/5-360/5+0:.3cm) -- (360*\x/5+0:.3cm)--(360*\x/5+0:.5cm) -- (360*\x/5+360/5+0:.5cm);
}
\end{tikzpicture}
}}\right),$$ and in a cubic category, this specializes to $$\Delta^\square(5, 1) = d^{11} P_{ABA}^3 P_{SO(3)}^5 P_{G_2} Q_{1,1}^{-2}.$$
A cubic category with $\dim \Span D^\square(5,1) \leq 10$ must have $\Delta^\square(5,1) = 0$. Since $d \neq 0$, we must have one of $P_{SO(3)} = 0$, $P_{ABA} = 0$, or $P_{G_2} = 0$. Since $P_{SO(3)} \neq 0$, the proposition is proved.
Proof of Proposition \[prop:5:uniqueness\] (Uniqueness) {#proof-of-proposition-prop5uniqueness-uniqueness .unnumbered}
-------------------------------------------------------
We fix a value of $(d,t)$ satisfying $P_{G_2}$ or $P_{ABA}$.
It is well-known that any planar trivalent graph has a pentagon or smaller face. We want an analogue of this result for open planar trivalent graphs, that is, planar trivalent graphs having boundary. (This is similar to the analysis in [@MR1403861].) In order to state this result we introduce some language.
\[defininggrowth\] An *open* planar trivalent graph is a planar trivalent graph in the disc which meets the boundary in $n \geq 1$ specified points. An open planar trivalent graph has two kinds of faces, *internal faces* and *boundary faces*, depending on whether they touch the boundary of the disc.
We say a planar trivalent graph is *connected* if the vertices and edges (whether internal edges or boundary edges) form a connected topological space.
We say that an open planar trivalent graph is *boundary connected* if every component meets the boundary. Note that a face need not be topologically a disc, unless the graph is boundary connected.
A *boundary region* of a connected planar trivalent graph is a small neighborhood of some contiguous proper subset of the boundary faces. A boundary region is called a *growth region* if the number of edges meeting the boundary is greater than the number of edges not meeting the boundary. Figure \[fig:growthregion\] illustrates these definitions.
(2,0) arc (0:180:2cm); (10:2cm) – (30:1cm) – (50:2cm); (170:2cm) – (150:1cm) – (130:2cm); (30:1cm)–(45:.5cm) – ++(-90:.5cm); (150:1cm)–(135:.5cm) – ++(-90:.5cm); (45:.5cm)–(90:.5cm)–(135:.5cm); (90:.5cm)–(90:2cm); at (30:1.6cm) [$D$]{}; at (60:1cm) [$C$]{}; at (120:1cm) [$B$]{}; at (150:1.6cm) [$A$]{};
We now restrict our attention to boundary connected open trivalent graphs. We assign a *charge* to each face as follows. Let $n$ be the number of edges meeting that face, and let $m$ be the number of disjoint boundary intervals meeting that face. Now assign the charge $6-n-2m$. In particular, an internal $n$-gon face is assigned $6-n$, so the only positively charged internal faces are pentagons or smaller. A boundary face that only meets the boundary once and touches $n$ edges is given charge $4-n$. A standard Euler characteristic argument shows that the total charge of the graph is $6$.
\[lem:boundarycharge\] In a connected open planar trivalent graph, any boundary region for which the sum of the charges on its constituent boundary faces is at least $2$ is a growth region.
Once we assume a graph is connected, each boundary face meets the boundary in a single interval. Now, if a boundary region has $I$ incoming edges and $O$ outgoing edges, then the total charge of the boundary faces is $O-I+1$.
We say that an internal face is *small* if it has $5$ or fewer sides.
Any connected open planar trivalent graph has an internal small face or a growth region.
Since the total charge is $6$, either the boundary charge is at least $6$, or there is a positively charged internal face, which must be a small face. If the boundary charge is 6 or more, then there are at least two boundary faces because a single boundary face has charge $4-n$ (where $n$ is the number of edges it touches). Thus, we can divide the boundary into two proper sub-regions. One of these has charge three or greater, so by Lemma \[lem:boundarycharge\] this is a growth region.
It follows that we can easily inductively enumerate all boundary connected open graphs with $n$ boundary points and no internal small faces by first enumerating the connected graphs by attaching growth regions to open graphs with strictly fewer boundary points and with no internal small faces, and then writing down the all the planar unions of such graphs.
\[cor:nopents\] If $n \leq 5$, then any boundary connected open planar trivalent graph with $n$ boundary points and no small faces is in $D(n,0)$.
Of course, when $n = 6$ there’s an open graph with a single hexagonal face which does not lie in $D(6,0)$.
\[lem:pentagon-reduction\] Suppose $\cC$ is a category generated by a trivalent vertex, with relations reducing $n$-gons for each $n \leq 4$. Suppose further there is some relation between the diagrams in $D^\square(5,1)$. Then there is a relation reducing the $5$-gon (as linear combination of diagrams in $D(5,0)$).
If the relation between the diagrams in $D^\square(5,1)$ already includes the pentagon, we are done. Otherwise there must be a relation only amongst the diagrams in $D(5,0)$. If this relation involves any of the diagrams of the form $\mathfig{0.04}{tree1heavy}$ we can add an $H$ to the boundary of this relation and obtain a relation writing a pentagon as a linear combination of diagrams without internal faces. Otherwise, if there’s a relation only involving the diagrams of the form $\mathfig{0.04}{forest1heavy}$, we find this diagram as a sub-diagram of a pentagon consisting of a vertex and its opposite edge. We then apply the relation inside the pentagon, and obtain another relation writing a pentagon as a linear combination of diagrams without internal faces. (One can readily verify no pentagons appear in other terms.)
\[cor:5:evaluation\] Given fixed relations reducing $n$-gons for each $n \leq 4$, and some relation between the diagrams in $D^\square(5,1)$, there is at most one trivalent category satisfying these relations.
As any closed trivalent graphs contains an $n$-gon with $n \leq 5$, and by the previous Lemma we can reduce this diagram, we see that the available relations suffice to evaluate all closed diagrams. By Corollary \[cor:reductions=>uniqueness\] we are done.
\[lem:5:dependent=>spans\] Suppose $\cC$ is a cubic category, with a relation between the diagrams in $D^\square(5,1)$. The $\cC_5$ is spanned by $D(5,0)$.
By Lemma \[lem:pentagon-reduction\], we can reduce any diagram in $\cC_5$ to a linear combination of diagrams without small faces. By Corollary \[cor:nopents\], these are in $D(5,0)$.
\[lem:ABA-relation\] In a cubic category where $D^\square(5,1)$ spans $\cC_5$ and $P_{ABA}=0$ there are relations: $$\begin{aligned}
\label{eq:free1}
\mathfig{0.1}{tree1} + \zeta \mathfig{0.1}{tree2} + \zeta^2 \mathfig{0.1}{tree3} + \zeta^3 \mathfig{0.1}{tree4} + \zeta^4 \mathfig{0.1}{tree5} = 0 \\
\label{eq:free2}
\mathfig{0.1}{tree1} + \zeta^{-1} \mathfig{0.1}{tree2} + \zeta^{-2} \mathfig{0.1}{tree3} + \zeta^{-3} \mathfig{0.1}{tree4} + \zeta^{-4} \mathfig{0.1}{tree5} = 0\end{aligned}$$ with $\zeta^5 = 1$ and $t+\zeta^2 +\zeta^3 = 0$ (so if $t = \frac{1+\sqrt{5}}{2}$, $\zeta = \exp(\pm 2\pi i/5)$, and if $t = \frac{1-\sqrt{5}}{2}$, $\zeta=\exp(\pm 4 \pi i /5)$).
By non-degeneracy and $D^\square(5,1)$ spanning, anything in the kernel of $M(5,1)$ must be a relation. When $P_{ABA} = t^2-t-1=0$, we obtain the relations above. (See the calculation in [code/ABA.nb]{}.)
Now we turn our attention to the case where $P_{G_2}=0$. The following Lemma is well-known [@MR1265145].
The curve $P_{G_2}=0$ is rational and can be parameterized by $$\begin{aligned}
d & =x^5+x^4-5x^3-4x^2+6x+3, \\
\intertext{and}
t & = - \frac{x-1}{x^2-2}\end{aligned}$$ where $x \neq \pm \sqrt{2}$.
It is more usual to change variables so that $x = q^2+q^{-2}$ in order to relate this to quantum groups with the usual variables. Note that this change of variables is typically $4$-to-$1$ with $\pm q^{\pm 1}$ all corresponding to the same pair $(d,t)$. In these variables we have $$\begin{aligned}
d & =q^{10} + q^8 + q^2 + 1 + q^{-2} + q^{-8} + q^{-10}, \\
\intertext{and}
t & = - \frac{q^2-1+q^{-2}}{q^4+q^{-4}},\end{aligned}$$ where $q$ is not a primitive $16$th root of unity. Recall that we also have that $q$ is not a primitive $3$rd or $6$th root of unity since $d+t+d t \neq 0$.
\[lem:G2-relation\] In a cubic category where $D^\square(5,1)$ spans $\cC_5$ and $P_{G_2}=0$ there is a relation: $${
\begin{tikzpicture}[baseline=0cm]
\foreach \x in {1, ..., 5}
\draw (360*\x/5+90:.8cm)--(360*\x/5+90:.5cm);
\foreach \x in {1, ..., 5}
\draw (360*\x/5+90:.5cm) .. controls +(360*\x/5+120+90:.3cm) and +(360*\x/5+360/5-120+90:.3cm) .. (360*\x/5+360/5+90:.5cm);
\end{tikzpicture}
} = \alpha \left(\mathfig{0.1}{tree1} + \text{rotations}\right) +\beta \left(\mathfig{0.1}{forest1} + \text{rotations}\right),$$ where $$\begin{aligned}
\alpha &= - \frac{1}{(q^2+1+q^{-2})(q^4+q^{-4})} \displaybreak[1] \\
\beta &=- \frac{1}{(q^2+1+q^{-2})^2(q^4+q^{-4})^2}
$$
When $P_{G_2}=0$ these relations are in the radical of the inner product on $M^\square(5,1)$. (See the calculation in [code/G2.nb]{}.)
We now prove the proposition. Suppose $D^\square(5,1)$ is dependent. Then Lemma \[lem:5:dependent=>spans\] shows that $D(5,0)$ spans, and hence trivially $D^\square(5,1)$ spans also. On the other hand, if $D^\square(5,1)$ is independent, it also must span, since we are assuming $\dim \cC_5 \leq 11$. Then Lemma \[lem:ABA-relation\] and Lemma \[lem:G2-relation\] ensure that there are relations amongst the diagrams in $D^\square(5,1)$. By Corollary \[cor:5:evaluation\] there is a unique cubic category at the given values of $d$ and $t$.
Proof of Proposition \[prop:5:realization:ABA\] (Realization) {#proof-of-proposition-prop5realizationaba-realization .unnumbered}
-------------------------------------------------------------
Note that for any $(d,t)$ with $P_{ABA} = t^2-t-1 = 0$ we can rewrite $(d,t) =
(\delta^2 \tau, \bar{\tau})$ where bar is the Galois conjugate $(1+\sqrt{5})/2
\leftrightarrow (1-\sqrt{5})/2$, by taking $\tau$ to be the Galois conjugate of $t$ and $\delta = \sqrt{d \tau^{-1}}$. This change of variables is generally 2-to-1, and there is a symmetry $\delta \leftrightarrow -\delta$.
Let $TL_\delta$ be the Temperley-Lieb category of planar tangles with loop value $\delta$, and let $\cG_{\tau}$ be the golden category with loop value $\tau$ (so that the triangle value will be $\bar{\tau}$). If $\cC$ and $\cD$ are two pivotal categories, then their free product $\cC * \cD$ consists of planar diagrams with connected components labelled blue and red, where the blue parts live in $\cC$ and the red parts live in $\cD$ (cf. [@1308.5723] and [@MR1950890 Section 8]). We’ve shown all blue components in what follows with dashed lines. Consider the free product $TL_\delta \ast \cG_{\tau}$. This is a category of planar diagrams with connected components labelled blue and red, where the blue strands have no vertices and have loop value $\delta$, while red strands allow trivalent vertices and have loop value $\tau$.
Inside the free product $TL_\delta \ast \cG_{\tau}$ we can find a trivalent category, which we call $ABA_{(\delta^2 \tau, \bar{\tau})}$ as follows. Given a trivalent graph interpret each strand as a red strand then a blue strand then a red strand (hence the name $ABA$), and think of each trivalent vertex as given by $$\frac{1}{\sqrt{\delta}}
\begin{tikzpicture}[baseline]
\draw[red] (0,0) -- (90:1);
\draw[red] (0,0) -- (210:1);
\draw[red] (0,0) -- (330:1);
\draw[blue,dashed] (100:1) to[out=-90,in=30] (200:1);
\draw[blue,dashed] (220:1) to[out=30,in=150] (320:1);
\draw[blue,dashed] (340:1) to[out=150,in=-90] (80:1);
\end{tikzpicture}.$$ Here the normalization factor ensures that the bigon factor $b$ is $1$. We call this category $ABA'_{(d,t)}$ and then define $ABA_{(d,t)}$ to be the quotient by the negligibles.
\[rem:generic\] As far as we know, it may be that $ABA'_{(d,t)}$ has no negligbles and so $ABA'_{(d,t)} = ABA_{(d,t)}$. Indeed, when $dt^{-1}$ is a positive number bigger than $4$ the pairing is positive definite because it is the restriction of an obviously positive definite pairing on the tensor product. Hence when $d$ is generic, $ABA'_{(d,t)}$ has no negligbles.
We can now prove the proposition. We first find a spanning set for the $n$-boundary point space for $ABA$. First note that the blue (A-labelled) lines of the $ABA$ diagram give a noncrossing partition of the red boundary points, where two red boundary points are in the same partition if you can get between them without crossing a blue line. Second, recall that the dimension of the space of $m$-boundary point red diagrams is $F_{m-1}$ (the $m-1$st Fibonacci number). There is a standard explicit basis given by fixing a trivalent tree connecting all the boundary vertices and then picking a subset of internal edges to delete, with the condition that each vertex have either two or three edges coming out of it.
Putting the above two steps together, for each noncrossing partition (specifying the location of the blue lines), we can find a spanning set for diagrams compatible with that partition consisting of $\prod_{p \in \pi} F_{| p |-1}$ diagrams (where $\pi$ is a partition, $p$ ranges over parts of the partition, and $|p|$ is the size of the part). This process gives spanning sets for the $n$-boundary point spaces for $0 \leq n \leq 5$ with sizes $1,0,1,1,4,8$. We illustrate this below by giving the spanning set for $n=5$.
(36:1cm) .. controls (54:.8cm) .. (72:1cm); (36:1cm) .. controls (54:.8cm) .. (72:1cm); (36:1cm) .. controls (54:.8cm) .. (72:1cm); (36:1cm) .. controls (54:.8cm) .. (72:1cm); (36:1cm) .. controls (54:.8cm) .. (72:1cm);
(18:1cm) .. controls (18:.6cm) .. (54:.5cm) arc (54:270:.5cm) .. controls (-54:.6cm) .. (-54:1cm); (90:1cm) – (90:.5cm); (162:1cm) – (162:.5cm); (234:1cm) – (234:.5cm);
(36:1cm) .. controls (54:.8cm) .. (72:1cm); (36:1cm) .. controls (54:.8cm) .. (72:1cm); (36:1cm) .. controls (54:.8cm) .. (72:1cm); (36:1cm) .. controls (54:.8cm) .. (72:1cm); (36:1cm) .. controls (54:.8cm) .. (72:1cm);
(18:1cm) – (0,0) – (162:1cm); (90:1cm) – (0,0); (234:1cm) .. controls (270:.4cm).. (306:1cm);
(36:1cm) .. controls (54:.8cm) .. (72:1cm); (36:1cm) .. controls (54:.8cm) .. (72:1cm); (36:1cm) .. controls (54:.8cm) .. (72:1cm); (36:1cm) .. controls (54:.8cm) .. (72:1cm); (36:1cm) .. controls (54:.8cm) .. (72:1cm);
(-54:1cm) – (0,0) – (162:1cm); (234:1cm) – (0,0); (18:1cm) .. controls (54:.4cm).. (90:1cm);
(36:1cm) .. controls (54:.8cm) .. (72:1cm); (108:1cm) .. controls (54:.2cm) .. (0:1cm); (180:1cm) .. controls (198:.8cm) .. (216:1cm); (252:1cm) .. controls (270:.8cm) .. (288:1cm); (324:1cm) .. controls (0,0) .. (144:1cm);
(-54:1cm) – (234:.2cm) – (162:1cm); (234:1cm) – (234:.2cm); (18:1cm) .. controls (54:.4cm).. (90:1cm);
(36:1cm) .. controls (54:.8cm) .. (72:1cm); (108:1cm) .. controls (54:.2cm) .. (0:1cm); (180:1cm) .. controls (198:.8cm) .. (216:1cm); (252:1cm) .. controls (270:.8cm) .. (288:1cm); (324:1cm) .. controls (0,0) .. (144:1cm);
(-54:1cm) – (234:.2cm) – (162:1cm); (234:1cm) – (234:.2cm); (18:1cm) .. controls (54:.4cm).. (90:1cm);
(36:1cm) .. controls (54:.8cm) .. (72:1cm); (108:1cm) .. controls (54:.2cm) .. (0:1cm); (180:1cm) .. controls (198:.8cm) .. (216:1cm); (252:1cm) .. controls (270:.8cm) .. (288:1cm); (324:1cm) .. controls (0,0) .. (144:1cm);
(-54:1cm) – (234:.2cm) – (162:1cm); (234:1cm) – (234:.2cm); (18:1cm) .. controls (54:.4cm).. (90:1cm);
(36:1cm) .. controls (54:.8cm) .. (72:1cm); (108:1cm) .. controls (54:.2cm) .. (0:1cm); (180:1cm) .. controls (198:.8cm) .. (216:1cm); (252:1cm) .. controls (270:.8cm) .. (288:1cm); (324:1cm) .. controls (0,0) .. (144:1cm);
(-54:1cm) – (234:.2cm) – (162:1cm); (234:1cm) – (234:.2cm); (18:1cm) .. controls (54:.4cm).. (90:1cm);
(36:1cm) .. controls (54:.8cm) .. (72:1cm); (108:1cm) .. controls (54:.2cm) .. (0:1cm); (180:1cm) .. controls (198:.8cm) .. (216:1cm); (252:1cm) .. controls (270:.8cm) .. (288:1cm); (324:1cm) .. controls (0,0) .. (144:1cm);
(-54:1cm) – (234:.2cm) – (162:1cm); (234:1cm) – (234:.2cm); (18:1cm) .. controls (54:.4cm).. (90:1cm);
By computing inner products we see that these spanning sets actually form bases unless $P_{SO(3)} = 0$ (in which case the nondegenerate quotient is just a golden category). Hence the ABA categories are cubic categories with $\dim
\cC_5 \leq 11$.
Finally, it is straighforward to check that the loop value is $\delta^2 \tau = d$. We see that the triangle value is $\bar{\tau}$: $$\begin{aligned}
{
\begin{tikzpicture}[baseline=0cm]
\foreach \x in {1, ..., 3}
\draw (360*\x/3+90:.8cm)--(360*\x/3+90:.5cm);
\foreach \x in {1, ..., 3}
\draw (360*\x/3+90:.5cm) .. controls +(360*\x/3+120+90:.3cm) and +(360*\x/3+360/3-120+90:.3cm) .. (360*\x/3+360/3+90:.5cm);
\end{tikzpicture}
}
& =
\left(\frac{1}{\sqrt{\delta}}\right)^3
\begin{tikzpicture}[baseline=0]
\draw[red] (90:1cm) -- (90:.7cm) -- (-30:.7cm) -- (-30:1cm);
\draw[red] (-150:1cm) -- (-150:.7cm) -- (-30:.7cm);
\draw[red] (90:.7cm)--(-150:.7cm);
\draw[blue,dashed] (-20:1cm) .. controls (30:.5cm) .. (80:1cm);
\draw[blue,dashed,rotate=120] (-20:1cm) .. controls (30:.5cm) .. (80:1cm);
\draw[blue,dashed,rotate=-120] (-20:1cm) .. controls (30:.5cm) .. (80:1cm);
\draw[blue,dashed] (0,0) circle (.2cm);
\end{tikzpicture}
=
\left(\frac{1}{\sqrt{\delta}}\right)^3 \delta
\begin{tikzpicture}[baseline=0]
\draw[red] (90:1cm) -- (90:.7cm) -- (-30:.7cm) -- (-30:1cm);
\draw[red] (-150:1cm) -- (-150:.7cm) -- (-30:.7cm);
\draw[red] (90:.7cm)--(-150:.7cm);
\draw[blue,dashed] (-20:1cm) .. controls (30:.5cm) .. (80:1cm);
\draw[blue,dashed,rotate=120] (-20:1cm) .. controls (30:.5cm) .. (80:1cm);
\draw[blue,dashed,rotate=-120] (-20:1cm) .. controls (30:.5cm) .. (80:1cm);
\end{tikzpicture} \displaybreak[1] \\
& =
\bar{\tau} \cdot
\frac{1}{\sqrt{\delta}}
\begin{tikzpicture}[baseline]
\draw[red] (0,0) -- (90:1);
\draw[red] (0,0) -- (210:1);
\draw[red] (0,0) -- (330:1);
\draw[blue,dashed] (100:1) to[out=-90,in=30] (200:1);
\draw[blue,dashed] (220:1) to[out=30,in=150] (320:1);
\draw[blue,dashed] (340:1) to[out=150,in=-90] (80:1);
\end{tikzpicture}
= \bar{\tau} \cdot
\begin{tikzpicture}[baseline=.1cm,scale=0.75]
\draw (0,0) -- (0,1);
\draw (0,0) -- (0.7,-0.5);
\draw (0,0) -- (-0.7,-0.5);
\end{tikzpicture}.\end{aligned}$$ This completes the proof of Proposition \[prop:5:realization:ABA\].
By Remark \[rem:generic\], when $d$ is generic there are no further relations, and so the dimension of the $n$-boundary point space is given by $\sum_\pi
\prod_{p \in \pi} F_{| p |-1}$. This sequence begins $1,0,1,1,4,8,25,64,\ldots$ and its ordinary generating function satisfies the relation $G(x) =
\frac{1-xG(x)}{1-xG(x)-x^2G(x)^2}$. Therefore it is given by OEIS A046736 [@EIS], which counts the number of ways to place non-intersecting diagonals on a $n+2$-gon so as to create no triangles. To find this generating function identity, we use a general recipe due to Speicher [@MR1268597] for any such weighted counting of non-crossing partitions. (This approach thus applies to the pivotal tensor category generated by $ABA$ in the free product of Temperley- Lieb with an arbitrary trivalent category.) Let $a_i$ be an arbitrary sequence with $a_0 = 1$, let $b_n = \sum_\pi \prod_{p \in \pi} a_{| p |}$, and let $A(x)$ and $B(x)$ be their ordinary generating functions. Then rewriting [@MR1676282 Exercise 5.35] yields $B(x) = A(xB(x))$. In our particular example, we use that the generating function for the shifted Fibonacci sequence is $\frac{1-x}{1-x-x^2}$.
It is not difficult to work out the simple objects in the ABA categories. They are of the form $A^{(n_1)}BA^{(n_2)}BA^{(n_3)}B\ldots B A^{(n_k)}$ where $A^{(n)}$ denotes the $n$th Jones–Wenzl made of blue strands (so for generic $\delta$ you allow the $n_i$ to be any positive number, while for $\delta =
\zeta+\zeta^{-1}$ there is a corresponding bound on $n_i$). The fusion rules are given by concatenation and applying the usual $SU(2)$ fusion rules for blue Jones-Wenzl’s and $B^2 = B+1$ for red strands. So, for example, $(ABA) (ABA) =
ABA^{(2)}BA + ABA+A^{(2)}+1$.
Proof of Proposition \[prop:5:realization:G2\] (Realization) {#proof-of-proposition-prop5realizationg2-realization .unnumbered}
------------------------------------------------------------
We recall Kuperberg’s skein theoretic description of the quantum $G_2$ spider categories [@MR1403861; @MR1265145] (warning, there is a sign error in [@MR1403861]). We change conventions in two ways: Kuperberg’s $q$ is our $q^2$ (which agrees with the usual quantum group conventions) and we normalize the trivalent vertex so that the bigon equals the strand (which is possible so long as $q$ is not a primitive $3$rd or $6$th or $16$th root of unity).
\[def:G2\] If $q$ is not a primitive $3$rd, $6$th, or $16$th root of unity, let $(G_2)_q'$ be the pivotal category generated by a trivalent vertex, modulo the following skein relations, and let $(G_2)_q$ be the nondegenerate quotient of $(G_2)_q'$ by its negligible ideal. $$\begin{aligned}
\bigcirc &= \Phi_7 \Phi_{14} \Phi_{24} = q^{10} + q^8 + q^2 + 1 + q^{-2} + q^{-8} + q^{-10} \displaybreak[1] \\
\begin{tikzpicture}[scale=.5, baseline=0]
\draw (0,.5) circle (.5cm);
\draw (0,0)--(0,-1);
\end{tikzpicture}
&= 0 \displaybreak[1] \\
{
\begin{tikzpicture}[baseline=0cm]
\foreach \x in {1, ..., 2}
\draw (360*\x/2+90:.8cm)--(360*\x/2+90:.5cm);
\foreach \x in {1, ..., 2}
\draw (360*\x/2+90:.5cm) .. controls +(360*\x/2+120+90:.3cm) and +(360*\x/2+360/2-120+90:.3cm) .. (360*\x/2+360/2+90:.5cm);
\end{tikzpicture}
} &= \begin{tikzpicture}[scale=.5, baseline=0]
\draw (0,1)--(0,-1);
\end{tikzpicture}
\displaybreak[1] \\
{
\begin{tikzpicture}[baseline=0cm]
\foreach \x in {1, ..., 3}
\draw (360*\x/3+90:.8cm)--(360*\x/3+90:.5cm);
\foreach \x in {1, ..., 3}
\draw (360*\x/3+90:.5cm) .. controls +(360*\x/3+120+90:.3cm) and +(360*\x/3+360/3-120+90:.3cm) .. (360*\x/3+360/3+90:.5cm);
\end{tikzpicture}
} &= - \frac{\Phi_{12}}{\Phi_{16}} \begin{tikzpicture}[scale=.5,baseline=0]
\draw (90:1cm)--(0,0)--(-30:1cm);
\draw (0,0) -- (-150:1cm);
\end{tikzpicture}
= - \frac{q^2-1+q^{-2}}{q^4+q^{-4}} \begin{tikzpicture}[scale=.5,baseline=0]
\draw (90:1cm)--(0,0)--(-30:1cm);
\draw (0,0) -- (-150:1cm);
\end{tikzpicture}
\displaybreak[1] \\
{
\begin{tikzpicture}[baseline=0cm]
\foreach \x in {1, ..., 4}
\draw (360*\x/4+45:.8cm)--(360*\x/4+45:.5cm);
\foreach \x in {1, ..., 4}
\draw (360*\x/4+45:.5cm) .. controls +(360*\x/4+120+45:.3cm) and +(360*\x/4+360/4-120+45:.3cm) .. (360*\x/4+360/4+45:.5cm);
\end{tikzpicture}
} &= \frac{\Phi_8}{\Phi_3 \Phi_6 \Phi_{16}} \left(\;{ \begin{tikzpicture}[baseline=0cm]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}
}\; +\; { \begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}}\; \right) + \frac{1}{\Phi_3 \Phi_6 \Phi_{16}^2} \left(\; {\begin{tikzpicture}[baseline=0cm]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\; + \; \identity \; \right) \displaybreak[1] \\
{
\begin{tikzpicture}[baseline=0cm]
\foreach \x in {1, ..., 5}
\draw (360*\x/5+90:.8cm)--(360*\x/5+90:.5cm);
\foreach \x in {1, ..., 5}
\draw (360*\x/5+90:.5cm) .. controls +(360*\x/5+120+90:.3cm) and +(360*\x/5+360/5-120+90:.3cm) .. (360*\x/5+360/5+90:.5cm);
\end{tikzpicture}
} &= - \frac{1}{\Phi_3 \Phi_6 \Phi_{16}} \left(\mathfig{0.1}{tree1} + \mathfig{0.1}{tree2} + \mathfig{0.1}{tree3} + \mathfig{0.1}{tree4} + \mathfig{0.1}{tree5} \right) \\
& \qquad -\frac{1}{\Phi_3^2 \Phi_6^2 \Phi_{16}^2} \left(\mathfig{0.1}{forest1} + \mathfig{0.1}{forest2} + \mathfig{0.1}{forest3} + \mathfig{0.1}{forest4} + \mathfig{0.1}{forest5}\right),
$$ where $\Phi_k$ is the $k$th symmetrized cyclotomic polynomial. That is, $\Phi_k = \prod_\zeta \left(q^{\frac{1}{2}}-\zeta q^{-\frac{1}{2}}\right)$ where the product is taken over all primitive $k$th roots of unity. Explicitly, $$\begin{aligned}
\Phi_3 & = q+1+q^{-1} \displaybreak[1] \\
\Phi_6 & = q-1+q^{-1} \displaybreak[1] \\
\Phi_7 & = q^3 + q^2 + q + 1 + q^{-1} + q^{-2} + q^{-3} \displaybreak[1] \\
\Phi_8 & = q^2+q^{-2} \displaybreak[1] \\
\Phi_{12} & = q^2-1+q^{-2} \displaybreak[1] \\
\Phi_{14} & = q^3 - q^2 + q - 1 + q^{-1} - q^{-2} + q^{-3} \displaybreak[1] \\
\Phi_{16} & = q^4+q^{-4} \displaybreak[1] \\
\Phi_{24} & = q^4-1+q^{-4}.\end{aligned}$$
Now, suppose that in addition, $q$ is not a primitive $7$th, $14$th, or $24$th root of unity. We want to show that $(G_2)_q$ is a trivalent category. We’ve already seen in Corollary \[cor:nopents\] that for $n \leq 5$ the $n$-boundary point space of $(G_2)_q$ is spanned by diagrams in $D(n,0)$ and hence has dimensions bounded by $1,0,1,1,4,10$. However, a priori these relations might collapse everything.
In [@MR2308953], Sikora and Westbury introduce the notion of confluence, and claim that the above relations are confluent (we have verified this calculation). By definition, confluence means that if we start with a graph and then use one of the above relations to simplify one face or another face, then we can apply more simplifications on faces until both expressions become equal. By the Diamond Lemma [@MR0007372] this shows that any two reductions give the same answer. This means that the inner product of diagrams is well-defined, and then taking inner products lets us prove that the obvious spanning set is a basis for $n \leq 4$, except when we are also on the $SO(3)$ curve (see the following remark). In particular, $(G_2)_q$ is a cubic category. Finally we observe that the formulas for $d$ and $t$ agree with the ones in the parameterization of $P_{G_2} = 0$.
If $q$ is not a $3$rd, $6$th, $16$th, $7$th, $14$th, or $24$th root of unity, then the dimension of the $n$-boundary point spaces of $(G_2)_q$ are $1,0,1,1,4,10$ unless $q$ is a primitive $20$th root of unity in which case they $1,0,1,1,4,9$.
Unless $(d,t)$ lies on the $Q_{1,2}$ curve, the $10$ diagrams in $D(5,0)$ are linearly independent. The only points on the intersection of the $Q_{1,2}$ and $G_2$ curves are $(-2,-2)$, $(-1,-1)$, $(2,0)$, $(\tau, \bar{\tau})$, and $(\bar{\tau},\tau)$. These correspond, respectively, to a primitive $3$rd or $6$th root of unity, a primitive $20$th root of unity, a primitive $12$th root of unity, and a primitive $30$th root of unity. The only case not excluded by our assumptions is $q$ a primitive $20$th root of unity. Calculating the determinant of a $9$-by-$9$ minor of $M(5,0)$ shows that the $4$-boundary point space is $9$ dimensional at this value.
\[rem:G2-i\] When $q = \pm i$ is a primitive $4$th root of unity $(G_2)_q$ can still be defined by the above relations and confluence argument. Since we have an explicit spanning set for the $4$-boundary point space, a direct calculation shows that the relation $${ \begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}}\; - \; { \begin{tikzpicture}[baseline=0cm]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}
}\; - \frac{1}{2}\; {\begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\; +
\frac{1}{2}\; {\begin{tikzpicture}[baseline=0cm]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}$$ does lie in the radical of the inner product, so in the nondegenerate quotient $(G_2)_{\pm i}$ the dimension of the $4$-box space drops from $4$ down to $3$. This gives an alternate proof that the point $(d,t) = (-1,3/2)$ on $P_{SO(3)}$ can be realized.
When $q$ is one of the bad values (a primitive $3$rd, $6$th, or $16$th root of unity) the above definition can be modified by normalizing the trivalent vertex differently to give a well-defined category. However, in this category the value of the bigon will be $0$ and so the trivalent vertex will be zero.
The category $(G_2)_q$ gets its name from its relationship to the quantum group $U_q(\mathfrak{g}_2)$. If $q$ is generic, then the category of maps between tensor powers of the standard $7$-dimensional representation of $U_q(\mathfrak{g}_2)$ are given by the above diagrams. In fact, by the results of this section it is clear that the subcategory of $\mathrm{Rep}(U_q(\mathfrak{g}_2))$ generated by the trivalent vertex must be $(G_2)_q$ and then Kuperberg showed that the dimensions match up so the subcategory is the whole category.
When $q$ is a root of unity, the correct algebraic category is the category of tilting modules. There is a map $(G_2)_q' \to \mathrm{Rep}^{\mathrm{tilting}}(U_q(\mathfrak{g}_2))$, but it is not clear whether this is surjective, or if it descends to a map from the nondegenerate quotient $(G_2)_q$ to the non-degenerate quotient of the category of tilting modules.
Diagrams with six boundary points {#sec:six}
=================================
We now move on to the diagrams with six boundary points. We have $$\begin{aligned}
D(6,0) & = \left\{
\mathfig{0.1}{urn_sha1_1efb25f6833fbe664615559abab36b703e35cb24} + \text{1 rotation},
\mathfig{0.1}{urn_sha1_448b57b212c25633e803562e5ab6433c11d56dd8} + \text{2 rotations},
\mathfig{0.1}{urn_sha1_df91ef8b1687dae1b2d9fe38719e6812d7201d49} + \text{5 rotations},
\right. \displaybreak[1] \\
& \qquad
\mathfig{0.1}{urn_sha1_a0b22b5762fa24ead060f3de9448c681b424e133} + \text{5 rotations},
\mathfig{0.1}{urn_sha1_9f7ce5c387d4ae2c3b7e9317f261376c005a92c9} + \text{5 rotations},
\mathfig{0.1}{urn_sha1_1baad19a7b800a28edd9aa033244492c73330c5b} + \text{2 rotations}, \displaybreak[1] \\
& \qquad
\left.
\mathfig{0.1}{urn_sha1_20aca0123680da1b425e3ab29d0fb3d3ceada0da} + \text{2 rotations},
\mathfig{0.1}{urn_sha1_b91a68952d267c8269410143311be8b4ce67a732} + \text{2 rotations},
\mathfig{0.1}{urn_sha1_f357e0b86229ce63736f2ab08989fc2a2f1db2f6} + \text{1 rotation}
\right\}
\displaybreak[1] \\
D^\square(6,1) & \setminus D(6,0) =
\left\{
\mathfig{0.1}{urn_sha1_36f2a66789ebdceb39fd251941990993ef82b625} + \text{5 rotations},
\mathfig{0.1}{urn_sha1_7de6a0854fa495f044e12f0708481932e9cb121e}
\right\}
\displaybreak[1] \\
D^\square(6,2) & \setminus D^\square(6,1) =
\left\{
\mathfig{0.1}{urn_sha1_55f53ebaabfb09b3d4727e4b45c3637b0ed8f84c.pdf} + \text{2 rotations}
\right\}\end{aligned}$$ so $\#D(6,0) = 34, \#D^\square(6,1) = 41$, and $\#D^\square(6,2) = 44$.
In this section we prove
\[thm:6\] If $\cC$ is a trivalent category with $\dim \cC_4 = 4$, $\dim \cC_5 = 11$, and $\dim \cC_6 \leq 40$ then $d^2-3d-1=0$, $t=-\frac{2}{3}d+\frac{5}{3}$, and $\cC$ is the $H3$ fusion category constructed by Grossman and Snyder [@MR2909758] (which is Morita equivalent to the even parts of the Haagerup subfactor [@MR1686551]) or its Galois conjugate.
This theorem follows from four propositions.
\[prop:6:preliminary-nonexistence\] There are no trivalent categories with $\dim \cC_4
= 4$, $\dim \cC_5 = 11$, and $D(6,0)$ linearly dependent.
\[prop:6:nonexistence\] For any $(d,t)$ not satisfying $d^2-3d-1=0$ and $t=-\frac{2}{3}d+\frac{5}{3}$ there are no trivalent categories with $\dim \cC_4 = 4$, $\dim \cC_5 = 11$, and $\dim \Span D^\square(6,2) \leq 40$.
\[prop:6:uniqueness\] For each pair $(d,t)$ satisfying $d^2-3d-1=0$ and $t=-\frac{2}{3}d+\frac{5}{3}$, there is at most one trivalent category with $\dim \cC_4 = 4$, with $\dim \cC_5 = 11$, and with $\dim \cC_6 \leq 40$.
\[prop:6:realization\] The $H3$ fusion category and its Galois conjugate are trivalent and have $\dim \cC_4 = 4$, $\dim \cC_5 = 11$, and $\dim \cC_6 = 37 \leq 40$.
These two categories exhaust the possibilities allowed by the first three propositions.
Proof of Proposition \[prop:6:preliminary-nonexistence\] (Non-existence) {#proof-of-proposition-prop6preliminary-nonexistence-non-existence .unnumbered}
------------------------------------------------------------------------
We have the following values of determinants.
\[fact:Delta6\_0\] In any cubic category, $$\begin{aligned}
\Delta(6,0) = -d^{34} Q_{1,1}^{-8} Q_{0,1}^2 Q_{0,2}^9 Q_{2,4,a} Q_{3,5}^2 Q_{6,9} P_{SO(3)}^{19}.\end{aligned}$$
\[fact:Delta6\_1\] In any cubic category, $$\begin{aligned}
\Delta^\square(6,1) = d^{41} Q_{1,1}^{-29} Q_{0,2}^{16} Q_{2,3} Q_{3,4}^2 Q_{7,11} P_{SO(3)}^{27} P_{G_2}^6.\end{aligned}$$
\[fact:Delta6\_2\] In any cubic category, $$\begin{aligned}
\Delta^\square(6,2) = d^{44} Q_{1,1}^{-44} Q_{0,2}^{19} Q_{2,3} Q_{4,5}^2 Q_{8,12} P_{SO(3)}^{33} P_{G_2}^9\end{aligned}$$
\[fact:Delta7\_0\] In any cubic category, $$\begin{aligned}
\Delta(7,0) & = -d^{112} Q_{1,1}^{-70} Q_{0,2}^{48} Q_{11,19} Q_{36,60}^2 P_{SO(3)}^{76}
Q_{2,4,b}\end{aligned}$$
If $D(6,0)$ is dependent, then $\Delta(6,0)$ must vanish, and indeed $\Delta^\square(6,1), \Delta^\square(6,2)$ and $\Delta(7,0)$ must vanish also. This only happens at finitely many points, all of which are on the $G_2$ or $SO(3)$ curves. (This calculation is curious; finding intersections of $\Delta(6,0)$ with the other varieties appears to be rather hard. However the Gröbner basis calculation showing $\Delta^\square(6,1)$ and $\Delta^\square(6,2)$ intersect at finitely many points besides $Q_{0,2} Q_{2,3}
P_{SO(3)} P_{G_2} = 0$ is quite manageable, and after that we can easily find the complete intersection.) This calculation can be found in the file [code/GroebnerBasisCalculations.nb]{} available with the [arXiv]{} source of this article. Now, using the full strength of Proposition \[prop:5:uniqueness\], we see that any trivalent category with $\cC_5 \leq 11$ at one of these points must actually be an $ABA$ or $(G_2)_q$ category, contradicting our assumption that $\dim \cC_5 = 11$.
Proof of Proposition \[prop:6:nonexistence\] (Non-existence) {#proof-of-proposition-prop6nonexistence-non-existence .unnumbered}
------------------------------------------------------------
Beyond the determinant calculations in the previous section, with high probability we have the following two determinants.
\[conj:Delta7\_1\] \[conj:Delta7\_2\] In any cubic category, $$\begin{aligned}
\Delta^\square(7,1) & = - d^{155} Q_{1,1}^{-242} Q_{0,2}^{91} Q_{2,3}^7
Q_{21,33} Q_{51,69}^2 P_{SO(3)}^{133} P_{G_2}^{35} \\
\intertext{and}
\Delta^\square(7,2) & = - d^{183} Q_{1,1}^{-403} Q_{0,2}^{119}
Q_{4,5}^{14} Q_{22,36} Q_{54,78}^2 P_{SO(3)}^{189} P_{G_2}^{63}.\end{aligned}$$
With a probability of $1- 10^{-500}$, each of these conjectures is correct.
We begin by explaining what we mean. The Schwartz-Zippel lemma [@MR594695; @MR575692; @Demillo-Lipton] (pointed out to us by Dylan Thurston) gives a method of probabilistically checking polynomial identities (we first clear denominators if necessary): if $P \in
k[x_1, \ldots, x_n]$ has total degree bounded by $D$, and the $x_i$ are drawn uniformly and independently from a finite subset $S
\subset k$ of size $N$, then either $P$ is identically zero or $P(x_1, \ldots, x_n) \neq 0$ with probability at least $1- \frac
{D}{N}$.
We can easily bound the total degree of $\Delta^\square (7,1)$ at 1611, and of $\Delta^\square(7,2)$ at 2664. Evaluating the determinant of $M^\square(7,1)$ at a pair of values $(d,t)$ drawn uniformly from positive integers at most $10^{40}$ takes on the order of 3 minutes (on a 12-core Xeon E5), while the determinant of $M^\square(7,2)$ takes 6 minutes. This gives us a probability of error of at most one part in $10^{12}$, for $\Delta^\square(7,1)$, or $10^{6}$, for $\Delta^\square(7,2)$, per minute of running time; we stopped after reaching $10^{500}$. These checks are implemented in [code/SchwartzZippel.nb]{}.
To guess these polynomials in the first place, we adopted the following ad-hoc strategy. Suppose we have some large matrix $M(d,t)$ with entries in $\bbQ(d,
t)$, and want to evaluate the determinant. Arithmetic in $\bbQ(d, t)$ is difficult, so we avoid this by first specializing one variable, $d$, to various different primes. We work with a set of primes $\cP$, which is chosen to be big enough that results we obtain below are successfully verified by the Schwartz-Zippel lemma argument given above!
At each prime $p \in \cP$ we can compute $\det M(p,t)$ as a rational function in $\bbQ(t)$ relatively quickly. We now want to recover $M(d,t)$. In our examples, these are not irreducible, and it turns out to be most efficient to first factorize each $\det M(p,t)$ into products of powers (possibly negative) of polynomials in $\bbZ[t]$. For large enough primes $p$, the factorization is uniform, in the sense that degrees and multiplicities of the irreducible factors of the different $\det M(p,t)$ are in bijection, and so we obtain $\det M(p, t) = \prod_{i \in \cI} \cK_{p,i}(t)^{n_i}$, for some fixed index set $\cI$ and exponents $n_i$ for each $i \in \cI$. Write $\cK_{p,i}(t) = \sum_r
\cL_{p,i,r} t^r$ for some integers $\cL_{p,i,r}$.
We now want to recover an irreducible polynomial $K_i(d,t) = \sum_r L_{i,r}(d) t^r = \sum_{r,s} L_{i,r,s} d^s t^r$ so $\cK_{p,i}(t) = K_i(p, t)$. This requires that $L_{i,r}(p) = \cL_{p,i,r}$ for each $p$. In particular, this says that the constant term $L_{i,r,0}$ of $L_{i,r}$ satisfies $$L_{i,r,0} \equiv L_{i,r}(0) \equiv \cL_{p,i,r}
\pmod{p}.$$ By the Chinese remainder theorem, we then know $L_{i,r,0} \pmod
{\prod_{p \in \cP} p}$, and we guess that it is actually equal to this residue. We then continue making guesses recursively, using the identities $$L_{i,r,s} \equiv \left(\cL_{p, i,r} - \sum_{t=0}^{s-1} L_{i,r,t} p^t \right) p^{-s} \pmod{p}$$ and the Chinese remainder theorem. This method is implemented in the notebook [code/GuessDeterminants.nb]{}.
We will give two separate proofs of Proposition \[prop:6:nonexistence\]. The first is easy to follow but depends on Conjecture \[conj:Delta7\_1\], while the second is more difficult but unconditional.
\[lem:conditional-proof\] Fix some $(d,t)$ not satisfying $d^2-3d-1=0$ and $t=-\frac{2}{3}d+\frac{5}{3}$. If Conjecture \[conj:Delta7\_1\] holds at this $(d,t)$ then there are no trivalent categories with $\dim \cC_4 = 4$ and $\dim \cC_5 = 11$, and $\dim \Span D^\square(6,2) \leq 40$.
Since $\dim \Span D^\square(6,2) \leq 40$ and there are $41$ diagrams in $D^\square(6,1)$, we must have a relation amongst $D^\square(6,1)$. Thus $(d,t)$ must give a solution to $\Delta^\square(6,1) = \Delta^\square(6,2) = \Delta^\square(7,1) = \Delta^\square(7,2) = 0$, and the possibilities are (see [code/GroebnerBasisCalculations.nb]{}):
(a) $(d,t)$ is on the $P_{ABA}$ or $P_{G_2}$ curve,
(b) $d^2-3d-1=0$ and $t=-\frac{2}{3}d+\frac{5}{3}$,
(c) \[item:smallbadpoint\] $d$ is a root of the degree 33 polynomial $S_a(d)$, and $t = T_a (d)$, or
(d) $d$ is a root of the degree 63 polynomial $S_b(d)$, and $t = T_b(d)$.
The polynomials $S_a, T_a, S_b,$ and $T_b$ (the last of which is stupendously large) are available in the file [code/BadPoints.nb]{}.
In the first case, the full strength of Proposition \[prop:5:uniqueness\] shows that $\dim \cC_5 < 11$. It remains to eliminate the last two cases. For both those values of $(d,t)$ the rank of $M^\square(6,2)$ is 43 which is incompatible with $\dim \Span D^\square(6,2) \leq 40$. (This calculation is done twice in the last section of [code/BadPoints.nb]{}. We check slowly and directly that the rank is exactly $43$ by doing arithmetic in the number field, but we also quickly see that the rank is at least $43$ by calculating the rank of the matrix modulo a prime in the number field, thereby reducing the question to calculating the rank over $\mathbb{Z}/11\mathbb{Z}$ and $\mathbb{Z}/41\mathbb{Z}$ respectively. The latter approach was suggested to us by David Roe.)
Now we turn to the unconditional proof of Proposition \[prop:6:nonexistence\], which follows immediately from the following lemma.
If $\cC$ is a trivalent category with $\dim \cC_4 = 4$ and $\dim \cC_5 = 11$, and $\dim \Span D^\square(6,2) \leq 40$, then Conjecture \[conj:Delta7\_1\] holds.
In order for there to be such a trivalent category, we must have that $(d,t)$ is a solution to $\Delta^\square(6,1) = \Delta^\square(6,2) = 0$. We consider each factor of $\Delta^\square(6,1)$ separately. The $P_{G_2}$ and $P_{ABA}$ factors contradict $\dim \cC_5 = 11$ by Proposition \[prop:5:uniqueness\].
On the elliptic curve $Q_{2,3}$ any rational function can be written in the form $\alpha(t) d + \beta(t)$ for some rational functions $\alpha$ and $\beta$, and algebra can be done efficiently on functions of this form just as it is done with ordinary rational functions. We can then compute the value of the determinants $\Delta^\square(7,1)$ and $\Delta^\square(7,2)$ exactly and verify Conjecture \[conj:Delta7\_1\] for these points. (This calculation is performed in [code/DeterminantsOnEllipticCurve.nb]{}.)
The other two factors $Q_{3,4}$ and $Q_{7,11}$ of $\Delta^\square(6,1)$ intersect $\Delta^\square(6,2) = 0$ in finitely many points (see [code/GroebnerBasisCalculations.nb]{} for this calculation). For each of these finitely many points we can compute the determinants $\Delta^\square(7,1)$ and $\Delta^\square(7,2)$ exactly (cf. [code/DeterminantsOnExtraPoints.nb]{}). Thus Lemma \[lem:conditional-proof\] applies, and we note that the points described in item of Lemma \[lem:conditional-proof\] all lie on $Q_{2,3}$.
Proof of Proposition \[prop:6:uniqueness\] (Uniqueness) {#proof-of-proposition-prop6uniqueness-uniqueness .unnumbered}
-------------------------------------------------------
In the proof of Proposition \[prop:5:uniqueness\] (Uniqueness) we used an analogue for open graphs of the well-known theorem that any closed planar trivalent graph has pentagonal or smaller face. That theorem plays a key role in the proof of the (easy) $5$-color theorem, and the study and eventual proof of the $4$-color theorem lead to an enormous number of similar results proved using the *discharging method*. The typical illustration of the discharging method is the following well-known lemma.
A *very small* face is a square, triangle, or bigon. A *pentapent* is a pair of adjacent pentagons. A *hexapent* is a pentagon and an adjacent hexagon.
\[lem:discharging-closed-pentapent\] Any closed trivalent graph contains either a very small face, a pentapent, or a hexapent.
Suppose that the graph has no very small faces. We assign a charge of $6-n$ to every $n$-gon face. By measuring Euler characteristic, the total charge is $12$, which is certainly positive. We now “discharge" the pentagons, distributing their charge equally among the neighboring faces. Since this does not change the total charge of the graph there must be a face with positive charge. Such a face must either be a pentagon next to a pentagon, a hexagon next to a pentagon, or a $7$-gon next to at least $6$ pentagons. In the final case, at least two of those six pentagons are adjacent so there’s a pentapent.
Just as before, in order to apply this technique to our setting, we need an analogue of this lemma for open graphs.
A *lonely pentagon* of an open planar trivalent graph is a pentagon which touches at most two internal faces. A lonely pentagon is either a corner pentagon or a bridge pentagon. A *corner pentagon* is a pentagon which touches at most two internal faces which are adjacent to each other, and a *bridge pentagon* is a pentagon which touches exactly two internal faces which are not adjacent to each other.
\[lem:planarsubgraphs\] Every connected open planar trivalent graph has either a very small face, a pentapent, a hexapent, a growth region, or a corner pentagon.
The proof of this Lemma below will use the discharging method following the same outline as in the closed case. We assign the usual charges of $6-n$ to each interior $n$-gon face and $4-n$ to each boundary face touching $n$ edges. By measuring Euler characteristic, the total charge is $6$. We then ‘discharge’ the internal pentagons, distributing their charge equally among their neighboring internal faces. Since this does not change the total charge of the graph, we go looking for faces with positive charge and find that positive charge indicates that we have one of the features listed in Lemma \[lem:planarsubgraphs\]. This last step turns out to be somewhat delicate, so we first prove a slightly weaker lemma:
Every connected open planar trivalent graph has either a very small face, a pentapent, a hexapent, a growth region, or a lonely pentagon. \[lem:weakerplanarsubgraphs\]
Suppose a connected open planar trivalent graph $T$ has no growth regions. By discharging we will show that $T$ has either a very small face, a pentapent, a hexapent, or a lonely pentagon.
Assign a charge as described above. Since $T$ is connected, a quick argument shows that its Euler characteristic is $6$. If the boundary has charge 6 or more, it has at least two boundary faces because a single boundary face has charge $4-n$ (where $n$ is the number of edges it touches). Thus, we can divide the boundary into two proper sub-regions. One of these will have charge 3 or more, and hence be a growth region by Lemma \[lem:boundarycharge\].
If the boundary has charge less than 6, then there must be a positive charge among the internal faces. We now have the internal pentagons discharge according to the following rule: each distributes its charge of 1 evenly among the adjacent internal faces. Suppose there are no lonely pentagons. Then each pentagon distributes its charge among at least 3 faces, and so faces receive at most $\frac{1}{3}$ charge from each neighboring pentagon. The total charge has not changed, so there must be some face with positive charge.
We now consider the ways in which an $n$-gon may end up with positive charge. If $n \geq 7$, either it neighbors a pair of adjacent pentagons and we are done, or it neighbors at most $\lfloor \frac{n}{2} \rfloor $ adjacent pentagons. In the latter case, the total charge is at most $6-n+\frac{1}{3} \lfloor
\frac{n}{2} \rfloor \leq 0$. If $n=5$ or $6$, the positively-charged face must have received some charge from an adjacent pentagon, showing the existence of a pentapent or hexapent. Finally, we may have had a very small face all along.
Lemma \[lem:weakerplanarsubgraphs\] is almost what we need, except it proves we must have a lonely pentagon, not necessarily a corner pentagon.
Suppose that a connected open planar trivalent graph $T$ has bridge pentagons. Each bridge pentagon, when removed, disconnects the graph into two parts which each have fewer bridge pentagons. By descent, there must be a subgraph which is bridge-pentagon-free and connected to the rest of the graph by a single bridge pentagon. Call this subgraph $T'$ and the bridge pentagon connecting it to the rest of the graph $B$. $$\begin{tikzpicture}[baseline=0]
\draw[dashed, gray] (3.4,-.7) circle (2cm);
\fill[white] (30:1cm) -- ++(18:1cm) coordinate (X) -- ++(-54:1cm) coordinate (Z) -- ++(-126:1cm) coordinate (Y) -- (-30:1cm)--(30:1cm);
\draw[gray] (30:1cm) -- (X);
\draw[gray] (-30:1cm) -- (Y);
\draw[gray] (X) -- (Z);
\draw[gray] (Y)--(Z);
\draw[gray] (X)-- ++(72:.5cm);
\draw[gray] (Y)-- ++(-72:.2cm);
\draw[gray] (Z)-- ++(0:.2cm);
\draw (30:1cm)--(-30:1cm);
\draw (30:1cm) -- ++(18:.3cm);
\draw (30:1cm) -- ++(144:.2cm);
\draw(-30:1cm) -- ++(-18:.3cm);
\draw (-30:1cm) -- ++(-144:.2cm);
\node (T') at (0,0) {$T'$};
\node[gray] at (1.5,0) {$B$};
\node[gray] (T) at (4,0) {rest of $T$};
\draw[dashed] (0,0) circle (1.3cm);
\end{tikzpicture}$$
Observe $T'$ is still connected, by virtue of containing an edge of $B$. Now consider the boundary of $T'$. If it has total charge of $6$ or greater, then after $B$ has been attached it still has charge of $2$ or greater. To see this, consider the boundary face of $T'$ which $B$ attaches to, and its two boundary neighbors. A small neighborhood of these three faces has four outgoing edges, and at least one incoming edge (otherwise $B$ would be a corner pentagon), hence its charge is at most $4$. Thus, the boundary complement of this region (the region enclosed in blue lines below) has charge at least $2$ and so is a growth region by Lemma \[lem:boundarycharge\].
$$\begin {tikzpicture}[baseline=0] \draw (90:1cm)--(30:1cm)--(-30:1cm) --(-90:1cm);
\draw[dotted, thick] (-90:1cm) -- ++(150:.35cm);
\draw[dotted, thick] (90:1cm) -- ++(-150:.35cm);
\draw (90:1cm)--(90:1.5cm);
\draw (-90:1cm)--(-90:1.5cm);
\draw (30:1cm) -- ++(18:.5cm);
\draw(-30:1cm) -- ++(-18:.5cm);
\node (T') at (0,0) {$T'$};
\node at (1.25,0) {$B$};
\node at (60:1.2cm) {$A$};
\node at (-60:1.2cm) {$C$};
\draw[red] (100:1.5cm) -- (100:1cm) .. controls (100:.8cm) .. (90:.8cm) arc (90:-90:.8cm) .. controls (-100:.8cm) .. (-100:1cm)--(-100:1.5cm);
\draw[red, dashed] (100:1.5cm) arc (100:-100:1.5cm);
\draw[blue] (103:1.5cm) -- (103:1cm) .. controls (103:.8cm) .. (110:.8cm) arc (110: 250:.8cm) .. controls (-103:.8cm) .. (-103:1cm) -- (-103:1.5cm);
\draw[blue, dashed] (103:1.5cm) arc (103:257:1.5cm);
\end{tikzpicture}$$
Alternately, if the boundary of $T'$ had total charge of $5$ or less, then there is a net positive charge and at least one small face in the interior. As above, we discharge any pentagons and conclude we must have a very small face, pentapent, hexapent, or corner pentagon in $T'$ (recall above we ensured that $T'$ had no bridge pentagons). The very small face, pentapent, or hexapent also appears in $T$. A corner pentagon of $T'$ is either also a corner pentagon of $T$, or is adjacent to $B$, hence part of a pentapent.
\[lem:removemaxgrowth\] Let $T$ be a connected open planar trivalent graph with no very small faces, pentapents, hexapents or corner pentagons. If we remove a maximal growth region from $T$, the remaining graph (call it $T'$) also has no very small faces, pentapents, hexapents or corner pentagons.
Since any face of $T'$ is a face of $T$, $T'$ has no very small faces, pentapents or hexapents. If we created a corner pentagon by removing a growth region from $T$, the growth region that we removed was not maximal. To see why, consider a corner pentagon in $T$ which was not in $T'$. It must look like the following diagram:
(100:4cm) .. controls (100:3cm) .. (162:2.5cm) .. controls (180:3cm) .. (-3,-2) – (-4,-2) – (180:4cm) arc (180:100:4cm); (18:2cm) – (90:2cm)–(162:2cm)–(234:2cm)–(306:2cm)–(18:2cm); (18:2cm) – (18:4cm); (90:2cm) – (90:4cm); (162:2cm) – (162:2.5cm); (234:2cm) – (234:2.5cm); (306:2cm) – (306:2.5cm); (0:4cm) .. controls (3:2cm) .. (18:1.5cm) – (90:1.5cm) .. controls (105:2cm) .. (97:4cm) arc (97:0:4cm); (4.05,-2) – (4.05,0) arc (0:180:4.05cm) – (-4.05,-2);
Here the dashed curve is the boundary of $T$, the shaded blue region is the growth region, and the graph (perhaps also the growth region) continues below the bottom of the picture. Then the union of the blue region with the region enclosed by red is a larger growth region.
\[lem:enumerategraphs\] We can enumerate connected planar trivalent graph with no very small faces, pentapents, or hexapents by starting with the empty diagram, and
- sequentially adding growth regions;
- in the final step, simultaneously adding some “H"s to create corner pentagons.
We can enumerate all planar trivalent graphs (not necessarily connected) with no very small faces, pentapents, or hexapents by taking planar disjoint unions of such connected planar trivalent graphs.
Consider a graph $T$ which has no very small faces, pentapents or hexapents. From each corner pentagon, remove an “H" neighborhood of one of its sides which touches an external face. Since $T$ has no pentapents, the regions we remove will not overlap.
Let $T'$ be the graph with an “H" removed from each corner pentagon. $T'$ has no corner pentagons, so by repeatedly applying Lemma \[lem:removemaxgrowth\], we get a sequence of growth regions building $T'$ up from the empty diagram.
This lemma shows in particular that there are a finite number of such planar trivalent graphs with any given number of trivalent vertices, or with any given number of boundary points and internal faces.
Every planar trivalent graph with no very small faces and at most 6 boundary points is in $D^\square(n,1)$ or has an internal pentapent or an internal hexapent.
This corollary is proved by having a computer write down all the graphs in $D^\square(n \leq 6,1)$ with no very small faces, pentapents, or hexapents, according to the algorithm of Lemma \[lem:enumerategraphs\].
\[cor:6:reductions=>span\] In a cubic category with reduction relations for pentapents and hexapents, $D^\square(n,1)$ spans $\cC_n$ for $n \leq 6$.
Inside $D^\square(6,1)$ there are 6 ‘pentafork’ diagrams. We next analyze relations amongst these diagrams, up to lower order terms (i.e. terms with strictly fewer vertices, which in this case is exactly the diagrams in $D(6,0)$).
\[lem:pentaforks=>reductions\] We let $\rho$ denote the operator which rotates an open graph by one click counterclockwise. A trivalent category with relations reducng $n$-gons for $n \leq 4$ and a relation amongst the 6 pentaforks modulo $\Span(D(6,0))$ must also have a relation reducing a pentapent to something in $\Span(D^\square(6,1))$ and a relation reducing a hexapent into $\Span(D^\square(7,1))$
If there is a relation amongst the 6 pentaforks, then there is one of the form $$\sum_{i=0}^6 \zeta^i \rho^i \left({
\begin{tikzpicture}[baseline=0, rotate=60]
\coordinate (F) at (30:.6cm);
\coordinate (P1) at (30:.4cm);
\coordinate (P2) at (120:.6cm);
\coordinate (P3) at (180:.7cm);
\coordinate (P4) at (240:.7cm);
\coordinate (P5) at (300:.6cm);
\draw (60:1cm)--(F)--(0:1cm);
\draw (F)--(P1)--(P2)--(P3)--(P4)--(P5)--(P1);
\draw (P2)--(120:1cm);
\draw (P3)--(180:1cm);
\draw (P4)--(240:1cm);
\draw (P5)--(300:1cm);
\draw[dotted, thin, gray] (0,0) circle (1cm);
\end{tikzpicture}
}\right) = 0 \mod{\Span(D(6,0))}$$ for some (not necessarily primitive) 6-th root of unity $\zeta$. Gluing an ‘H’ diagram onto the upper right boundary face of each diagram, we obtain
$$\rho^2\left({\begin{tikzpicture}[baseline=0]
\draw (-30:.3cm)--(150:.3cm)--(110:.6cm)--(60:.7cm)--(10:.6cm)--(-30:.3cm)--(-70:.6cm)--(-120:.7cm)--(-170:.6cm)--(150:.3cm);
\draw (10:.6cm)--(0:1cm);
\draw (60:.7cm)--(60:1cm);
\draw (110:.6cm)--(120:1cm);
\draw (-70:.6cm)--(-60:1cm);
\draw (-120:.7cm)--(-120:1cm);
\draw (-170:.6cm)--(-180:1cm);
\end{tikzpicture}}\right) = - \zeta^{-2} \left({\begin{tikzpicture}[baseline=0]
\draw (-30:.3cm)--(150:.3cm)--(110:.6cm)--(60:.7cm)--(10:.6cm)--(-30:.3cm)--(-70:.6cm)--(-120:.7cm)--(-170:.6cm)--(150:.3cm);
\draw (10:.6cm)--(0:1cm);
\draw (60:.7cm)--(60:1cm);
\draw (110:.6cm)--(120:1cm);
\draw (-70:.6cm)--(-60:1cm);
\draw (-120:.7cm)--(-120:1cm);
\draw (-170:.6cm)--(-180:1cm);
\end{tikzpicture}}\right) \mod{\Span(D^\square(6,1))}.$$ Applying this relation three times we see that $${\begin{tikzpicture}[baseline=0]
\draw (-30:.3cm)--(150:.3cm)--(110:.6cm)--(60:.7cm)--(10:.6cm)--(-30:.3cm)--(-70:.6cm)--(-120:.7cm)--(-170:.6cm)--(150:.3cm);
\draw (10:.6cm)--(0:1cm);
\draw (60:.7cm)--(60:1cm);
\draw (110:.6cm)--(120:1cm);
\draw (-70:.6cm)--(-60:1cm);
\draw (-120:.7cm)--(-120:1cm);
\draw (-170:.6cm)--(-180:1cm);
\end{tikzpicture}}= \rho^6 \left({\begin{tikzpicture}[baseline=0]
\draw (-30:.3cm)--(150:.3cm)--(110:.6cm)--(60:.7cm)--(10:.6cm)--(-30:.3cm)--(-70:.6cm)--(-120:.7cm)--(-170:.6cm)--(150:.3cm);
\draw (10:.6cm)--(0:1cm);
\draw (60:.7cm)--(60:1cm);
\draw (110:.6cm)--(120:1cm);
\draw (-70:.6cm)--(-60:1cm);
\draw (-120:.7cm)--(-120:1cm);
\draw (-170:.6cm)--(-180:1cm);
\end{tikzpicture}}\right) = - {\begin{tikzpicture}[baseline=0]
\draw (-30:.3cm)--(150:.3cm)--(110:.6cm)--(60:.7cm)--(10:.6cm)--(-30:.3cm)--(-70:.6cm)--(-120:.7cm)--(-170:.6cm)--(150:.3cm);
\draw (10:.6cm)--(0:1cm);
\draw (60:.7cm)--(60:1cm);
\draw (110:.6cm)--(120:1cm);
\draw (-70:.6cm)--(-60:1cm);
\draw (-120:.7cm)--(-120:1cm);
\draw (-170:.6cm)--(-180:1cm);
\end{tikzpicture}}\mod{\Span(D^\square(6,1))}.$$ This is only possible if the pentapent is zero modulo $\Span(D^\square(6,1))$.
Now we turn our attention to the hexapent. Gluing lower two points of the tree $\tikz[scale=0.4]{\draw (0:1) arc (0:180:1); \draw (60:1) -- (60:1.5); \draw
(90:1) -- (90:1.5); \draw (120:1) -- (120:1.5); }$ to the upper right boundary face of the pentafork relation gives two hexapents, three pentapents, and a pentagon connected via an edge to a square. Applying the square reduction relation and the new pentapent relation, we get the following relation among the hexapents modulo lower order terms: $$\rho^2\left({\begin{tikzpicture}[baseline=0,rotate=180]
\draw (-20:.4cm)--(150:.3cm)--(110:.6cm)--(60:.7cm)--(10:.6cm)--(-20:.4cm)--(-45:.7cm)--(-90:.7cm)--(-135:.7cm)--(-170:.6cm)--(150:.3cm);
\draw (10:.6cm)--(0:1cm);
\draw (60:.7cm)--(60:1cm);
\draw (110:.6cm)--(120:1cm);
\draw (-170:.6cm)--(180:1cm);
\draw (-90:.7cm)--(-90:1cm);
\draw (-45:.7cm)--(-45:1cm);
\draw (-135:.7cm)--(-135:1cm);
\end{tikzpicture}}\right) = -
\zeta^{-2} {\begin{tikzpicture}[baseline=0,rotate=180]
\draw (-20:.4cm)--(150:.3cm)--(110:.6cm)--(60:.7cm)--(10:.6cm)--(-20:.4cm)--(-45:.7cm)--(-90:.7cm)--(-135:.7cm)--(-170:.6cm)--(150:.3cm);
\draw (10:.6cm)--(0:1cm);
\draw (60:.7cm)--(60:1cm);
\draw (110:.6cm)--(120:1cm);
\draw (-170:.6cm)--(180:1cm);
\draw (-90:.7cm)--(-90:1cm);
\draw (-45:.7cm)--(-45:1cm);
\draw (-135:.7cm)--(-135:1cm);
\end{tikzpicture}}\mod{\Span(D^\square(7,1))}.$$ Applying this relation seven times, we see that $${\begin{tikzpicture}[baseline=0,rotate=180]
\draw (-20:.4cm)--(150:.3cm)--(110:.6cm)--(60:.7cm)--(10:.6cm)--(-20:.4cm)--(-45:.7cm)--(-90:.7cm)--(-135:.7cm)--(-170:.6cm)--(150:.3cm);
\draw (10:.6cm)--(0:1cm);
\draw (60:.7cm)--(60:1cm);
\draw (110:.6cm)--(120:1cm);
\draw (-170:.6cm)--(180:1cm);
\draw (-90:.7cm)--(-90:1cm);
\draw (-45:.7cm)--(-45:1cm);
\draw (-135:.7cm)--(-135:1cm);
\end{tikzpicture}}= \rho^{14}\left({\begin{tikzpicture}[baseline=0,rotate=180]
\draw (-20:.4cm)--(150:.3cm)--(110:.6cm)--(60:.7cm)--(10:.6cm)--(-20:.4cm)--(-45:.7cm)--(-90:.7cm)--(-135:.7cm)--(-170:.6cm)--(150:.3cm);
\draw (10:.6cm)--(0:1cm);
\draw (60:.7cm)--(60:1cm);
\draw (110:.6cm)--(120:1cm);
\draw (-170:.6cm)--(180:1cm);
\draw (-90:.7cm)--(-90:1cm);
\draw (-45:.7cm)--(-45:1cm);
\draw (-135:.7cm)--(-135:1cm);
\end{tikzpicture}}\right) = -
\zeta^{-2} {\begin{tikzpicture}[baseline=0,rotate=180]
\draw (-20:.4cm)--(150:.3cm)--(110:.6cm)--(60:.7cm)--(10:.6cm)--(-20:.4cm)--(-45:.7cm)--(-90:.7cm)--(-135:.7cm)--(-170:.6cm)--(150:.3cm);
\draw (10:.6cm)--(0:1cm);
\draw (60:.7cm)--(60:1cm);
\draw (110:.6cm)--(120:1cm);
\draw (-170:.6cm)--(180:1cm);
\draw (-90:.7cm)--(-90:1cm);
\draw (-45:.7cm)--(-45:1cm);
\draw (-135:.7cm)--(-135:1cm);
\end{tikzpicture}}\mod{\Span(D^\square(7,1))}.$$ But $\zeta^{-2} \neq -1$ because $\zeta$ is a sixth root of unity and so cannot be a primitive fourth root of unity. Hence, we see that the hexapent is zero modulo $\Span(D^\square(7,1))$.
If $\dim \cC_4 = 4, \dim \cC_5 = 11$, and $\dim \Span D^\square(6,1) \leq 39$, then there is a relation amongst the 6 pentaforks modulo $\Span(D(6,0))$.
By Proposition \[prop:6:preliminary-nonexistence\], the 34 diagrams in $D(6,0)$ must be linearly independent. Hence, modulo $\Span(D(6,0))$, there must either be two relations amongst the pentaforks, or a relation writing the hexagon as a linear combination of pentaforks along with a relation amongst the pentaforks.
\[lem:40=>spans\] If $\dim \cC_4 = 4, \dim \cC_5 = 11$, and $\dim \cC_6 \leq 40$, then $D^\square(6,1)$ spans.
Suppose $\dim \Span D^\square(6,1) \leq 39$. By the previous relation, there’s a relation among the pentaforks modulo lower terms. Thus, Lemma \[lem:pentaforks=>reductions\] shows there are reduction relations for pentapents and hexapents, and finally Corollary \[cor:6:reductions=>span\] shows that $D^\square(6,1)$ spans.
Alternatively, if $\dim \Span D^\square(6,1) = 40$, then clearly $D^\square(6,1)$ spans because $\dim \cC_6 \leq 40$.
We’re now ready to give the proof of Proposition \[prop:6:uniqueness\]. Its statement assumes the hypotheses of Lemma \[lem:40=>spans\], and so we can now assume that $D^\square(6,1)$ spans. In particular, any element of the kernel of $M^\square(6,1)$, evaluated at the given values of $d$ and $t$, must be a relation in the category. By explicit calculation, we see the kernel of $M^\square(6,1)$ is four dimensional. By Proposition \[prop:6:preliminary-nonexistence\] we know that $D(6,0)$ is linearly independent, so there must be $4$ relations among the pentaforks and the hexagon modulo $D(6,0)$. In particular, there must be at least $3$ relations among the pentaforks modulo $D(6,0)$. Then Lemma \[lem:pentaforks=>reductions\] implies that there are relations reducing pentapents and hexapents, Lemma \[lem:discharging-closed-pentapent\] implies that we have enough relations to evaluate all closed diagrams, and Corollary \[cor:reductions=>uniqueness\] says that there is a unique cubic category at the given values of $d$ and $t$.
Although we don’t need to know the form of the pentafork, pentapent, and hexapent relations to prove uniqueness, we can obtain them explicitly as follows. By looking at the radical of the inner product on $D^\square(6,1)$, we obtain three relations amongst the pentaforks (modulo lower order terms), and one relation giving a hexagon in terms of the pentaforks and lower order terms. The pentafork relations have rotational eigenvalues $-1, \omega, \omega^2$, for $\omega$ a primitive cube root of unity. One can then use the argument from Lemma \[lem:pentaforks=>reductions\] to obtain explicit pentapent and hexapent reductions. See [code/H3\_relations.nb]{}.
The pentafork relations guarantee pentapent and hexapent relations. We do not know whether one can use non-degeneracy to go the other way and recover the pentafork relations from the pentapent and hexapent relations.
Proof of Proposition \[prop:6:realization\] (Realization) {#proof-of-proposition-prop6realization-realization .unnumbered}
---------------------------------------------------------
First we recall the definition of the $H3$ category. Then we must show it is trivalent, has $d = \frac{3+\sqrt{13}}{2}$ and $t=-\frac{2}{3}d+\frac{5}{3}$, and has dimension sequence $\dim \cC_4 = 4$, $\dim \cC_5 = 11$, and $\dim \cC_6
= 37 \leq 40$. The construction of the Haagerup fusion category and hence $H3$ involves a choice of square root of $13$. Thus taking Galois conjugation $\sqrt{13} \mapsto -\sqrt{13}$ gives another category which realizes the other value of $(d,t)$.
Recall that $H2$ is the Haagerup category with three invertible objects ($1$, $h$, and $h^2$) and three non-invertible objects ($Y$, $hY$ and $h^2Y$) with fusion rules $hY = Yh^2$ and $Y^2 = 1+Y+hY+h^2Y$. Since the associator on the $1$, $h$, $h^2$ subcategory is trivial, there’s a canonical algebra $\mathbb{C}[\mathbb{Z}/\mathbb Z3]$ in $H2$. As defined in [@MR2909758], $H3$ is the tensor category of bimodule objects in $H2$ over this algebra. Recall that $H3$ has the same Grothendieck ring as category $H2$, with $g^3 =
1$, $gX=Xg^2$ and $X^2 = 1+ X + gX + g^2X$. We will need two additional facts about $H3$. First, as shown in [@MR2909758], there’s no algebra structure on $1+X$ in $H3$. (In fact, this is the property that was used to show $H3$ is not the same as $H2$, because the Haagerup subfactor gives an algebra structure on $1+Y$ in $H2$. The second fact is that $H3$ is isomorphic to its complex conjugate. This follows from $H2$ being isomorphic to its complex conjugate and the structure constants for the algebra $\mathbb{C}[\mathbb{Z}/\mathbb Z3]$ being real.
From the fusion rules for $H3$ we see that $\dim \Inv(X^{\otimes 3}) = 1$, and thus that there is a map $f: X \tensor X \to X$. We need to see that this is a trivalent vertex, and that it generates the category. We know that $f$ must be a rotational eigenvector, but we do not yet know the eigenvalue. Note that since conjugation by $g$ permutes $X$, $gX$, and $g^2 X$ we must have the same rotational eigenvalue for each of the three maps $X \otimes X \rightarrow X$, $gX \otimes gX \rightarrow gX$, and $g^2 X \otimes g^2 X \rightarrow g^2 X$. Since $H3$ is isomorphic to its complex conjugate, we see that these eigenvalues must all be $1$.
Next, we consider the pivotal subcategory $H3' \subset H3$ generated by the trivalent vertex $f$. Since $1+X$ does not have the structure of an algebra object in $H3$, we see that $H3$ cannot take a functor from the $SO(3)$ category at this value of $(d,t)$. Hence, $\dim H3'_4$ cannot be smaller than $4$.
If $\dim \Inv_{H3'}(X^{\otimes 5}) < 11$, by Theorem \[thm:5\], $H3'$ is either a $(G_2)_q$ category or an ABA category. In both cases we compute $\tr{y_+} = \dim g X = \frac{3+\sqrt{13}}{2} = \dim g^2 X = \tr{y_-}$ using the formulas from Theorem \[thm:idempotents-and-traces\] and get a contradiction. If $H3'$ were an $ABA$ category then $d=\frac{3+\sqrt{13}}{2}$ and $t^2-t-1=0$. This splits into two cases based on the choice of $t$: either $\tr{y_+} \approx
-6.344$ while $\tr{y_-} \approx 12.9496$ if $t=\frac{1+\sqrt{5}}{2}$ (which is a contradiction) or $\tr{y_+} \approx 1.04123$ while $\tr{y_-} \approx 5.56432$ if $t=\frac{1-\sqrt{5}}{2}$ which is again a contradiction.
Similarly, we can rule out $H3'$ being a $(G_2)_q$ category, although the calculations are messier. We first find the possible values of $q$ so $$\frac{3+\sqrt{13}}{2} = q^{10} + q^{8} + q^{2} + 1 + q^{-2} + q^{-8} + q^{-10}$$ and for each, verify $\xi$ (with $t = - \frac{q^2-1+q^{-2}}{q^4+q^{-4}}$) is invertible and $\tr{y_+} \neq \tr{y_-}$. (This calculation is contained in [code/idempotents.nb]{}.)
It is clear from the fusion rules that $d = \frac{3+\sqrt{13}}{2}$. Since $\dim
\Inv_{H3'}(X^{\otimes 6}) \leq \dim \Inv_{H3}(X^{\otimes 6}) = 37$, then by Proposition \[prop:6:nonexistence\] we must have $t=-\frac{2}{3}d+\frac{5}{3}$.
Our final task is to show that $H3' = H3$. Since the principal graph for $H3$ is depth 3, $H3$ is generated by its morphisms in $\Inv_{H3}(X^{\otimes k})$ for $k \leq 6$. Hence it is enough to show that $\dim \Inv_{H3'}(X^{\otimes 6}) =
37$. Consider the diagrams in $D^\square(6,1)$, leaving out (any) 4 of the pentaforks, and compute the determinant of the corresponding 37-by-37 matrix. This is the nonzero number $$\begin{gathered}
-\frac{12874212105079943047176987387755967947399861}{
278128389443693511257285776231761} \\ -
\frac{3570663990466532246521487414951846015270252}{
278128389443693511257285776231761} \sqrt{13}\end{gathered}$$ when $d= \frac{3+\sqrt{13}}{2}$ and $t=-\frac{2}{3}d+\frac{5}{3}$. (See [code/H3\_relations.nb]{} for this calculation.) Hence these 37 diagrams must in fact be linearly independent in $H3'_6$. Thus $\dim \Inv_{H3'}(X^{\otimes 6}) \geq 37$, and in fact $H3'_6 = H3_6$, with $D^\square(6,1)$ spanning. Thus $H3' = H3$, so $H3$ is a trivalent category satisfying the conditions of the theorem.
Non-trivial rotational eigenvalues {#sec:rotationalev}
==================================
Suppose that $\cC$ is a pivotal category with the sequence $\dim \cC_n$ beginning $1,0,1,1$ which is generated by the map $1 \to X \tensor X \tensor X$. This map must be an eigenvector for rotation, whose eigenvalue must be a cube root of unity. Thus far we have considered the case where the rotational eigenvalue is $1$, and in this section we consider the case where the rotational eigenvalue for counterclockwise rotation is a primitive cube root of unity $\omega$. We call such a category a twisted trivalent category. Similarly to before, a twisted trivalent category gives an invariant of planar trivalent graphs which are decorated with a choice of direction at each vertex (which we denote by placing a dot in one of the three regions adjacent to the vertex) subject to the following skein relation $${
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (330:.8cm);
\coordinate (E2) at (210:.8cm);
\coordinate (E3) at (90:.8cm);
\coordinate (P1) at (0,0);
\draw (E1) -- (P1) -- (E2);
\draw (P1) -- (E3);
\path (P1) ++(150:.2cm) node {$\bullet$};
\end{tikzpicture}
}=
\rho\left({
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (330:.8cm);
\coordinate (E2) at (210:.8cm);
\coordinate (E3) at (90:.8cm);
\coordinate (P1) at (0,0);
\draw (E1) -- (P1) -- (E2);
\draw (P1) -- (E3);
\path (P1) ++(270:.2cm) node {$\bullet$};
\end{tikzpicture}
}\right) = \omega {
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (330:.8cm);
\coordinate (E2) at (210:.8cm);
\coordinate (E3) at (90:.8cm);
\coordinate (P1) at (0,0);
\draw (E1) -- (P1) -- (E2);
\draw (P1) -- (E3);
\path (P1) ++(270:.2cm) node {$\bullet$};
\end{tikzpicture}
}.$$ As before we normalize the bigon (with dots inward) to be $1$.
In a twisted trivalent category, the triangle with all dots pointing inward is some multiple of the trivalent vertex, but it’s manifestly rotationally invariant and hence must be zero. This significantly simplifies the analysis, because all the determinants considered above become polynomials in just one variable, the loop value $d$. It is then easy to detect intersections between the corresponding varieties, by factoring into irreducible polynomials.
We now quickly run through the analogues of all the above results in the twisted case. Let $D_\omega^\square(n,k)$ be defined as before as diagrams with $n$ boundary points and no more than $k$ faces, none of which are squares or smaller. Note that there is an ambiguity here; for each diagram you must fix the location of the dots. We let $M_\omega^\square(n,k)$ be the matrix of inner products and $\Delta_\omega^\square(n,k)$ be the determinant of this matrix. Note that $M_\omega^\square(n,k)$ is well-defined only up to rescaling rows by a power of $\omega$, and thus $\Delta_\omega^\square(n,k)$ is only well-defined up to an overall rescaling by a power of $\omega$. Below we always fix this normalization in a way that makes the determinant a polynomial with real coefficients (though it is not clear that this is possible in general).
Many of the small determinants used below can be calculated by hand. The larger ones, however, rely on a newer software implementation of our methods, which unfortunately is not ready for release. (The older implementation, used and described above, was not designed to keep track of rotations of vertices.) Our plans for further investigations of small skein theories will make use of the newer implementation, so we defer a description and release until a later paper.
For any $d \neq 2$ there are no twisted trivalent categories with $\dim \Span D_\omega(4,0) \leq 3$.
If $D_\omega(4,0) \leq 3$ then the determinant $\Delta_\omega(4,0) = d^5 (d - 2)$ vanishes. Since $d \neq 0$, we see that this forces $d=2$.
When $d=2$ there is at most one trivalent category with $\dim \cC_4 \leq 4$.
If $D_\omega(4,0)$ is linearly dependent then we get a relation of the form $$\;
{ \begin{tikzpicture}[baseline=0cm]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\node at (0,.4) {$\bullet$};
\node at (0,-.4) {$\bullet$};
\end{tikzpicture}
}\; = \alpha \;
{ \begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\node at (0,.4) {$\bullet$};
\node at (0,-.4) {$\bullet$};
\end{tikzpicture}}\;
+ \beta
\;
{\begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\; + \gamma \;
{\begin{tikzpicture}[baseline=0cm]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\;.$$ As before, this relation shows that $D_\omega(n,0)$ spans $\cC_n$. On the other hand if $D_\omega(4,0)$ is linearly independent then it is a basis of $\cC_4$. Thus computing the kernel of the inner product we see that $$\left( \;
{ \begin{tikzpicture}[baseline=0cm]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\node at (0,.4) {$\bullet$};
\node at (0,-.4) {$\bullet$};
\end{tikzpicture}
}\; - \;
{ \begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\node at (0,.4) {$\bullet$};
\node at (0,-.4) {$\bullet$};
\end{tikzpicture}}\; \right)
=
\left( \;
{\begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\; - \;
{\begin{tikzpicture}[baseline=0cm]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\; \right).$$ In either case, the relation shows that $D_\omega^\square(n,0)$ spans $\cC_n$ and hence the relation is enough to evaluate all closed diagrams. Thus there is at most one such category.
Note that this uniqueness statement applies for a particular choice of primitive cube root of unity for the rotational eigenvalue; typically, as below, examples will appear in pairs corresponding to both choices.
For realization we use [@MR2098028] which gives Chmutova’s classification of fusion categories of global dimension $6$. We recall the following notation from [@MR1976233 §3]. Suppose that $H$ is a subgroup of $G$, that $\xi
\in Z^3(G, \mathbb{C}^\times)$ is a $3$-cocycle and that $\psi \in C^2(H,
\mathbb{C}^\times)$ is a $2$-cochain whose coboundary is the restriction of $\xi$. Then we have a fusion category $\mathrm{Vec}(G,\xi)$ of twisted $G$-graded vector spaces, and the twisted group ring $\mathbb{C}_\psi[H]$ is an algebra object. Let $\cC(G,H,\xi,\psi)$ denote the category of bimodules over the twisted group ring. Recall that $H^3(S_3, \mathbb C^\times) =
\mathbb{Z}/6\mathbb{Z}$, $H^3(S_2, \mathbb C^\times) = \mathbb{Z}/2 \mathbb{Z}$, and $H^2(S_2, \mathbb C^\times) = 0$. If $\xi$ is an element of order $3$ in $H^3(S_3, \mathbb C^\times)$, then its restriction to $H^3(S_2, \mathbb
C^\times)$ is trivial. So there’s a 2-cochain $\psi$ on $S_2$ (which is unique up to homology) such that $d \psi = \xi$ on $S_2$.
If $\xi$ is a $3$-cocycle of order $3$ in $H^3(S_3, \mathbb C^\times)$ and $\psi$ a $2$-cochain on $S_2$ such that $d
\psi = \xi$ on $S_2$, then $\cC(S_3, S_2, \xi, \psi)$ gives a trivalent category with $d=2$ and $\dim \cC_4 = 3$.
These two categories (one for each choice of $\xi$) each have three objects $1$, $X$, $g$ with $g^2 = 1$, $g X = X g$, and $X^2 = 1+2X+g$. (In other words, they are near group categories [@MR1997336; @MR3167494; @1512.04288] for the group $\mathbb{Z}/2\mathbb{Z}$.) So a direct calculation shows that the dimensions of the Hom spaces are given by $1,0,1,1,3, \ldots$. Since these are distinct from $\mathrm{Rep}(S_3) = SO(3)_{\zeta_{12}}$, our previous classification shows that the rotational eigenvalue cannot be $1$ so they must both be twisted trivalent categories.
Combining these results we get the following classification.
A twisted trivalent category $\cC$ with $\dim \cC_4 \leq 3$ must be one of the two $\cC(S_3, S_2, \xi, \psi)$ categories.
(Here we haven’t specified which values of $\xi$ correspond to which rotational eigenvalues, although it must be a bijective correspondence; it would be interesting to work this out.)
Now, a twisted cubic category $\cC$ (that is, one with $\dim \cC_4 = 4$) must (by the analogue of Theorem \[prop:cubic\]) have $d \neq 2$, $D_\omega(4,0)$ a basis of $\cC_4$ and satisfy the relation $${
\begin{tikzpicture}[baseline=0cm]
\foreach \x in {1, ..., 4}
\draw (360*\x/4+45:.8cm)--(360*\x/4+45:.5cm);
\foreach \x in {1, ..., 4}
\draw (360*\x/4+45:.5cm) .. controls +(360*\x/4+120+45:.3cm) and +(360*\x/4+360/4-120+45:.3cm) .. (360*\x/4+360/4+45:.5cm);
\foreach \x in {1, ..., 4}
\node at (360*\x/4+45:.3cm) {$\bullet$};
\end{tikzpicture}
} = - \frac{1}{d}
\left( \;
{ \begin{tikzpicture}[baseline=0cm]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\node at (0,.4) {$\bullet$};
\node at (0,-.4) {$\bullet$};
\end{tikzpicture}
}\; + \;
{ \begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\node at (0,.4) {$\bullet$};
\node at (0,-.4) {$\bullet$};
\end{tikzpicture}}\; \right)
+ \frac{1}{d}
\left( \;
{\begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\; + \;
{\begin{tikzpicture}[baseline=0cm]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\; \right)$$
Using only this relation (along with the known values $d$ for loops, $1$ for bigons, and $0$ for triangles), we can readily compute all the following determinants. $$\begin{aligned}
\Delta_\omega(5,0) & = d^{10} (d - 2)^5 \\
\Delta_\omega^\square(5,1) & = d^9 (d - 2)^6\\
\Delta_\omega(6,0) & = - d^{26} (d-2)^{23} (d-1) \left(d^2-d-1\right) \left(d^3-2 d^2-3 d+1\right) \left(d^4-4 d^3+3 d^2-d-1\right)\\
\Delta_\omega^\square(6,1) & = - d^{12} (d-2)^{31} (d-1)^2 (d+1)^2 \left(d^2-3 d-1\right)^4 \left(d^4-2 d^3-3 d^2-d+2\right)\\
\Delta_\omega^\square(7,1) & = - d^{-86} (d-2)^{141} (d+1)^{16} \left(d^2-3 d-1\right)^{35} Q_{\omega,9} Q_{\omega,60}
$$ (The polynomials $Q_{\omega,i}$ appear in the appendix.)
In any twisted cubic category $D_\omega^\square(5,1)$ is linearly independent.
Since $d$ is not $0$ or $2$, we have $\Delta_\omega^\square(5,1) \neq 0$.
It follows that there are no twisted cubic categories with $\dim \cC_5 \leq 10$. It also follows that if $\dim \cC_5 = 11$, then $D_\omega^\square(5,1)$ is a basis for $\cC_5$.
If $\cC$ is a twisted cubic category with $\dim \cC_5 = 11$ and $\dim \cC_6 \leq 40$, then $d = -1$ or $d = \frac{3\pm \sqrt{13}}{2}$.
First, $\dim \cC_6 \leq 40$ implies that $D_\omega^\square(6,1)$ is linearly dependent and hence $\Delta_\omega^\square(6,1)$ and $\Delta_\omega^\square(7,1)$ both vanish. But their only shared factors are $d+1$ and $d^2 -3d- 1$.
There is at most one twisted cubic category with $\dim \cC_5 = 11$ and $\dim \cC_6 \leq 40$ for each of the three points $d = -1$ or $d = \frac{3 \pm \sqrt{13}}{2}$.
Again we must have a pentafork relation. The method of Lemma \[lem:pentaforks=>reductions\] can still be used to guarantee pentapent and hexapent reductions and thus uniqueness, as follows. The first argument there shows that we have a relation for reducing the pentapent unless $1 =
(-\zeta^{-2} \omega)^3 = -1$. The second argument there shows that we have a relation for reducing the hexapent unless $1 = (-\zeta^{-2} \omega)^7 =
-\zeta^{-2} \omega$ which can only occur when $\zeta^2 = -\omega$. Here $\zeta^2$ is a third root of unity, while $-\omega$ is a primitive sixth root of unity, so this can not happen.
The two cases with $d = \frac{3 + \sqrt{13}}{2}$ are realized by the twisted Haagerup fusion categories conjectured in [@MR2837122]. Ostrik observed that these can be constructed as follows. Start with the construction of two $\mathbb{Z}/9\mathbb{Z}$ Izumi near group categories in [@MR3167494]. Following Izumi and Evans-Gannon, the center of one of these categories contains a copy of $\mathrm{Rep}(\mathbb{Z}/3\mathbb{Z})$ as a symmetric tensor subcategory. One can then de-equivariantize [@MR2609644] by this to get a new category which has objects $1, g, g^2, X, gX, g^2 X$ but where the three invertible elements have nontrivial associator. We will denote the categories obtained this way $H_\omega$ (one for each primitive cube root of unity). The two with $d = \frac{3 -
\sqrt{13}}{2}$ are realized by the Galois conjugates of the $H_\omega$.
Note that the untwisted $H2$ (one of the even parts of the Haagerup subfactor) and $H3$ can be constructed in a similar way. Start with the unique $\mathbb
{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$ Izumi near-group category. The center of this category has two different copies of $\mathrm{Rep}(\mathbb {Z}/3\mathbb{Z})$ (one is the diagonal and the other the anti-diagonal). De-equivariantizing by each of these gives $H2$ and $H3$ (although it’s not clear which is which).
The two categories $H_\omega$ are twisted cubic categories with $\dim \cC_5 = 11$, $\dim \cC_6 = 37$, and $d = \frac{3 + \sqrt{13}}{2}$.
This argument closely follows the argument from \[prop:6:realization\]. Again let $H'$ be the subcategory generated by the trivalent vertex. The same argument as before shows that $H'$ must be a trivalent or twisted trivalent category. By the fusion rules we know that $d = \frac{3+\sqrt{13}}{2}$. We need to see that it is twisted and show that $H' = H_\omega$ which would show that $H_\omega$ is trivalent and, from the fusion rules, that the dimensions of the invariant spaces begin $1,0,1,1,4,11,37$. If $H'$ were untwisted, then it would have to be on our list of untwisted trivalent categories. First, $H'$ cannot be $SO(3)_q$ because $1+X$ is not an algebra (if it were then $g$ would be in the normalizer of that algebra contradicting the nontrivial associator). Second, $H'$ cannot be an $ABA$ or $(G_2)_q$ category for the same dimensional considerations that showed $H3'$ couldn’t lie in those familes. Third, $H'$ can’t be $H3$ because of the nontriviality of the associator. Hence it is a twisted trivalent category. Finally, in order to show that $H_\omega = H'$ it is enough to show that $\dim \Inv_{H'}(X^{\otimes 6}) = 37$ which follows calculating that the $41$-by-$41$ matrix $M_\omega^\square(6,1)$ has rank at least $37$ at this value of $d$. This rank calculation can be easily done modulo a prime sitting above $3$ in $\mathbb{Z}[d]$.
Does there exist a twisted trivalent category $\cQ_\omega$ with $d=-1$, $\dim \cC_5 = 11$ and $\dim \cC_6 \leq 40$?
Such a category is unique if it exists. It would have $\dim \cC_6 = 39$ and would satisfy the following two relations[^3] (where $\zeta$ is the primitive sixth root of unity which is a square root of $\omega$): $$\begin{gathered}
\nonumber
{
\begin{tikzpicture}[baseline=0cm]
\foreach \x in {1, ..., 6}
\draw (360*\x/6+0:.8cm)--(360*\x/6+0:.5cm);
\foreach \x in {1, ..., 6}
\draw (360*\x/6+0:.5cm) .. controls +(360*\x/6+120+0:.3cm) and +(360*\x/6+360/6-120+0:.3cm) .. (360*\x/6+360/6+0:.5cm);
\foreach \x in {1, ..., 6}
\node at (360*\x/6+0:.3cm) {$\bullet$};
\end{tikzpicture}
} \; + \; \sum_{i=0}^5 \rho^i\left( {
\begin{tikzpicture}[baseline=0]
\coordinate (F) at (90:.5cm);
\coordinate (P1) at (0:.5cm);
\coordinate (P2) at (90:.35cm);
\coordinate (P3) at (180:.5cm);
\coordinate (P4) at (240:.5cm);
\coordinate (P5) at (300:.5cm);
\coordinate (E1) at (0:.8cm);
\coordinate (E2) at (60:.8cm);
\coordinate (E3) at (120:.8cm);
\coordinate (E4) at (180:.8cm);
\coordinate (E5) at (240:.8cm);
\coordinate (E6) at (300:.8cm);
\draw (E2)--(F)--(E3);
\draw (F)--(P2)--(P3)--(P4)--(P5)--(P1)--(P2);
\draw (P1)--(E1);
\draw (P3)--(E4);
\draw (P4)--(E5);
\draw (P5)--(E6);
\path (F) ++(90:.15) node {$\bullet$};
\path (P1) ++(190:.2) node {$\bullet$};
\path (P2) ++(270:.175) node {$\bullet$};
\path (P3) ++(-10:.2) node {$\bullet$};
\path (P4) ++(60:.15) node {$\bullet$};
\path (P5) ++(120:.15) node {$\bullet$};
\end{tikzpicture}
}\right) \; + \; \sum_{i=0}^5 \rho^i\left( {
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (0:.8cm);
\coordinate (E2) at (60:.8cm);
\coordinate (E3) at (120:.8cm);
\coordinate (E4) at (180:.8cm);
\coordinate (E5) at (240:.8cm);
\coordinate (E6) at (300:.8cm);
\coordinate (P1) at (90:.5cm);
\coordinate (P2) at (90:.1cm);
\draw (E1)--(P2)--(E4);
\draw (P1)--(P2);
\draw (E2)--(P1)--(E3);
\draw (E5) .. controls (270:.2cm) .. (E6);
\path (P1) ++(90:.15) node {$\bullet$};
\path (P2) ++(270:.15) node {$\bullet$};
\end{tikzpicture}
}\right) \; + \; \sum_{i=0}^1 \rho^i\left( {
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (0:.8cm);
\coordinate (E2) at (60:.8cm);
\coordinate (E3) at (120:.8cm);
\coordinate (E4) at (180:.8cm);
\coordinate (E5) at (240:.8cm);
\coordinate (E6) at (300:.8cm);
\draw (E2) .. controls (90:.15) .. (E3);
\draw (E4) .. controls (210:.15) .. (E5);
\draw (E6) .. controls (320:.15) .. (E1);
\end{tikzpicture}
}\right) \\ = \sum_{i=0}^2 \rho^i\left( {
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (0:.8cm);
\coordinate (E2) at (60:.8cm);
\coordinate (E3) at (120:.8cm);
\coordinate (E4) at (180:.8cm);
\coordinate (E5) at (240:.8cm);
\coordinate (E6) at (300:.8cm);
\coordinate (P1) at (60:.3cm);
\coordinate (P2) at (240:.3cm);
\draw (E1)--(P1)--(E3);
\draw (P1)--(E2);
\draw (E4)--(P2)--(E6);
\draw (P2)--(E5);
\path (P1) ++(240:.15) node {$\bullet$};
\path (P2) ++(60:.15) node {$\bullet$};
\end{tikzpicture}
}\right) \; + \; \sum_{i=0}^2 \rho^i\left({
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (0:.8cm);
\coordinate (E2) at (60:.8cm);
\coordinate (E3) at (120:.8cm);
\coordinate (E4) at (180:.8cm);
\coordinate (E5) at (240:.8cm);
\coordinate (E6) at (300:.8cm);
\coordinate (P1) at (45:.5cm);
\coordinate (P2) at (90:.3cm);
\coordinate (P3) at (270:.3cm);
\coordinate (P4) at (225:.5cm);
\draw (E1)--(P1)--(E2);
\draw (E3)--(P2)--(P3)--(E6);
\draw (E4)--(P4)--(E5);
\draw (P1)--(P2)--(P3)--(P4);
\path (P1) ++(20:.2cm) node {$\bullet$};
\path (P4) ++(200:.2cm) node {$\bullet$};
\path (P2) ++(200:.2cm) node {$\bullet$};
\path (P3) ++(20:.2cm) node {$\bullet$};
\end{tikzpicture}
}) \right) \; + \; \sum_{i=0}^2 \rho^i\left( {
\begin{tikzpicture}[baseline=0, xscale=-1]
\coordinate (E1) at (0:.8cm);
\coordinate (E2) at (60:.8cm);
\coordinate (E3) at (120:.8cm);
\coordinate (E4) at (180:.8cm);
\coordinate (E5) at (240:.8cm);
\coordinate (E6) at (300:.8cm);
\coordinate (P1) at (45:.5cm);
\coordinate (P2) at (90:.3cm);
\coordinate (P3) at (270:.3cm);
\coordinate (P4) at (225:.5cm);
\draw (E1)--(P1)--(E2);
\draw (E3)--(P2)--(P3)--(E6);
\draw (E4)--(P4)--(E5);
\draw (P1)--(P2)--(P3)--(P4);
\path (P1) ++(20:.2cm) node {$\bullet$};
\path (P4) ++(200:.2cm) node {$\bullet$};
\path (P2) ++(200:.2cm) node {$\bullet$};
\path (P3) ++(20:.2cm) node {$\bullet$};
\end{tikzpicture}
}\right) \end{gathered}$$ and $$\begin{gathered}
\nonumber
0 = \sum_{i=0}^5 \zeta^i \rho^i\left( {
\begin{tikzpicture}[baseline=0]
\coordinate (F) at (90:.5cm);
\coordinate (P1) at (0:.5cm);
\coordinate (P2) at (90:.35cm);
\coordinate (P3) at (180:.5cm);
\coordinate (P4) at (240:.5cm);
\coordinate (P5) at (300:.5cm);
\coordinate (E1) at (0:.8cm);
\coordinate (E2) at (60:.8cm);
\coordinate (E3) at (120:.8cm);
\coordinate (E4) at (180:.8cm);
\coordinate (E5) at (240:.8cm);
\coordinate (E6) at (300:.8cm);
\draw (E2)--(F)--(E3);
\draw (F)--(P2)--(P3)--(P4)--(P5)--(P1)--(P2);
\draw (P1)--(E1);
\draw (P3)--(E4);
\draw (P4)--(E5);
\draw (P5)--(E6);
\path (F) ++(90:.15) node {$\bullet$};
\path (P1) ++(190:.2) node {$\bullet$};
\path (P2) ++(270:.175) node {$\bullet$};
\path (P3) ++(-10:.2) node {$\bullet$};
\path (P4) ++(60:.15) node {$\bullet$};
\path (P5) ++(120:.15) node {$\bullet$};
\end{tikzpicture}
}\right) \; + \; \sum_{i=0}^5 \zeta^i \rho^i\left( {
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (0:.8cm);
\coordinate (E2) at (60:.8cm);
\coordinate (E3) at (120:.8cm);
\coordinate (E4) at (180:.8cm);
\coordinate (E5) at (240:.8cm);
\coordinate (E6) at (300:.8cm);
\coordinate (P1) at (60:.2cm);
\coordinate (P2) at (120:.2cm);
\draw (E1)--(P1)--(E2);
\draw (P1)--(P2);
\draw (E3)--(P2)--(E4);
\draw (E5) .. controls (270:.2cm) .. (E6);
\path (P1) ++(20:.2) node {$\bullet$};
\path (P2) ++(160:.2) node {$\bullet$};
\end{tikzpicture}
}\right) \\ + \; \sum_{i=0}^5 \zeta^i \rho^i\left({
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (0:.8cm);
\coordinate (E2) at (60:.8cm);
\coordinate (E3) at (120:.8cm);
\coordinate (E4) at (180:.8cm);
\coordinate (E5) at (240:.8cm);
\coordinate (E6) at (300:.8cm);
\coordinate (P1) at (90:.5cm);
\coordinate (P2) at (90:.1cm);
\draw (E1)--(P2)--(E4);
\draw (P1)--(P2);
\draw (E2)--(P1)--(E3);
\draw (E5) .. controls (270:.2cm) .. (E6);
\path (P1) ++(90:.15) node {$\bullet$};
\path (P2) ++(270:.15) node {$\bullet$};
\end{tikzpicture}
}\right) \; + \; \sum_{i=0}^5 \zeta^i \rho^i\left( {
\begin{tikzpicture}[baseline=0, xscale=-1]
\coordinate (E1) at (0:.8cm);
\coordinate (E2) at (60:.8cm);
\coordinate (E3) at (120:.8cm);
\coordinate (E4) at (180:.8cm);
\coordinate (E5) at (240:.8cm);
\coordinate (E6) at (300:.8cm);
\coordinate (P1) at (30:.5cm);
\coordinate (P2) at (150:.5cm);
\coordinate (P3) at (240:.4cm);
\coordinate (P4) at (300:.4cm);
\draw (E1)--(P1)--(E2);
\draw (E3)--(P2)--(E4);
\draw (E5)--(P3);
\draw (E6)--(P4);
\draw (P2)--(P3)--(P4)--(P1);
\path (P1) ++(180:.15cm) node {$\bullet$};
\path (P2) ++(0:.15cm) node {$\bullet$};
\path (P3) ++(60:.15cm) node {$\bullet$};
\path (P4) ++(120:.15cm) node {$\bullet$};
\end{tikzpicture}
}\right).\end{gathered}$$
Such a $\cQ_\omega$ cannot come from any operator algebraic construction since $d=-1 < 0$. As we will see in the next section, $\cQ_\omega$ is not braided and so $\cQ_\omega$ is not a Drinfel’d-Jimbo quantum group. Furthermore, the dimensions of the objects at depth $2$ are not real (they are conjugate primitive sixth roots of unity), so $\cQ_\omega$ must have infinitely many simple objects. We do not think it comes from any well understood construction, but it has also passed every test we have attempted to use to rule it out. We also note that the above relations are particularly nice as they involve only 24 terms each.
In conclusion, we have the following classification of twisted trivalent categories.
dimension bounds new examples
------------------- ----------------------------
1,0,1,1,3,… $\cC(S_3, S_2, \xi, \psi)$
1,0,1,1,4,10,… nothing
1,0,1,1,4,11,37,… $H_\omega$
1,0,1,1,4,11,39,… $\cQ_\omega$, if it exists
1,0,1,1,4,11,40,… nothing more
Braided Trivalent Categories {#sec:braided}
============================
We call a trivalent or twisted trivalent category braided if there is an element in the $4$-boundary point space, which we write using a crossing, which satisfies the following relations (in the untwisted case ignore the dots): $${
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (45:.8cm);
\coordinate (E2) at (135:.8cm);
\coordinate (E3) at (225:.8cm);
\coordinate (E4) at (315:.8cm);
\draw[white, line width = 4pt] (E1) .. controls (180:1cm) .. (E4);
\draw (E1) .. controls (180:1cm) .. (E4);
\draw[white, line width = 4pt] (E2) .. controls (0:1cm) .. (E3);
\draw (E2) .. controls (0:1cm) .. (E3);
\end{tikzpicture}
}= {\begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}},$$ $$\label{eq:pullthrough1} {
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (90-72:.8cm);
\coordinate (E2) at (90:.8cm);
\coordinate (E3) at (90+72:.8cm);
\coordinate (E4) at (90+2*72:.8cm);
\coordinate (E5) at (90+3*72:.8cm);
\coordinate (P1) at (270:.2cm);
\draw (E4) -- (P1) -- (E5);
\draw (P1) -- (E2);
\draw[white, line width = 4pt] (E1) -- (E3);
\draw (E1) -- (E3);
\path (P1) ++(270:.2cm) node {$\bullet$};
\end{tikzpicture}}= {
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (90-72:.8cm);
\coordinate (E2) at (90:.8cm);
\coordinate (E3) at (90+72:.8cm);
\coordinate (E4) at (90+2*72:.8cm);
\coordinate (E5) at (90+3*72:.8cm);
\coordinate (P1) at (90:.2cm);
\draw (E4) .. controls (180:.2cm) .. (P1) .. controls (0:.2cm) .. (E5);
\draw (P1) -- (E2);
\draw[white, line width = 4pt] (E1) .. controls (270:.4cm) .. (E3);
\draw (E1) .. controls (270:.4cm) .. (E3);
\path (P1) ++(270:.2cm) node {$\bullet$};
\end{tikzpicture}},$$ $$\label{eq:pullthrough2} {
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (90-72:.8cm);
\coordinate (E2) at (90:.8cm);
\coordinate (E3) at (90+72:.8cm);
\coordinate (E4) at (90+2*72:.8cm);
\coordinate (E5) at (90+3*72:.8cm);
\coordinate (P1) at (270:.2cm);
\draw (E4) -- (P1) -- (E5);
\draw (E1) -- (E3);
\draw[white, line width = 4pt] (P1) -- (E2);
\draw (P1) -- (E2);
\path (P1) ++(270:.2cm) node {$\bullet$};
\end{tikzpicture}
}= {
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (90-72:.8cm);
\coordinate (E2) at (90:.8cm);
\coordinate (E3) at (90+72:.8cm);
\coordinate (E4) at (90+2*72:.8cm);
\coordinate (E5) at (90+3*72:.8cm);
\coordinate (P1) at (90:.2cm);
\draw (E1) .. controls (270:.4cm) .. (E3);
\draw[white, line width = 4pt] (E4) .. controls (180:.2cm) .. (P1) .. controls (0:.2cm) .. (E5);
\draw (E4) .. controls (180:.2cm) .. (P1) .. controls (0:.2cm) .. (E5);
\draw (P1) -- (E2);
\path (P1) ++(270:.2cm) node {$\bullet$};
\end{tikzpicture}
}.$$
Note that the latter two relations above imply the Reidemeister $3$ relations also hold via the Kauffman trick (since the category is generated by the trivalent vertex and so the crossing can be written in terms of trivalent vertices). As usual in quantum topology, the Reidemeister $1$ relation need not hold.
There are no braided twisted trivalent categories.
By dimensional considerations we have the following relations $${
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (315:.8cm);
\coordinate (E2) at (225:.8cm);
\coordinate (P1) at (45:.2cm);
\coordinate (P2) at (135:.2cm);
\coordinate (C1) at (0:.6cm);
\coordinate (C2) at (180:.6cm);
\coordinate (C3) at (90:.16cm);
\draw (E1) .. controls (C2) .. (P2);
\draw[white, line width = 4pt] (P2) .. controls (C3) .. (P1) .. controls (C1) .. (E2);
\draw (P2) .. controls (C3) .. (P1) .. controls (C1) .. (E2);
\end{tikzpicture}
}= \alpha {
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (225:.8cm);
\coordinate (E2) at (315:.8cm);
\draw (E1) .. controls (90:.2cm) .. (E2);
\end{tikzpicture}
}$$ $${
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (330:.8cm);
\coordinate (E2) at (210:.8cm);
\coordinate (E3) at (90:.8cm);
\coordinate (P1) at (90:.2cm);
\coordinate (C1) at (0:.6cm);
\coordinate (C2) at (180:.6cm);
\draw (E1) .. controls (C2) .. (P1);
\draw[white, line width=4pt] (P1) .. controls (C1) .. (E2);
\draw (P1) .. controls (C1) .. (E2);
\draw (P1) -- (E3);
\path (P1) ++(270:.2cm) node {$\bullet$};
\end{tikzpicture}
}= \beta {
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (330:.8cm);
\coordinate (E2) at (210:.8cm);
\coordinate (E3) at (90:.8cm);
\coordinate (P1) at (0,0);
\draw (E1) -- (P1) -- (E2);
\draw (P1) -- (E3);
\path (P1) ++(270:.2cm) node {$\bullet$};
\end{tikzpicture}
}$$ for some numbers $\alpha$ and $\beta$. If we use the other crossing in these pictures we get the same relations with $\alpha^{-1}$ and $\beta^{-1}$. We can now compute the action of rotation on the twisted trivalent vertex in two different ways: $${
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (45:.8cm);
\coordinate (E2) at (225:.8cm);
\coordinate (E3) at (270:.8cm);
\coordinate (P1) at (0,0);
\coordinate (P2) at (120:.6cm);
\coordinate (P3) at (.4cm,.2cm);
\coordinate (C1) at (100:.9cm);
\coordinate (C2) at (135:.5cm);
\coordinate (C3) at (.8,.1);
\draw (E1) -- (P1) -- (E3);
\draw (P1) .. controls (C1) .. (P2);
\draw[white, line width=4pt] (P2) .. controls (C2) .. (P3);
\draw[white, line width=4pt] (P3) .. controls (C3) .. (E2);
\draw (P2) .. controls (C2) .. (P3);
\draw (P3) .. controls (C3) .. (E2);
\path (P1) ++(330:.15cm) node {$\bullet$};
\end{tikzpicture}
}= {
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (45:.8cm);
\coordinate (E2) at (225:.8cm);
\coordinate (E3) at (270:.8cm);
\coordinate (P0) at (0,0);
\coordinate (C1) at (-.1,.7);
\coordinate (C2) at (-.3,-.2);
\coordinate (C3) at (.2,-.2);
\draw (P0) .. controls (C1) .. (E2);
\draw (P0) .. controls (C2) .. (E3);
\draw (P0) .. controls (C3) .. (E1);
\path (P0) ++(270:.2cm) node {$\bullet$};
\end{tikzpicture}
}= {
\begin{tikzpicture}[baseline=0]
\coordinate (E1) at (45:.8cm);
\coordinate (E2) at (225:.8cm);
\coordinate (E3) at (270:.8cm);
\coordinate (P1) at (0,0);
\coordinate (P2) at (120:.6cm);
\coordinate (P3) at (.4cm,.2cm);
\coordinate (C1) at (100:.9cm);
\coordinate (C2) at (135:.5cm);
\coordinate (C3) at (.8,.1);
\draw (P2) .. controls (C2) .. (P3);
\draw (P3) .. controls (C3) .. (E2);
\draw[white, line width=4pt] (E1) -- (P1) -- (E3);
\draw[white, line width=4pt] (P1) .. controls (C1) .. (P2);
\draw (E1) -- (P1) -- (E3);
\draw (P1) .. controls (C1) .. (P2);
\path (P1) ++(330:.15cm) node {$\bullet$};
\end{tikzpicture}
}.$$ Thus, $\alpha^{-1}\beta^2 = \omega = \alpha \beta^{-2}$, and so $\omega = \omega^{-1}$ which is a contradiction.
The same argument shows that if $X$ is a simple object in a braided tensor category and $f: 1 \rightarrow X^{\otimes n}$ is an eigenvector for both rotation and braiding, then the rotational eigenvalue is $\pm 1$.
\[lem:braiding-C5-bound\] If $\cC$ is a braided trivalent category with $\dim \cC_4 \leq 4$, then $\dim \cC_5 \leq 10$.
Since $\dim \cC_4 \leq 4$ we get that $D(4,0)$ forms a spanning set for $\cC_4$. Look at Equations and and expand every crossing as a sum of diagrams in $D(4,0)$. These each give a relation between diagrams in $D^\square(5,1)$. We claim that at least one of these relations is nontrivial. Since the crossings are rotations of each other, at least one of the crossings has a nontrivial coefficient of either ${\begin{tikzpicture}[rotate=90,scale=.25]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}$ or ${ \begin{tikzpicture}[scale=.25]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}
}$. In the former case the expanded relation has a nontrivial coefficient of ${
\begin{tikzpicture}[scale=.25]
\coordinate (E1) at (90-72:.8cm);
\coordinate (E2) at (90:.8cm);
\coordinate (E3) at (90+72:.8cm);
\coordinate (E4) at (90+2*72:.8cm);
\coordinate (E5) at (90+3*72:.8cm);
\coordinate (P1) at (0,0);
\draw (E1)--(P1)--(E3);
\draw (P1)--(E2);
\draw (E4) .. controls (270:.2cm) .. (E5);
\end{tikzpicture}
}$, and in the latter case the expanded relation has a nontrivial coefficient of the pentagon. Thus we have a nontrivial relation among $D^\square(5,1)$, so by Lemma \[lem:5:dependent=>spans\], we get that $D(5,0)$ spans $\cC_5$. In particular, $\dim \cC_5 \leq 10$.
This is enough to give a complete classification of braided trivalent categories, but before stating this classification we list the examples that occur.
The standard braiding on $SO(3)_q$ is
$$\begin{aligned}
\overcrossing & = (q^2 - 1) {\begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}+ q^{-2} {\begin{tikzpicture}[baseline=0cm]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}- (q^2+q^{-2}) { \begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}}.\end{aligned}$$
Note that although $SO(3)_{\pm q^{\pm 1}}$ are the same fusion category, there are two distinct braided tensor categories corresponding to $\pm q$ and $\pm q^{-1}$. Finally, note that when $q= \pm i$ this formula gives the standard symmetric braiding on $OSp(1|2)$.
Let $X$ be the standard $2$-dimensional representation of $S_3$ and $X \otimes X \rightarrow X$ be a non-zero map (which is unique up to scalar), then this generates a trivalent category. The standard symmetric braiding is
$$\begin{aligned}
\symmetriccrossing & = {\begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}- {\begin{tikzpicture}[baseline=0cm]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}+ 2 { \begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}}\\
& = { \begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}}+ { \begin{tikzpicture}[baseline=0cm]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}
}.\end{aligned}$$
It is easy to see from our classification that this category agrees with $SO(3)_q$ for $q$ a primitive $12$th root of unity as a trivalent category. However, the standard symmetric braiding on representations of $S_3$ does not agree with the standard braiding for $SO(3)_q$. We will denote this braided trivalent category by $S_3$, to distinguish it from $SO(3)_{\zeta_{12}}$.
The standard braiding for $(G_2)_q$ for $q \neq \pm i$ is
$$\begin{aligned}
\overcrossing & = \frac{1}{q+q^{-1}} \left(q^3 {\begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}+ q^{-3} {\begin{tikzpicture}[baseline=0cm]
\draw (45:.8cm) .. controls (0,0) .. (135:.8cm);
\draw (-45:.8cm) .. controls (0,0) .. (-135:.8cm);
\end{tikzpicture}}\right) \\
& \qquad
- \frac{q^6+q^4+q^2+q^{-2}+q^{-4}+q^{-6}}{q+q^{-1}}
\left( q { \begin{tikzpicture}[baseline=0cm,rotate=90]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}}+ q^{-1} { \begin{tikzpicture}[baseline=0cm]
\draw (0,.2) .. controls +(30:.3cm) .. (45:.8cm);
\draw (0,.2) .. controls +(150:.3cm) .. (135:.8cm);
\draw (0,.2) -- (0,-.2);
\draw (0,-.2) .. controls +(-30:.3cm) .. (-45:.8cm);
\draw (0,-.2) .. controls +(-150:.3cm) .. (-135:.8cm);
\end{tikzpicture}
}\right).\end{aligned}$$
Note that although $(G_2)_{\pm q^{\pm 1}}$ give the same fusion category, there are two distinct braided tensor categories corresponding to $\pm q$ and $\pm q^{-1}$.
The only braided trivalent categories with $\dim \cC_4 \leq 4$ are $SO(3)_q$ (for any $q$ including $SO(3)_{\pm i} = OSp(1|2)$), $S_3$, and $(G_2)_q$ for $q \neq \pm i$.
By our classification, we need only classify all braidings for the $SO(3)_q$, ABA, and $(G_2)_q$ categories. In the $G_2$ case, when $q$ is not a fourth or twentieth root of unity, this was done by Kuperberg in [@MR1265145]. Following Kuperberg, you write down a general element of the $4$-boundary point space and check whether it satisfies the braiding relations. Since this is somewhat tedious and since the hardest case was already done by Kuperberg, we will skip much of the details here.
For the $SO(3)_q$ categories (which agree for $\pm q^{\pm 1}$), generically there are exactly two braidings on corresponding to one corresponding to the standard braiding of $SO(3)_{\pm q}$ and the other to $SO(3)_{\pm q^{-1}}$. When $q = \pm 1$ or $q= \pm i$, these two braidings agree and yield the symmetric braidings on $SO(3)$ and $OSp(1|2)$. When $q$ is a primitive $12$th root of unity, there are three braidings, two corresponding to the $SO(3)_q$ braidings and one corresponding to the symmetric braiding which corresponds to the standard braiding on $S_3$.
A direct calculation shows that the ABA categories do not have a braiding. When $\delta$, the loop value for $A$, is not the golden ratio or its conjugate, there is a simple more conceptual approach. Namely, the fusion rules are noncommutative since $A^{(2)}(ABA) = A^{(3)} B A + ABA$ while $(ABA) A^{(2)} = A B A^{(3)} + ABA$. (When $\delta$ is the golden ratio, the fusion rules are commutative so one must use the brutal approach.)
Finally, for $(G_2)_q$, Kuperberg proved that when the $5$-boundary point space is $10$-dimensional, there are exactly two braidings for such a category, one corresponding to the standard braiding of $(G_2)_{\pm q}$ and the other to $(G_2)_{\pm q^{-1}}$. The only remaining cases are $q$ is a primitive $4$th or $20$th root of unity. The former case was already dealt with above, and it turns out in the latter case there are still only the two standard braidings.
We expect this theorem to give Wenzl-style recognition results [@MR1237835; @MR2132671] for $(G_2)_q$ and Deligne’s $S_t$, showing that they are the only braided tensor categories with their Grothendieck rings.
The only symmetric trivalent categories with $\dim \cC_4 \leq 4$ are $SO(3)$, $S_3$, $(G_2)$, and $OSp(1|2)$.
Note that by Deligne’s theorem [@MR1944506], any symmetric abelian category of exponential growth must be the category of representations of a supergroup, so this corollary is not surprising. The exponential growth condition is satisfied in our setting, but this corollary does not follow directly from Deligne’s theorem because [*a priori*]{} there could be symmetric trivalent categories which do not come from abelian categories.
Prospects
=========
There are several obstacles to pushing the above techniques further in the study of trivalent categories. First, even though in principal just knowing $d$ and $t$ should be enough to calculate the determinants $D^\square(n,k)$ for $n+2k <
12$, as we saw in Conjecture \[conj:Delta7\_1\], in practice arithmetic of two variable rational functions is sufficiently difficult that we cannot compute all of these determinants exactly. If one is willing to accept probabilistic proofs then we can compute more of these determinants. Second, if we want to go beyond $n+2k = 12$ then we need to introduce the dodecahedron as a third variable, which will make things quite a bit more complicated. Finally, we are already pushing up against the limits of practical Gröbner basis calculations: for example we already cannot directly intersect $\Delta(7,0)$ and $\Delta^\square(7,1)$. For all these reasons, it is unlikely that we will be able to push the classification of trivalent categories much further without new ideas.
Instead we plan to continue investigating these ideas in other settings than the trivalent setting. There are numerous good candidates for investigation, including the following.
- Braided trivalent categories with dimensions bounded by $1,0,1,1,5$. This includes the conjectured exceptional series of Deligne and Vogel [@MR1378507; @MR2769234].
- Skein theoretic invariants of planar graphs together with a $2$-coloring of the vertices and a $2$-coloring of the faces. These correspond to quadrilaterals of subfactors and we hope to strengthen the classification results of Grossman, Izumi, and Jones [@MR2257402; @MR2418197].
- Categories generated by a 4-valent vertex with a checkerboard shading. These were studied by Bisch, Jones, and Liu in [@MR1733737; @MR1972635; @1410.2876], and we hope to push their techniques beyond what can be done by hand.
- Categories generated by a $2n$-valent vertex with a checkerboard shading. For $n=2$ this is the previous example, and for $n=3$ they were studied by Dylan Thurston [@0405482].
- Skein theoretic invariants of virtual knots. This includes the representation theory of the Higman–Sims sporadic finite simple group [@MR1188082; @MR1469634].
Skein theoretic invariants and pivotal categories {#sec:local}
=================================================
The goal of this section is to provide background so that this paper is accessible to knot theorists, graph theorists, and other readers unfamiliar with tensor categories or planar algebras.
Suppose we want to study certain numerical invariants $f$ of planar trivalent graphs. Assume that $f$ of the empty diagram is $1$, that $f \left( \tikz[baseline=-1mm] \draw (0,0) circle (3mm); \right)$ and $f \left( \begin{tikzpicture}[baseline=-1mm]
\draw (0,0) circle (3mm);
\draw (.3,0)--(-.3,0);
\end{tikzpicture}
\right)$ are nonzero, and that $f$ satisfies the following multiplicative conditions:
1. $f \left( \begin{tikzpicture}[baseline=-1mm]
\node[circle,draw,dotted] (X) at (0,0) {X};
\node[circle,draw,dotted] (Y) at (1,0) {Y};
\end{tikzpicture} \right)
= f(X) \cdot f(Y)$
2. $f \left( \begin{tikzpicture}[baseline=-1mm]
\node[circle,draw,dotted] (X) at (0,0) {X};
\node[circle,draw,dotted] (Y) at (1,0) {Y};
\draw (X.east)--(Y.west);
\end{tikzpicture} \right)
= 0$
3. $f \left( \begin{tikzpicture}[baseline=-1mm]
\node[circle,draw,dotted] (X) at (0,0) {X};
\node[circle,draw,dotted] (Y) at (1,0) {Y};
\draw (X.north east) .. controls (.5,.4) .. (Y.north west);
\draw (X.south east) .. controls (.5,-.4) .. (Y.south west);
\end{tikzpicture} \right)
= f \left( \begin{tikzpicture}[baseline=-1mm]
\node[circle,draw,dotted] (X) at (0,0) {X};
\draw (X.north east) .. controls (.5,.4) and (.7,.2).. (.7,0) .. controls (.7,-.2) and (.5,-.4) .. (X.south east);
\end{tikzpicture} \right)
\cdot
f \left( \begin{tikzpicture}[baseline=-1mm]
\node[circle,draw,dotted] (X) at (0,0) {Y};
\draw (X.north west) .. controls (-.5,.4) and (-.7,.2).. (-.7,0) .. controls (-.7,-.2) and (-.5,-.4) .. (X.south west);
\end{tikzpicture} \right) /
f \left( \tikz[baseline=-1mm] \draw (0,0) circle (3mm); \right)
$
4. $f \left( \begin{tikzpicture}[baseline=-1mm]
\node[circle,draw,dotted] (X) at (0,0) {X};
\node[circle,draw,dotted] (Y) at (1,0) {Y};
\draw (X.east) -- (Y.west);
\draw (X.north east) .. controls (.5,.4) .. (Y.north west);
\draw (X.south east) .. controls (.5,-.4) .. (Y.south west);
\end{tikzpicture} \right)
= f \left( \begin{tikzpicture}[baseline=-1mm]
\node[circle,draw,dotted] (X) at (0,0) {X};
\draw (X.north east) .. controls (.5,.4) and (.7,.2).. (.7,0) .. controls (.7,-.2) and (.5,-.4) .. (X.south east);
\draw (X.east)--(.7,0);
\end{tikzpicture} \right)
\cdot
f \left( \begin{tikzpicture}[baseline=-1mm]
\node[circle,draw,dotted] (X) at (0,0) {Y};
\draw (X.north west) .. controls (-.5,.4) and (-.7,.2).. (-.7,0) .. controls (-.7,-.2) and (-.5,-.4) .. (X.south west);
\draw (X.west)--(-.7,0);
\end{tikzpicture} \right) /
f \left( \begin{tikzpicture}[baseline=-1mm]
\draw (0,0) circle (3mm);
\draw (.3,0)--(-.3,0);
\end{tikzpicture}
\right)
$
Thus the invariant of any $k$-disconnected graph for $k \leq 3$ is determined by the invariants of the pieces.
\[ex:vertex-counting-invariant\] An almost trivial example of a multiplicative invariant of graphs is $a^{\# V}$, for some number $a$, where $\# V$ denotes the number of trivalent vertices in the graph.
\[ex:chromatic-invariant\] An important example of a multiplicative invariant of graphs is the number of $n$-colorings of the faces of the graph, divided by $n$. (The division by $n$ is a normalization factor ensuring that the empty graph is assigned $1$ instead of $n$.) This example can be generalized by considering non-integer specializations of the chromatic polynomial.
What examples are there of such multiplicative invariants of trivalent planar graphs?
While the question appears to be an elementary question about planar trivalent graphs, we discover that the examples are actually related to quite distant subjects in mathematics. In particular, we are able to identify each of the small examples we encounter with some suprising or exotic object coming from representation theory or the theory of subfactors!
In order to understand the main results of the paper in the language of graph invariants, we first want to extend this invariant of closed trivalent graphs to an invariant of planar graphs with boundary. That is, we extract a sequence of vector spaces, the ‘open graphs, modulo negligibles’. We now describe how these vector spaces have the structure of a pivotal tensor category (or planar algebra).
Let $\hat{\cC}_n$ denote the (infinite dimensional) vector space with basis the planar trivalent graphs drawn in the disc, with $n$ fixed boundary points, up to isotopy rel boundary. This vector space has a natural bilinear pairing, given by gluing two open graphs together (starting at a preferred boundary point), to obtain a closed planar graph, which we then evaluate to a number using our multiplicative invariant $f$. The kernel of this bilinear pairing is called ‘the negligible elements’. Let $\cC^f_n$ denote the quotient vector space of $\hat{\cC}_n$ by negligible elements.
One may assemble these vector spaces into a single algebraic structure, variously axiomatized as an (unshaded) planar algebra [@math.QA/9909027], a spider [@MR1403861] or a pivotal tensor category [@MR1686423]. We’ll only describe the last in any detail. The category, which we’ll call $\cC^f$, has as objects the natural numbers. We’ll first describe a bigger category of trivalent graphs, which we call $\hat{\cC}$ and which does not depend at all on our multiplicative invariant. In $\hat{\cC}$, the morphisms from $n$ to $m$ are simply the formal linear combinations of planar graphs drawn in a rectangle with $n$ points along the bottom edge and $m$ points along the top edge, i.e. the vector space $\hat{\cC}_{n+m}$. We can compose morphisms in the obvious way, by stacking rectangles. This category is a tensor category, with the tensor product given by drawing diagrams side by side. Finally it is a pivotal category, with the evaluation and coevaluation maps given by caps and cups.
Inside $\hat{\cC}$, the negligible (with respect to $f$) elements form a planar ideal — if some (linear combination of) graphs pair with arbitrary other graphs to give zero, then glueing more graph to the boundary preserves this property. We thus define the category $\cC^f$ to be the quotient of $\hat{\cC}$ by the negligible ideal. This “ideal" property says that we can treat the negligible elements as skein relations: they can be applied locally in any part of a graph. Furthermore, typically this ideal is finitely generated by a few particular skein relations.
Thus, in $\cC^f$, the objects are still the natural numbers and the morphisms from $n$ to $m$ are just $\cC^f_{n+m}$. The category $\cC^f$ is still a pivotal tensor category, and now it is *evaluable* (i.e. $\dim \cC^f_0 = 1$, and in fact $\cC^f_0$ may be identified with the ground field by sending the empty diagram to $1$) and *non-degenerate* (i.e. for every morphism $x: a \to b$, there is another morphism $x': b \to a$ so $\langle x, x' \rangle \neq 0 \in
\cC^f_0$). Writing $X$ for the generating object in $\cC^f$ (i.e. 1 in the natural numbers!), we see that $X$ is a symmetrically self-dual object, with duality pairings and copairings given by the cap and cup diagrams. Moreover, the trivalent vertex is a rotationally symmetric map $1 \to X \tensor X \tensor X$.
If the invariant is the normalized number of $n$-colorings described in Example \[ex:chromatic-invariant\], then a linear combinations of graphs is negligible if and only if for any coloring of the boundary faces the given linear combination of the numbers of ways of extending that coloring to the interior is zero. For example, the following element of $\cC_3$ is negligible: $${
\begin{tikzpicture}[baseline=0cm]
\foreach \x in {1, ..., 3}
\draw (360*\x/3+90:.8cm)--(360*\x/3+90:.5cm);
\foreach \x in {1, ..., 3}
\draw (360*\x/3+90:.5cm) .. controls +(360*\x/3+120+90:.3cm) and +(360*\x/3+360/3-120+90:.3cm) .. (360*\x/3+360/3+90:.5cm);
\end{tikzpicture}
}
- (n-3) \cdot
\begin{tikzpicture}[baseline=.1cm,scale=0.75]
\draw (0,0) -- (0,1);
\draw (0,0) -- (0.7,-0.5);
\draw (0,0) -- (-0.7,-0.5);
\end{tikzpicture}$$ In particular, this gives a skein relation in $\cC^f$ which says that you can remove a triangle and multiply by $(n-3)$. There are also other negligible elements; in fact after renormalizing the trivalent vertex, $\cC^f$ becomes equivalent to the pivotal category $SO(3)_q$ coming from quantum groups where $q$ is a number satisfying $ (q+q^{-1})^2=n$ (see Section \[sec:four\] for a description of $SO(3)_q$).
The construction of $\cC^f$ from $f$ gives a bijective correspondence between trivalent categories and multiplicative invariants of planar graphs.
First we prove that the category $\cC^f$ constructed from a multiplicative invariant $f$ is trivalent. Consider $\cC^f_0$. The empty diagram is not negligible, so we need only show that any closed diagram is a multiple of the empty diagram. If $\alpha$ is a closed diagram and $\beta$ is the empty diagram, then $\alpha - f(\alpha) \beta$ is negligible, so in $\cC^f_0$ we have that $\alpha = f(\alpha) \beta$. Now we look at $\cC^f_1$. By multiplicativity we have that any diagram with one boundary point is negligible, so $\dim\cC^f_1
= 0$. The remaining cases are similar.
Given a trivalent category $\cC$, we need to construct a multiplicative invariant of planar graphs. The usual diagrammatic calculus for pivotal categories shows that any trivalent category gives an invariant of closed graphs just by interpreting the graphs as elements of $\cC_0$ and sending the empty diagram to $1$.
We want to check that this invariant is multiplicative, in which case it is clear that it provides an inverse to $f \mapsto \cC^f$. We first check that the loop and the theta are nonzero. The single strand in $\cC_2$ must be nonzero, because if it were zero then all nonempty diagrams would be zero. Since $\dim
\cC_2 = 1$, we see that any diagram in $\cC_2$ is a multiple of the single strand, hence nondegeneracy says that the inner product of the strand with itself is nonzero, hence the loop value is nonzero. Similarly, by considering $\cC_3$ we see that the theta value is nonzero. Next we want to prove the multiplicative properties. Each of these are similar, so we only prove (2). We have that
(0,-.5)–(0,.5); (X) at (0,0) [X]{};
is some multiple of the single strand, so we see that $\begin{tikzpicture}[baseline=-1mm]
\draw (0,-.5)--(0,.5);
\node[circle,draw,dotted, fill=white] (X) at (0,0) {X};
\end{tikzpicture}
=
\left( \begin{tikzpicture}[baseline=-1mm]
\node[circle,draw,dotted] (X) at (0,0) {X};
\draw (X.north east) .. controls (.5,.4) and (.7,.2).. (.7,0) .. controls (.7,-.2) and (.5,-.4) .. (X.south east);
\end{tikzpicture}
/
\begin{tikzpicture}[baseline=-1mm]
\draw (0,0) circle (.3cm);
\end{tikzpicture}
\right)
\cdot
\begin{tikzpicture}[baseline=-1mm]
\draw (0,-.5)--(0,.5);
\end{tikzpicture}
$ (by pairing with the strand). Substituting this into the LHS of (2) gives the RHS.
Polynomials appearing in determinants
=====================================
This appendix contains some of the irreducible factors of determinants appearing in this paper. The other irreducible factors, which are very large, are contained in text files packaged with the [arXiv]{} source of this paper, and described here. Each polynomial is named as $Q_{i,j}$, where $i$ is the largest exponent of $d$ and $j$ is the largest exponent of $t$. Where two polynomials have the same pair of largest exponents, we name them with an additional character in the subscript, as in $Q_{2,4,a}$ and $Q_{2,4,b}$.
$$\begin{aligned}
P_{SO(3)} & = d (t-1)-t+2 \\\displaybreak[1]
P_{ABA} & = t^2-t-1 \\\displaybreak[1]
P_{G_2} & = d^2 t^5+d \left(2 t^5-4 t^4-t^3+6 t^2+4 t+1\right)+t^5-4 t^4+t^3+7 t^2-2 \\\displaybreak[1]
Q_{0,1} & = t+1 \\\displaybreak[1]
Q_{1,1} & = d (t+1)+t \\\displaybreak[1]
Q_{1,2} & = d \left(2 t^2+2 t+1\right)+3 t^2-2 \\\displaybreak[1]
Q_{2,3} & = d^2 \left(t^3+t^2-2 t-1\right)+d \left(2 t^3-2 t^2+t\right)+t^3-3 t^2+t+4 \\\displaybreak[1]
Q_{3,4} & = d^3 \left(t^4+3 t^3-t^2-3 t-1\right)+d^2 \left(2 t^4+t^2+2 t+1\right)+ \\
& \qquad + d \left(t^4-3 t^3+3 t^2+6 t+1\right)-t^2+2 t+2 \displaybreak[1]\\
Q_{3,5} & = d^3 \left(3 t^5+4 t^4-2 t^3-6 t^2-4 t-1\right)+d^2 \left(8 t^5+2 t^4-11 t^3-5 t^2+5 t+3\right)+\\
& \qquad + d \left(7 t^5-6 t^4-6 t^3+7 t^2+3 t-1\right)+2 t^5-4 t^4+t^3+5 t^2-2 t-2 \displaybreak[1]\\
Q_{2,4,a} & = d^2 \left(t^4-t^3-4 t^2-3 t-1\right)+d \left(2 t^4-6 t^3-7 t^2+t+3\right)+t^4-5 t^3+t^2+2 t-2 \\\displaybreak[1]
Q_{2,4,b} & = d^2 \left(t^4+2 t^3-t^2-2 t-1\right)+d \left(2 t^4-2 t^3-2 t^2+3 t+4\right)+t^4-4 t^3+5 t^2+2 t-4 \\\displaybreak[1]
Q_{4,5} & = d^4 t^5+d^3 \left(3 t^5-3 t^4-3 t^3+7 t^2+5 t+1\right)+d^2 \left(3 t^5-5 t^4-5 t^3+10 t^2+12 t+2\right)+\\
& \qquad + d \left(t^5-t^4-5 t^3+3 t^2+9 t+5\right)+t^4-3 t^3+4 t+1 \displaybreak[1]\\
Q_{6,9} & = d^6 \left(4 t^8+t^7-15 t^6-20 t^5-6 t^4+8 t^3+10 t^2+5 t+1\right)+\\
& \qquad + d^5 \left(2 t^9+12 t^8-19 t^7-54 t^6-17 t^5+21 t^4-11 t^3-43 t^2-30 t-7\right)+\\
& \qquad + d^4 \left(6 t^9-6 t^8-31 t^7+11 t^6-119 t^4-130 t^3-21 t^2+35 t+14\right)+\\
& \qquad + d^3 \left(2 t^9-32 t^8+72 t^7+59 t^6-227 t^5-258 t^4+59 t^3+164 t^2+43 t-3\right)+\\
& \qquad + d^2 \left(-10 t^9+10 t^8+123 t^7-136 t^6-305 t^5+103 t^4+225 t^3+23 t^2-38 t-13\right)+\\
& \qquad + d \left(-12 t^9+56 t^8-9 t^7-149 t^6-16 t^5+175 t^4+46 t^3-89 t^2-17 t+16\right) + \\
& \qquad -4 t^9+28 t^8-49 t^7-4 t^6+69 t^5-54 t^4-9 t^3+54 t^2-14 t-20 \displaybreak[1]\\
Q_{\omega,9} & = d^9-7 d^8+15 d^7-2 d^6-14 d^5-16 d^4+41 d^3-23 d^2+d+5 \displaybreak[1]\\
Q_{\omega,60} & = d^{60}-42 d^{59}+825 d^{58}-10050 d^{57}+84827 d^{56}-524435 d^{55}+2444075 d^{54}\displaybreak[1]\\
&\qquad -8680920 d^{53}+23364055 d^{52}-46267136 d^{51}+62172868 d^{50}\displaybreak[1]\\
&\qquad -43026307 d^{49}-10724689 d^{48}+19327948 d^{47}+113757871 d^{46}\displaybreak[1]\\
&\qquad -289556454 d^{45}+161677043 d^{44}+403173198 d^{43}-822414523 d^{42}\displaybreak[1]\\
&\qquad +340360209 d^{41}+658154819 d^{40}-734499791 d^{39}-499750302 d^{38}\displaybreak[1]\\
&\qquad +1417408819 d^{37}-680996389 d^{36}-701113119 d^{35}+1161482902 d^{34}\displaybreak[1]\\
&\qquad -934417344 d^{33}+751648667 d^{32}-23523738 d^{31}-1359642298 d^{30}\displaybreak[1]\\
&\qquad +1528218917 d^{29}+342409869 d^{28}-1836361788 d^{27}+946900947 d^{26}\displaybreak[1]\\
&\qquad +763927401 d^{25}-1172104767 d^{24}+652553812 d^{23}-193252562 d^{22}\displaybreak[1]\\
&\qquad -352541742 d^{21}+857069723 d^{20}-561108191 d^{19}-289399926 d^{18}\displaybreak[1]\\
&\qquad +602082003 d^{17}-186224613 d^{16}-206339296 d^{15}+185432097 d^{14}\displaybreak[1]\\
&\qquad -10906225 d^{13}-54265030 d^{12}+26840191 d^{11}+547786 d^{10}\displaybreak[1]\\
&\qquad -5118901 d^9+1967134 d^8-218389 d^7-37050 d^6+47054 d^5\displaybreak[1]\\
&\qquad -35063 d^4+10325 d^3-903 d^2-49 d+7 \displaybreak[1]\\
$$
The other factors, $Q_{7,11}, Q_{8,12}, Q_{11,19}, Q_{21,33}, Q_{22,36}, Q_{51,69}, Q_{54,78}$, and $Q_{36,60}$ are available in LaTeX and [Mathematica]{} formats in the [polynomials/]{} subdirectory of the [arXiv]{} source as files [Q\_i,j.tex]{} and [Q\_i,j.m]{}, and also in the [Mathematica]{} notebook
[^1]: Indeed, N.S. initially did such calculations by hand. Due to human error this initial version missed the ABA case, but the error was easily caught by the more reliable computer.
[^2]: In [@1202.4396] the point $(d,t)=(-1,3/2)$ was not discussed, since in the subfactor context $d>0$.
[^3]: Although the computer alerted us to the existence of these relations, we actually computed them by hand, since it is difficult to read off from our computer program where the dots belong. This by-hand calculation following [@MR1265145] took two people-days.
|
---
address:
- |
Cavendish Laboratory, University of Cambridge[^1], Madingley Road,\
Cambridge, CB3 0HE, U.K.
- 'Department of Physics, University of Durham, Durham, DH1 3LE, U.K.'
- 'Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, U.K.'
author:
- 'R.S. Thorne'
- 'A.D. Martin and W.J. Stirling'
- 'R.G. Roberts'
title: MRST Global Fit Update
---
There has recently been a great deal of updated data which help to determine parton distributions, particularly on small $x$ structure functions from HERA[@H1; @ZEUS] and on inclusive jets from the Tevatron[@D0; @CDF]. In both cases these new data are both more precise and extend the kinematic range, and thus determine the parton distributions more accurately than ever before. The MRST global fit is performed by minimizing the $\chi^2$ for all data adding statistical and systematic errors in quadrature (this procedure will be improved before producing finalized distributions), except for the jet data where the dominant correlated errors render a proper treatment absolutely essential. The evolution begins at $Q_0^2=1\Gev^2$, and data are cut for $Q^2 < 2 \Gev^2$ and $W^2 < 10 \Gev^2$ in order to exclude regions where higher twist and/or higher orders in $\alpha_S$ are expected to play an important role.
The main effect of the new data is to constrain the gluon distribution, and hence, via evolution the small $x$ quark distributions. The jet data are important in determining the high $x$ gluon. Previously[@MRST98; @MRST99] MRST used prompt photon data to perform this role, but it has become clear that there are both large theoretical uncertainties and conflicts between data sets, and we now drop this approach. The best fit to jet data produces a high $x$ gluon in between our previous $g$ and $g \uparrow$ distributions, i.e. $xg(x) \sim (1-x)^{5.5}$. If the gluon becomes smaller at high $x$ the fit to the jets becomes unacceptable. If it becomes larger, then although the jet data are initially supportive, the fit to DIS data deteriorates dramatically. Hence, there is an uncertainty of $\sim 20 \%$ at $Q^2 = 20 \Gev^2$, $x=0.4$.
\[fig:mrst2001\]
The improved HERA data also significantly affect the gluon. In order to obtain an acceptable fit the parameterization at $Q_0^2$ is extended to the form $$xg(x,Q_0^2)=A_g(1-x)^{\eta_g}(1+\epsilon_gx^{0.5}+\gamma_g x)x^{\delta_g}
-A_-(1-x)^{\eta_-}x^{-\delta_-},
\label{eq:gluon}$$ which allows the gluon to become negative at small $x$. $\eta_-$ is fixed at $\sim 10$ so that the large $x$ form is unchanged, and $\delta_- \sim 0.2$ - the gluon becoming quite negative at the smallest $x$. It becomes positive for all $x>10^{-5}$ for $Q^2 > 2-3 \Gev^2$. A good fit is obtained for the HERA structure function data as seen in fig. 1. Compared to the last MRST fit $F_2(x,Q^2)$ is flatter in $Q^2$ at the smallest $x$, but steeper at $0.05 < x < 0.0004$. In this intermediate range the data would prefer an even higher $dF_2(x,Q^2)/d\ln Q^2$. Also, we find that the jet and DIS data push in opposite directions for $\alpha_S(M_Z^2)$. Once the high $x$ constraint on the gluon is imposed, the DIS data prefer $\alpha_S(M_Z^2)=0.121$ while the jet data prefer $\alpha_S(M_Z^2)=0.117$, a compromise of $0.119$ being reached.
In order to improve the analysis of physical quantities sensitive to structure functions and make truly quantitative predictions there are a number of issues to address. Many important ones are theoretical, such as higher orders, resummation of $\ln(1-x)$ and $\ln(1/x)$ terms, higher twist, etc.. However, a direct issue which has been a recent focus of attention is the uncertainty due to the experimental errors on current data. Rather than obtain parton distributions with errors[@Botje; @CTEQH; @Gielea], we follow our original suggestion[@MRST99] and look at the error on a physical quantity determined by the parton distributions[@CTEQL; @Gieleb], in practice the $W$ cross-section at hadron colliders.
We study $\sigma_W$ for the Tevatron and LHC, which probes mainly the quarks in the region $0.5 < x < 0.005$ for the former and $0.5 <x< 0.00007$ for the latter. For our best fit we find $$\sigma_W({\rm Tev}) = 22.3nb, \qquad \sigma_W({\rm LHC}) = 192nb.
\label{eq:sigma}$$ In our previous study we examined the variation in $\sigma_W$ due to variation in normalization of data, $\alpha_S(M_Z^2)$, form of high $x$ gluon etc, rather than the uncertainty on the data. In order to do this the best way to proceed is to perform the global fit whilst constraining the value of the quantity in question, i.e. to use the Lagrange multiplier method[@CTEQL]. However, this previous study[@CTEQL] produced a large value for the uncertainty in $\sigma_W$ of order $5-8\%$. We have now performed an analogous study using the most up-to-date data, and also using a different method of determining the limits of an unacceptable fit. We impose the rough criterion that no data set has a less than $1\%$ confidence level. This leads to an estimated uncertainty on $\sigma_W$ of about $\pm 2\%$. In short, for the upper limit on $\sigma_W({\rm LHC})$ the fit to H1 data fails, for the lower limit on $\sigma_W({\rm LHC})$ the fit to ZEUS data fails, for the upper limit on $\sigma_W({\rm Tev})$ the fit to NMC proton data fails and for the lower limit on $\sigma_W({\rm Tev})$ the fit to NMC $n/p$ data fails. Often at least one other data set has a deteriorating quality of fit and the increase in global $\chi^2$ is about 80. A roughly symmetric deterioration in the fit quality to CCFR, for example, is shown in fig. 2.
\[fig:ccfr\]
We conclude that the uncertainty in $\sigma_W$, and similarly $\sigma_Z$, due to experimental errors is rather small. We can repeat the procedure for a variety of other observables, e.g. Higgs production will be directly sensitive to the small $x$ gluon rather than small $x$ quarks, and the uncertainty due to experimental errors will obviously be largest for quantities sensitive to the large $x$ gluon. However, we also need to consider the uncertainty due to theoretical errors. As was shown in [@MRSTNNLO], the perturbative series seems fairly convergent for quark sensitive processes (though an improvement is perhaps needed in $dF_2(x,Q^2)/d\ln Q^2$) probably because it is the quarks that are directly tied down by experiment at present. However, the expansion is less reliable for gluon sensitive quantities - especially in the region where resummations are expected to be important. Hence, although we will continue detailed work into uncertainties due to experimental errors, we believe that the great precision of data now leads to the theoretical errors being dominant in most cases, and suggest that this should be a major area of study in the immediate future.
[99]{}
H1 collaboration: C. Adloff, [*et al.*]{}, [*Eur. Phys. J.*]{} C [**13**]{}, 609 (2000);\
H1 collaboration: C. Adloff, [*et al.*]{}, [hep-ex/0012052]{};\
H1 collaboration: C. Adloff, [*et al.*]{}, [hep-ex/0012053]{}.
ZEUS collaboration: S. Chekanov [*et al.*]{}, [hep-ex/0105090]{}.
D0 collaboration: B. Abbott [*et al.*]{}, .
CDF collaboration: T. Affolder [*et al*]{}, [hep-ex/0102074]{}.
A.D. Martin [*et al.*]{}, [*Eur. Phys. J.*]{} C [**4**]{}, 463 (1998). A.D. Martin [*et al.*]{}, [*Eur. Phys. J.*]{} C [**14**]{}, 133 (2000).
M. Botje, [*Eur. Phys. J.*]{} C [**14**]{}, 285 (2000).
J. Pumplin [*et al.*]{}, [hep-ph/0101032]{}.
W.T. Giele, S.A. Keller and D.A. Kosower, [hep-ph/0104052]{}.
D. Stump [*et al.*]{}, [hep-ph/0101051]{}.
W.T. Giele, S.A. Keller, [hep-ph/0104053]{}.
A.D. Martin [*et al.*]{}, [*Eur. Phys. J.*]{} C [**18**]{}, 117 (2000).
[^1]: Royal Society University Research Fellow
|
---
abstract: |
We study an one-dimensional stochastic model of vehicular traffic on open segments of a single-lane road of finite size $L$. The vehicles obey a stochastic discrete-time dynamics which is a limiting case of the generalized Totally Asymmetric Simple Exclusion Process. This dynamics has been previously used by Bunzarova and Pesheva \[Phys. Rev. E 95, 052105 (2017)\] for an one-dimensional model of irreversible aggregation. The model was shown to have three stationary phases: a many-particle one, MP, a phase with completely filled configuration, CF, and a boundary perturbed MP+CF phase, depending on the values of the particle injection ($\alpha$), ejection ($\beta$) and hopping ($p$) probabilities.
Here we extend the results for the stationary properties of the MP+CF phase, by deriving exact expressions for the local density at the first site of the chain and the probability P(1) of a completely jammed configuration. The unusual phase transition, characterized by jumps in both the bulk density and the current (in the thermodynamic limit), as $\alpha$ crosses the boundary $\alpha =p$ from the MP to the CF phase, is explained by the finite-size behavior of P(1). By using a random walk theory, we find that, when $\alpha$ approaches from below the boundary $\alpha =p$, three different regimes appear, as the size $L\rightarrow \infty$: (i) the lifetime of the gap between the rightmost clusters is of the order $O(L)$ in the MP phase; (ii) small jams, separated by gaps with lifetime $O(1)$, exist in the MP+CF phase close to the left chain boundary; and (iii) when $\beta =p$, the jams are divided by gaps with lifetime of the order $O(L^{1/2})$. These results are supported by extensive Monte Carlo calculations.
address:
- 'Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia'
- 'Institute of Mechanics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria'
author:
- 'J. G. Brankov'
- 'N. Zh. Bunzarova'
- 'N. C. Pesheva'
- 'V. B. Priezzhev'
title: A model of irreversible jam formation in dense traffic
---
Non-equilibrium phenomena, One-dimensional stochastic process, Stationary states, Congested traffic, Irreversible jam formation
Introduction
============
Our aim here is to study a discrete-time discrete-space model of congested traffic on finite segments of a single-lane road, which allows a detailed description of the growth of clusters of jammed vehicles under different boundary conditions. We assume that the vehicles (particles) obey a stochastic dynamics with the following properties: (a) Existing clusters of jammed vehicles do not break into parts during their motion along the selected segment of the road; (b) With given probability, such clusters are translated as a whole entity one site to the right, provided the target site is empty, in the same way as isolated vehicles do; (c) Any two vehicles or clusters of vehicles, occupying consecutive positions on the chain, may become nearest-neighbors and merge irreversibly into a single cluster. Here we note that property (b) is characteristic of the synchronized vehicle movement in high density traffic. Due to the property (c), the size of the moving jam grows monotonically with time until it spreads on the whole road segment, or leaves the system before that happens. Properties (a) and (b) correspond to the case when the drivers in a jam closely follow the vehicles in front of them, thus keeping the minimum safe distance between the vehicles. This is a simplified picture of synchronously moving vehicles in a jam.
It is the balance between the typical time interval separating the appearance of gaps at the entrance of the road, and the average portion of time that a gap between consecutive clusters persists, that determines the stationary probability P(1) of a completely jammed configuration of the system. This dynamics has already been used in a one-dimensional model of irreversible aggregation [@BP16]. The model was shown to have three stationary phases: a many-particle one, MP, a phase with completely filled configuration, CF, and a mixed MP+CF phase. The term ’cluster’ is used to denote successively occupied sites. The evolution of clusters in continuous models was described in the review by Mahnke et al [@MKL05].
Various generalizations of the totally asymmetric simple exclusion process (TASEP) have been used to model traffic flow with a special interest in the formation and spreading of traffic jams, see, e.g. the reviews [@N96; @CSS00; @H01]. The jams are known to occur either spontaneously in congested traffic, or due to different inhomogeneities on the road. In this context, the dynamics of the clusters of jammed vehicles is of primary interest. The problem is of interest also for various transport processes taking place in the living cells. From the viewpoint of physics, molecular motors are proteins or macromolecular complexes that utilize chemical energy to move collectively on a single track in a way resembling vehicular traffic [@TC08]. Detailed models of such motor traffic systems describe the mechanochemistry of individual motors with the aim to investigate their collective dynamics [@GGNSC07]. Advances of experimental methods have allowed to follow the transport of individual molecules through nanochannels. The effect of crowding of particles in such channels in an experimentally relevant non-equilibrium steady state regime was analyzed in [@ZPB09]. A method for analysis of one-dimensional cellular automata models, based on the cluster-size distribution was proposed by Ezaki and Nishinari [@EN12]. A comprehensive presentation of traffic flow dynamics and modeling, including microscopic many-particle models, can be found in the book [@TK13].
The model we study here is designed to be simple rather than realistic. In the framework of particle hopping models of traffic, we model jams of vehicles which closely follow the motion of the leading vehicle.
We note that the random-sequential, forward-ordered, sublattice-parallel, and parallel updates of the TASEP configurations, for the definitions see [@RSSS98], violate the integrity of the existing clusters of nearest neighbor particles with nonvanishing probability. The only dynamics that can move forward clusters as a whole entity is the backward-ordered one. However, the probability for translation of a cluster of $k$ particles one site to the right is $p^k$, while such a cluster is broken into two parts with probability $p - p^k$. Thus, in a stationary state of the model on an infinite chain, a specific cluster-size distribution with finite mean cluster size takes place. Remarkably, different types of cluster statistics may appear depending on the dynamic rules and the boundary conditions. For example, exact and asymptotic results on the cluster statistics generated by TASEP on finite rings, with two different dynamics, were obtained in [@PM01] by mapping on the zero range process. The results obtained for the standard TASEP show that in the limit of infinite system size, the mean cluster size converges to a finite value, depending on the particle density, while the expectation value of the size of the longest cluster diverges logarithmically with the number of sites on the infinite ring. Since in our model the cluster growth is unlimited on the infinite chain, we consider the stationary cluster size distributions appearing on finite open chains. At that, we focus on the probability $P(1)$ of a completely jammed configuration on segments containing large number of sites $L \gg 1$.
The paper is organized in five sections. In Section 2 we formulate the model, present the phase diagram and those properties which were obtained in [@BP16]. In Section 3 we derive some new results about the local density of cars at the ends of the chain. Here the role of the probability P(1) of finding completely jammed configurations on the road segment is elucidated and an exact (in the thermodynamic limit) analytic expression for P(1) is derived. A central result of the paper is the theoretical establishment of three regimes of asymptotic growth of P(1) on approaching the threshold of complete jamming. A discussion of the results and some perspectives for further applications are given in the concluding Section 4.
The Model
=========
Actually, our model is a special case of a more general TASEP kinetics. First it was promoted as an exactly solvable generalization of the TASEP on a ring by Wölki in 2005 [@W05]. We consider it on an open chain of $L$ sites, labeled consecutively by the index $i = 1,2,\dots ,L$. Each site of the lattice can be empty or occupied by just one particle.
![(Color online) A sketch of the algorithm for update of the gTASEP configurations, illustrated for three consecutive updates. The bulk hopping rules imply that a single particle, as well as a whole cluster of particles, hops one site to the right with probability $p$, provided the target site was vacant. Such a hop is shown by black curved arrow with $p$ above it. Red dashed arrows with $\tilde{\rm p}$ above it illustrate a hop with the modified probability $\tilde{p}$ (see text). Our model corresponds to the limit case of $\tilde{p} = 1$. The left boundary condition also depends on whether the first site was vacant (black solid arrow with $\alpha$ above it) or it was occupied but became vacant during the backward update (red dashed arrow with $\tilde{\alpha}$ above it).[]{data-label="Sketch"}](Fig1.eps){width="100mm"}
The dynamics of the model corresponds to the discrete-time backward-ordered update with probabilities $p$ and $\tilde{p}$ defined as follows. A particle can hop to a vacant nearest-neighbor site in the forward direction, or stay at its place. During each moment of time $t$, the probability of a hop along the bond $(i,i+1)$ depends on whether a particle has jumped from site $i+1$ to site $i+2$ in the previous step, when the bond $(i+1,i+2)$ was updated, or not. The update of the whole configuration comprises the following steps.
\(1) When the last site $L$ is occupied, the particle is removed from it with probability $\beta$, and stays in place with probability $1-\beta$.
\(2) Next, the configuration of the system passes through $L-1$ consecutive updates of all the pairs of nearest-neighbor sites in the backward order $(L-1,L), \dots, (i,i+1),\dots, (1,2)$. If site $i+1$ remains empty after the update of $(i+1,i+2)$, then the jump of a particle from site $i$ to site $i+1$ takes place with probability $p$, and the particle stays immobile with probability $1-p$; if site $i+1$ remains occupied, no jump takes place and the configuration of the bond $(i,i+1)$ is conserved. If in the previous step a particle has jumped from site $i+1$ to site $i+2$, thus leaving $i+1$ empty, then the jump of a particle from site $i$ to site $i+1$ in the next step takes place with a different probability $\tilde{p}$, and the particle stays immobile with probability $1-\tilde{p}$.
\(3) Finally, the first site is updated by applying the following boundary condition: a particle is injected at the first site of the chain with probability $\alpha >0$, if the site was vacant at that moment of time, or with probability $\tilde{\alpha} = \min\{\alpha \tilde{p}/p, 1\}$, if the site was initially occupied but became vacant after the update of the bond $(1,2)$ at the same moment of time.
The different possible updates are shown schematically in Fig. \[Sketch\] for three consecutive moments of time, along with the corresponding occurrence probability. Note that when $\tilde{p} = p$ one has the standard TASEP with backward-sequential update, and when $\tilde{p} = 0$ one has the TASEP with parallel update. In the general case of $0<\tilde{p}<1$, the model was studied on a ring in [@DPPP; @PhD; @DPP15; @AB16].
We believe that even simple particle hopping models, such as gTASEP, can be useful in describing essential features of real traffic flow [@TK13]. For example, the second hopping rate $\tilde{p}$ describes a kind of kinematic attraction ($\tilde{p} > p$), or repulsion ($\tilde{p} < p$), between the vehicles. The case of attraction reflects the natural tendency of the driver to catch up with the car ahead. Thus, clusters of synchronously moving cars appear. Under the gTASEP dynamics, a cluster of $k$ cars is translated by one site as a whole entity with probability $p\tilde{p}^{k-1}$, and is broken into two parts with probability $p(1-\tilde{p}^{k-1})$. Obviously, the case of stochastic translation of clusters as a whole, combined with their irreversible growth, can be described by the gTASEP in the limit $\tilde{p} =1$. At that, clusters of cars translate by one site forward with the same probability $p$ with which single vehicles move.
A special attention should be paid to the interpretation of the open boundary conditions at the ends of a finite segment of the road. We can suggest one to consider the endpoints of a portion of a single-lane road as toll pay points, operating independently with different efficiency, say, proportional to $\tilde{\alpha}$ at the entrance, and to $\beta$ at the exit of the considered road segment. We remind the reader that the introduction of the $\tilde{p}/p$-dependent injection probability $\tilde{\alpha}$, instead of simply $\alpha$, is necessary for consistency with the special cases of backward-ordered sequential algorithm ($\tilde{p}=p$) and the parallel one ($\tilde{p}=0$). This left boundary condition has been introduced first by Hrab[á]{}k in [@PhD].
As shown in [@BP16], the model with irreversible aggregation (jam formation) has three stationary phases: a many-particle one, MP, a phase with completely filled configuration, CF, and a mixed MP+CF phase, see Fig. \[PhaseDiag\].
![(Color online) Phase diagram in the plane of injection ($\alpha$) - ejection ($\beta$) probabilities. The many-particle phase MP occupies two regions, MP I and MP II; it contains a macroscopic number of single cars or clusters of jammed cars of size $O$(1) as $L\rightarrow \infty$; MP I and MP II differ only by the shape of the local density profile. The configurations of the boundary perturbed phase MP+CF contain a macroscopic number of cars and small jams of size $O$(1) close to the left end of the chain, or a single jam completely filling the whole road. The stationary non-equilibrium phase CF consists of a completely jammed configuration and carries the current $J = \beta$. The small circle with the letter T denotes the triple point. The unusual phase transition, discussed in Ref. [@BP16], takes place across the boundary $\alpha = p$ between the MP I and CF phases.[]{data-label="PhaseDiag"}](Fig2.eps){width="80mm"}
Note that in the case of $\tilde{p} = 1$, properties (a) and (b) are completely fulfilled. Property (c) is related to the *symmetric* random walk behavior of the gap between two consecutive clusters *in the bulk* of the chain: at each time step it decreases or increases with the same probability $p(1-p)$ and does not change with probability $r = 1-2p(1-p)$. However, when a cluster reaches the last site of the chain, it starts hopping forward with probability $\beta$, instead of $p$. Then the random walk behavior of the gap between this cluster and the one on the left of it changes from symmetric to asymmetric one.
Local density
-------------
From the viewpoint of traffic, it is important to know the magnitude of the flow, and the car density both at the entrance and exit of the road. In the completely jammed phase CF, the car density profile is completely flat, $\rho_i \equiv 1$, $i = 1,2,\dots,L$, and the current depends trivially on the ejection probability, $J = \beta$. The local densities at the ends of the road were derived in [@BP16] only for the MP phase ($\alpha <p$ and $\beta >\alpha$), where the profile is flat from the first site up to the bulk, $\rho_1^{\rm MP} = \rho_{\rm b}^{\rm MP}$. From the local balance of the average inflow and outflow of particles at the first site of the chain, the result $\rho_1^{\rm MP} = \alpha/p$ was obtained; by using the global stationarity condition it was found in addition that $\rho_L^{\rm MP} = \alpha/\beta$. Thus, on the line $\beta = p$ between the subregions MPI and MPII the density profile is completely flat; it bends downwards in MPI, where $\beta >p$, and upwards in MPII, where the opposite inequality $\beta < p$ holds.
In the phase MP+CF ($\beta < \alpha < p$) the road is completely jammed from the bulk up to the last site, $\rho_{\rm b}^{\rm MP+CF} =\rho_L^{\rm MP+CF} =1$. The particle current is given by $J = \beta$, as in the pure phase CF. However, the road near the entrance is not completely jammed and the local density $\rho_1^{\rm MP+CF}$ decreases with increasing $\beta$ down to $\rho_1^{\rm MP+CF}=\alpha/p$ at $\beta =\alpha$. Here we prove the result $$\rho_1^{\rm MP+CF}=1 - (1/\alpha -1/p)\beta,
\label{ro1}$$ which completes the description of the local density profiles.
The derivation of Eq. (\[ro1\]), despite its simplicity, needs some reflection. Consider the input-output balance of particles at the first site. Clearly, the probability that the empty site $i=1$ will become occupied by a particle is $\alpha (1- \rho_1)$. The problem is that the probability with which the occupied first site will become empty at the end of an update depends on whether the configuration is completely jammed, or not. In the former case, which takes place with probability P(1), the particle at the first site belongs to the cluster occupying all the sites in the chain, including the exit one $i=L$. Hence, the site $i=1$ will become vacant with probability $\beta$, when the whole cluster moves one site to the right, and will remain vacant with probability $(1-\tilde{\alpha})$ at the end of that update. On the other hand, with probability 1 - P(1) the particle at the first site is isolated, or belongs to a cluster which is separated by vacant site(s) from the end of the chain. In this case, the first site becomes vacant with probability $p$, with which the cluster it belongs to moves one site to the right, and remains vacant again with probability $(1-\tilde{\alpha})$. Thus, the particle balance at the first site cannot be satisfied during each single configuration update.
To solve the problem, we take into account that the stationary current $J_{i,i+1}$ through a bond $(i,i+1)$ is constant along the chain, $J_{1,2}= J_{2,3}=\dots = J_{L-1,L} = J_{\rm out}$, where $J_{\rm out} =\beta \rho_L$ is the current of particles leaving the chain. Since in our case $\rho_L =1$, the global stationarity of the occupation number at the first site requires the equality $\alpha (1- \rho_1)= (1-\tilde{\alpha})\beta$, which implies the result (\[ro1\]).
Probability of complete jamming
-------------------------------
An expression for the probability P(1) of completely jammed configurations of the system in the mixed MP+CF phase can be derived from the global stationarity condition $J_{\rm in} = J_{\rm out}$, where $J_{\rm in}$ ($J_{\rm out}$) is the input (output) current. A particle may enter the system in two cases: (i) If the system is in a completely jammed configuration, which happens with probability P(1), the first site becomes empty with probability $\beta$, as a result a particle ejection from the last site of the chain and a deterministic shift of the whole cluster one site to the right; then a particle may enter the system with probability $\tilde{\alpha} = \alpha/p$. The total probability of this event is P(1)$(\alpha/p)\beta$. (ii) When the system is not completely jammed, which takes place with probability 1-P(1), a particle enters at the first site with probability $\alpha$, as it was proven in [@BP16]. By equating the input current $$J_{\rm in}= P(1)\frac{\alpha \beta}{p} + [1-P(1)]\alpha
\label{inputprob}$$ to the output current $J_{\rm out}= \beta$ (we recall that $\rho_L^{\rm MP+CF} =1$), one solves for P(1) and obtains: $$P(1) = \frac{p(\alpha - \beta)}{\alpha (p-\beta)}, \qquad \beta \le \alpha <p.
\label{P1Rational}$$
This result is in excellent agreement with the Monte Carlo simulation data, see Fig. \[P1betaFig\].
![(Color online) Comparison of the analytic result (\[P1Rational\]), solid red line, as a function of the ejection probability $\beta$, to the computer simulation, shown by empty blue squares, for a lattice of size $L=400$, injection probability $\alpha = 0.5$ and hopping probability $p = 0.6$. \[P1betaFig\]](Fig3.eps){width="100mm"}
Some details of our Monte Carlo simulation techniques are given in the next subsection.
Details of the Monte Carlo simulations
--------------------------------------
The configuration update rules, described at the beginning of the section, were implemented in a code for Monte Carlo simulations, written in Fortran 90. The code was ran on a personal computer with an Intel Core i5-3350P 3.1 GHz processor and 8 GB RAM. The standard generator of pseudo-random numbers ran2 was used [@NRec96].
Typically, each data point was obtained by averaging over 100 independent temporal series (runs) of length $4\times 10^6$ configuration updates for chains of length 400 sites. In every time run, $2\times 10^6$ lattice updates were omitted before the data collection starts. Beforehand, tests were made, starting from different initial configurations (e.g., a completely empty lattice or with a given initial density), and averaging over time series of different length. The time behavior of the bulk density was recorded, and the number of runs, their length and the number of omitted updates between them, were chosen as to ensure that the system reaches a stationary state with acceptable accuracy. Stationarity was controlled also by monitoring the input and output currents: they were allowed to differ by less than $10^{-5}$ in absolute units.
To assess the finite-size effects we have studied lattice sizes from $L= 200$ to $L= 1200$. Optimal balance between CPU time and relative accuracy within $10^{-3}$ for local particle densities, and $10^{-4}$ for global ones (bulk density and current) was achieved for chains with $L=400$ sites.
Random walk theory
==================
Evolution of the system configurations
--------------------------------------
In the particle hopping discrete-space traffic models, one often uses the dual representation of configurations in terms of empty sites positions, instead of particle coordinates. In the case of our model, such a representation leads to a very peculiar dynamics of the inter-cluster gaps. First, because existing clusters cannot break up, gaps may appear only at the first site of the chain. Second, gaps disappear when two consecutive clusters merge or when the rightmost cluster leaves the system and the one following it reaches the last site of the chain. Third, and most interesting feature of the gap dynamics is that, as long as two consecutive gaps exist, the distance between them remains constant. Indeed, because of particle conservation in the bulk, the number of particles in a cluster between two gaps remains fixed. Thus, in the space-time picture, the neighboring edges of two coexisting gaps are forced to move in parallel, see a computer illustration in Fig. \[Fig4\], and gaps may not cross or merge. Finally, the width of each gap performs a random walk, which begins with an initial state containing one or several neighboring empty sites, and ends up when the random walk reaches to the origin. If a configuration has two or more gaps, the random walks performed by their edges are not independent.
![\[Fig5\] Simulated time evolution of clusters and gaps (white strips), in the boundary perturbed MP+CF phase, at $L=200$, $\alpha =0.31$, $\beta =0.3$, and $p=0.6$.](Fig4.eps){width="40mm"}
![\[Fig5\] Simulated time evolution of clusters and gaps (white strips), in the boundary perturbed MP+CF phase, at $L=200$, $\alpha =0.31$, $\beta =0.3$, and $p=0.6$.](Fig5.eps){width="40mm"}
In general, when $\beta \not= p$, the size of the rightmost gap performs an *asymmetric* random walk: its right edge belongs to the cluster which extends to the exit site $i=L$, hence it moves one site to the right with the ejection probability $\beta$ and stays in place with probability $1-\beta$; on the other hand, its left edge moves one site to the right with probability $p$, or stays in place with probability $1-p$. Therefore, after each update, the gap width increases by one site with probability $p_g$, decreases by one site with probability $q_g$, and remains the same with probability $r$, where $$p_g = \beta (1-p),\quad q_g = p(1-\beta), \quad r = 1 - q_g - p_g = 1-\beta -p +2\beta p.
\label{assRW}$$
In contrast, the size of all gaps on the left of the rightmost one performs a *symmetric* random walk: both edges of each of these gaps execute a directed random walk with hopping probability $p$. Hence, for the corresponding gap width one has $$p_g = q_g =p(1-p),\quad r = 1- 2p(1-p).
\label{symRW}$$ However, since the inner edges of all pairs of coexisting consecutive gaps are forced to move synchronously, we have a set of *interacting random walks*: independent is only the random walk of the external edges: the right edge of the rightmost gap and the left edge of the leftmost gap.
We note that there is a close relationship between the asymmetry of the random walk, performed by the width of the rightmost gap, and the shape of the stationary local density profile in the MP phase. When $\beta >p$ ($\beta < p$) and the width of that gap increases (decreases), the stationary local density profile bends down (up) close to the chain end, see Fig. 3 in Ref. [@BP16].
A considerable simplification of this picture takes place when $\beta =p$: the widths of all the existing gaps perform symmetric random walks, starting from the corresponding initial conditions. In this case, the local density profile is completely flat.
In principle, the lifetime probability distribution for all the gaps can be calculated, provided the initial conditions are given. However, the problem of calculating the stationary probability P(1) of a completely filled configuration by averaging over the time evolution of the system remains hardly solvable.
Evolution of single-gapped configurations
-----------------------------------------
We can estimate analytically the probability P(1) of a completely filled configuration in the case when the appearance of a vacancy at the first site of a completely filled configuration is a rare event. That event happens when: (i) a particle of the completely filling cluster leaves the system from its right end, which takes place with probability $\beta$, and leads to the deterministic translation of all the remaining $L-1$ particles by one site to the right; (ii) the resulting vacancy of the first site is not immediately filled by a particle from the left reservoir, which takes place with probability $(1-\tilde{\alpha})\ll 1$. Thus, the appearance of a vacancy at the first site of a completely filled configuration will take, on the average, $$\bar{N} = [(1-\tilde{\alpha})\beta]^{-1}\gg 1
\label{Nc}$$ updates (discrete time steps).
In the course of time, the single vacancy may give rise to gaps of size one or more empty sites between the clusters of filled sites. Thus, we have to evaluate the average lifetime (in number of updates) $\bar{n}$ of the gap in the different asymptotic regimes. If $\bar{n}< \bar{N}$, an estimate of the probability $P(1)$ is given by the ratio $$P(1) \simeq (\bar{N} - \bar{n})/\bar{N}.
\label{P1est}$$
Growing gap
-----------
First, consider the case when the boundary $\alpha = p$ of the CF phase is approached from region MPI, i.e., at $\beta >p$. In this case $p_g > q_g$ and the gap asymptotically grows with the number $n\gg 1$ of time steps as $W(n) \propto (p_g - q_g)n$. The gap will exist until its lower edge reaches the end of the chain, that is for $L/p$ time steps on the average. The two edges of the gap coexist until the time moment $L/\beta$ when the upper edge reaches the end of the chain. According to the asymmetric random walk theory, the probability that a gap with unit initial width collapses, before the time moment $L/\beta$, tends to $q_g/p_g < 1$, as $L\rightarrow \infty$. Therefore, the probability that the gap, which has opened at site $i=1$, survives until the end of the chain is $$P_{surv}= 1-q_g/p_g =\frac{\beta-p}{\beta(1-p)}
\label{surv}$$ Then, a long gap, propagating through the whole chain, appears on the average after $$\bar{N_c}=\frac{\bar{N}}{P_{surv}}=\frac{p(1-p)}{(p-\alpha)(\beta-p)}
\label{long}$$ updates. Hence, expression (\[P1est\]) becomes $$P(1) \simeq (\bar{N_c} - \bar{n})/\bar{N_c}.
\label{P1ex}$$.
![(Color online) Data collapse of the computer simulation data for the probability P(1) of completely jammed configuration at $\beta = 0.9$, as a function of the finite-size scaling variable $x=L(p-\alpha)$ for different chain lengths $L$. The linear fit to the small $x$ asymptotic behavior for $L=800$ is shown by solid red line.[]{data-label="CollP109"}](Fig6.eps){width="110mm"}
The crucial assumption is that $\bar{n} = L/p \ll \bar{N_c}$, and no other long-living gap appears during the considered time interval $\bar{N_c}$. In such a case, from Eqs. (\[long\]) and (\[P1ex\]) we obtain the result, $$P(1) \simeq 1 - a_1 L(p-\alpha),
\label{P1upper}$$ where $$a_1 = \frac{\beta-p}{p^2(1-p)}
\label{a1}$$ The important result here is the appearance of the finite-size scaling variable $L(p-\alpha)$, which is confirmed by applying the data collapse method to the results of our computer simulations for different chain lengths $L$, see Fig. \[CollP109\]. Inserting $\beta = 0.9$ and $p = 0.6$ into (\[a1\]) we obtain $a_1\simeq 2.08$, which agrees very well with $a_1^{\rm sim} \simeq 2.063$ evaluated from Fig. \[CollP109\] for $L=400$.
Note that the neglect of short-living gaps implies that Eqs. (\[P1upper\]) and (\[a1\]) actually give an upper bound for P(1). The lower bound corresponds to $P_{surv}= 1$, which leads to $a_1 = \beta/p^2$.
Next, using the stationary condition $J_{\rm in}=J_{\rm out}$ and the value of $J_{\rm out}=\beta \rho_L$ at the end of the chain, we obtain from Eq. (\[inputprob\]), in the vicinity of $\alpha = p$, $$\rho_L = P(1)+ [1-P(1)]\frac{\alpha}{\beta}
\label{L}$$ and $$J=P(1)\beta + [1-P(1)]\alpha
\label{J}$$
![(Color online) Illustration of the change in the behavior of the current and the local particle density $\rho_L$ in the MP phase, when $\alpha$ increases, approaching the boundary $\alpha = p$ with the CF phase. Computer simulation data for $L=200$, $p=0.6$ and three values of $\alpha$ are shown. At $\alpha = 0.5$ one observes the usual first-order phase transition that takes place at $\beta = \alpha <p$. The unusual phase transition, discussed in Ref. [@BP16], takes place across the boundary $\alpha = p$ between the MP and CF phases.[]{data-label="RoJPout"}](Fig7.eps){width="110mm"}
Expressions (\[L\]) and (\[J\]) allow us to explain the so called ’unusual’ phase transition between the MP and CF phases, found in Ref. [@BP16]. We recall that this non-equilibrium ’zeroth-order’ phase transition at $\beta > p$ manifests itself by jumps both in the density $\rho_L(\alpha)$ and the current $J(\alpha)$ at the boundary $\alpha=p$, $\beta>p$, between the MPI and CF phases, see Fig. 7 in [@BP16]. This transition appears instead of the usual non-equilibrium second-order transition between the low-density (high-density) and the maximal current phases in the standard TASEP. The MPI phase is characterized, in general, by vanishing probability P(1) of completely jammed configurations. However, close to the boundary $\alpha = p$, $\beta>p$, the value of P(1) increases rapidly as $(p-\alpha)\rightarrow 0$ and the jumps (in the limit $L\rightarrow \infty$) in the $\alpha$-dependence of $\rho_L(\alpha)$ and $J(\alpha)$ are a direct consequence of Eqs. (\[L\]) and (\[J\]). The effect of the growing P(1) on the current and local particle density $\rho_L$, when $\alpha$ approaches the boundary between the MP and CF phases, is illustrated in Fig. \[RoJPout\].
Critical gap
------------
When the boundary $\alpha = p$ of the CF phase is approached along the line $\beta = p$, from Eq. (\[assRW\]) we obtain that the gap width performs a symmetric random walk with $p_g = q_g = p(1-p)$, $r = 1 -2p(1-p)$. The calculation of $\bar{n}$ in this case becomes involved, since the average length of a symmetric random walk on the infinite chain, with initial state at $i=1$ and one absorbing state at $i=0$, diverges. That is why, we have to take into account that the lifetime of the symmetric random walk on the finite chain is limited, on the average, by $L/p$. Let us introduce the average lifetime $\bar{n}_M$ of the gap during $M$ updates, $$\bar{n}_M = \sum_{m=1}^M m f_{1,0}^{(m)},$$ where $f_{1,0}^{(m)}$ is the probability that the initial gap of unit size will vanish at update $m$. By using the generating function of $f_{1,0}^{(m)}$, see Eq. (53) in Ref. [@CM01],
![(Color online) Data collapse of the computer simulation data for the probability P(1) of completely jammed configuration at $\beta = p = 0.6$, as a function of the finite-size scaling variable $x=L^{1/2}(p-\alpha)$ for different chain lengths $L$. The linear fit to the small $x$ asymptotic behavior for $L=1200$ is shown by solid red line.[]{data-label="CritGapColl"}](Fig8.eps){width="110mm"}
$$F_{1,0}(s) = \sum_{m}f_{1,0}^{(m)}s^m = \left\{1-r s - \left[(1-r s)^2 -4 p^2(1-p)^2 s^2\right]^{1/2}\right\}/[2p(1-p)s],$$
we obtain the exact result $$\bar{n}_M = \frac{p(1-p)}{r}\sum_{n=0}^{M/2-1}\left(\frac{p(1-p)}{r}\right)^{2n} \left(\begin{array}{c} 2n+1 \\ n \end{array}\right)
\sum_{k=2n+1}^M r^k \left(\begin{array}{c} k \\ 2n+1 \end{array}\right).$$ In the case of $M\gg 1$, we use the approximation $$\sum_{k=2n+1}^M r^k \left(\begin{array}{c} k \\ 2n+1 \end{array}\right)\simeq \sum_{k=2n+1}^{\infty} r^k \left(\begin{array}{c} k \\ 2n+1 \end{array}\right) = \frac{r^{2n+1}}{(1-r)^{2n+2}},$$ which leads to $$\bar{n}_M \simeq \frac{1}{4p(1-p)}\sum_{n=0}^{M/2}\frac{1}{4^{n}} \left(\begin{array}{c} 2n+1 \\ n \end{array}\right)\simeq \frac{1}{2\sqrt{\pi}p(1-p)}\int_0^{M/2} \frac{{\rm d} x}{\sqrt{x}} = \frac{\sqrt{M}}{\sqrt{2\pi}p(1-p)}.$$ Finally, by setting $M = L/p$, from Eqs. (\[Nc\]) and (\[P1est\]) at $\beta =p$, we obtain $$P(1) \simeq 1 - b_1 L^{1/2}(p-\alpha), \qquad b_1 = \frac{1}{\sqrt{2\pi p}p(1-p)}.
\label{P1Crit}$$ The important feature here is the appearance of the critical finite-size scaling variable $x=L^{1/2}(p-\alpha)$, which is confirmed by the results of our computer simulations, see Fig. \[CritGapColl\]. In spite of the crude approximations, the estimate of the constant $b_1$ is rather good: at $p = 0.6$ Eq. (\[P1Crit\]) yields $b_1 \simeq 2.15$, while the corresponding values obtained from the small $x$ asymptotic behavior shown in Fig. \[CritGapColl\] is $b_1^{\rm sim} \simeq 2.51$.
Short-living gap
----------------
When the boundary of the CF phase is approached from the mixed MP+CF phase, i.e., at $\beta < p$, one has $p_g < q_g$ and the gap between the newly growing cluster and the cluster which is leaving the system from its right boundary closes in a finite number of time steps. An upper estimate can be given by the result for a random walk on an infinite chain with initial state at site $i=1$ and one absorbing state at the origin $i=0$: $$\bar{n} < (q_g - p_g)^{-1} = (p-\beta)^{-1}.
\label{P1finite}$$ Obviously, in this case $P(1) \rightarrow 1$ as $\alpha \rightarrow p$ at $\beta < p$.
Discussion
==========
We have reformulated the one-dimensional model of irreversible aggregation of particles, proposed in Ref. [@BP16], as a model of irreversible jam formation in dense traffic on finite segments of a single-lane road. In the above reference, the stationary state values of the local density at the endpoints of the road segment were derived only in the MP phase. Here we have completed this study by deriving an exact expression for the stationary density at the first site of the chain in the mixed MP+CF phase, see Eq. (\[ro1\]). From the bulk up to the last site, the road segment is completely jammed, $\rho_{\rm b}^{\rm MP+CF} =\rho_L^{\rm MP+CF} =1$. We have also derived an exact expression for the probability P(1) of complete jamming in the mixed MP+CF phase, see Eq. (\[P1Rational\]) and Fig. \[P1betaFig\]. The appearance of the unusual ’zeroth order’ phase transition, found in Ref. [@BP16], has been explained by the rapid increase of the P(1) contribution to both the local density $\rho_L$ and the current $J$, see Eqs. (\[L\]) and (\[J\]), as the particle injection probability $\alpha$ approaches from the left the boundary $\alpha =p$, $\beta >p$, between the MP and CF phases.
In addition, we have analyzed, within the random walk theory, the evolution of a single gap between two large clusters of vehicles. Three qualitatively different regimes were established when the injection rate approaches from the left the boundary $\alpha =p$ with the CF phase: (i) the lifetime of the rightmost gap in the jammed configuration is of size $O(L)$ in the MP phase; (ii) macroscopic jams, separated by short-living gaps of length $O(1)$, exist in the MP+CF phase; and (iii) there is a critical regime when the macroscopic jams are divided by gaps of intermediate lifetime of the order $O(L^{1/2})$ when the triple point $\alpha =\beta =p$ is approached. These results are supported by extensive Monte Carlo calculations.
The microstructure of traffic flow is usually described in terms of the probability distribution of the distance gap and time-headway, see a survey in [@LC11]. For example, the exact analytical expression for the probability distribution of the distance headways in the steady state of Nagel-Schreckenberg model of vehicular traffic at $V_{\rm max} = 1$ was obtained by Chowdhury et al [@CMGSS97]. The time- and distance-headways in the general case of that model were studied in Ref. [@CPS98]. A microscopic theory of spatial-temporal congested traffic patterns in heterogeneous traffic flow with a variety of driver behavioral characteristics and vehicle parameters was presented by Kerner and Klenov in [@KK04]. The time-headway distribution for the TASEP under various updates, including the generalized one, but under periodic boundary conditions only, was reported by Hrabák and Krbálek in Ref. [@HK16]. However, their results for the gTASEP break down in our limiting case of $\tilde{p} =1$ ($p\gamma =1$ in their notation). We are not aware of analogous results for open chains. Surprisingly, the authors of Ref. [@KS15] demonstrated that the vehicular gap distributions in the vicinity of a signalized intersection are a consequence of the general stochastic nature of queueing systems, rather than a consequence of traffic rules and drivers behavior.
One may speculate about the usefulness of the gap representation of the system configurations, as compared to the cluster one. In a generic point in the MP phase, the two representations seem equivalently complicated, see Fig. \[Fig4\]. Nevertheless, the gap representation seems promising, because one has an ensemble of equivalent critical gaps, besides the growing rightmost one (when $\beta >p$), which start at the left boundary, do not merge or cross and their probability distribution is known. In a forthcoming paper we are going to use the gap presentation for calculation of the density profile over the whole chain interval $[0,L]$.
Acknowledgement {#acknowledgement .unnumbered}
===============
The partial support by a grant of the Plenipotentiary Representative of the Bulgarian Government at the Joint Institute for Nuclear Research, Dubna, is gratefully acknowledged. VBP acknowledges the support of the RFBR, grant 16-02-00252.
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---
abstract: 'For any bipartite quantum system the Schmidt decomposition allows us to express the state vector in terms of a single sum instead of double sums. We show the existence of the Schmidt decomposition for tripartite system if the partial inner product of a basis (belonging to a Hilbert space of smaller dimension) with the state of the composite system gives a disentangled basis. In this case the reduced density matrix of each of the subsystem has equal spectrum in the Schmidt basis.'
address: 'School of Informatics, Dean Street, University of Wales, Bangor LL 57 1UT, UK.'
author:
- 'Arun Kumar Pati[^1]'
title: Existence of the Schmidt decomposition for tripartite systems
---
=10000
.5cm
The key to quantum information processing is the entangled nature of quantum states which has no classical counter part in the theory. These states are central to the study of quantum non-locality, quantum teleportation, quantum cryptography, dense coding and so on [@bd]. In the simplest term a pure entangled state is a one which cannot be expressed as a direct product of the states of two or more subsystems. If we have a quantum system consisting of two subsystems $A$ and $B$, then the state of the combined system in general can be expressed as $\ver \psi \ra_{AB} = \sum_{ij} a_{ij} \ver u_i \ra \otimes
\ver v_j \ra$, where $\{ \ver u_i \ra \}, (i=1,2,...N_A) \in {\cal H}_A$ and $\{ \ver v_i \ra \}, (i=1,2,...N_B) \in {\cal H}_B$ are the complete set of orthonormal basis vectors in their respective Hilbert spaces. The expansion coefficients in the above state of the combined system contain $N_A N_B$ terms and is very difficult to manipulate with. However, the Schmidt decomposition (SD) theorem [@sch] comes to rescue us. It says that any arbitrary state of a bipartite (two-subsystem) quantum system can be expressed as [@ap]
$$\ver \psi \ra_{AB} = \sum_{i=1}^{N_A} \sqrt p_i \ver x_i \ra_A \otimes \ver y_i \ra_B,$$
where $\{ \ver x_i \ra_A \}$ and $\{ \ver y_i \ra_B \}$ are two orthonormal basis sets belonging to Hilbert spaces ${\cal H}_A$ and ${\cal H}_B$, respectively and $N_A \le N_B$. The complex phases can be absorbed in the basis states and the expansion coefficients $\sqrt p_i$ can be chosen to be real and positive. This simplifies the expression to a great extent.
The Schmidt decomposition theorem has been applied in many worlds interpretation of quantum theory [@eve], in reproducing Clebsch-Gordon expansion of angular momentum states [@bcrs], in proving Bell’s inequality [@ng; @spdr] and is quite successful. Recently, in quantum optics context this has been applied [@aepk] and a geometric approach [@pka] to Schmidt decomposition of two-spin of particles is given in relation to Hardy’s proof of quantum non-locality. However, all these discussions pertains strictly to bipartite systems only.
If we have a composite system consisting of more than two subsystems there does not exist a Schmidt decomposition in general. But if it would exist, it will be quite useful simply because the number of terms one deals with in the expansion of the state vector will be comprehensibly small. For example, if we have a quantum system comprising of three subsystems (a tripartite system) then the general state of the system is given by
$$\ver \psi \ra_{ABC} = \sum_{ijk} a_{ijk} \ver u_i \ra_A \otimes
\ver v_j \ra_B \otimes \ver w_k \ra_C,$$
where $\{ \ver u_i \ra \} \in {\cal H}_A= C^{N_A}$, $\{ \ver v_j \ra \}
\in {\cal H}_B = C^{N_B}$ and $\{ \ver w_k \ra \} \in {\cal H}_C= C^{N_C}$. In this case there are $N_A N_B N_C$ terms to be dealt with. On the other hand if a Schmidt decomposition for tripartite system exists , then we can write the state in (2) as
$$\ver \psi \ra_{ABC} = \sum_{i} \sqrt p_i \ver x_i \ra_A \otimes \ver y_i \ra_B
\otimes \ver z_i \ra_C,$$
where $i = 1,2,... N_A$ (say) if $N_A = \dim{\cal H}_A$ is the smallest of all the three and $\{ \ver x_i \ra_A \}, \{ \ver y_i \ra_B \}$ and $\ver w_i \ra_C$ are again orthonormal basis sets belonging to their respective Hilbert spaces. The general argument which goes against the existence of Schmidt decomposition for a tripartite system such as (3) is that the “equal-spectrum” condition for reduced density matrices does not hold [@jp]. Nevertheless, it is worth exploring under what conditions SD can exist. If the Schmidt decomposition for a tripartite system exists it would be useful in modal interpretation of quantum theory [@sk; @ejs; @rh], for example. This possibility has been explored by Peres [@per] and he found a necessary and sufficient condition for the existence of a Schmidt decomposition for a tripartite system. However, his condition does not give insight why does it fail for some tripartite systems and why does it [*always work*]{} for a bipartite system. Recently, Thapliyal [@at] has discussed Schmidt decomposability for a multipartite systems. He found that if the density matrix is multiseparable after tracing out any party, then the state is Schmidt decomposable.
In this letter we find a simple criterion for the existence of Schmidt decomposition for tripartite system. This gives insight to the question: why does it work always for a bipartite not for a tripartite system. To state it simply, we prove that [*if the partial inner product of a basis of any one of the subsystems (belonging to a Hilbert space of smaller dimension) with the state of the composite system gives a disentangled basis, then Schmidt decomposition for a tripartite system exists*]{}. This condition turns out to be sufficient as well as necessary for the existence of SD. If the partial inner product gives an entangled basis the Schmidt decomposition in terms of a single sum does not exist, though the triple sum can be converted to a double sum. Using our existence criterion we show that the reduced density matrix of each subsystem (by taking partial traces over any two subsystems) has the same eigenvalues, i.e. the “equal-spectrum” requirement holds. Our criterion is also consistent with the existence of Schmidt decomposition for a bipartite system. When we take the partial inner product of any one of the basis with the state of an arbitrary bipartite system, then the resulting basis belong to a Hilbert space of a single subsystem (no question of an entangled basis). This is the main reason why the Schmidt decomposition always works for a bipartite system. We also discuss the connection between our criterion and Thapliyal’s multiseparability criterion for the existence of SD.
Now we prove the following theorem.\
: For any state $ \ver \psi \ra_{ABC} \in
{\cal H}_A \otimes {\cal H}_B \otimes {\cal H}_C $ of a tripartite system let dim ${\cal H}_A = N_A$ is smallest of $N_B$ and $N_C$. If the “partial inner product” of the basis $\ver u_i \ra_A$ with the state $\ver \psi \ra_{ABC}$, i.e. ${_A\la} u_i \ver \psi \ra_{ABC} = \ver \psi_i \ra_{BC}$ has Schmidt number one then the Schmidt decomposition for a tripartite system exists.
Proof: Let $ \ver \psi \ra_{ABC} = \sum_{ijk} a_{ijk} \ver u_i \ra_A \otimes
\ver v_j \ra_B \otimes \ver w_k \ra_C$ and the partial inner product of the basis $\ver u_i \ra$ and state $\ver \psi \ra_{ABC}$ is a basis vector in the Hilbert space ${\cal H}_B \otimes {\cal H}_C$ spanned by basis vectors $\{ \ver v_j \ra_B \otimes \ver w_k \ra_C \}$. It is given by
$$\ver \psi_i \ra_{BC} = \sum_{jk} a_{ijk}
\ver v_j \ra_B \otimes \ver w_k \ra_C,$$
where $\{ \ver \psi_i \ra_{BC} \}$ is an orthogonal basis set but need not be normalised. We know that any vector in a bipartite Hilbert space can be written as a Schmidt decomposition form, i.e.
$$\ver \psi_i \ra_{BC} = \sum_{\mu} \sqrt{ p_{\mu}^{(i)} }
\ver y_{\mu} \ra_B \otimes \ver z_{\mu} \ra_C,$$
where $\{ \ver y_{\mu} \ra_B \}$ and $\{ \ver z_{\mu} \ra_C \}$ are orthonormal basis for ${\cal H}_B$ and ${\cal H}_C$ which can be unitarily related to the basis $\{ \ver v_j \ra_B \}$ and $\{ \ver w_k \ra_C \}$, respectively. Therefore, the arbitrary state in general can be written as
$$\ver \psi \ra_{ABC} = \sum_{i \mu} \sqrt{ p_{\mu}^{(i)} }
\ver u_i \ra_A \otimes \ver y_{\mu} \ra_B \otimes \ver z_{\mu} \ra_C,$$
Now there can be two situations: (i) either the basis (we call bi-Schmidt basis) $\{ \ver \psi_i \ra_{BC} \}$ is entangled or (ii) it is separable. Here, by entangled or separable basis we mean a basis all of whose members are so. We can apply the pure state entanglement criterion, i.e., if the Schmidt number is equal to one then the state is separable and if it is greater than one it is entangled. Here, the Schmidt number is nothing but the number of non-zero eigenvalues in the reduced density matrix of a bipartite system and is the same as the number of terms in the Schmidt decomposition of a bipartite state. This is a good measure of entanglement for pure states. Now imposing this condition on bi-Schmidt basis, we write it as a separable form. So, if $\ver \psi_i \ra_{BC}$ has Schmidt number one we can write $\ver \psi_i \ra_{BC} = \ver \beta_i \ra_{B} \otimes \ver \gamma_i \ra_{C}$. Since $\{ \ver \psi_i \ra_{BC} \}$ is not normalised $\{ \ver \beta_i \ra_{B} \}$ and $\{ \ver \gamma_i \ra_{C} \}$ need not be orthonormal though they satisfy orthogonality condition. Therefore, the tripartite system can be written as
$$\ver \psi \ra_{ABC} = \sum_{i}
\ver u_i \ra_A \otimes \ver \beta_i \ra_B \otimes \ver \gamma_i \ra_C.$$
Now we calculate the reduced density matrix of each subsystem. The reduced density matrix for $A$ can be obtained by taking partial traces over $B$ and $C$. Thus,
$$\rho_A = {\rm tr}_B({\rm tr}_C(\rho_{ABC} )) = {\rm tr}_B [ \sum_{i}
q_i \ver u_i \ra_A {_A}\la u_i \ver \otimes \ver \beta_i \ra_B {_B}\la \beta_i \ver ],$$
where we have used the trace equality ${\rm tr}_C( \ver \gamma_i \ra_C \la \gamma_j \ver) =
{_C}\la \gamma_j \ver \gamma_i \ra_C = q_i \delta_{ij}$ and $q_i =
\parallel \gamma_i \parallel^2 $ is the (squared) norm of the basis vector $\ver \gamma_i \ra_C$. Performing the second trace we can write the above one as
$$\rho_A = \sum_{i}
q_i r_i \ver u_i \ra_A {_A}\la u_i \ver,$$
where we have used ${\rm tr}_B( \ver \beta_i \ra_B \la \beta_j \ver) =
{_B}\la \beta_j \ver \beta_i \ra_B = r_i \delta_{ij}$ and $r_i =
\parallel \beta_i \parallel^2 $ is the (squared) norm of the basis vector $\ver \beta_i \ra$. Similarly, we can obtain the reduced density matrix $\rho_B$ and $\rho_C$ as
$$\begin{aligned}
&&\rho_B = \sum_{i}
q_i r_i \ver \beta_i' \ra_B {_B}\la \beta_i' \ver \nonumber\\
&& \rho_C = \sum_{i}
q_i r_i \ver \gamma_i' \ra_C {_C}\la \gamma_i' \ver ,
\end{aligned}$$
where we have defined the orthonormal basis vectors $\ver \beta_i' \ra_B$ and $\ver \gamma_i' \ra_C$ for ${\cal H}_B$ and ${\cal H}_C$ as $\ver \beta_i \ra_B = \sqrt r_i \ver \beta_i' \ra_B$ and $\ver \gamma_i \ra_C = \sqrt q_i \ver \gamma_i' \ra_C$. By comparing all the density matrices, we see that they have same eigenvalue spectrum $\{ q_i r_i \}$ in the Schmidt basis . Now we can redefine the state of the tripartite system as
$$\begin{aligned}
\ver \psi \ra_{ABC} & = & \sum_{i} \sqrt{ q_i r_i}
\ver u_i \ra_A \otimes \ver \beta_i' \ra_B \otimes \ver \gamma_i' \ra_C \nonumber\\
& = & \sum_{i} \sqrt d_i \ver i \ra_A \ver i \ra_B \ver i \ra_C,
\end{aligned}$$
with $d_i = q_ir_i$ this is the Schmidt decomposition for a tripartite system and hence the proof.
The above proof shows that the disentangled-partial-inner product crietrion is a sufficient one. But it can be checked that it is also a necessary condition, because if SD exists then the partial inner product of a basis in the smallest Hilbert space with the joint state will give a disentangled basis in other two tensor product Hilbert spaces. It should be remarked that $\rho_A, \rho_B$ and $\rho_C$ have the same number of distinct non-zero eigenvalues (non-degenerate spectrum), however, the number of zero-eigen values of $\rho_A, \rho_B$ and $\rho_C$ can be different as ${\cal H}_A , {\cal H}_B$ and ${\cal H}_C $ have different dimensions. The Schmidt decomposition (11) is unique for non-degenerate spectrum of reduced density matrices. The same is true for a bipartite system. Further, when we have a bipartite system then the “partial inner product” of any of the basis with the state of the system gives a single (disentangled) basis for the other one. Hence, the SD is always posible for a bipartite system.
Once we know that the SD exists, then no local unitary operation of the form $U_B \otimes U_C$, local general measurements and classical communication can disprove the existence of Schmidt decomposition. We know that any measure of entanglement $E(\rho_{i,BC})$, with $\rho_{i,BC} = \ver
\psi_i \ra_{BC} {_BC}\la \psi_i \ver $ satisfies the requirements [@pk] (i) $E(\rho_{i,BC}) = 0$ when $\rho_{i,BC}$ is separable, (ii) $E(\rho_{i,BC}) = E(U_B \otimes U_C \rho_{i,BC} U_B^{\dagger} \otimes
U_C^{\dagger})$, and (iii) $E(\rho_{i,BC})$ cannot increase under local general measurement and classical communication, the Schmidt number of the bi-Schmidt basis does not change and it is impossible to disprove the existence of Schmidt decomposition.
Next we discuss the situation when Schmidt decomposition does not exist for a tripartite system. This is the case when the bi-Schmidt basis is an entangled basis. Physically, this means there exists “[*entanglement within entanglement”*]{}. When bi-Schmidt basis is separable there is entanglement between each subsystem $A, B$ and $C$ and there is no “ entanglement within entanglement”. When there is “entanglement within entanglement” the “equal-spectrum” requirement breaks down. However, the “equal-spectrum” holds within the subsystems $B$ and $C$, i.e., $\rho_B$ and $\rho_C$ have same non-zero eigenvalues. To see this consider the state of a tripartite system as $\ver \psi \ra_{ABC} = \sum_{i} \ver u_i \ra_A \otimes
\ver \psi_i \ra_{BC}$. The density matrix of the tripartite system is given by
$$\rho_{ABC} = \sum_{i j}
\ver u_i \ra_A {_A}\la u_j \ver \otimes \ver \psi_i \ra_{BC} {_{BC}}\la \psi_j \ver.$$
On tracing over subsystem $B$ and $C$ we have the reduced density matrix $\rho_A$ given by
$$\rho_{A} = \sum_{i} p_i
\ver u_i \ra_A {_A}\la u_i \ver,$$
where we have used the trace equality ${\rm tr}_{BC}(\ver \psi_i \ra_{BC}
{_{BC}}\la \psi_j \ver ) = {_{BC}\la} \psi_j \ver \psi_i \ra_{BC} = p_i \delta_{ij}$ and $_{BC}\la \psi_i \ver \psi_i \ra_{BC} = \sum_k p_k^{(i)} = p_i$ is the (squared) norm of the bi-Schmidt basis. The reduced density matrix $\rho_{AB}$ given by
$$\rho_{AB} = {\rm tr}_C(\rho_{ABC}) = \sum_{ij \mu} \sqrt{p_{\mu}^{(i)} p_{\mu}^{(j)}}
\ver u_i \ra_A {_A}\la u_j \ver \otimes \ver y_{\mu} \ra_{B} {_B}\la y_{\mu} \ver.$$
On tracing over the subsystem $A$ we get the reduced density matrix for $\rho_B$ given by
$$\rho_{B} = \sum_{i \mu} p_{\mu}^{(i)} \ver y_{\mu} \ra_{B} \la y_{\mu} \ver
= \sum_{\mu} s_{\mu} \ver y_{\mu} \ra_{B} {_B}\la y_{\mu} \ver,$$
where we have defined $\sum_{i} p_{\mu}^{(i)} = s_{\mu}$ and each of them are some positive numbers. To obtain the reduced density matrix for $C$ we first note that $\rho_{AC}$ is given by
$$\rho_{AC} = {\rm tr}_B(\rho_{ABC}) = \sum_{ij \mu} \sqrt{p_{\mu}^{(i)} p_{\mu}^{(j)}}
\ver u_i \ra_A {_A}\la u_j \ver \otimes \ver z_{\mu} \ra_{C} {_C}\la z_{\mu} \ver.$$
From the above one we get the reduced density matrix for $\rho_C$ given by
$$\rho_{C} = \sum_{i \mu} p_{\mu}^{(i)} \ver z_{\mu} \ra_{C} {_C}\la z_{\mu} \ver
= \sum_{\mu} s_{\mu} \ver z_{\mu} \ra_{C} {_C}\la z_{\mu} \ver.$$
From (15) and (17) these we can see that $\rho_B$ and $\rho_C$ have same eigenvalue spectrum $\{ s_{\mu} \}$ whereas the eigenvalue of $\rho_A$ has different spectrum $\{ p_i \}$. Thus, the “equal-spectrum” requirement breaks down when the bi-Schmidt basis $\{ \ver \psi_i \ra_{BC} \}$ has Schmidt number greater than one. Interestingly, if we look the subsystems $B$ and $C$ as a single subsystem $BC$, then the “equal-spectrum” requirement holds for subsystems $A$ and $BC$. This is because we have
$$\rho_{BC} = \sum_{i} \ver \psi_i \ra_{BC} {_{BC}}\la \psi_i \ver.$$
On defining an orthonormal basis $\ver \psi_i' \ra_{BC}$ as $\ver \psi_i \ra_{BC} = \sqrt p_i \ver \psi_i' \ra_{BC}$, we have
$$\rho_{BC} = \sum_{i} p_i \ver \psi_i' \ra_{BC} {_{BC}}\la \psi_i' \ver$$
This shows that $\rho_A$ and $\rho_{BC}$ have equal-spectrum as expected intuitively.
Before concluding we briefly discuss the link between our criterion and Thapliyal’s [@at]. We show that the multiseparability criterion discussed in [@at] is a special case of our partial-inner product criterion when each subsystem has equal dimension. When we have a tripartite system and all the Hilbert spaces have equal dimension, then choosing a Hilbert space of lowest dimension becomes degenerate. Therefore, one has to check if the partial-inner product gives disentangled basis with respect to all the basis vectors and all the parties concerned. For example, if ${_A\la} u_i \ver \psi \ra_{ABC} = \ver \psi_i \ra_{BC}$, ${_B\la} v_i \ver \psi \ra_{ABC} = \ver \phi_i \ra_{AC}$ and ${_C\la} w_i \ver \psi \ra_{ABC} = \ver \chi_i \ra_{AB}$ all gives disentangled basis then there will be a Schmidt decomposition. However, as can be seen if the above conditions hold the density matrices such as $\rho_{BC}$, $\rho_{AC}$ and $\rho_{AB}$ are multiseparable.
In conclusion, we have found a simple criterion for the existence of Schmidt decomposition for tripartite system and discussed why does it fail in some cases. This also answers why does it always work for a bipartite system. The existence of Schmidt decomposition might be useful in quantifying entanglement content of a pure tripartite system. For example, if the SD exists then the von Neumann entropy of any of the reduced density matrix would give the entanglement content of a pure tripartite system. Our criterion can be generalised to multipartite (more than three) entangled states. Recently, generalised Schimdt-like decomposition has been found for three qubit [@acin] and multpartite cases [@tony].
I thank S. L. Braunstein and A. Peres for going through my paper. I also thank C. H. Bennett for useful remarks and interesting questions. The financial support from EPSRC is gratefully acknowledged.
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[^1]: email:akpati@sees.bangor.ac.uk
|
---
abstract: 'Processing-in-memory (PIM) turns out to be a promising solution to breakthrough the memory wall and the power wall. While prior PIM designs yield successful implementation of bitwise Boolean logic operations locally in memory, it is difficult to accomplish the multiplication (MUL) instruction in a fast and efficient manner. In this paper, we propose a new stochastic computing (SC) design to perform MUL with in-memory operations. Instead of using the stochastic number generators (SNGs), we harness the inherent stochasticity in the memory write behavior of the magnetic random access memory (MRAM). Each memory bit serves as an SC engine, performs MUL on operands in the form of write voltage pulses, and stores the MUL outcome in-situ. The proposed design provides up to 4x improvement in performance compared with conversational SC approaches, and achieves 18x speedup over implementing MUL with only in-memory bitwise Boolean logic operations.'
author:
- |
Xin Ma$^{1,2}$, Liang Chang$^{1,3}$, Shuangchen Li$^{1}$, Lei Deng$^{1}$, Yufei Ding$^{2}$, Yuan Xie$^{1}$\
\
$^1$Department of Electrical and Computer Engineering, UCSB, California, USA\
$^2$Department of Computer Science, UCSB, California, USA\
$^3$School of Electronic and Information Engineering, Behang University, Beijing, China
title: 'In-memory multiplication engine with SOT-MRAM based stochastic computing\'
---
Stochastic computing, PIM, SOT-MRAM
Introduction
============
The processing-in-memory (PIM) paradigm has been considered as a promising alternative to break the bottlenecks of conventional von-Neumann architecture. In the era of big data, data movement between the processor and the memory results in huge power consumption (power wall) and performance degradation (memory wall), known as the von-Neumann bottleneck[@koo2017summarizer]. By placing the processing units inside/near the memory, PIM remarkably reduces the energy and performance overhead induced by data transport[@zhang2014top][@ahn2016scalable]. In the recent advancement of PIM designs, it also allows fully leverage the large memory internal bandwidth and embrace massive parallelism by simultaneously activating multiple rows, subarrays, and banks in memory arrays for bit-wise operations [@li2016pinatubo]. These performance gains are all achieved at a minimal cost of slightly modifying the memory array peripheral circuits [@chi2016prime] [@li2017drisa].
Multiplication (MUL) is always a complex task to accomplish in efficient PIM designs, despite that MUL instructions are frequently used in Neuro Network (NN) algorithms and linear transforms (e.g. Discrete Fourier Transform). As shown by the recent developed DRISA [@li2017drisa], it takes 143 cycles to calculate an 8-bit multiplication, which deviates from its original motivation to achieve high performance with in-memory bitwise operations. The situation may be even worse with operands composing of more bits, as the cycle count can increase exponentially with the operand’s bit length. The challenge mainly lies in the fact that MUL can not be effectively decomposed into a small serial of bitwise Boolean logic operations which can be performed locally in memory.
To tackle such challenge, prior efforts propose to either approximate the MUL or utilize the analog computing features of hardware devices. On one hand at the algorithm level, binary NN with approximate binary weights and activations has been developed [@courbariaux2016binarized]. As such, the MUL is simplified into bitwise XNOR operations that become PIM friendly [@angizi2018imce]. Unfortunately, such simplification comes at the cost of the undesired and significant degradation in the classification accuracy of NN. On the other hand at the hardware level, ReRAM is implemented to ease MUL in novel PIM designs, taking advantage of ReRAM’s analog storage. The analog resistance/conductance of ReRAM encodes the weights in NN. By activating one entire row/column simultaneously in a ReRAM crossbar, the dot product between a matrix and a vector in NN can be easily achieved using Ohm’s law[@chi2016prime]. Nevertheless, ReRAM itself suffers from the long write latency, high programming voltage and limited endurance, which hinders its application in high-speed and energy efficient architecture design.
In this work, we propose a new stochastic computing (SC) design to effectively perform MUL with in-memory operations, in light of the simplicity to implement MUL with SC. In order to tightly couple SC with PIM, we embrace the inherent stochasticity of the memory bit in spin-orbit-torque magnetic random access memory (SOT-MRAM). Specifically, the stochastic number generation and massive AND operations in the conventional SC-based MUL are implemented with simple memory write operations in SOT-MRAM. Consequently, each bit serves as an SC engine, and the large supporting circuits for stochastic number generation and logic operations can be effectively saved. Finally, the MUL outcome is represented by the probability distribution of the binary storage states among MRAM bits, and can be converted back to its binary form with pop-count. The contributions of this paper are summarized as follows:
- We propose the idea of employing the inherent stochastic write in SOT-MRAM to promote SC in the PIM design.
- We develop an efficient approach to implement MUL in the way of memory write, by converting the binary multipliers to the write voltage pulse with varied duration.
- We propose two strategies of pop-count to convert the MUL result back to its binary format, offering flexibility to further trade performance with area.
- The proposed design provides up to 4x improvement in performance and significant reduction in area occupancy compared with conversational SC approaches, and achieves 18x speedup over implementing MUL with only in-memory bitwise Boolean logic operations.
Preliminaries
=============
This section introduces the motivation to combine SC with PIM and the preliminary design with the stochastic switching behavior of SOT-MRAM.
SC and PIM
----------
SC provides an alternative approach to implement the MUL function. SC is an approximate computing method, which has been studied for decades and widely applied to image/signal processing, control systems and general purpose computing[@hayes2015introduction][@alaghi2013survey]. SC method essentially trades the data representation density for simpler logic design and lower power. For instance, SC represents a n-bit binary number with a stochastic bitstream ( $2^n$-bit). The value of the binary number $X$ equals to the probability of the appearance of “1”s in the bitstream $(\{x_i\})$: $$X=2/6(binary)\rightarrow \{x_i\}=\{1,0,0,0,1,0\}(stoch.),$$ Benefiting from such data representation, the MUL between two numbers can be converted to simple bitwise AND operations, which dramatically reduces the complexity of logic design. $$\begin{split}
&Y=3/6(binary)\rightarrow \{x_i\}=\{0,1,0,1,1,0\}(stoch.),\\
&X \bullet Y=1/6 \rightarrow \{x_i\&y_i\}=\{0,0,0,0,1,0\}(stoch.).
\end{split}$$ However, SC is not friendly to conventional von-Neumann architecture. The data explosion of SC aggravates data movement between processor and memory, which offsets the simplicity brought by SC.
Instead, SC tightly couples with PIM from multi-fold aspects, leading to significant performance gain: First, the many bits of stochastic bitstream can be stored in off-chip memory with large capacity. Second, the logic operation with reduced complexity can be implemented by the processing units locally in memory. Finally, the stochastic feature of bitstream allows parallel computing on the individual bit, so that the internal memory bandwidth can be fully leveraged. Therefore, the MUL instruction can be significantly accelerated by combining SC with PIM.
Several challenges still exist towards combine SC with PIM. The random bitstream still relies on stochastic number generators (SNGs), which incurs large area overhead for the supporting circuits. In addition, those stochastic bits can be hardly generated in parallel and with eliminated correlations, resulting in degradation of performance and accuracy in computing MUL. In our design, we overcome these drawbacks by utilizing the inherent stochasticity in MRAM bit.
SOT-MRAM and its stochastic switching
-------------------------------------
SOT-MRAM utilizes the spin-orbit torques to write the memory cell, overcoming the drawbacks of Spin Transfer Torque-MRAM (STT-MRAM) in terms of high write latency, and large write energy dissipation[@wang2018high] [@chang2017prescott]. Fig. \[MRAMbit\] compares the similarity and difference between SOT-MRAM and STT-MRAM cells. Similarly, both types of MRAM cells store the bit value in a magnetic tunnerling junction (MTJ). The bit value “0” or “1” is read out electrically as high or low tunneling magneto-resistance, which is controlled by the antiparallel (AP) or parallel (P) alignment of magnetization in the free layer (FL) and the reference layer (RL). Although the write of MRAM bit is always fulfilled by controlling the magnetization direction of the FL, the mechanisms used are different between STT-MRAM and SOT-MRAM. In STT-MRAM, the write current passes through MTJ and the spin polarized current exerts notable STT to switch the FL magnetization[@wang2018high] [@chang2016evaluation]. Differently in SOT-MRAM, SOTs are generated by transversing write current though an additional heavy metal layer (HML) to switch the magnetization in the adjunct FL. As a result, SOT-MRAM does not suffer from the asymmetric of write latency between “AP $\rightarrow$ P” and “P $\rightarrow$ AP” in STT-MRAM, speeding up the write procedure. Moreover, the energy efficiency of write is fundamentally higher in SOT-MRAM. That’s because each electron can be reused multiple times to exert SOTs after bounced back from HML and FL interface, while it can be used once at most in STT.
![Illustration of the memory cells and the write strategies of STT-MRAM (a) and SOT-MRAM (b).[]{data-label="MRAMbit"}](MRAMbit.jpg){width="3"}
We harness the stochastic behavior within the memory write of SOT-MRAM to perform SC. The probability $P_{usw}$ of MRAM bit remains **not** switched under the appliance of electrical current $I$ is described [@seki2011switching] $$P_{usw}=exp(-\tau exp(-\Delta (1-I/I_c))).
%\lable{Probability}$$ Here, $\tau$ denotes the pulse duration of the applied $I$ in nanosecond, $\Delta$ represents the thermal stability parameter of the MTJ, and $I_c$ is the critical current strength required to switch the FL magnetization. Fig. \[Probability\] plots the $P_{usw}$ as functions of $I$ and $\tau$, with $\Delta=60.9$ and $I_c=80\mu A$ estimated from previous micromagnetic simulations on SOT driven magnetization dynamics[@chang2017prescott]. By finely controlling the parameters in the write of SOT-MRAM, each memory bit can serve as a stochastic bit generator with the desired probability of holding either “0” or “1”. Utilizing this feature of SOT-MRAM, the large amount of stochastic bits in SC can be generated in parallel and in-situ stored in memory with a simple write operation.
![The probability of a MRAM bit remaining unswitched under different pulse duration and strength of the applied electrical current[]{data-label="Probability"}](uswP.jpg){width="3"}
Data conversion and hardware design
===================================
To implement the MUL operation with the stochastic switching of MRAM bit, the binary operands have to be translated into certain parameter of the write voltage pulse. The flow of our proposed sequential data conversion can be summarized as: $$\begin{split}
&X(binary), Y(binary) \rightarrow ln(X)(binary), ln(Y)(binary)\\
&\rightarrow ln(X)(time), ln(Y)(time) \rightarrow X*Y (stoch.) \\
&\rightarrow X*Y (Binary)
\end{split}$$ In this section, we will introduce them and the related hardware design step by step.
Binary numbers to logarithmic timing signals
--------------------------------------------
We first perform logarithmic operation on the digital numbers stored in memory, i.e. $X(binary), Y(binary) \rightarrow ln(X)(binary), ln(Y)(binary)$. The multiple bits of the operand $X$ are read out by sensing amplifiers (SAs) and decoded to find their logarithmic values using a lookup table (LUT) (Fig. \[LUT\]). The LUT method is usually used in logarithm multiplication, and has been demonstrated to be fast and accurate[@nandan201865]. This conversion step is necessary, since an exponential operation is inherently included in the following stochastic switching of the MRAM bit.
Afterwards, we convert the $ln(X)$ to timing signals with a digital-to-time converter (DTC). The DTC outputs a voltage square pulse $V_{tX}$, where the pulse duration $\tau_{X}$ in Eq. 3 is proportional to the value of input $ln(X)$. The magnitude of the $V_{tX}$ pulse is normalized and fixed to drive SOT-MRAM bit in its non-deterministic switching region.
![Data conversion from binary to logarithmic timing signals[]{data-label="LUT"}](Circuits.jpg){width="3"}
Logarithmic timing signals to stochastic bitstream
--------------------------------------------------
The write voltage pulse $V_{tX}$ is subsequently applied onto the source lines (SLs) of multiple rows of SOT-MRAM bits, and drives their stochastic switching behaviors. The entire row of MRAM array can be written simultaneously with a cross-point design (Fig. \[Crosspoint\])[@chang2017prescott]. The MTJs in a row share a set of driving transistors, and are directly linked to the BLs and SLs without additional transistors for individual bit. As a result, minimal area and energy overhead are introduced to enable such simultaneous write.
![Schematics of the cross-point SOT-MRAM array, with stochastic bits in-situ stored under a pulsed voltage[]{data-label="Crosspoint"}](Memoryarray.jpg){width="3"}
Fig. \[SCexample\] shows how the SC-based MUL is performed. **Initialization**: a preset operation is required to initialize all the bits to “1” with reversed current $I_c$. **Input first operand**: the converted write voltage pulse $V_{tX}$ is input onto the MRAM array, resulting in partial switching of the bits. The probability of remaining “1”s equals to $P_X$, where $P_X$ is proportional to the value of operand $X$. **MUL with the second operand input**: The MUL operation is performed by inputing a subsequent voltage pulse $V_{tY}$ (similarly converted from operand $Y$) onto the same MRAM array. As a result, the remaining “1”s survive from not switched by neither pulse $V_{tX}$ nor $V_{tY}$, and they are distributed among the MRAM arrays with a probability equaling $P_X*P_Y$ (proportional to $X*Y$).
![Illustration for the SC-based MUL in SOT-MRAM[]{data-label="SCexample"}](SCexample.jpg){width="3"}
Stochastic bits to Binary numbers
---------------------------------
At last, we perform bit counting to convert the outcome from the stochastic representation to its binary format. Either approximate pop-count (APC) [@kim2015approximate] or PIM-based ADD operations [@li2017drisa] can be employed to bit counting. APC method can be performed with one clock cycle, but introduces much area overhead. Alternatively, PIM-based ADD is area-efficient, but takes many clock cycles to perform the pop-count.
Specifically, we can accelerate the PIM-based pop-count for the vectored multiply-and-accumulate (MAC) in NN. Fig. \[Popcount\] shows the two-step strategy, where the sum is performed after several MULs have been done. In the first step, we perform row-wise sum with a carry-save addition (CSA). Then in the second step, the intermediate sum results undergo a column-wise additions with full adder (FA). Our motivation here is to lessen the usage of FA for column-wised addition, since it takes more clock cycles than the lock step bitwise operations of CSA. As shown in Fig. \[Popcount\], the delay from FA can be averaged out, and the pop-count related cycle count converges to that of CSA after many MULs.
![The two steps of the PIM-based pop-count strategy.[]{data-label="Popcount"}](Bitcount.jpg){width="3.5"}
Put them all together
---------------------
After putting all the pieces together, we point out strategies to further improve the performance and accuracy, and explain certain considerations in the design.
The sequential flow of data conversion can be separated and pipelined to improve the throughput and performance. For example, the LUT operation on the second operand can be performed simultaneously with the stochastic memory write for the first operand. Moreover, the bit counting can work in parallel with MUL operations for NN applications. There is no need for the relative slow pop-count to start until all the fast MULs between $w_i$ and $x_i$ have been finished in the computation of $\sum_iw_ix_i$. Furthermore, one could pre-convert certain frequently used data (e.g. weight $w_i$ in NN) into stochastic bits, which can be stored non-volatilely in MRAM arrays. Once other multipliers (e.g. inputs $x_i$) come, their converted timing signals can be directly input onto the corresponding MRAM arrays to perform MUL operations.
There are several normalization units in the circuits that can be used to fine tune the accuracy and performance. For example, the pulse duration of $V_t$ can be scaled to a range where $P_{usw}\approx 0.5$. Through such scaling, the switching voltage pulse can not be longer than the usual time required to switch MRAM bit, avoiding unnecessary slowdown in computing. Moreover, the bitstream can be tuned neither sparse nor dense to guarantee the accuracy of MUL, so that more bits are effectively involved in SC. This is fundamentally similar to the improved classification accuracy of NN with more neurons involved.
Multiple rows can be simultaneously activated and wrote to generate more stochastic bits in parallel. This situation happens when performing MUL on operands with more bits. In the cross-point MRAM design, we limit the number of memory cells in each row due to the concern of IR drop. The MTJs farther away from the driving transistors in the row would suffer from a lower switching voltage[@liang2010cross], and would likely undergo stochastic switching with undesired and incorrect probability.
Finally, we note that the pulse duration $\tau$ is used here for computing, instead of the magnitude of switching voltage pulse $V_t$ (equivalent to $I$ in Eq. 3). That’s because the usage of the magnitude $I$ requires more complicated circuits design for data conversion, owing to the complex dependence of $P_{usw}$ on $I$. In addition, the two inputs $I_X$ and $I_Y$ has to be input simultaneously onto the MRAM arrays. This is not friendly to the pipeline strategies mentioned above, but will introduce large area overhead onto the driven transistors to enable higher current write instead.
Monte Carlo simulations
=======================
To estimate the accuracy and its dependence on hardware variance from statistics, we performe the Monte Carlo simulations on the stochastic switching of MRAM bits. In the following, $nbit$ denotes the number of stochastic bits per MUL, $P$ represents the probability $bitcount(B_i=1, i=1:nbit)/nbit$ that the bit remains not switched under certain input voltage pulse. For one MUL operation, we test the proposed SC with 1000 iterations and make statistics on the results among iterations.
Accuracy
--------
Fig. \[Statadd\](a) shows the distribution of the error $P_{XY}-P_X*P_Y$ among the 1000 iterations, where the the probability $P_{XY}$ is stochastically computed (with $\tau_{X}=0.3 ns, \tau_{Y}=0.4 ns$) and $P_X, P_Y$ are theoretically calculated from the two operands. The error distribution is centered to zero, indicating that there is no intrinsic bias in the SC arithmetic. The distribution can be well fitted with a Gaussian function (red line), with the standard deviation $\sigma \approx 1.6\%$. This indicates that the MUL is with about $3.2\%$ uncertainty for $nbit=1000$.
![(a) The probability distribution of the deviation from theoretical calculated MUL among 1000 iterations. (b) The MUL uncertainty as a function of stochastic bits number and input values.[]{data-label="Statadd"}](Statadd.jpg){width="3"}
We further investigate the dependence of $\sigma$ on the inputs $\tau_Y$ and the number of stochastic bits $nbit$. As shown in Fig. \[Statadd\](b), $\sigma$ is almost independent on the inputs $\tau_Y$, but decreases with larger $nbit$. Therefore, we can improve the accuracy of SC by using more MRAM bits, despite that the improvement becomes more gradual with larger $nbit$.
The impact of hardware variance
-------------------------------
We also investigate the impact of hardware variance on the accuracy of MUL operation, by introducing random fluctuations on the devices’ parameters in Monte Carlo simulation.
The critical currents of MRAM bits $I_c$ may be slightly varied, since the many MRAM bits can not be manufactured identically and they may also experience different thermal fluctuations when in use[@an2016current]. Therefore, we introduce 0% to 10% random fluctuations $\sigma(I_c)$ on the $I_c$. As shown in Fig. \[Faulttorr\](a), the accuracy of SC remains almost unchanged under different strength of fluctuations.
We also compare the fault tolerance of our design with that of logarithm multiplication. To implement logarithm multiplication[@nandan201865], we replace the DTC and SOT-MRAMs with an antilogarithm amplifier. Then we introduce 4% to 10% random fluctuations $\sigma(Circuits)$ on DTC and antilogarithm respectively for the two cases. As shown in Fig. \[Faulttorr\](b), the accuracy of our SC+PIM design remains almost unchanged, while logarithm multiplication suffers from severe degradation in accuracy with stronger fluctuations.
![The dependence of MUL uncertainty on the variance of critical current among SOT-MRAM bits. The MUL uncertainties as functions of circuit variance in our SC+PIM design and logarithm multiplication[]{data-label="Faulttorr"}](Faulttorr.jpg){width="3"}
Evaluation
==========
In this section, we evaluate the performance, power and area overhead of the proposed SC+PIM design, and compare them with that of other approaches using either SC or PIM.
Experimental setup
------------------
We adopt the cross-point design of SOT-MRAM arrays similar to PRESCOTT[@chang2017prescott], to enable the parallel memory write. The low-power DTC generates voltage pulses with 22 ps time resolution and occupies $75\mu m*25\mu m$ in area[@wang2015digital]. For the APC, we design one-cycle fully parallel circuit synthesized with 45nm FreePDK[@stine2009freepdk], integrating parameters from[@kim2015approximate]. Our evaluation is based on the multiplication between two 10-bit operands that represented by $2^{10}$ stochastic bits.
In the following, different configurations have been compared: **SC+PIM (with APC)** denotes our SC+PIM design with pop-count conducted by APC. **SC+PIM (with CSA)** is our SC+PIM design with pop-count performed with CSA+FA. Specially, the evaluation is averaged onto each MUL for the situation of performing 100 MULs in a MAC. **SC** represents the usage of a built multiplier with the state-of-the-art SNG [@kim2016energy] and popcount with APC. **PIM** is the situation that we only use in-memory Boolean logic operations to implement MUL.
Performance
-----------
Fig. \[Performance\](a) compares the cycle count used to perform each MUL operation with different designs. Evidently, our SC+PIM approach outperforms prior approaches using either SC or PIM. The boost of performance in our design benefits from the parallel generation of stochastic bits. In contrast, prior SC approaches requires additional cycles to generate stochastic bitstreams or to shuffle the existing pseudo-stochastic or deterministic bitstreams [@kim2016energy].
In addition, we investigate the dependence of MUL cycle count on the operands’ bit length as shown in Fig. \[Performance\](b). The cycle count remains unchanged in our SC+PIM design, since different amount of stochastic bits ($2^n$ for n-bit operand) can be generated in parallel. As a comparison, the cycle count required for MUL increases exponentially for the operands’ bits length in prior PIM design. Therefore, the speedup of SC+PIM over PIM becomes more attractive for MUL between operands with more bits.
![(a) The cycle count to implement each MUL with different approaches. (b) The MUL latency as a function of multipliers’ bit length with SC+PIM and PIM approaches[]{data-label="Performance"}](Perfcomp2.jpg){width="3.5"}
Energy consumption
------------------
Our SC+PIM design consumes 58% less energy compared with the SC method (Fig. \[Energy\]), thanks to the low write energy of SOT-MRAM[@jabeur2014spin]. In our design, most energy is spent through memory write, such as in the generation/computing of stochastic bits and the pop-count with bitwise addition (CSA). The situation is similar to prior SC approaches, where 88% of the energy is consumed in data buffering related operations.
As shown by the breakdown of the energy consumption in Fig. \[Energy\], the initialization step costs more energy than the following steps performing SC for MUL. That’s because a write voltage pulse with a higher magnitude and a longer pulse duration needs to be applied to guarantee the initialization. Afterwards, the memory bits are mainly driven in a non-deterministic switching region which consumes less energy.
![The energy consumption for each MUL with different approaches and their breakdown.[]{data-label="Energy"}](Powercomp2.jpg){width="3"}
Area overhead
-------------
The area overhead of different designs is compared in Fig. \[Area\]. The area overhead is smaller by about one order of magnitude for our SC+PIM design than conventional SC. The improvement originates from the removal of the additional circuits for SNG, which occupies 95% of the area in the conventional SC approach.
As shown by the breakdown of area overhead in Fig. \[Area\], the memory space required for the LUT table is comparable to the DTC and APC in our design, for the case of 10-bit multiplication. The LUT table size will shrink for regular 8-bit multiplication, since it depends exponentially on the bit length of the operands.
![The area overhead in different approaches with their breakdown. []{data-label="Area"}](Areacomp2.jpg){width="3"}
Conclusion
==========
In this paper, we propose a new SC design to perform MUL with in-memory operations. The stochastic random generation and AND operation in conventional SC are implemented by the simple write operations onto the SOT-MRAM. Such design is enabled by converting the binary multipliers to the varied pulse duration of the write voltage for SOT-MRAM. Consequently, the stochastic bits for the MUL outcome are in-situ stored. Two strategies of pop-count (APC or PIM-based ADD) have been proposed to convert the MUL result back to its binary format, offering flexibility to further trade off performance with area. Our approach improves the performance to compute MUL with PIM, in synergy with the mitigation of area overhead for supporting circuits of SC.
|
---
abstract: |
Lidar sensors are widely used in various applications, ranging from scientific fields over industrial use to integration in consumer products. With an ever growing number of different driver assistance systems, they have been introduced to automotive series production in recent years and are considered an important building block for the practical realisation of autonomous driving. However, due to the potentially large amount of Lidar points per scan, tailored algorithms are required to identify objects (e.g. pedestrians or vehicles) with high precision in a very short time.\
In this work, we propose an algorithmic approach for real-time instance segmentation of Lidar sensor data. We show how our method leverages the properties of the Euclidean distance to retain three-dimensional measurement information, while being narrowed down to a two-dimensional representation for fast computation. We further introduce what we call *skip connections*, to make our approach robust against over-segmentation and improve assignment in cases of partial occlusion. Through detailed evaluation on public data and comparison with established methods, we show how these aspects enable state-of-the-art performance and runtime on a single CPU core.
author:
- 'Frederik Hasecke$^{1,2}$, Lukas Hahn$^{1,2}$ and Anton Kummert$^{1}$[^1] [^2]'
bibliography:
- 'root.bib'
title: ' **Fast Lidar Clustering by Density and Connectivity** '
---
INTRODUCTION
============
Precise segmentation of object instances is an important tool for a variety of applications ranging from object detection to computer-aided data labelling. Lidar sensor data is usually represented as a three-dimensional point cloud in Cartesian coordinates. Hence it makes sense to consider clustering algorithms to fulfil the task of object segmentation for this type of sensor. Classic spatial or multidimensional point-based approaches [@10.5555/3001460.3001507][@Ankerst1999] are suitable to solve this problem, but suffer from long and fluctuating runtime. This is a problem, especially for automotive applications on embedded devices which are geared towards reliable and fixed timings of operations. Other methods, like the approach by Moosmann et al. [@Moosmann2009], are targeted towards Lidar point clouds, but are also too complex for real-time application, since they work on finding clusters in the three-dimensional domain. More recent works in this field [@Bogoslavskyi2016][@bogoslavskyi17pfg][@Zermas2017] use lower dimensional representations to segment clusters, but still work sequentially in several iterations, resulting in a potentially high and/or fluctuating runtime.\
This work presents a Lidar point cloud segmentation approach which provides (1) a high level of accuracy in point cloud segmentation, while (2) being able to run in real time, faster than usual sensor recording frequencies at (3) a constant rate with very little fluctuation independent of the scene’s context. We do so by avoiding the creation of a three-dimensional point cloud from the range measurements provided by the Lidar scanner and work directly on the laser range values of the sensor. Alternatively, computations can be applied to a cylindrical range image projection of the three-dimensional point cloud. This approach circumvents the problem of sparsity in the point cloud by forcing a two-dimensional neighbourhood on each measurement and thus offers the advantage of working with dense, two-dimensional data with clearly defined neighbourhood relationships between adjacent measurements. We avoid a loop based implementation for the computation to bypass fluctuations in the runtime depending on the number of objects and complexity of the scene. A Python implementation of our approach runs in real time on a single CPU core.
RELATED WORK {#chapter:relatedwork}
============
There is a large number of previous works on Lidar instance segmentation especially in, but not limited to, automotive application. The main focus for most clustering approaches is on improving segmentation accuracy and execution time. Most of these separate objects in the three-dimensional space, resulting in high accuracy but comparatively long runtime. Implementations of these can be found in the DBSCAN [@10.5555/3001460.3001507], Mean Shift [@1055330][@Comaniciu2002] and OPTICS [@Ankerst1999] algorithm.\
Other approaches use voxelization to reduce the complexity of the point cloud and find clusters in this representation [@5548059] or bird’s eye view projection coupled with the height information to separate overlapping objects [@inproceedings].\
Moosmann et al. [@Moosmann2009] proposed the use of a local convexity criterion on the spanned surface of three-dimensional Lidar points in a graph-based approach. While showing good results in urban environments and avoiding the additional step of a ground plane extraction, the algorithm fails to achieve real-time capability. Based on this work, Bogoslavskyi and Stachniss [@Bogoslavskyi2016] used a similar criterion - the spanned angle between adjacent Lidar measurements in relation to the Lidar sensor origin - to define the convexity of the range image surface as a measure to segment objects. They further utilise the neighbourhood conditions in the range image to display a very fast execution time. Zermas et al. [@Zermas2017] exploit the same relationship in scan lines of Lidar sensors to find break points in each and merge the separate lines of channels into three-dimensional objects in a subsequent step.\
Other current methods use machine learning directly on three-dimensional point clouds [@Lahoud2019][@Zhang][@Yang2019], projections into a camera image [@Wang2019a], or on spherical projections of Lidar points in the range image space [@Wang2019], to segment object instances in point clouds. These results look very promising in some cases, but suffer from a longer runtime, which currently prevents application on embedded automotive hardware.
METHODS
=======
The raw data of Lidar sensors is usually provided as a list of range measurements, each coupled with a number relating to the origin channel and the lateral position. These two values are corresponding to the $y$ and $x$ position of the measurement in the range image shown in <span style="font-variant:small-caps;">Figure</span> \[pointcloudAndRangeImage\]. These values can also be used to create a three-dimensional representation of the measurements, but doing so will increase the computation time for the proposed method. Therefore, we work directly on the raw two-dimensional representation. <span style="font-variant:small-caps;">Figure</span> \[pointcloudAndRangeImage\] shows an example of a range image as well as a three-dimensional representation of Lidar measurements.\
![*Top:* Range image representation of the raw distance measurements. *Bottom:* Three-dimensional visualisation of the range measurements coupled with the respective channel of origin and lateral position.[]{data-label="pointcloudAndRangeImage"}](LidarFormat.png){width="50.00000%"}
For the application of object segmentation we assume that the Lidar is part of a ground-based vehicle. Therefore we want to extract and discard the points belonging to the ground plane from the computation. This will prevent the algorithm from connecting two instances via this plane. Here, a simple height based threshold is not sufficient, as the road surface itself can be uneven. Pitching and rolling of the ego vehicle can also influence the way the ground is perceived in the sensor data. Since we have detailed information about the Lidar sensor itself, we can use the angle position of each given channel to determine the angle in which the laser beam would hit a horizontal surface. We use this information to exclude all range image values belonging to a horizontal plane below a certain height. To do so, we compare each cell of the range image with the neighbouring one above using the equation $$\beta = \arctan{(\frac{d_2 \cdot \sin{\alpha}}{d_1 - d_2 \cdot \cos{\alpha}})},
\label{equation1}$$ in which the selected cell value is $d_2$ and the one from the cell above is $d_1$ corresponding to the respective depth measurement. For the implementation, we use the function atan2 instead of the arctan to ensure proper handling of all quadrants of the Euclidean plane. As a result, we obtain an image representing the angle values of the entrance angle in relation to the point cloud surface. Together with a lookup table of the channel angles $\delta_r$ with respect to the channel $r$, we can now exclude all range image cells that have a high probability of belonging to the ground. A similar metric has been described by Chu et al. [@Chu2017] to separate the ground without a ground plane estimation. The left drawing in <span style="font-variant:small-caps;">Figure</span> \[groundAngle\] depicts the relationship of the channel angles $\delta_r$ to the surface angles $\beta_r$.\
![Trigonometrical relations used in the proposed clustering method. *Left:* The angle measurement of Lidar points in the vertical direction can be used to define a horizontal orientation for the ground plane extraction. *Right:* With the Lidar Sensor in **O**, the lines **OA** and **OB** show two neighbouring distance measurements. The distance between the two measurements can be calculated using the spanned angle **$\alpha$** between the points.[]{data-label="groundAngle"}](angles_combined.png){width="45.00000%"}
After removing the Lidar measurements belonging to the ground plane from the range image, we use a systematic approach to create an object instance segmentation for Lidar sensor data through clustering in the image space of the range image. Inspired by the direct connectivity of Bogoslavskyi and Stachniss [@Bogoslavskyi2016], we exploit the neighbourhood relationship of adjacent measurements in the range image. As visualised in the right illustration of <span style="font-variant:small-caps;">Figure</span> \[groundAngle\], we compare the given range values $\vert\vert OA\vert\vert$ and $\vert\vert OB\vert\vert$ for each pair of Lidar measurements and apply the cosine law $$\begin{aligned}
D & = \sqrt{\vert\vert OA\vert\vert^2 + \vert\vert OB\vert\vert^2 - 2 \cdot \vert\vert OA\vert\vert \cdot \vert\vert OB\vert\vert \cdot \cos{(\alpha)}}\\
D & = \sqrt{d_1^2 + d_2^2 - 2 \cdot d_1 \cdot d_2 \cdot \cos{(\alpha)}}\\
\end{aligned}
\label{equation2}$$ to calculate the Euclidean distance $D$ of the measured points. The $\alpha$ angle is required for the calculation and usually provided by the manufacturer of the Lidar sensor for both the horizontal and vertical direction. Using the physical distance between two measured points, we can define a threshold value between those which are close enough together to belong to the same object, or too far apart to be considered neighbours on the same object. The distance of neighbouring points on a given object is in general relatively dense. The distances of those points in the range image from two separate objects are substantially larger.\
With use of the Euclidean distance as a threshold value, we provide a single parameter implementation with a clear physical meaning which is adaptable to different sensors. By exclusively using variables which are given by the range measurements and reducing the computational effort by pre-calculating the cosine of the given angles, we can reduce the calculation of the squared Euclidean distance to a total of four multiplications, one addition and one subtraction. $$\begin{aligned}
D & = \sqrt{d_1^2 + d_2^2 - 2 \cdot d_1 \cdot d_2 \cdot \cos{(\alpha)}}\\
D^2 & = d_1 \cdot d_1 + d_2 \cdot d_2 - \underbrace{2 \cdot \cos{(\alpha)}}_{Constant} \cdot d_1 \cdot d_2\\
D^2 & = d_1 \cdot d_1 + d_2 \cdot d_2 - c \cdot d_1 \cdot d_2\\
\end{aligned}
\label{equation3}$$ These operations are present and optimised in most embedded devices and allow a much more efficient runtime for an automotive application of this approach.\
With the calculated threshold between each measurement, we are able to connect all Lidar points in the range image into separate clusters and background points. Here we intentionally decided against a loop based implementation to iterate over all rows and columns of the range image to connect or separate the cells in the image, as this tends to be susceptible to disruptions in the runtime. Especially in scenes of multiple small objects, which force many iterations of the outer loop. We recommend the work by Bogoslavskyi and Stachniss [@Bogoslavskyi2016] for further reading on a loop based implementation of a threshold calculation between measurements in a range image implementation.\
Our approach exploits the vectorised nature of the range image by applying the connection or separation of neighbouring points, without the use of a loop over the rows and columns in the image. To do so, we create two virtual copies of the range image and shift one copy over the $x$ axis and the other one over the $y$ axis, thus shifting each measurement by one cell in the $x$ and $y$ direction respectively. These shifted images enable us to stack them on a third axis and compare each value with his vertical and horizontal neighbour over the whole multidimensional array.\
By applying the threshold on these distance values calculated for each measurement and its direct neighbourhood, we end up with the original range image and two binary images representing the connection or separation between two points in the range image. For further use of these images and to reduce the computational expenditure, the range image can be reduced to a binary image representing the presence and absence of Lidar measurements for the corresponding pixels in the range image.\
The three created binary images contain all required information to segment the Lidar measurements of the whole frame into clusters and background points. For this we utilise a simple and efficient image processing algorithm; the pixel connectivity. The 4-connected pixel connectivity, also known as von Neumann neighbourhood, is defined as a two-dimensional square lattice composed of a central cell and its four adjacent cells. To apply the pixel connectivity on our data, we need to combine the binary Lidar measurements with the binary threshold images of connected measurements. By arranging these three images as shown in <span style="font-variant:small-caps;">Figure</span> \[supersize\], we are able to apply 4-connectivity on the resulting image to label each island of interconnected measurements as a different cluster. The usage of this condition allows us to avoid a loop based implementation once again. The resulting segmented image can now be subsampled to the original range image.
![ Combination of defined image representations for instance segmentation. The red squares represent the Lidar range values, the yellow and blue squares represent the horizontal and vertical connections of these measurements respectively. The black cross depicts how a 4-connectivity kernel is applied to the resulting combined image.[]{data-label="supersize"}](supersize_resize_color_friendly.png){width="30.00000%"}
In a subsequent step we can apply a threshold on the labelled clusters for objects below a certain number of Lidar measurements to reduce false clusters resulting from noise in the sensor, in our case we decided on a minimum of 100 points to be considered a cluster candidate. We have thus segmented the measurements to connected components of separate objects and non-segmented points which correspond to the ground plane and background noise.
Skip connections
----------------
Due to the characteristics of Lidar sensors, namely the sparsity (especially in vertical direction) and missing measurements resulting from deflected laser beams, which have no remission value back to the sensor, segmentation algorithms like the proposed one can be prone to under- or over-segmenting. Missing values can result in missing connections between areas of the same object, due to which the direct neighbourhood approach described above will over-segment a single object into multiple clusters. Examples of such challenging instances are shown in <span style="font-variant:small-caps;">Figure</span> \[SkipResults\].\
To overcome the limitations of the direct neighbourhood approach and to ensure a more robust segmentation, we have extended the two-dimensional Euclidean clustering by what we have named *skip connections*. For this, we reduce the combined image shown in <span style="font-variant:small-caps;">Figure</span> \[supersize\] to a sparse matrix, connecting only every second point in the vertical and horizontal direction, thus connecting a subset of original points. The schematic visualisation in <span style="font-variant:small-caps;">Figure</span> \[SkipConnections\] displays a connection of each measurement with its second neighbour. Due to the known $\alpha$ angle between all measurements, we can extend the Euclidean distance calculation from each measurement to any other using the cosine law described above. This allows us to robustly connect segments of the same object, which have no direct connection due to missing measurements in the range image. An example of this improved segmentation can be seen in <span style="font-variant:small-caps;">Figure</span> \[SkipResults\]. These additional *skip connections* can be adjusted according to the utilised sensor depending on the sparsity and proneness to errors. In the results shown in <span style="font-variant:small-caps;">Section</span> \[chapter:evaluation\] we have added *two-skip connections* as visualised in <span style="font-variant:small-caps;">Figure</span> \[SkipConnections\].
![Additional *skip connections* (dotted lines) between non-neighbouring Lidar points on top of the direct connections via neighbouring points (yellow and blue squares).[]{data-label="SkipConnections"}](Skip_connections_color_friendly.png){width="40.00000%"}
The combined use of the direct connectivity of neighbouring measurements and the *skip connections* enables a pseudo three-dimensional Euclidean clustering while exploiting the fast runtime of two-dimensional pixel connectivity. Thus, we are able to improve the quality of the segmentation without sacrificing our real-time ability.
EXPERIMENTAL EVALUATION {#chapter:evaluation}
=======================
We have set up the evaluation to prove our three claims of (1) a high level of accuracy in point cloud segmentation, (2) being able to run in real time at usual sensor recording frequencies and (3) offering a constant processing rate with very little fluctuation independent of the scene’s context. The first experimental evaluation refers to the second and third claim, the second experiment is mainly concerned with a qualitative metric of the segmentation results.
Runtime
-------
Following the experimental setup of [@Bogoslavskyi2016][@bogoslavskyi17pfg] we designed our first experiment on the provided data by Moosmann et al. [@Moosmann2013_1000032359] to support the claim that the proposed approach can be used for online segmentation. All listed methods have been evaluated on the same Intel^^ Core i7-6820HQ CPU @ 2.70 GHz. <span style="font-variant:small-caps;">Figure</span> \[fig:timeplot\] shows the execution time of the four methods over the 2500 Frames dataset. Please note, that we have removed the empty frames of the dataset from the evaluation. The proposed method runs as fast as the algorithm of [@Bogoslavskyi2016][@bogoslavskyi17pfg] while suffering from less fluctuation due to the vectorised implementation. A box-plot of the average runtime of [@Bogoslavskyi2016] and the proposed method can be seen in <span style="font-variant:small-caps;">Figure</span> \[fig:boxplot\] which shows the fluctuating nature of the loop based method as opposed to ours.
![Frame-wise execution timings for a 64-beam Velodyne dataset [@Moosmann2013_1000032359]. Please note the logarithmic scale for the runtime.[]{data-label="fig:timeplot"}](all_runs.png){width="48.00000%"}
![Averaged timings for segmenting approximately 2,500 scans from a 64-beam Velodyne dataset [@Moosmann2013_1000032359] with our approach and the method by Bogoslavskyi and Stachniss [@Bogoslavskyi2016].[]{data-label="fig:boxplot"}](BoxPlotRuntime.png){width="46.00000%"}
As can be seen in <span style="font-variant:small-caps;">Figure</span> \[fig:timeplot\], the proposed method together with the additional *skip connections* suffers from a slightly longer runtime, while still running at a frequency above 30 Hz. This is three times faster than the recording frequency of the used sensor.
Segmentation Results
--------------------
For our evaluation, we use the dataset “SemanticKITTI” by Behley et al. [@Behley2019] as ground truth for object instance segmentation. For each ground truth object with more than 100 Lidar point measurements, we select the cluster of each algorithm with the most overlap with the ground truth. Using these two lists of points, we calculate the Intersection over Union (IoU). We compute this metric for each algorithm listed below, over all ten sequences with Lidar instance ground truth in the dataset. We present the mean and standard deviation of the IoUs with the best parameter for each method. With our threshold parameter as $\unit{0.8m}$ on the direct neighbourhood implementation without *skip connections*, we outperform [@Bogoslavskyi2016][@bogoslavskyi17pfg] with the same average runtime. Together with the proposed *skip connections* and our distance threshold as $\unit{0.6m}$ we manage to come very close to the more general three-dimensional Euclidean distance clustering algorithm DBSCAN as shown in <span style="font-variant:small-caps;">Table</span> \[table:iou\] while being on average faster by a factor of 17 and a maximum-value difference by a factor of 60 due to the fluctuating nature of the DBSCAN algorithm (80 and 225 times faster without *skip connections*).\
For a further evaluation of the instance-level performance, we compute the average precision in a similar fashion as established benchmarks for instance segmentation for image data. We define 10 bins of point-wise overlap of the ground truth and proposed clusters ranging from an IoU of 0.5 to 0.95 in steps of 0.05. We average the precision of all bins into one single metric score, the Average Precision (AP) which is shown in <span style="font-variant:small-caps;">Table</span> \[table:ap\] for each method. We also additionally list the AP for the overlap values of 50%, 75% and 95% in which the evaluation is restricted to objects above the denoted IoU. We show in <span style="font-variant:small-caps;">Table</span> \[table:ap\] that we matched as many ground truth instances as the DBSCAN algorithm for an IoU over 50%, while we are still above 60% AP for all instances with an IoU over 75%. This is particularly important in the context of driver assistance systems, since a missed instance is a bigger problem than an object that was not matched perfectly. We further proved, that the proposed *skip connections* improve the results of our algorithm and help to find otherwise missed objects.
------------------------------------------------------------ ------------- ---------------- ------------------- ----------------------
**Method** $IoU_{\mu}$ $IoU_{\sigma}$ $IoU_{\mu}^{0.5}$ $IoU_{\sigma}^{0.5}$
Bogoslavskyi et al.[@Bogoslavskyi2016][@bogoslavskyi17pfg] 63.75 32.19 81.31 11.58
Ours 65.29 33.07 83.37 11.72
Ours 66.64 32.74 83.70 11.30
(*Skip Connections*)
DBSCAN**** 67.61 34.92 86.59 11.04
------------------------------------------------------------ ------------- ---------------- ------------------- ----------------------
: Comparison of the Intersection over Union of Bogoslavskyi et al. [@Bogoslavskyi2016][@bogoslavskyi17pfg], DBSCAN [@10.5555/3001460.3001507], and the proposed method[]{data-label="table:iou"}
![*Left:* Direct connectivity between neighbouring Lidar points. *Right:* Additional *skip connections* between every second Lidar measurement. The proposed *skip connections* enable a more accurate segmentation of the car and reduce the over-segmentation of occluded objects as the truck in the bottom figures.[]{data-label="SkipResults"}](SkipSuccess.png "fig:"){width="48.00000%"} ![*Left:* Direct connectivity between neighbouring Lidar points. *Right:* Additional *skip connections* between every second Lidar measurement. The proposed *skip connections* enable a more accurate segmentation of the car and reduce the over-segmentation of occluded objects as the truck in the bottom figures.[]{data-label="SkipResults"}](skipsuccess02.png "fig:"){width="48.00000%"}
------------------------------------------------------------ ------- ------------- ------------- -------------
**Method** $AP$ $AP^{50\%}$ $AP^{75\%}$ $AP^{95\%}$
Bogoslavskyi et al.[@Bogoslavskyi2016][@bogoslavskyi17pfg] 49.66 73.45 54.12 4.66
Ours 52.80 73.59 57.45 8.66
Ours 54.72 75.57 60.11 8.23
(*Skip Connections*)
DBSCAN**** 58.54 74.86 63.27 17.92
------------------------------------------------------------ ------- ------------- ------------- -------------
: Comparison of the Average Precision of Bogoslavskyi et al. [@Bogoslavskyi2016][@bogoslavskyi17pfg], DBSCAN [@10.5555/3001460.3001507], and the proposed method[]{data-label="table:ap"}
CONCLUSION
==========
We have presented an algorithm for real-time instance segmentation of Lidar sensor data by using raw range images. To make this approach more robust against over-segmentation, we introduced what we call *skip connections*, to use the larger neighbouring context for a more precise assignment of measured points to an instance, especially in cases of partial occlusion. These properties of our method facilitate the preservation of three-dimensional information in the measurements when being reduced to a two-dimensional representation for fast computation. In a detailed evaluation, we have shown, that our approach is as fast as comparable state-of-the-art methods, while being more stable in its runtime, and more importantly, providing an overall better performance in instance segmentation. The experiments show, that not only our accuracy in separating objects is higher than with comparable approaches, but we are able to match most ground truth instances that only the far more complex DBSCAN algorithm was able to match. This is particularly important in the context of driver assistance systems, since a missed instance is a bigger problem than an object that was not matched perfectly. We have also shown, that the proposed *skip connections* improve the results of our algorithm and help to find otherwise missed objects. The further use of segmented point clouds for classification and to remove false positives, is outside of the scope of this work and will be covered in future publications.
[^1]: $^{1}$ University of Wuppertal, Department of Electical Engineering
[^2]: $^{2}$ Aptiv, Wuppertal, Germany
|
---
abstract: |
We study the nuclear $\mu^--e^-$ conversion in the general framework of the effective Lagrangian approach without referring to any specific realization of the physics beyond the standard model (SM) responsible for lepton flavor violation (LFV). We analyze the role of scalar meson exchange between the lepton and nucleon currents and show its relevance for the coherent channel of conversion. We show that this mechanism introduces modifications in the predicted conversion rates in comparison with the conventional direct nucleon mechanism, based on the contact type interactions of the nucleon currents with the LFV leptonic current. We derive from the experimental data lower limits on the mass scales of the generic LFV lepton-quark contact terms and demonstrate that they are more stringent than the similar limits existing in the literature.
.3cm
[*PACS:*]{} 12.60.-i, 11.30.Er, 11.30.Fs, 13.10.+q, 23.40.Bw
[*Keywords:*]{} Lepton flavor violation, $\mu -e$ conversion in nuclei, scalar mesons, hadronization, physics beyond the standard model.
address: |
Institut für Theoretische Physik, Universität Tübingen,\
Auf der Morgenstelle 14, D-72076 Tübingen, Germany\
Departamento de Física, Universidad Técnica Federico Santa María,\
Casilla 110-V, Valparaíso, Chile\
author:
- |
Amand Faessler , Thomas Gutsche , Sergey Kovalenko ,\
Valery E. Lyubovitskij , Ivan Schmidt
title: 'Scalar meson mediated nuclear $\mu^--e^-$ conversion'
---
Introduction
============
The study of lepton flavor violating (LFV) processes offers a good opportunity for shedding light on the possible physics beyond the Standard Model (SM). Muon-to-electron ($\mu^--e^-$) conversion in nuclei $$\begin{aligned}
\mu^- + (A,Z) \longrightarrow e^- \,+\,(A,Z)^\ast
\label{I.1} \end{aligned}$$ is commonly recognized as one of the most sensitive probes of lepton flavor violation and of the related physics behind it (for reviews, see [@Kosmas:1993ch; @mu-e; @theory-exp]).
At present, on the experimental side there is one running conversion experiment, SINDRUM II [@Honecker:1996zf], and two planned ones, MECO [@Molzon; @MECO] and PRIME [@PRIME]. So far this LFV process has not been observed and experimental results correspond to the upper limits on the conversion branching ratio $$\begin{aligned}
\label{Ti}
&&R_{\mu e}^{A} = \frac{\Gamma(\mu^- + (A,Z) \rightarrow e^- +
(A,Z))} {\Gamma(\mu^- + (A,Z)\rightarrow \nu_{\mu} + (A,Z-1))}\,. \end{aligned}$$ The current and expected limits from the above mentioned experiments are presented in Table I. As is known from previous studies (see, for instance, Ref. [@mu-e; @theory-exp; @Faessler:2004ea] and references therein) and will be discussed later in the present paper, these experimental bounds allow to set stringent limits on the mechanisms of conversion and the underlying theories of LFV.
The theoretical studies of conversion, presented in the literature, cover various aspects of this LFV process, elaborating adequate treatment of the structure effects [@Kosmas:1993ch; @Kosmas:2001mv; @Faessler:pn] of the nucleus participating in the reaction and, considering underlying mechanisms of LFV at the level of quarks within different scenarios of physics beyond the SM (see [@mu-e; @theory-exp] and references therein).
In general the conversion mechanisms can be classified as photonic and non-photonic. In the former case the conversion is mediated by the photon exchange between the LFV leptonic vertex and the ordinary electromagnetic nuclear vertex. The non-photonic mechanisms are based on the 4-fermion contact lepton-quark LFV interactions. These two categories of mechanisms differ significantly from each other since they receive different contributions from the new physics and require different treatment of the effects of the nucleon and the nuclear structure.
In the present paper we continue studying the non-photonic meson exchange mechanism of conversion. Previously [@Faessler:2004ea; @Faessler:2004jt] we analyzed the vector-meson mediation of the conversion. The contribution of this mechanism to the coherent conversion results in some important issues for the physics beyond the SM absent in the case of the conventional direct lepton-nucleon interaction. Here, we extend our analysis to the scalar-meson exchange mechanism, which completes the study of meson exchange contributions to the coherent mode of the conversion [^1].
General Framework
=================
The effective Lagrangian ${\cal L}_{eff}^{lq}$ describing the coherent conversion at the quark level can be written in the form [@Kosmas:2001mv; @Faessler:2004jt] $$\begin{aligned}
{\cal L}_{eff}^{lq}\ =\ \frac{1}{\Lambda_{LFV}^2}
\left[(\eta_{VV}^{q} j_{\mu}^V\ + \eta_{AV}^{q}
j_{\mu}^A )J_{q}^{V\mu} +
(\eta_{SS}^{q} j^S\ + \eta_{PS}^{q} j^P\ )J_{q}^{S}\right],
\label{eff-q}\end{aligned}$$ where the lepton and quark currents are defined as: $$\begin{aligned}
\label{lepton-currents}
&&j_{\mu}^V = \bar e \gamma_{\mu} \mu\,, \,\,\,
j_{\mu}^A = \bar e \gamma_{\mu} \gamma_{5} \mu\,, \,\,\,
j^S = \bar e \ \mu\,, \\ \nonumber
&& j^P = \bar e \gamma_{5} \mu\,, \,\,\,
J_{q}^{V\mu} = \bar q \gamma^{\mu} q\,, \,\,\,
J_{q}^{S} = \bar q \ q \,.\end{aligned}$$ In Eqs. (\[eff-q\]) and (\[lepton-currents\]) the summation runs over all the quark species $q= \{u,d,s,b,c,t\}$. The parameter $\Lambda_{LFV}$ with the dimension of mass is the characteristic high energy scale of lepton flavor violation attributed to new physics. The dimensionless LFV parameters $\eta^{q}$ in Eq. (\[eff-q\]) depend on a concrete LFV model. In the present study we treat these parameters as phenomenological parameters to be constrained from the experiment.
The quark level Lagrangian (\[eff-q\]) generates the effective lepton-nucleon interactions that can be specified in terms of an effective Lagrangian on the nucleon level $$\begin{aligned}
\hspace*{-.5cm}
{\cal L}_{eff}^{lN} = \frac{1}{\Lambda_{LFV}^2}
\left[j_{\mu}^a (\alpha_{aV}^{(0)} J^{V\mu \, (0)} +
\alpha_{aV}^{(3)} J^{V\mu \, (3)}) + j^b (\alpha_{bS}^{(0)}
J^{S \, (0)} + \alpha_{bS}^{(3)} J^{S \, (3)})\right]\,.
\label{eff-N} \end{aligned}$$ Here, the isoscalar $J^{(0)}$ and isovector $J^{(3)}$ nucleon currents are $J^{V\mu \, (k)} = \bar N \, \gamma^\mu \, \tau^k \, N\,$ and $J^{S \, (k)} = \bar N \, \tau^k \, N\,,$ where $N$ is the nucleon isospin doublet, $ k = 0,3 $ and $\tau_0\equiv\hat I$. The summation over the double indices $a = V,A$ and $b = S,P$ is implied in (\[eff-N\]). The nucleon Lagrangian (\[eff-N\]) is the basis for the derivation of the nuclear transition operators.
Naturally, the Lagrangian in terms of effective nucleon fields (\[eff-N\]) is equivalent to the quark level Lagrangian (\[eff-q\]). The former Lagrangian is supposed to appear after the hadronization from the quark level Lagrangian (\[eff-q\]) and, therefore, must correspond to the same order $1/\Lambda_{LFV}^{2}$ in inverse powers of the LFV scale.
To make a bridge between the underlying LFV physics and observables one needs to relate the lepton-nucleon LFV parameters $\alpha$ in Eq. (\[eff-N\]) to the lepton-quark LFV parameters $\eta$ in Eq. (\[eff-q\]). This implies a certain hadronization prescription which specifies the way in which the effect of quarks is simulated by hadrons. In the absence of a true theory of hadronization we rely on some reasonable assumptions and models. In Refs. [@Faessler:2004ea; @Faessler:2004jt] we considered the two mechanisms of nuclear $\mu^- - e^-$ conversion: direct nucleon mechanism and vector-meson exchange between nucleon and lepton currents. We found that the vector-meson exchange plays an important role in the coherent muon-electron conversion. Now, we extend this analysis to the scalar sector of the LFV Lagrangians in Eqs. (\[eff-q\]) and (\[eff-N\]).
As in the case of the vector currents, considered in Refs. [@Faessler:2004ea; @Faessler:2004jt], here we distinguish the following two hadronization mechanism. The first one is the direct embedding of the quark currents into the nucleon (Fig.1a), which we call direct nucleon mechanism (DNM). The second mechanism consists of two stages (Fig.1b). First, the quark currents are embedded into the interpolating scalar meson fields which then interact with the nucleon currents. We call this possibility meson-exchange mechanism (MEM).
In general one expects all the mechanisms to contribute to the coupling constants $\alpha$ in Eq. (\[eff-N\]). However, at present the relative amplitudes of different mechanisms are unknown. In view of this problem we assume for the first approximation, that only one mechanism is operative and estimate its contribution to the process in question. This allows us to evaluate the importance of a specific mechanism.
Let us update the contribution of the direct nucleon mechanism derived in Ref. [@Kosmas:2001mv]. The relation between the quark-lepton and nucleon-lepton LFV parameters in Eqs. (\[eff-q\]) and (\[eff-N\]) takes in this case the form $$\begin{aligned}
\label{alpha}
\hspace*{-.65cm}
&&\alpha_{bS[DNM]}^{(3)} = \frac{1}{2}\eta_{bS}^{(3)}
(G_{S}^{u} - G_{S}^{d}),
\\ \nonumber
&&\alpha_{bS[DNM]}^{(0)} = \frac{1}{2}\eta_{bS}^{(0)}
(G_{S}^{u} + G_{S}^{d}) + \eta_{bS}^{s} G_{S}^{s},\end{aligned}$$ where $b=S,P$ and $\eta^{(0,3)}=\eta^{u}\pm\eta^{d}$ are the isoscalar and isovector quark couplings. The form factors $G_{S}^{q}$ are related to the scalar condensates in the nucleon [@Inoue:2003bk] $$\begin{aligned}
\label{mat-el1}
&&\langle p|\bar{u}\ u|p\rangle =
G_{S}^{u} \bar{p}\ p, \ \ \
\langle p|\bar{d}\ d|p\rangle =
G_{S}^{d} \bar{p}\ p, \ \ \
\langle p|\bar{s}\ s|p\rangle =
G_{S}^{s} \bar{p}\ p,
\\ {\nonumber}&&\langle n|\bar{u}\ u|n\rangle =
G_{S}^{d} \bar{n}\ n, \ \ \
\langle n|\bar{d}\ d|n\rangle =
G_{S}^{u} \bar{n}\ n, \ \ \
\langle n|\bar{s}\ s|n\rangle =
G_{S}^{s} \bar{n}\ n . {\nonumber}\end{aligned}$$ Since the maximal momentum transfer $q$ in $\mu^- -e^-$ conversion is much smaller than the typical scale of the nucleon structure we can safely neglect the $q^2$-dependence of the nucleon form factors $G_{S}^{q}$. At $q^2=0$ these form factors are related to the corresponding meson-nucleon sigma-terms: $$\begin{aligned}
\label{sigma}
\sigma_{\pi N} = \hat m \, [ G_{S}^{u} + G_{S}^{d} ]\,, \,\,\,
\sigma_{K N}^{I=1} &=& \frac{\hat m + m_s}{4} \, [ G_{S}^{u} - G_{S}^{d} ]
\,, \,\,\,
y_N = \frac{2 G_{S}^{s}}{G_{S}^{u} + G_{S}^{d}},\end{aligned}$$ where $\hat m=(m_u+m_d)/2$ and $m_s$ are the masses of current quarks; $y_N$ is the strangeness of the nucleon. For these parameters we use the following values [@Gasser:1982ap; @Gasser:1990ce] in our analysis $$\hat m = 7 \;{\rm MeV},\; m_s/\hat m=25, \; y_N = 0.2\, .$$ The canonical value of the $\pi N$ sigma term $\sigma_{\pi N} = 45 \pm 8$ MeV [@Gasser:1990ce] was originally extracted from the dispersional analysis of $\pi N$ scattering data taking into account chiral symmetry constraints. In particular, the value of the sigma-term, $\sigma_{\pi N} = 45 \pm 8$ MeV, has been deduced from the analysis of two quantities: $\sigma_{\pi N}(t=2M_{\pi}^2) = 60 \pm 8 $ MeV, the scalar nucleon form factor at the Cheng-Dashen point $t=2M_\pi^2$, and the difference $\Delta_\sigma =\sigma_{\pi N}(2M_{\pi}^2) -\sigma_{\pi N}(0)=15.2 \pm 0.4$ MeV [@Gasser:1990ce]
as induced by explicit chiral symmetry breaking. The value of the isovector kaon-nucleon sigma-term $\sigma_{KN}^{I=1}$ was estimated in Ref. [@Gasser:2000wv] using the baryon mass formulas: $$\begin{aligned}
\sigma_{KN}^{I=1} \sim \frac{m_s + \hat{m}}{m_s - \hat{m}} \,
\frac{m_\Xi^2 - m_\Sigma^2}{8 m_P} = 48 \,\, {\rm MeV}
\sim 50 \,\, {\rm MeV} \, .\end{aligned}$$ Substituting the above values of the hadronic parameters to Eq. (\[sigma\]) we obtain for the scalar nucleon form factors: $$\begin{aligned}
\label{sffn}
G_{S}^{u} = 3.74\,, \ G_{S}^{d} = 2.69\,, \ G_{S}^{s} = 0.64\,.\end{aligned}$$ We have to point out that these values contain appreciable theoretical and experimental uncertainties, as seen from their derivation (for the possible error bars see, for instance, Ref. [@Corsetti:2000yq]). In our analysis of the DNM contribution to we take the numbers from Eq. (\[alpha\]) as central values of the scalar nucleon form factors. Note, in Ref. [@Kosmas:2001mv] the different set of the values for the nucleon scalar form factors was derived on the basis of the QCD sum rules input parameters, which overestimates the pion-nucleon sigma-term and the strangeness of the nucleon.
Scalar Meson Contribution
=========================
In the following we turn to the scalar meson-exchange mechanism of conversion. Although the status of scalar mesons is still unclear [@Eidelman:2004wy] we think it is reasonable to study their effect in the conversion since their contribution is associated with the experimentally most interesting coherent mode of this exotic process.
The lightest unflavored scalar mesons are the isoscalar $f_0(600)$ and the isotriplet $a_0(980)$ states. The former in the context of the nonlinear realization of chiral symmetry can be treated as a resonance in the $\pi\pi$ system (see detailed discussion, e.g. in Refs. [@Colangelo:2001df; @Oset:2000gn]). For simplicity we neglect a possible small strange content of the isoscalar meson and treat this state as $\bar u u + \bar d d$.
We derive the LFV lepton-meson effective Lagrangian in terms of the interpolating $f_0$ and $a_0^0$ fields. Retaining all the interactions consistent with Lorentz invariance, we obtain the general form of this Lagrangian: $$\begin{aligned}
\label{eff-LS}
{\cal L}_{eff}^{lS}\ &=& \ \frac{\Lambda_H^2}{\Lambda_{LFV}^2}
\biggl[ ( \xi_S^{f_0} j^S \, + \, \xi_P^{f_0}j^P ) \, f_0
+ ( \xi_S^{a_0} j^S \, + \, \xi_P^{a_0} j^P ) \, a_0^0 \biggr] \end{aligned}$$ with the unknown dimensionless coefficients $\xi$ to be determined from the hadronization prescription. We assume that the Lagrangian to be generated by the quark-lepton Lagrangian (\[eff-q\]), and, therefore, all its terms have the same suppression $\Lambda_{LFV}^{-2}$ with respect to the large LFV scale $\Lambda_{LFV}$. Another scale in the problem is the hadronic scale $\Lambda_H \sim 1$ GeV which adjusts the physical dimensions of the terms in Eq. (\[eff-LS\]). In the Lagrangian in Eq. (\[eff-LS\]) we neglect derivative terms since their contribution to conversion is suppressed by a factor $(m_{\mu}/\Lambda_H)^2\sim 10^{-2}$.
In order to relate the parameters $\xi$ of the Lagrangian (\[eff-LS\]) to the “fundamental" parameters $\eta$ of the quark-lepton Lagrangian (\[eff-q\]) we use an approximate method based on the standard on-mass-shell matching condition [@Faessler:1996ph] $$\label{match}
\langle \mu^+ \, e^-|{\cal L}_{eff}^{lq}|S\rangle \approx
\langle \mu^+ \, e^-|{\cal L}_{eff}^{lS}|S \rangle ,$$ where $|S= f_0, a_0 \rangle$ are the on mass-shell scalar meson states. We solve equation (\[match\]) using the quark current matrix elements $$\begin{aligned}
\label{mat-el2}
&&\langle 0|\bar u \, u|f_0(p)\rangle \, = \,
\langle 0|\bar d \, d|f_0(p)\rangle
\, = \, m_{f_0}^2 \, f_{f_0}\,,
\\
&& \langle 0|\bar u \, u|a_0^0(p)\rangle =
\,- \, \langle 0|\bar d \, d|a_0^0(p)\rangle
\, = \, m_{a_0}^2 \, f_{a_0^0} \,.
\nonumber\end{aligned}$$ Here $p$, $m_S$ and $f_{S}$ are the scalar-meson four-momentum, mass and decay constant, respectively. The quark operators in Eq. (\[mat-el2\]) are taken at $x=0$. In the numerical calculations we use the following values of scalar meson masses [@Eidelman:2004wy]: $$\begin{aligned}
\label{constants}
m_{f_0} = 500 \,\, \mbox{MeV}\,, \hspace*{1cm}
m_{a_0} = 984.7 \,\, \mbox{MeV} \,. \end{aligned}$$ The coupling constants $f_{S}$ in Eqs. (\[mat-el2\]) can be estimated using the linear $\sigma$-model in the case of the $f_0$ meson [@Delbourgo:1993dk] and by QCD sum rules in the case of the $a_0$ [@Maltman:1999jn]. In the linear $\sigma$-model one has the following relationship [@Delbourgo:1993dk]: $$\begin{aligned}
\label{mat-el3}
\langle 0|\bar u \, u|f_0(p)\rangle = m_{f_0}^2 \, \frac{\sqrt{N_c}}{2\pi}, \end{aligned}$$ where $N_c=3$ is the number of quark colors. Comparing Eqs. (\[mat-el2\]) and (\[mat-el3\]) we get $$\begin{aligned}
f_{f_0} = \frac{\sqrt{N_c}}{2\pi} = 0.28 \,. \end{aligned}$$ The coupling constant $f_{a_0^0}$ of the neutral $a_0^0$ meson is related to the coupling constant $f_{a_0^\pm}$ of the charged $a_0^\pm$ state due to isospin invariance: $$\begin{aligned}
\label{a0-apm}
f_{a_0^\pm} \, = \, f_{a_0^0} \, \sqrt{2}\,,\end{aligned}$$ with the definition $\langle 0|\bar d \, u|a_0^-(p)\rangle =
m_{a_0}^2 f_{a_0^\pm}\,$. The value of $f_{a_0^\pm}$ was estimated in Ref. [@Maltman:1999jn]: $$\begin{aligned}
\label{apm}
f_{a_0^\pm} = \frac{0.0447 \ {\rm GeV}^3}{m_{a_0}^2 \ (m_s - \hat{m})} \,. \end{aligned}$$ Combining Eqs. (\[apm\]) and (\[a0-apm\]) we have $$\begin{aligned}
f_{a_0^0} \, = \, 0.19 \,. \end{aligned}$$ Solving Eq. (\[match\]) with the help of Eqs. (\[mat-el2\]), we obtain the expressions for the coefficients $\xi$ of the lepton-meson Lagrangian (\[eff-LS\]) in terms of the generic LFV parameters $\eta$ of the initial (\[eff-q\]) lepton-quark effective Lagrangian: $$\begin{aligned}
\xi_b^{a_0} \, = \, \left(\frac{m_{a_0}}{\Lambda_H}\right)^2
f_{a_0^0} \,\eta_{b S}^{(3)}\,, \hspace*{1cm}
\xi_b^{f_0} \, = \, \left(\frac{m_{f_0}}{\Lambda_H}\right)^2
\, f_{f_0} \, \eta_{b S}^{(0)}\,, \end{aligned}$$ where $b = S, P$ and $\eta^{(0,3)}=\eta^{u}\pm\eta^{d}$.
For our analysis we also need the effective Lagrangian describing the interactions of the scalar mesons with nucleons. We take it in the following form: $$\begin{aligned}
\label{MN}
{\cal L}_{SN} \, = \,
\bar{N} [ g_{_{a_0 NN}} \, \vec{a_0} \,
\vec{\tau} \, + \, g_{_{f_0 NN}} \, f_0] N\,. \end{aligned}$$ For the meson-nucleon couplings $g_{SNN}$ we adopt the central values $$\begin{aligned}
\label{SN-couplings}
g_{_{a_0 NN}} \simeq g_{_{f_0 NN}} \simeq 5 \, \end{aligned}$$ used in phenomenological description of nucleon-nucleon interactions and recently also calculated in the chiral unitary approach [@Oset:2000gn]. Since our analysis does not pretend to high accuracy we do not supply the error bars for the meson-nucleon couplings $g_{SNN}$.
Now, having specified the interactions of the scalar mesons $f_0, a_0$ with leptons and with nucleons we can derive the scalar meson-exchange contributions to the conversion. This contribution can be expressed in the form of the nucleon-lepton effective Lagrangian (\[eff-N\]) which arises in second order in the Lagrangian ${\cal L}_{eff}^{lS} + {\cal L}_{SN}$ and corresponds to the diagram in Fig.1b. We estimate this contribution only for the coherent $\mu^- - e^-$ conversion process. In this case we disregard all the derivative terms of nucleon and lepton fields. Neglecting the kinetic energy of the final nucleus, the muon binding energy and the electron mass, the square of the momentum transfer $q^2$ to the nucleus has a constant value $q^2 \approx - m_{\mu}^2$. In this approximation the meson propagators convert to $\delta$-functions leading to effective lepton-nucleon contact type operators. Comparing them with the corresponding terms in the Lagrangian (\[eff-N\]), we obtain the scalar meson exchange contribution to the coupling constants of this Lagrangian: $$\begin{aligned}
\label{alpha-S-ex}
\alpha_{bS[MEM]}^{(3)} &=& \beta_{a_0}\eta_{bS}^{(3)}\,
\hspace*{1cm}
\alpha_{bS[MEM]}^{(0)} = \beta_{f_0}\eta_{bS}^{(0)}\end{aligned}$$ with $b=S,P$ and the coefficients $$\begin{aligned}
\label{beta}
\beta{_{a_0}} = \frac{g_{_{a_0 NN}} \, f_{a_0^0} \,
m_{a_0}^2}{m_{a_0}^2
+ m_{\mu}^2}\,, \hspace*{1cm}
\beta{_{f_0}} = \frac{g_{_{f_0 NN}} \, f_{f_0} \, m_{f_0}^2 }
{m_{f_0}^2 + m_{\mu}^2}\,.\end{aligned}$$ Substituting the values of the meson coupling constants and masses we obtain for these coefficients $$\begin{aligned}
\label{beta-num}
\beta{_{a_0}} = 0.93 \,, \hspace*{1cm} \beta{_{f_0}} = 1.32 \,,\end{aligned}$$ These values should be considered as rough estimates in view of the uncertainties in the scalar meson masses and couplings.
Constraints on LFV parameters from conversion
=============================================
From the Lagrangian (\[eff-N\]), following the standard procedure, one can derive the formula for the branching ratio of the coherent $\mu^- - e^-$ conversion. To leading order in the nonrelativistic reduction the branching ratio takes the form [@Kosmas:1993ch] $$R_{\mu e^-}^{coh} \ = \
\frac{{\cal Q}} {2 \pi \Lambda_{LFV}^4} \ \
\frac{p_e E_e \ ({\cal M}_p + {\cal M}_n)^2 }
{ \Gamma_{\mu c} }
\, ,
\label{Rme}$$ where $p_e, E_e$ are 3-momentum and energy of the outgoing electron, ${{\cal M}}_{p,n}$ are the nuclear $\mu^- - e^-$ transition matrix elements and $\Gamma_{\mu c}$ is the total rate of the ordinary muon capture. The factor ${\cal Q}$ takes the form $$\begin{aligned}
\hspace*{-1cm}
{\cal Q} &=& |\alpha_{VV}^{(0)}+\alpha_{VV}^{(3)}\ \phi|^2 +
|\alpha_{AV}^{(0)}+\alpha_{AV}^{(3)} \phi|^2 +
|\alpha_{SS}^{(0)}+\alpha_{SS}^{(3)} \phi|^2 +
|\alpha_{PS}^{(0)} + \alpha_{PS}^{(3)} \phi|^2
\nonumber \\
\hspace*{-1cm}
&+& 2{\rm Re}\{(\alpha_{VV}^{(0)}+\alpha_{VV}^{(3)}
\phi)(\alpha_{SS}^{(0)}+ \alpha_{SS}^{(3)} \phi)^\ast
+ (\alpha_{AV}^{(0)}+\alpha_{AV}^{(3)}\ \phi)(\alpha_{PS}^{(0)} +
\alpha_{PS}^{(3)}\ \phi)^\ast\}\,
\label{Rme.1} \end{aligned}$$ in terms of the parameters of the lepton-nucleon effective Lagrangian (\[eff-N\]) and the nuclear structure factor $$\begin{aligned}
\label{phi}
\phi = ({\cal M}_p - {\cal M}_n)/({\cal M}_p + {\cal M}_n) \, ,\end{aligned}$$ that is typically small for the experimentally interesting nuclei.
The nuclear matrix elements ${\cal M}_{p,n}$ have been calculated in Refs. [@Kosmas:2001mv; @Faessler:pn] for the nuclear targets $^{27}$Al, $^{48}$Ti and $^{197}$Au. We show their values in Table II together with the experimental values of the total rates $\Gamma_{\mu c}$ of the ordinary muon capture [@Suzuki:1987jf] and the 3-momentum $p_e$ of the outgoing electron. Using the quantities from Table II we find for the dimensionless scalar lepton-nucleon couplings of the Lagrangian (\[eff-N\]) the following limits $$\begin{aligned}
\label{alpha-lim}
\alpha_{bS}^{(k)} \left(\frac{1 \mbox{GeV}}{\Lambda_{LFV}}\right)^2
\leq 1.2 \times 10^{-12} \left[B^{(k)}(A)\right]^{-2}, \ \ \ \end{aligned}$$ with $k=0,3$. Here the scaling factors $B^{(k)}(A)$ depend on the target nucleus $A$ used in an experiment setting the upper limit $R_{\mu e}^{A}(Exp)$ on the branching ratio of conversion: $R_{\mu e}^{A}\leq R_{\mu e}^{A}(Exp)$. The numerical values of these factors for the experiments discussed in the introduction have been calculated with the help of Table II and are given in Table I.
From the limits in Eq. (\[alpha-lim\]) one can deduce individual limits on the terms contributing to the coefficients $\alpha_{bS}^{(0,3)}$, assuming the absence of significant cancellations (unnatural fine-tuning) between the different terms. In this way, using Eqs. (\[alpha\]), (\[alpha-S-ex\]), we derive constraints on the 4-fermion quark-lepton LFV couplings of the Lagrangian (\[eff-q\]) for the two considered mechanisms of hadronizations: the direct nucleon mechanism (DNM) and the meson-exchange mechanism (MEM). The corresponding limits are shown in Table 3. Following the common practice, we presented these limits in terms of the individual mass scales, $\Lambda_{ij}$, of the scalar quark-lepton contact operators in Eq. (\[eff-q\]). In the conventional definition [@Eichten:1983hw] these scales determine the couplings $z_{ij}=g^2/\Lambda^2_{ij}$ of the 4-fermion contact terms of the form $z_{ij} (\bar{l_{i}}l_j)(\bar{q}q)$ with a fixed $g^2=4\pi$. Historically [@Eichten:1983hw] this definition of $\Lambda_{ij}$ originates from substructure models and corresponds to the compositeness scale in the strong coupling regime $g^2/4\pi=1$. Thus, the individual mass scales $\Lambda_{ij}$, introduced in this way, are related to our notations according to the formula: $$\begin{aligned}
\label{Lambda-LFV}
\frac{\eta_{bS}^{(k)}}{\Lambda^2_{LFV}} =
\frac{4\pi}{\left(\Lambda^{(k,b)}_{\mu e}\right)^2}\end{aligned}$$ with $k=0, 3, s$ and $b=P,S$.
Let us compare our limits for these mass scales with the corresponding limits existing in the literature. The limits on $\Lambda_{\mu e}$ can be derived from the experimental bounds on the rates of the $\pi^+\rightarrow \mu^+ \nu_e$, $\pi^0\rightarrow
\mu^\pm e^\mp$ decays [@Kim:1997rr]. Despite the scalar quark current does not directly contribute to these processes the limits come from the gauge invariance with respect to the SM group which relates the couplings of the scalar and pseudoscalar lepton-quark contact operators. Typical limits from these processes are of $\Lambda_{\mu e} \geq~\mbox{few TeV}$. In the future experiments at the LHC it is also planned to set limits on the mass scales of various contact quark-lepton interactions from the measurement of Drell-Yan cross sections in the high dilepton mass region [@Krasnikov:2003ef; @Gupta:1999iy]. In this case typical expected limits for the scales of the lepton flavor diagonal contact terms are $\Lambda_{l l} \geq 35~\mbox{TeV}$. We are not aware of the corresponding analysis for the limits on the LFV scales, like $\Lambda_{\mu e}$, expected from the experiments planned at the LHC. However, one may expect these limits to be significantly stronger (typically by an order of magnitude) than the above cited limits for the lepton flavor diagonal mass scales $\Lambda_{l l}$. This is motivated by the fact that, usually a flavor diagonal process have much more SM background than the LFV processes. A comparison of the above limits with the limits in Table III, extracted from , shows that the latter are more stringent.
The following comment is in order. As follows from Eq. (\[Rme.1\]) and Table II, the contribution of the isovector couplings $\alpha^{(3)}$ to the conversion rate, Eqs. (\[Rme\]) and (\[Rme.1\]), is suppressed by a small nuclear factor $\phi$. On the other hand, the information on the isovector couplings may be important for the phenomenology of the physics beyond the SM allowing one to distinguish the contribution of d and u quarks. In this respect, the contribution of the scalar mesons is important, since it is comparable on magnitude with the DNM mechanism. Taking this contribution into account we extracted reasonable constraints on mass scales of the isovector quark-lepton contact terms $\Lambda^{(3)}_{\mu e}$ presented in Table 3. The strange quark contributions are not affected by the meson exchange because we neglected the possible strange quark component of the scalar mesons.
Summary
=======
We analyzed the nuclear $\mu^--e^-$ conversion in the general framework the effective Lagrangian approach without referring to any specific realization of the physics beyond the standard model responsible for lepton flavor violation. The two mechanisms of the hadronization of the underlying effective quark-lepton LFV Lagrangian have been studied: the direct nucleon (DNM) and the scalar meson-exchange (MEM) mechanisms. We showed that the scalar meson-exchange contribution is comparable on magnitude with the DNM mechanism and, therefore, can modify the limits on the LFV lepton-quark couplings derived on the basis of the conventional direct nucleon mechanism. These results lead us to the conclusion that the meson exchange mechanism may have an appreciable impact on the phenomenology of the LFV physics beyond the standard model and, therefore, should be taken in to account in the analysis of the LFV effects in hadronic and nuclear semileptonic processes.
From the experimental upper bounds on the conversion rate we extracted the lower limits on the mass scales of the LFV lepton-quark contact terms involved in this process and showed that they are more stringent than the similar limits existing in the literature.
[**Acknowledgments**]{}
This work was supported by the FONDECYT projects 1030244, 1030355, by the DFG under contracts FA67/25-3, GRK683. This research is also part of the EU Integrated Infrastructure Initiative Hadronphysics project under contract number RII3-CT-2004-506078 and President grant of Russia “Scientific Schools” No.1743.2003.
[99]{} T. S. Kosmas, G. K. Leontaris and J. D. Vergados, Prog. Part. Nucl. Phys. [**33**]{}, 397 (1994) \[arXiv:hep-ph/9312217\]. W.J. Marciano, Lepton flavor violation, summary and perspectives, Summary talk in the conference on “New initiatives in lepton flavor violation and neutrino oscillations with very intense muon and neutrino beams”, Honolulu-Hawaii, USA, Oct. 2-6, 20002, River Edge, NJ: World Scientific, 2000 (see also $http://meco.ps.uci.edu/lepton\_workshop.$); Y. Kuno and Y. Okada, Rev. Mod. Phys. [**73**]{}, 151 (2001) \[arXiv:hep-ph/9909265\]. W. Honecker [*et al.*]{} \[SINDRUM II Collaboration\] , Phys. Rev. Lett. [**76**]{}, 200 (1996); P. Wintz, in Status of muon to electron conversion at PSI, Invited talk at [@mu-e; @theory-exp]; A. van der Schaaf, Private communication. W. Molzon, The improved tests of muon and electron flavor symmetry in muon processes, Spring. Trac. Mod. Phys. [**163**]{}, 105 (2000). J. Sculli, The MECO experiment, Invited talk at [@mu-e; @theory-exp]. Y. Kuno, The PRISM: Beam-Experiments, Invited talk at [@mu-e; @theory-exp]. A. Faessler, T. Gutsche, S. Kovalenko, V. E. Lyubovitskij, I. Schmidt and F. Simkovic, Phys. Rev. D [**70**]{} (2004) 055008 \[arXiv:hep-ph/0405164\]. T. S. Kosmas, S. Kovalenko and I. Schmidt, Phys. Lett. B [**511**]{}, 203 (2001); Phys. Lett. B [**519**]{}, 78 (2001). A. Faessler, T. S. Kosmas, S. Kovalenko and J. D. Vergados, Nucl. Phys. B [**587**]{}, 25 (2000). A. Faessler, T. Gutsche, S. Kovalenko, V. E. Lyubovitskij, I. Schmidt and F. Simkovic, Phys. Lett. B [**590**]{} (2004) 57 \[arXiv:hep-ph/0403033\]. T. Inoue, V. E. Lyubovitskij, T. Gutsche and A. Faessler, Phys. Rev. C [**69**]{} (2004) 035207 \[arXiv:hep-ph/0311275\]. J. Gasser and H. Leutwyler, Phys. Rept. [**87**]{}, 77 (1982). J. Gasser, H. Leutwyler and M. E. Sainio, Phys. Lett. B [**253**]{} (1991) 252, 260. J. Gasser and M. E. Sainio, arXiv:hep-ph/0002283; M. E. Sainio, PiN Newslett. [**16**]{}, 138 (2002) \[arXiv:hep-ph/0110413\]. A. Corsetti and P. Nath, Phys. Rev. D [**64**]{}, 125010 (2001) \[arXiv:hep-ph/0003186\]. S. Eidelman [*et al.*]{} \[Particle Data Group Collaboration\], Phys. Lett. B [**592**]{}, 1 (2004). G. Colangelo, J. Gasser and H. Leutwyler, Nucl. Phys. B [**603**]{} (2001) 125 \[arXiv:hep-ph/0103088\]; J. A. Oller, E. Oset and J. R. Pelaez, Phys. Rev. Lett. [**80**]{}, 3452 (1998) \[arXiv:hep-ph/9803242\]. E. Oset, H. Toki, M. Mizobe and T. T. Takahashi, Prog. Theor. Phys. [**103**]{}, 351 (2000) \[arXiv:nucl-th/0011008\]. A. Faessler, S. Kovalenko, F. Simkovic and J. Schwieger, Phys. Rev. Lett. [**78**]{}, 183 (1997). R. Delbourgo and M. D. Scadron, Mod. Phys. Lett. A [**10**]{}, 251 (1995) \[arXiv:hep-ph/9910242\]; R. Delbourgo, M. D. Scadron and A. A. Rawlinson, Mod. Phys. Lett. A [**13**]{}, 1893 (1998) \[arXiv:hep-ph/9807505\]. K. Maltman, Phys. Lett. B [**462**]{}, 14 (1999) \[arXiv:hep-ph/9906267\]. T. Suzuki, D. F. Measday and J. P. Roalsvig, Phys. Rev. C [**35**]{}, 2212 (1987). E. Eichten, K. D. Lane and M. E. Peskin, Phys. Rev. Lett. [**50**]{}, 811 (1983); V. D. Barger, K. Cheung, K. Hagiwara and D. Zeppenfeld, Phys. Lett. B [**404**]{}, 147 (1997) \[arXiv:hep-ph/9703311\]; Phys. Rev. D [**57**]{}, 391 (1998) \[arXiv:hep-ph/9707412\]. J. E. Kim, P. Ko and D. G. Lee, Phys. Rev. D [**56**]{}, 100 (1997) \[arXiv:hep-ph/9701381\]. N. V. Krasnikov and V. A. Matveev, Phys. Usp. [**47**]{}, 643 (2004) \[Usp. Fiz. Nauk [**174**]{}, 697 (2004)\] \[arXiv:hep-ph/0309200\]. A. K. Gupta, N. K. Mondal and S. Raychaudhuri, arXiv:hep-ph/9904234.
[**Table I.**]{} The experimental upper limits on the conversion branching ratio $R_{\mu e}^{A}$ and the values of the scaling factors $B^{(0,3)}(A)$ from Eq. (\[alpha-lim\]) for the running and forthcoming experiments.
Target, Experiment $R_{\mu e}^{A}$ $B^{(0)}(A)$ $B^{(3)}(A)$
------------------------------------------ -------------------------------------- -------------- --------------
$^{48}$Ti, SINDRUM II[@Honecker:1996zf] $6.1\times 10^{-13}$(90% C.L.) 1 0.3
$^{27}$Al, MECO[@MECO] $\sim 2\times 10^{-17}$ $^{\dagger}$ 11.5 1.7
$^{197}$Au, SINDRUM II[@Honecker:1996zf] $\sim 6\times 10^{-13}$ $^{\dagger}$ 1.27 0.46
$^{48}$Ti, PRIME[@PRIME] $\sim 10^{-18}$ $^{\dagger}$ 28 8.8
$^{\dagger}$ expected upper limits.
[**Table II.**]{} Transition nuclear matrix elements ${\cal M}_{p,n}$ from Eqs. (\[Rme\]), (\[phi\]) and other useful quantities (see the text).
------------ -------------------- ------------------------------------------------ ------------------------- -------------------------
Nucleus $p_e \, (fm^{-1})$ $\Gamma_{\mu c} \, ( \times 10^{6} \, s^{-1})$ ${\cal M}_p(fm^{-3/2})$ ${\cal M}_n(fm^{-3/2})$
$^{27}$Al 0.531 0.71 0.047 0.045
$^{48}$Ti 0.529 2.60 0.104 0.127
$^{197}$Au 0.485 13.07 0.395 0.516
------------ -------------------- ------------------------------------------------ ------------------------- -------------------------
[**Table III.**]{} Lower limits on the individual mass scales, $\Lambda_{\mu e}$, of the scalar quark-lepton contact operators in Eq. (\[eff-q\]) inferred from the experimental upper bounds on the branching ratio of conversion for the direct nucleon mechanism (DNM) and the meson exchange mechanism (MEM). The superscript notation is $b = P,S$. The values of the scaling factors $B^{(0,3)}(A)$ for the running and some planned conversion experiments are given in Table 2.
LFV Mass Scale DNM MEM
--------------------------- ----------------------------------- -----------------------------------
$\Lambda^{(0,b)}_{\mu e}$ $5.8\times 10^{3} B^{(0)}(A)$ TeV $3.7\times 10^{3} B^{(0)}(A)$ TeV
$\Lambda^{(3,b)}_{\mu e}$ $2.3\times 10^{3} B^{(3)}(A)$ TeV $3.1\times 10^{3} B^{(3)}(A)$ TeV
$\Lambda^{(s,b)}_{\mu e}$ $2.6\times 10^{3} B^{(0)}(A)$ TeV no limits
[**Fig.1:**]{} Diagrams contributing to the nuclear $\mu^--e^-$ conversion in the scalar channel: direct nucleon mechanism (a) and meson-exchange mechanism (b).
**Fig.1**
[^1]: Pseudoscalar and axialvector mesons do not contribute to the coherent conversion.
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---
abstract: 'The transmission through a quantum point contact (QPC) in the quantum Hall regime usually exhibits multiple resonances as a function of gate voltage and high non-linearity in bias. Such behavior is unpredictable and changes sample by sample. Here, we report observation of sharp transition of the transmission through an open QPC at finite bias which was consistently observed for all the tested QPCs. It is found that the bias dependence of the transition can be fitted to the Fermi-Dirac distribution function through universal scaling. The fitted temperature matches quite nicely to the electron temperature measured via shot noise thermometry. While the origin of the transition is unclear, we propose a phenomenological model based on our experimental results, which may help to understand such a sharp transition. Similar transitions are observed in the fractional quantum Hall regime and it is found that the temperature of the system can be measured by rescaling the quasiparticle energy with the effective charge ($e^*=e/3$). We believe that the observed phenomena can be exploited as a tool for measuring the electron temperature of the system and for studying the quasiparticle charges of the fractional quantum Hall states.'
author:
- Changki Hong
- Jinhong Park
- Yunchul Chung
- Hyungkook Choi
- Vladimir Umansky
title: Nontrivial transition of transmission in a highly open quantum point contact in the quantum Hall regime
---
A quantum point contact (QPC) [@van1988quantized; @wharam1988one] is the most essential building block of the quantum devices such as quantum dots [@meirav1990single], electron interferometers [@yacoby1994unexpected; @ji2003electronic], etc [@venkatachalam2012local; @bocquillon2013coherence]. In the quantum Hall (QH) regime [@klitzing1980new], it is used to control tunneling between counter propagating edge states, and it provides a useful tool to measure the fractional charge via the shot noise measurement [@picciotto1997direct; @kane1997observation; @chung2003scattering; @bid2009shot]. However, the transmission through a QPC usually exhibits multiple resonances as a function of QPC gate voltages and high non-linearity in bias. Such resonances and non-linear behaviors often hamper to develop ideal quantum devices.
The resonance peaks observed near pinch-off region resemble those of a quantum dot and are explained by the tunneling through localized states in the QPC gap [@jain1988quantum; @furusaki1998resonant; @martins2013coherent]. The exstence of such localized states in an almost closed QPC can be measured as coulomb diamonds and were used as an electron thermometry by Altimiras and coworkers [@altimiras2012chargeless]. However, when the QPC is partially open, the resonant peaks are superposed by many other resonances randomly, hence making it very difficult to study their transport behavior systematically. Such random overlap between resonant peaks were usually regarded as the resonant tunneling through multiple localized states which can exist in a rather open QPC. Hence, not much attention has been paid until now. Here, we report sharp transition of the transmission at finite bias for a highly open QPC, which is observed consistently for all the QPCs we have tested. Moreover, it cannot be explained by overlap of multiple resonant peaks. We found that the shape of the transition step is exactly proportional to the Fermi-Dirac distribution function (not the derivative of the distribution function). Also, all the sharp transition traces can be rescaled into a single transition trace, which fits well to the Fermi-Dirac distribution function. The estimated temperature in the fitting is very close to the electron temperature measured via the shot noise thermometry [@spietz2003primary].
The experiments were conducted with three QPCs, QPC A and B with 75nm width and 150nm gap, and QPC C with 150nm width and 300nm gap. The QPC gaps used in this experiment are the typical QPC gaps used for Fabry-Perot interferometer [@choi2015robust] and Mach-Zehnder interferometer [@weisz2014an] devices. QPCs with wider gap tend to show less resonance peaks but the same chaotic resonances were observed.
The QPCs were fabricated on two different MBE-grown GaAs/AlGaAs heterojunction wafers (for a fractional QH sample: mobility $\mu=$ 3.0$\times$10$^6$cm$^2$/Vs, electron density $n=$ 1.0$\times$10$^{11}$cm$^{-2}$ with 100nm of 2DEG depth; for an integer QH sample: $\mu=$ 3.2$\times$10$^6$cm$^2$/Vs, $n=$ 2.3$\times$10$^{11}$cm$^{-2}$ with 70nm of 2DEG depth). The measurements were performed by using two different dilution refrigerators, refrigerator A with an electron temperature of 22mK and refrigerator B with an electron temperature of 55mK. Both electron temperatures were measured by using shot noise thermometry [@spietz2003primary].
![(color online) (a) The transmission through a QPC measured at $\bm{\nu=1}$ (B=9.345T). (b) The transmission of a QPC as a function of bias voltage for various QPC gate voltages. The traces are measured at the white dotted lines in figure (a) and shifted by 0.2 for clarity (bottom to top). Traces marked as A, C, E show sharp transitions at finite biases. Inset shows the measurement set-up. (c) The transmission measured at zero bias as a function of gate voltage. Points A, C, E marks where the trace A, C, E in figure (b) is measured. Note that the transmission at zero bias is very close to unity.[]{data-label="fig:short"}](figure1.pdf){width="1\linewidth"}
Fig. 1 shows the transmission, $t$ of a QPC A measured as a function of bias and QPC gate voltage at filling factor $\nu=1$ in the refrigerator A (electron temperature of 22mK). The transmission was measured by applying a modulation voltage, $V_{ac}$ of 750kHz, $0.5\mu V_{rms}$ with a bias voltage $V_{SD}$ to the source S and measuring the differential conductance $g=dI_t/dV_{ac}$ at the drain D (Fig. 1(b) inset), where $I_t$ is the transmitted current. The current was measured by monitoring the voltage development over quantum Hall edge using home-made cryogenic low-noise voltage amplifier. The transmission is defined by $g/(\nu e^2/h)$, where $\nu$ is the corresponding filling factor of the quantum Hall states. At zero bias, full transmission ($t=1$) through QPC is observed for the QPC gate voltages above 0.1V (see Fig. 1(a) and (c)). However, the edge state starts to get reflected at finite bias and the transmission drops below 1, as shown in Fig. 1(b). The shape of the peak around zero bias is rather random and slightly asymmetric in bias voltage. The transition slope, $\left|dt/dV_{SD}\right|$ from full to partial transmission varies randomly. Some peaks show slow transitions (trace B, D, F) while others show very sharp transitions (A, C, E). For sharp transitions, we found that the transition does not get infinitely sharper and there is an upper limit for the transition slope. Such sharp transitions were consistently observed for other QPCs we have tested (more than 10, not shown here).
We took the trace A in the Fig. 1(b), which shows the steepest transition slope, and compared with the Fermi-Dirac distribution using Eq.(1), as shown in Fig. 2(a). $$\label{eq : 1}
F(V_{SD}) = 1-C\frac { 1 }{e^{(eV_0-eV_{SD})/k_BT_0}+1}$$ Here, $e$ is the electron charge, $k_B$ is the Boltzmann constant, $V_{SD}$ is the bias voltage. $C$, $T_0$ and $V_0$ are the fitting parameters, which is related to the transition amplitude, the temperature and the bias voltage of the transition center, respectively. The trace was fitted with temperature of 18mK, which is close to the electron temperature of 22mK measured via shot noise thermometry. A similar fit was made with the data measured with the QPC B in a dilution refrigerator B (with an electron temperature of 55mK) as shown in Fig. 2(b). The figure shows almost perfect fit, except for the region at the end of the transition ($V_{SD}>140\mu$V). Note that using different parameters $C$, $T_0$ and $V_0$ do not give any reasonable fit. After the sharp transition, the transmission usually starts to increase slightly, as it is shown in the Fig. 2. Most of the transition traces show similar behavior at the end of the transition region, which will be discussed later. In both cases, the electron temperatures measured with shot noise thermometry are slightly higher than the fitted temperatures. However, we believe that such discrepancies are reasonable considering the measurement uncertainty of both techniques.
![(color online) (a) The trace (circles) showing a sharp transition (trace A in the Fig. 1(b)) is fitted with Fermi-Dirac distribution by using Eq. (1) with the temperature of 18mK (solid line). The electron temperature measured via shot noise thermometry was around 22mK. (b) Similar fitting but with a trance measured with a QPC B in a refrigerator B (with an electron temperature of 55mK). (c) Sharp transitions (10 traces) measured with QPC A at 3 different magnetic fields (B=9.165T, 9.354T, 9.4T) at $\bm{\nu=1}$. (d) The traces (circles) in (c) were rescaled and shifted to position around zero bias voltage and fitted with Eq. (1) (solid line) with temperature of 15.6mK.[]{data-label="fig:short"}](figure2.pdf){width="1\linewidth"}
Fig. 2(c) shows 10 traces of sharp transitions measured with QPC A at three different magnetic fields (in $\nu=1$ plateau) in the refrigerator A. The traces were individually fitted and the average temperature was found to be 15.6mK with the standard deviation of 2.1mK. By shifting and renormalizing the traces using their individual fitting parameters $C$ and $V_0$, the traces were rescaled and plotted in Fig. 2(d). The solid line is the fit using Eq. (1) with the average temperature of 15.6mK. The figure clearly shows that the transition traces can be universally scaled into a single trace which can be represented with the energy distribution of the Fermi-Dirac function.
Since the resonant behavior and the non-linear conductance of a QPC under high magnetic fields are usually attributed to the tunneling through localized states [@jain1988quantum; @furusaki1998resonant; @martins2013coherent], we compare our results with the above model. Fig. 3(a) is a schematic diagram showing the tunneling through localized states in a QPC. Let’s assume that a droplet of localized states exists between counter propagating edge states with some finite energy $E_C$ due to charging energy. This allows us to adopt quantum dot analogy except for some subtle differences, which will be discussed later. The tunneling from the transmitting edge T to the reflecting edge R is suppressed at zero bias, due to the charging energy, which allows perfect transmission through the QPC. When the bias is applied larger than the charging energy, then the tunneling from transmitting edge T to the reflecting edge R is allowed via the localized state, hence reduces the transmission.
![(color online) (a) The localized states (with the charging energy $E_C$) formed between QPC gates, which allow tunneling between the transmitting edge channel T and reflecting edge channel R (b) Left Fermi sea is biased with a voltage $V_{SD}$ and is coupled to the localized state with A discrete, B constant and C energy dependent tunneling density energy state in energy. The right Fermi sea is grounded. $\Gamma_L,\Gamma_R$ is the tunneling rate between the localized states and the left and the right Fermi sea, respectively. Using a lattice model \[14\], we calculate the transmissions (denoted as solid lines in (c), (d) and (e)) through the localized states with the three different energy dependencies ((c) 0D, (d) 2D, (e) 1D) of the transmission probability. We used parameters $\hbar\Gamma_L/k_BT=10$ and $E_C/k_BT =10$ for drawing the solid lines of (c), (d), and (e). Solid lines are fitted to Eq. (1) (dashed lines). We used fitting parameters $C=\pi\hbar\Gamma_L/k_BT$, $\frac{eV_0}{k_BT}=10$, $\frac{T_0}{T}=1$ for Fig. 3(c), $C = 0.0175$, $\frac{eV_0}{k_BT}=11.8$, $\frac{T_0}{T}=1.3$ for Fig. 3(d), and $C = 0.018$, $\frac{eV_0}{k_BT}=9.5$, $\frac{T_0}{T}=1$ for Fig. 3(e). The schematic diagrams showing possible localized states (green island) formed in (f) relatively open, (g) intermediate, (h) relatively closed QPC.[]{data-label="fig:short"}](figure3.pdf){width="1\linewidth"}
The sharpness of the transition can be naively explained by the energy level broadening of the localized state. If the energy level broadening in the localized state is much smaller than the temperature broadening ($\hbar\Gamma \ll k_BT$, where $\Gamma = \Gamma_L+\Gamma_R$) then the transition will be sharp and the slope will be only determined by the temperature broadening, whereas, at the opposite limit, the transition will be slow and the slope will be determined by the energy level broadening which is bigger than the temperature broadening. Hence the maximum transition slope is limited by the temperature broadening, which explains why the observed transition does not get infinitely sharper. Up to this point, the quantum dot analogy works perfectly to explain our experimental results. However, the quantum dot analogy fails to explain why the transition reflects the Fermi-Dirac distribution function itself, not the derivative of the distribution function, considering that the transmission is determined by measuring the differential conductance. In general, the differential conductance for a tunneling device can be written as follows [@buttiker1992scattering]. $$\label{eq : 2}
g = \frac{ dI_t }{dV_{SD}}=\frac{ e^2 }{h}\int_{-\infty}^{\infty}dE\left(\frac{-\partial f(E-eV_{SD})}{\partial E}\right)T(E)$$ Here, $h$ is the plank constant, $f(E)$ is the Fermi-Dirac distribution function and $T(E)$ is the transmission probability. It can be easily seen from the equation that the differential conductance will be proportional to $\frac{\partial f(E-eV_{SD})}{\partial E}|_{E=E_C}$ when the transmission probability is given by a delta function (which is the case for an ideal quantum dot with discrete energy states) while it will be proportional to $f(E_C-eV_{SD})$ when the transmission probability has a sudden jump from zero to a finite value at $E_C$ (which corresponds to the case of an electron transport through a two-dimensional electron puddle).
Fig. 3(c), (d) and (e) are the calculated transmissions through the localized states with the three different energy dependencies of the transmission probability shown in Fig. 3(b). We theoretically consider a tight binding model of a lattice $1\times 1$ (Fig. 3(c)), $40\times 40$ (Fig. 3(d)), and $100\times 1$ (Fig. 3(e)) with hopping $t_0$ to describe various localized states in a QPC. We attach the edge channels T and R to the sites (1, 1) (Fig. 3(c)), (20, 1) and (20, 40) (Fig. 3(d)), and (33, 1) and (66, 1) (Fig.3(e)) with coupling strengths $\Gamma_L=\Gamma_R$. We used parameters $\hbar\Gamma_L/k_BT$= 0.1 and $E_C/k_BT$=10 for drawing (c), (d), and (e), and parameters $t_0/k_BT$=2 for (d) and $t_0/k_BT$=10 for (e). As it can be seen from the figures, the calculated transmission through the one-dimensional lattice shown in Fig. 3(e) is quite consistent with the observed experimental results. A monotonic increase in the transmission after the sharp transition shown in the experiments may be attributed to the van Hove singularity of finite size 1D wire. We speculate that the localized states might be formed in a highly open QPC because the local filling factor in the QPC area is effectively lowered by the QPC potential [@hashisaka2015shot]. This lowered filling factor may lead to formation of an unwanted electron puddle in the QPC area by electrostatic Coulomb interaction [@chklovskii1992electrostatics] and the composite edges between the QPC area and vacuum [@rosenow2010signatures].
In Fig. 1(a), the width of the Coulomb diamond in bias voltage is shrinking in the region i, as the QPC closes. The result can be explained by considering the charging energy between the edges and the localized states. When QPC start to pinch, the tunneling is taking place mainly between the upper and the lower edges through the localized states in the gap of a QPC, as shown in Fig. 3(f). As the QPC closes more, the capacitance between the edges and the localized states becomes larger (because the distances between them become closer), hence reducing the charging energy, which reduces the width of the Coulomb diamonds. In the strongly pinched regime, where electrons tunnel from the left edge to the right edge as shown in Fig. 3(h), both edges are pushed away, as the QPC closes. This increases the charging energy hence making Coulomb diamond wider, which can be seen in the region iii of the Fig. 1(a).
In the intermediate regime (region ii in Fig. 1(a)), the localized states experience the capacitance from all the surrounding edges, as shown in Fig. 3(g). The widths of Coulomb diamonds are roughly the same, regardless of QPC gate voltages. In this regime, the capacitance cannot be determined solely by the geometrical distance between the edges and the localized states because the QPC gate voltage also modifies the quasiparticle populations in the edge states, which also alters the capacitance [@16]. As the QPC closes, the upper and lower edges are getting closer to the localized states and depopulated while it is the opposite for the left and right edges. Such counteraction may keep the overall capacitance stay roughly the same, hence keeping the Coulomb diamond widths constant. Here, the quasiparticle can tunnel to any edge states, which might be responsible for the chaotic resonances observed. Overall, the tunneling through localized states (with a finite charging energy) roughly explains observed non-linearity and resonances in all QPC voltage ranges, including the sharp transition observed in a highly open quantum point contact.
![(color online) Two sharp transition traces observed at $\nu=1/3$ showing energy gap $14\mu$eV (blue circles) in (a) and $5.7\mu$eV (red circles) in (b) respectively. Both are fitted with Fermi-Dirac distribution by using Eq. (1) with the temperature of 18mK (solid line). (c) The reflections measured at zero bias. Blue circles are corresponding to the trace of (a) and the red ones to that of (b). The black solid lines are the fitting for equation $r\propto e^{-\Delta/k_BT}$ and the green dashed lines are the fitting for equation $r\propto \cosh^{-2}(\Delta/2.5k_BT)$ (details are in the main text).[]{data-label="fig:short"}](figure4.pdf){width="1\linewidth"}
Similar sharp transitions were also observed for the $\nu = 1/3$ fractional quantum Hall state. Two sharp transition traces were shown in Fig. 4(a) and (b). To fit the data with a temperature which is reasonably close to the electron temperature of the refrigerator A (22mK), the electron charge $e$ in Eq. (1) has to be replaced to the effective charge $e^*=e/3$, which is the quasiparticle charge of $\nu = 1/3$ fractional quantum Hall state. With the replacement, both traces were fitted well with the temperature of 18mK, which is close to the temperature measured with shot noise thermometry (22mK). It calls for further theoretical investigation because the $\nu = 1/3$ fractional quantum Hall edge state is believed to be described by the chiral Luttinger liquid theory, where the state does not follow the Fermi-Dirac distribution function [@luttinger1963exactly; @haldane1981luttinger; @wen1990chiral].
We measured the reflected $r$ at zero bias as a function of a temperature at the same fractional filling factor $\nu=1/3$. We estimate the energy gap $\Delta$ by two different equations $r\propto e^{-\Delta/k_BT}$ and $r\propto \cosh^{-2}(\Delta/2.5k_BT)$. The first equation assumes simple thermal excitation and the second equation assumes tunneling through thermally broadened continuous energy levels [@altimiras2012chargeless; @beenakker1990theory]. The blue and red circles in Fig. 4(c) are the reflections measured at zero bias of the traces in Fig. 4(a) and (b) and the corresponding energy gaps are 9.5$\mu$eV (11.9$\mu$eV) and 4.3$\mu$eV (5.4$\mu$eV) by using the first (second) equation, respectively. The fitting was more reasonable with the first equation, as it can be seen from the figure. The energy gap can be also obtained from the transition traces shown in Fig. 4(a) and (b) by extracting the fitting parameter $V_0$ from Eq. (1), which roughly corresponds to the bias voltage of the transition mid-point. Again, multiplying the effective charge $e/3$ rather than $e$ to the bias voltage gives comparable energy gaps measured with the thermal activation experiment, which are $14\mu$eV and $5.7\mu$eV for Fig. 4(a) and (b), respectively. The energy gaps measured with thermal excitation experiment(with the first equation) were slightly smaller ($\sim$ 3/4) than the values estimated from the transition traces. Note that the temperature in Fig. 4 (c) is the lattice temperature measured with a thermometer in the refrigerator and is generally slightly lower than the electron temperature, which may results in energy gap to be smaller. Thus, the results are consistent with the tunneling through localized states with a finite charging energy gap.
The above analysis shows that the quasiparticle charge of a fractional quantum Hall state can be estimated by fitting the non-linear conductance transition, which agrees well with the thermal activation measurement in a highly open QPC. These measurement techniques can be used to confirm the reported shot noise measurement results, for example the observation of super Poissonian noise in fractional quatum Hall regime [@rodriguez2002super] and the observation of e/3 charge through a local fractional quantum Hall state in a QPC at $\nu=1$ quantum Hall state [@hashisaka2015shot].
To summarize, we have observed a sharp transition of QPC transmission at finite bias in a highly open QPC in the quantum Hall regime. The transition traces in bias voltage fit almost perfectly with Fermi-Dirac distribution function. Also, it was found that all the sharp transition traces can be rescaled into a single transition trace, which fits to the Fermi-Dirac distribution function. Similar transition is observed in a fractional quantum Hall regime and the temperature of the system is measured by rescaling the quasiparticle energy with the effective charge ($e^*=e/3$). We believe that the observed phenomena can be exploited as a handy tool for measuring the electron temperature of the system.
We thank M. Heiblum for discussion and D. Mahalu, N. Ofek, E. Weisz, I. Sivan, R. Sabo, and I. Gurman for their help in experiment. This work was partially supported by the National Research Foundation of Korea grant (NRF-2014R1A2A1A11053072).
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abstract: 'In this paper, we compute and verify the positivity of the Li coefficients for the Dirichlet $L$-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125 (2007) no.1, 50-58; J. Number Theory 130 (2010) no.4, 1109-1114. Furthermore, we formulate a criterion for the partial Riemann hypothesis and we provide some numerical evidence for it using new formulas for the Li coefficients.'
address:
- 'Faculty of Science of Tunis, Department of Mathematics, 2092 Tunis, Tunisia'
- 'Faculty of Science of Tunis, Department of Mathematics, 2092 Tunis, Tunisia'
- 'Faculty of Science of Monastir, Department of Mathematics, 5000 Monastir, Tunisia'
author:
- 'Sami Omar, Raouf Ouni, Kamel Mazhouda'
title: 'On the zeros of Dirichlet $L$-functions'
---
[^1]
[Introduction]{}\[sec.1\] The Li criterion for the Riemann hypothesis (see. [@6]) is a necessary and sufficient condition that the sequence $$\lambda_{n}=\sum_{\rho}\left[1-\left(1-\frac{1}{\rho}\right)^{n}\right]$$ is non-negative for all $n\in{\mathbb {N}}$ and where $\rho$ runs over the non-trivial zeros of $\zeta(s)$. This criterion holds also for the Dirichlet $L$-functions and for a large class of Dirichlet series, the so called the Selberg class as given in [@10]. More recently, Omar and Bouanani [@9] extended the Li criterion for function fields and established an explicit and asymptotic formula for the Li coefficients.\
Numerical computation of the first 100 of the Li coefficients $\lambda_{n}$ which appear in this criterion was made by Maslanka [@8] and later by Coffey in [@2], who computed and verified the positivity of about 3300 of the Li coefficient $\lambda_{n}$. The main empirical observation made by Maslanka is that these coefficients can be separated in two parts, where one of them grows smoothly while the other one is very small and oscillatory. This apparent smallness is quite unexpected. If it persisted until infinity then the Riemann hypothesis would be true. As we said above, this criterion was extended to a large class of Dirichlet series [@10] and no calculation or verification of the positivity to date in the literature made for other $L$-functions.\
In this paper, we compute and verify the positivity of the Li coefficients for the Dirichlet $L$-functions using an arithmetic formula established in [@10; @11]. Furthermore, we formulate a criterion for the partial Riemann hypothesis. Additional results are presented, including new formulas for the Li coefficients. Basing on the numerical computations made below, we conjecture that these coefficients are increasing in $n$. Should this conjecture hold, the validity of the Riemann hypothesis would follow.\
Next, we review the Li criterion for the case of the Dirichlet $L$-functions. Let $\chi$ be a primitive Dirichlet character of conductor $q$. The Dirichlet $L$-function attached to this character is defined by $$L(s, \chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}, \ \ \ \ (Re(s) > 1).$$ For the trivial character $\chi = 1$, $L(s, \chi)$ is the Riemann zeta function. It is well known [@3] that if $\chi\neq1$ then $L(s,\chi)$ can be extended to an entire function in the whole complex plane and satisfies the functional equation $$\xi(s,\chi)= \omega_{\chi}\xi(1-s,\overline{\chi}),$$ where $$\xi(s,\chi)=\left(\frac{q}{\pi}\right)^{(s+a)/2}\Gamma\left(\frac{s+a}{2}\right)L(s,\chi),$$ $$a=\left\{\begin{array}{crll}0&\hbox{if}&\chi(-1)= 1\\1&\hbox{if}&\chi(-1) =-1,\end{array}\right.\ \ \ \ \hbox{and}\ \ \ \ \omega_{\chi}=\frac{\tau(\chi)}{\sqrt{q}i^{a}},$$ where $\tau(\chi)$ is the Gauss sum $$\tau(\chi)=\sum_{m=1}^{q}\chi(m)e^{2\pi im/q}.$$ The function $\xi(s,\chi)$ is an entire function of order one. The function $\xi(s,\chi)$ has a product representation $$\label{eq.1}\xi(s,\chi) =\xi(0,\chi)\prod_{\rho}\left(1-\frac{s}{\rho}\right),$$ where the product is over all the zeros of $\xi(s,\chi)$ in the order given by $|Im(\rho)|<T$ for $T\rightarrow\infty$. If $N_{\chi} (T)$ counts the number of zeros of $L(s,\chi)$ in the rectangle $0\leq Re(s)\leq1$, $0<Im(s)\leq T$ (according to multiplicities) one can show by standard contour integration the formula $$N_{\chi}(T)=\frac{1}{2\pi}T\log T+c_{1}T+O\left(\log T\right),$$ where $$c_{1}=\frac{1}{2\pi}\left(\log q-\left(\log(2\pi)+1\right)\right).$$ We put $$\lambda_{\chi}(n)=\sum_{\rho}\left[1-\left(1-\frac{1}{\rho}\right)^{n}\right],$$ where the sum over $\rho$ is $\sum_{\rho}=\lim_{T\mapsto\infty}\sum_{|Im\rho|\leq T}$.\
[**Li’s criterion**]{} says that $\lambda_{\chi}(n)> 0$ for all $n = 1, 2, .$ . . if and only if all of the zeros of $\xi(s,\chi)$ are located on the critical line $Re(s) = 1/2$.\
The paper is organized as follows. In Section \[sec.2\], we recall the arithmetic formula for the Li coefficients for the Dirichlet $L$-functions and we give an estimate for the error term of $\lambda_{\chi}(n)$. In Section \[sec.3\], we show that $\lambda_{\chi}(n)\geq0$ if every non-trivial zero of $L(s,\chi)$ with $|Im(\rho)|<\sqrt{n}$ satisfies $Re(\rho)=1/2$ (that is partial Riemann hypothesis) and we give an estimate for the difference $|\lambda_{\chi}(n)-\lambda_{\chi}(n,T)|$, where $\lambda_{\chi}(n,T)$ are the partial Li coefficients. In Section \[sec.4\], we prove new formulas (integral and summation formula) for the Li coefficients $\lambda_{\chi}(n)$. Finally, in Section \[sec.5\], we report numerical computations of the Li coefficients using different formulas established in the previous sections unconditionally or under the Riemann hypothesis.
Li’s coefficients {#sec.2}
=================
Applying [@10 Theorem 2.2] for the case of the Dirichlet $L$-functions, we get the following arithmetic formula.
\[th.1\]Let $\chi$ be a primitive Dirichlet character of conductor $q>1$. We have $$\begin{aligned}
\label{eq.2}
\lambda_{\chi}(n)&=&-\sum_{j=1}^{n}(_{j}^{n})\frac{(-1)^{j-1}}{(j-1)!}\sum_{k=1}^{+\infty}\frac{\Lambda(k)}{k}\chi(k)(\log k)^{j-1}\nonumber\\
&&\ \ +\ \ \frac{n}{2}\left(\log\frac{q}{\pi}-\gamma\right)+\tau_{\chi}(n)\end{aligned}$$ where $$\tau_{\chi}(n)=\left\{\begin{array}{crll}\sum_{j=2}^{n}(_{j}^{n})(-1)^{j}\left(1-\frac{1}{2^{j}}\right)\zeta(j)-\frac{n}{2}\sum_{l=1}^{+\infty}\frac{1}{l(2l-1)}&\hbox{if}&\chi(-1)=1,\\ \sum_{j=2}^{n}(_{j}^{n})(-1)^{j}2^{-j}\zeta(j)&\hbox{if}&\chi(-1)=-1.\end{array}\right.$$
Theorem \[th.1\] is also proved by Coffey in [@2] and Li in [@7]. The arithmetic formula above can be written as $$\begin{aligned}
\label{eq.3}
\lambda_{\chi}(n)&=&\left[\log\frac{q}{\pi}+\psi\left(\frac{a+1}{2}\right)\right]\frac{n}{2}+\sum_{j=2}^{n}(_{j}^{n})\frac{1}{(j-1)!}2^{-j}\psi^{(j-1)}\left(\frac{a+1}{2}\right)\nonumber\\
&&-\ \sum_{j=1}^{n}(_{j}^{n})\frac{(-1)^{j-1}}{(j-1)!}\sum_{k=1}^{+\infty}\frac{\Lambda(k)\chi(k)}{k}(\log k)^{j-1},\nonumber\end{aligned}$$ where $a=0$ if $\chi(-1)=1$ and 1 if $\chi(-1)=-1$, with $\psi(\frac{1}{2})=-\gamma-2\log2$, $\psi(1)=-\gamma$, $\gamma$ is the Euler constant, $\psi^{(j-1)}(1)=(-1)^{j}(j-1)!\zeta(j)$ and $\psi^{(j-1)}(\frac{1}{2})=(-1)^{j+1}j!(2^{j+1}-1)\zeta(j+1).$ Here, $\psi=\frac{\Gamma'}{\Gamma}$ denotes the digamma function.\
An asymptotic formula for the number $\lambda_{\chi}(n)$ was proved in [@12 Theorem 3.1] using the arithmetic formula. Furthermore, it is equivalent to the Riemann hypothesis.
\[th.2\] We have $$RH\Leftrightarrow\lambda_{\chi}(n)=\frac{1}{2}n\log n+c_{\chi}n+O(\sqrt{n}\log n),$$ where $$c_{\chi}=\frac{1}{2}(\gamma-1)+\frac{1}{2}\log(q/\pi),$$ and $\gamma$ is the Euler constant.
Here, we estimate the error term for $\lambda_{\chi}(n)$ using the arithmetic formula (\[eq.2\]). Writing (\[eq.2\]) in the form $$\lambda_{\chi}(n)=\tilde{\lambda}_{\chi}(n,M)+E_{M},$$ where $$\label{eq.4}\tilde{\lambda}_{\chi}(n,M)=\frac{n}{2}\left(\log\frac{q}{\pi}-\gamma\right)+\tau_{\chi}(n)-\sum_{j=1}^{n}(_{j}^{n})\frac{(-1)^{j-1}}{(j-1)!}\sum_{k\leq M}\frac{\Lambda(k)}{k}\chi(k)(\log k)^{j-1}$$ and $$E_{M}=-\sum_{j=1}^{n}(_{j}^{n})\frac{(-1)^{j-1}}{(j-1)!}\sum_{k>M}\frac{\Lambda(k)}{k}\chi(k)(\log k)^{j-1}=-\sum_{k>M}\frac{\Lambda(k)}{k}\chi(k)L_{n-1}^{1}(\log k),$$ where $L_{n-1}^{1}$ is an associated Laguerre polynomial of degree $n-1$.\
Next, for our computation in the section \[sec.5\], we need to find $M$ such that $|E_{M}|\leq 10^{-\nu}$. An estimate for the Laguerre polynomials is due to Koepf and Schmersau [@5 Theorem 2]. Actually, they have shown that $$|L_{n}^{\alpha}(x)|<e^{x/2}\left[(n+\alpha)/x\right]^{\alpha/2}$$ for $x\in{[0,4(n+\alpha)]}$ when $n+\alpha>0$ and $\alpha$ is an integer. Then, we obtain $$\begin{aligned}
\label{eq.33}|E_{M}|&\leq&\left|\sqrt{\frac{n}{\log M}}\sum_{m>M}^{+\infty}\frac{\Lambda(m)}{\sqrt{m}}\chi(m)\right|\nonumber\\
&\leq&\sqrt{\frac{n}{\log M}}\left\{\left|\sum_{p>M}^{+\infty}\frac{\log p}{\sqrt{p}}\chi(p)\right|+\left|\sum_{p^{2}>M}^{+\infty}\frac{\log p}{p}\chi(p^{2})\right|\right\}\nonumber\\
&&+\ \ \sqrt{\frac{n}{\log M}}\left|\sum_{p^{j}>M,j>2}^{+\infty}\frac{\log p}{p^{j/2}}\chi(p^{j})\right|.
\end{aligned}$$ We have $|\chi(p^{j})|<1$. Therefore, $$\left|\sum_{p>M}^{+\infty}\frac{\log p}{\sqrt{p}}\chi(p)\right|+\left|\sum_{p^{2}>M}^{+\infty}\frac{\log p}{p}\chi(p^{2})\right|\leq\left|\sum_{p>M}^{+\infty}\frac{\log p}{\sqrt{p}}\chi(p)\right|+\left|\sum_{p>\sqrt{M}}^{+\infty}\frac{\log p}{p}\chi(p)\right|\leq\frac{2}{\sqrt{M}}$$ and $$\sum_{p^{j}>M,j>2}^{+\infty}\frac{\log p}{p^{j/2}}\leq\left\{\begin{array}{crll} &\frac{\log M}{\sqrt{M}}& \hbox{if}\ \ M+1 \hbox{ is prime,}\\ &\frac{1}{\sqrt{M}}& \hbox{otherwise.}\end{array}\right.$$ Then $$\left\{\begin{array}{crll}|E_{M}|&\leq&\sqrt{\frac{n}{\log M}}\ \frac{\log M+2}{\sqrt{M}}\leq \sqrt{n}\ \frac{\log M+2}{\sqrt{M}}& \hbox{if}\ M+1 \hbox{ is prime,}\\ |E_{M}|&\leq&\sqrt{\frac{n}{\log M}}\frac{3}{\sqrt{M}}\leq3\frac{\sqrt{n}}{\sqrt{M}}&\ \hbox{otherwise.}\end{array}\right.$$ Let $M$ be such that $|E_{M}|\leq 10^{-\nu}$. Then, using the theory of the Lambert $W$ function, we choose $M$ $$\left\{\begin{array}{crll}M&=&\frac{n}{4}\ \left[W_{-1}\left(-\frac{10^{-\nu}}{\sqrt{n}}\right)\right]^{2}+4n\ 10^{2\nu}& \hbox{if}\ M+1\ \hbox{ is prime,}\\ M&=&9n\ 10^{2\nu}&\hbox{otherwise,}\end{array}\right.$$ where $W_{-1}$ denotes the branch satisfying $W(x)\leq-1$ and $W(x)$ is the Lambert $W$ function, which is defined to be the multivalued inverse of the function $w\longmapsto we^{w}$.
[Partial Li criterion]{}\[sec.3\] In the following proposition, we propose a partial Li criterion which relates the partial Riemann hypothesis to the positivity of the Li coefficients up to a certain order.
\[prop.1\] For sufficiently large $T\geq T_{1}\geq1$, if every non-trivial zero $\rho$ of $L(s,\chi)$ with $|Im(\rho)|<T$ satisfies $Re(\rho)=1/2$, then $\lambda_{\chi}(n)\geq0$ for all $n\leq T^{2}$.
We have $$\lambda_{\chi}(n)=\sum_{\rho}\left[1-\left(1-\frac{1}{\rho}\right)^{n}\right].$$ Then $$\begin{aligned}
\label{eq.8}
\lambda_{\chi}(n)&=&\sum_{\rho}\left(1-Re\left[\left(1-\frac{1}{\rho}\right)^{n}\right]\right)\nonumber\\
&=&\sum_{\rho; |Im(\rho)|<T}\left(1-Re\left[\left(1-\frac{1}{\rho}\right)^{n}\right]\right)+\sum_{\rho; T<|Im(\rho)|}\left(1-Re\left[\left(1-\frac{1}{\rho}\right)^{n}\right]\right)\nonumber\\\end{aligned}$$ Let $\rho=\beta+i\gamma$, then we obtain $$1-\left(1-\frac{1}{\rho}\right)^{n}=1-\left(\frac{1+\frac{\beta-1}{i\gamma}}{1+\frac{\beta}{i\gamma}}\right)^{n},\ \frac{1+\frac{\beta-1}{i\gamma}}{1+\frac{\beta}{i\gamma}}=\left(1-\frac{\beta}{\gamma^{2}}\right)+\frac{i}{\gamma}+O\left(\frac{1}{\gamma^{3}}\right)$$ and using binomial identity, we get $$Re\left[\left(\frac{1+\frac{\beta-1}{i\gamma}}{1+\frac{\beta}{i\gamma}}\right)^{n}\right]=\left(1-\frac{\beta}{\gamma^{2}}\right)^{n}+O\left(\frac{1}{\gamma^{3}}\right),$$ where the $O$-symbol depends on $n$. Therefore $$\begin{aligned}
\label{eq.9}\sum_{ |\gamma|>T}\left(1-Re\left[\left(1-\frac{1}{\rho}\right)^{n}\right]\right)&=&\sum_{ |\gamma|>T}\left[1-\left(1-\frac{\beta}{\gamma^{2}}\right)^{n}+O\left(\frac{1}{\gamma^{3}}\right)\right]\nonumber\\
&=&\sum_{ |\gamma|>T}\frac{n\beta}{\gamma^{2}}+O\left(\frac{1}{\gamma^{3}}\right).\end{aligned}$$ Then, the second sum in goes to 0 in absolute value as $T \to \infty$ by conditional convergence (see also Proposition \[prop.2\] below). Finally, it suffices to prove that, under the Riemann hypothesis, there exists a positive constant $c_{0}$ such that for large $T$ we have $$\label{eq.11}
\sum_{|Im(\rho)|<T}\left(1-Re\left[\left(1-\frac{1}{\rho}\right)^{n}\right]\right)\geq c_{0} \frac{\log T}{T}.$$ Thus, for $1\leq n\leq T^{2}$, we get $$\begin{aligned}
\label{eq.12}\sum_{ |\gamma|<T}\left(1-Re\left[\left(1-\frac{1}{\rho}\right)^{n}\right]\right)&\geq&\sum_{ |\gamma|<T}\frac{n^{2}}{2\gamma^{2}}\nonumber\\
&\geq&\frac{n^{2}}{2}\sum_{ |\gamma|<T}\frac{1}{\gamma^{2}}\nonumber\\
&\geq&\frac{n^{2}}{2T^{2}}\sum_{ |\gamma|<T}1\nonumber\\
&\geq&\frac{1}{2T^{2}}N_{\chi}(T).\end{aligned}$$ Recall that $$N_{\chi}(T)=\frac{1}{2\pi}T\log T+c_{1}T+O\left(\log T\right).$$ Then, equation is proved.
[**Remark**]{}
- Recall that the $10^{13}$ first zeros of the Riemann zeta function lie on the line $Re(s)=1/2$ (see. [@4]). Then, from Proposition \[prop.1\], we might expect that the first $10^{26}$ Li coefficients $\lambda_{\zeta}(n)$ are non-negative.
- In the section \[sec.5\], we will use the first $10^{4}$ critical zeros of the Dirichlet $L$-functions to compute the first Li coefficients $\lambda_{\chi}(n)$. Then, from Proposition \[prop.1\] above, we also might expect that the first $10^{8}$ Li coefficients are non-negative.
[**Conversely.**]{} From the work of Brown [@1], the first observation is that the first “non-trivial” inequality $\lambda_{\chi}(2)\geq0$ is sufficient to establish the non-existence of a Siegel zero for $\xi(s,\chi)$ (see [@1 Corollary 1]).\
Let $r>1$ be a real number. By the invariance of the zeros $\rho$ of $\xi(s,\chi)$ under the map $\rho\longmapsto1-\overline{\rho}$, $$\forall \ \rho,\ \ \left|\frac{\rho}{\rho-1}\right|\leq r\ \Leftrightarrow\ \forall\ \rho,\ \rho\in{D_{r}},$$ where $D_{r}$ is the closed region bounded by the lines $\{z\in{\Bbb C}:\ Re(z)=0,1\}$ and the arcs of tow circles. The second observation (see [@1 Theorem 3]) is that, for large $N\in{\Bbb N}$, the equality $\lambda_{\chi}(1)\geq0,...,\lambda_{\chi}(n)\geq0$ imply the existence of a certain zero-free region for $\xi(s,\chi)$, that is, there exist constants $N,\mu, \nu$ depending only on $q$ such that if $\lambda_{\chi}(1)\geq0,...,\lambda_{\chi}(n)\geq0$ hold, and $n\geq N$, then the zeros of $\xi(s,\chi)$ belong to $D_{r}$, where $r=\sqrt{1+T^{-2}}$ and $T=\left(\frac{n}{\mu\log^{2}(\nu n)}\right)^{1/3}$.\
Let us define the partial Li coefficients by $$\lambda_{\chi}(n,T)=\sum_{\rho;\ |Im\rho|\leq T}1-\left(1-\frac{1}{\rho}\right)^{n}$$ with a parameter $T$. An estimate for the error term $|\lambda_{\chi}(n)-\lambda_{\chi}(n,T)|$ is stated in the following proposition.
\[prop.2\] For sufficiently large $T\geq1$, we have $$\label{eq.12}
\left|\lambda_{\chi}(n)-\lambda_{\chi}(n,T)\right|\leq\frac{3n^{2}}{2T^{2}}\left[\frac{1}{2\pi}T\log T+\left(\frac{1}{\pi}+\log\left(\frac{q}{2\pi e}\right)\right)T+\frac{1}{2}\right].$$
Note that $\rho=\beta+i\gamma$, where $\beta$, $\gamma\in{\rb}$ and $0\leq\beta\leq1$. We have $$\lambda_{\chi}(n)-\lambda_{\chi}(n,T)=\frac{1}{2}Re\left[ \sum_{|\gamma|> T}\left(2-\left(\frac{\rho-1}{\rho}\right)^{n}-\left(\frac{\rho}{\rho-1}\right)^{n}\right)\right].$$ Using a binomial expansion of the inner term in the sum in the right-hand side, we obtain $$\begin{aligned}
\label{eq.13}
Re\left[2-\left(\frac{\rho-1}{\rho}\right)^{n}-\left(\frac{\rho}{\rho-1}\right)^{n}\right]&=&Re\left[n\left(\frac{1}{\rho}+\frac{1}{1-\rho}\right)-\frac{n(n-1)}{2}\left(\frac{1}{\rho^{2}}+\frac{1}{(1-\rho)^{2}}\right)\right]\nonumber\\
&&+\ Re\left[\sum_{k=3}^{n}\left(_{k}^{n}\right)(-1)^{k-1}\left(\rho^{-k}+(1-\rho)^{-k}\right)\right].\end{aligned}$$ We have $$\frac{1}{1+\gamma^{2}}\leq Re\left(\frac{1}{\rho}+\frac{1}{1-\rho}\right)\leq\frac{1}{\gamma^{2}}$$ and $$\frac{1}{1+\gamma^{2}}-\frac{2}{\gamma^{4}}\leq Re\left(\frac{1}{\rho^{2}}+\frac{1}{(1-\rho)^{2}}\right)\leq\frac{2}{\gamma^{2}}.$$ Suppose now that $|\gamma|\geq T\geq n$, then $\frac{(n-3)}{|\gamma|}...\frac{(n-k)}{|\gamma|}\leq1$ for all $n\geq k\geq3$. Then $$\sum_{k=3}^{n}\left(_{k}^{n}\right)\frac{1}{|\gamma|^{k}}\leq2\frac{n^{3}}{|\gamma|^{3}}\sum_{k=3}^{\infty}\frac{1}{k!}=(2e-5)\frac{n^{3}}{|\gamma|^{3}}\leq\frac{n^{3}}{2|\gamma|^{3}}.$$ Therefore, $$\begin{aligned}
\label{eq.14}
Re\left[2-\left(\frac{\rho-1}{\rho}\right)^{n}-\left(\frac{\rho}{\rho-1}\right)^{n}\right]&\leq&\frac{n}{\gamma^{2}}+\frac{n^{2}-n}{\gamma^{2}}+\sum_{k=3}^{n}\left(_{k}^{n}\right)\frac{2}{|\gamma|^{k}}\nonumber\\
&\leq&\frac{n}{\gamma^{2}}+\frac{n^{3}}{2|\gamma|^{3}}\nonumber\\
&\leq&\frac{3n^{2}}{2|\gamma|^{2}}.\end{aligned}$$ Then $$\left|\lambda_{\chi}(n)-\lambda_{\chi}(n,T)\right|\leq\frac{3}{4}n^{2}\sum_{|\gamma|> T}\frac{1}{\gamma^{2}}.$$ We have $$\label{eq.15}
\frac{1}{2}\sum_{|\gamma|>T}\frac{1}{\gamma^{2}}\leq\int_{T}^{\infty}-\frac{d}{dt}\left[t^{-2}\right]_{t=x}\left(N_{\chi}(x)-N_{\chi}(T)\right)dx
=\int_{T}^{\infty}x^{-2}dN_{\chi}(x).$$ Furthermore, $$\begin{aligned}
\int_{T}^{\infty}x^{-2}dN_{\chi}(x)&=&\int_{T}^{\infty}x^{-2}\left[\frac{1}{2\pi}\log x+\frac{1}{2\pi}+\log\left(\frac{q}{2\pi e}\right)+\frac{1}{x}\right]dx\nonumber\\
&=&T^{-2}\left[\frac{1}{2\pi}\log T+\frac{1}{2\pi}T\log T+\frac{1}{2\pi}T+\log\left(\frac{q}{2\pi e}\right)T+\frac{1}{2}\right].\nonumber\end{aligned}$$ Finally, we get $$\left|\lambda_{\chi}(n)-\lambda_{\chi}(n,T)\right|\leq\frac{3}{2}\frac{n^{2}}{T^{2}}\left[\frac{1}{2\pi}T\log T+\left(\frac{1}{\pi}+\log\left(\frac{q}{2\pi e}\right)\right)T+\frac{1}{2}\right]$$ and Proposition \[prop.2\] follows.
For our computations at the end of this paper, we need to find $T_{0}$ such that $\left|\lambda_{\chi}(n)-\lambda_{\chi}(n,T)\right|\leq 10^{-k}$. To do so, it suffices to find $T_{0}$ such that $$\frac{3n^{2}}{4\pi}\frac{\log T}{T}\leq \frac{10^{-k}}{3}\ \ \ \Leftrightarrow \ \ \ \frac{\log T}{T}\leq \frac{4\pi 10^{-k}}{9n^{2}}.$$ Using the theory of the Lambert $W$ function, we get $$T_{0}=-\frac{9n^{2}}{4\pi}W_{-1}\left(-\frac{4\pi}{9n^{2}}10^{-k}\right),$$ where $W_{-1}$ denotes the branch satisfying $W(x)\leq-1$ and $W(x)$ is the Lambert $W$ function which is defined to be the multivalued inverse of the function $w\longmapsto we^{w}$.
[New formulas for the Li coefficients]{}\[sec.4\] In this section, we give new formulas for the Li coefficients under the Riemann hypothesis (integral and summation formula) which will be used to compute and verify the positivity of the Li coefficients $\lambda_{\chi}(n)$ under the Riemann hypothesis.\
From (\[eq.1\]) we have $$\label{eq.16}
\log\xi\left(\frac{z}{z-1},\chi\right)=\log\xi\left(\frac{1}{1-z},\chi\right)=\log \xi(0,\chi)+ \sum_{n=1}^{\infty}\lambda_{\chi}(n)\frac{z^{n}}{n}.$$ The number $\lambda_{\chi}(n)$ does not depend on the choice of the logarithm. Rewrite (\[eq.16\]) at the point $z=-1$. Note that the region of convergence for this is an open disk of radius 2 centered at $z=-1$ and it encloses in particular the entirety of the closed unit disk, except for the point $z=1$ that is a pole of $\xi(\frac{1}{1-z},\chi)$.\
Assume that the Riemann hypothesis holds. Then, we have $$\begin{aligned}
\label{eq.17}
\log\xi\left(\frac{1}{1-z},\chi\right)&=&\log \xi(0,\chi)+ \sum_{n=1}^{\infty}\lambda_{\chi}(n)\frac{z^{n}}{n}\nonumber\\
&=&C_{\chi}(0)+\sum_{n=1}^{\infty}C_{\chi}(n)(z+1)^{n}.\end{aligned}$$ Expanding $(z+1)^{n}$, we obtain $$\label{eq.18}
\lambda_{\chi}(n)=n\sum_{j=1}^{\infty}\left(_{n}^{j}\right)C_{\chi}(j).$$ We have $$\label{eq.19}
N_{\chi}(T)=\frac{1}{\pi}Im\left(\log\xi_{\chi}\left(\frac{1}{2}+iT\right)\right)=\sum_{n=1}^{\infty}\frac{C_{\chi}(n)}{\pi}\left(Im\left(\frac{2\gamma+i}{2\gamma-i}+1\right)\right)^{n},$$ where we have used the substitution $\frac{1}{2}+i\gamma=\frac{1}{1-z}$ or $z=\frac{2\gamma+i}{2\gamma-i}$. Then, since $$\left(Im\left(\frac{2\gamma+i}{2\gamma-i}+1\right)\right)^{n}=\frac{(4\gamma)^{n}}{(4\gamma^{2}+1)^{n/2}}\sin\left(n\tan^{-1}\frac{1}{2\gamma}\right)=2^{n}\cos^{n}\theta\sin(n\theta),$$ $$\cos\theta:=\frac{2\gamma}{\sqrt{4\gamma^{2}+1}},$$ we get $$\pi N_{\chi}(\gamma)=\sum_{n=1}^{\infty}C_{\chi}(n)2^{n}\cos^{n}\theta\sin(n\theta).$$ Using the identity $$\int_{0}^{\pi/2}\cos^{n}\theta\sin(n\theta)\sin(2m\theta)d\theta=\frac{\pi}{2^{n+2}}\left(_{m}^{n}\right),\ \ m, n\in{\nb},$$ we deduce $$\int_{0}^{\pi/2}\pi N_{\chi}(\gamma)\sin(2m\theta)d\theta=\sum_{n=1}^{\infty}C_{\chi}(n)\frac{\pi}{4}\left(_{m}^{n}\right).$$ Hence, $$\sum_{n=1}^{\infty}C_{\chi}(n)\left(_{m}^{n}\right)=4\int_{0}^{\pi/2}N_{\chi}(\gamma)U_{m-1}(\cos(2\theta))\sin(2\theta)d\theta,$$ where $U_{m-1}$ are the Chebyschev polynomial of the second kind. Using that $$U_{m-1}(\cos\theta):=\frac{\sin(m\theta)}{\sin(\theta)},\ \ \ \cos(2\theta)=\cos^{2}\theta-\sin^{2}\theta=\frac{4\gamma^{2}-1}{4\gamma^{2}+1},$$ $$\sin(2\theta)d\theta=-2\cos\theta d(\cos\theta)$$ and that as $\gamma$ proceeds from 0 to $\infty$, $\theta$ subtends an angle from $\pi/2$ to $0$, we obtain $$\sum_{n=1}^{\infty}C_{\chi}(n)\left(_{m}^{n}\right)=8\int_{0}^{\infty}N_{\chi}(\gamma)U_{m-1}\left(\frac{4\gamma^{2}-1}{4\gamma^{2}+1}\right)\times\frac{2\gamma}{\sqrt{4\gamma^{2}+1}}\times\frac{2}{(4\gamma^{2}+1)^{3/2}}d\gamma.$$ Therefore, from (\[eq.18\]) we get for all $n\in{\nb}$ the following proposition.
\[prop.3\] Under the Riemann hypothesis, for $n\geq1$, we have $$\label{eq.20}
\lambda_{\chi}(n)=32\ n\ \int_{0}^{\infty}\frac{\gamma}{(4\gamma^{2}+1)^{2}}N_{\chi}(\gamma)U_{n-1}\left(\frac{4\gamma^{2}-1}{4\gamma^{2}+1}\right)d\gamma.$$
Next, we give another formula for the Li coefficient. Recall that the function $N_{\chi}(T)$ is a real step function, increasing by unity each time a new critical zero is counted: $$\label{eq.21}N_{\chi}(T)=\sum_{\rho, Im(\rho)>0}\phi(T-Im(\rho))=\sum_{k=1}^{\infty}\alpha_{k}\phi(T-\gamma_{k}),$$ where $\rho_{j}=\beta_{k}+i\gamma_{k},\ \gamma_{k}>0$ and $\phi(x-a)=1$ if $x\geq a$ and 0 if $x<a$. The zeros are ordered so that $\gamma_{k+1}>\gamma_{k}$ and the $\alpha_{k}$ counts the number of zeros with imaginary part $\gamma_{k}$ including the multiplicities. Simplification of the integral formula (\[eq.20\]) is stated in the following proposition.
\[prop.4\] Under the Riemann hypothesis, we have $$\lambda_{\chi}(n)=2\sum_{k=1}^{\infty}\alpha_{k}\left(1-T_{n}\left(\frac{4\gamma_{k}^{2}-1}{4\gamma_{k}^{2}+1}\right)\right),\ \ n\in{\nb}.$$
By (\[eq.21\]), the formula (\[eq.20\]) can be written as follows: $$\begin{aligned}
\lambda_{\chi}(n)&=&32n\sum_{k=1}^{\infty}\alpha_{k}\int_{0}^{\infty}\phi(\gamma-\gamma_{k})\frac{\gamma}{(4\gamma^{2}+1)^{2}}U_{n-1}\left(\frac{4\gamma^{2}-1}{4\gamma^{2}+1}\right)d\gamma\nonumber\\
&=&2n\sum_{k=1}^{\infty}\alpha_{k}\int_{\gamma_{k}}^{\infty}\frac{16\gamma}{(4\gamma^{2}+1)^{2}}U_{n-1}\left(\frac{4\gamma^{2}-1}{4\gamma^{2}+1}\right)d\gamma\nonumber\\
&=&2n\sum_{k=1}^{\infty}\alpha_{k}\left[\frac{1}{n}T_{n}(y)\right]_{\frac{4\gamma_{k}^{2}-1}{4\gamma_{k}^{2}+1}}^{1}\nonumber\\
&=&2\sum_{k=1}^{\infty}\alpha_{k}\left(1-T_{n}\left(\frac{4\gamma_{k}^{2}-1}{4\gamma_{k}^{2}+1}\right)\right),\nonumber\end{aligned}$$ using the following relation between the Chebyshev polynomials of the second kind and the first kind $$\int U_{n}(x)dx=\frac{1}{n+1}T_{n+1}(x).$$
This is remarkable summation expression for the Li coefficients. We numerically evaluate some of the first terms by the right hand side expression and find them to be indeed close to the required values of the Li coefficients. This is reassuring, and the results are presented in the tables below.\
Under the Riemann hypothesis, from the above arguments used in the proof of Propositions \[prop.3\] and \[prop.4\], one can derive the following formula $$\lambda_{\chi}(n,T)=2\sum_{k=1}^{N}\alpha_{k}\left(1-T_{n}\left(\frac{4\gamma_{k}^{2}-1}{4\gamma_{k}^{2}+1}\right)\right),$$ where $N=[N_{\chi}(T)]$ with $[x]=x-\{x\}$ and $\{x\}$ denotes the fractional part of $x$ (the last formula will be denoted $\lambda_{\chi}(n,N)$). Therefore, the latter formula allows one to estimate the error term $|\lambda_{\chi}(n)-\lambda_{\chi}(n,N)|$ in Proposition \[prop.4\] by evaluating directly the partial Li coefficients as in Proposition \[prop.2\].
[Numerical computations]{}\[sec.5\]
In this section, we compute and verify the positivity of the values of $\lambda_{\chi}(n)$ unconditionally or under the Riemann hypothesis. We first compute unconditionally (without assuming the Riemann hypothesis) $\tilde{\lambda}_{\chi}(n,M)$ by using equation (\[eq.3\]) and computing prime numbers up to $M$ (see. Section \[sec.2\]). We also compute under the Riemann hypothesis $$\label{eq.22}\lambda_{\chi}(n,N)=2\sum_{k=1}^{N}\alpha_{k}\left(1-T_{n}\left(\frac{4\gamma_{k}^{2}-1}{4\gamma_{k}^{2}+1}\right)\right),\ \hbox{with}\ N=10^{4}.$$ Furthermore, we carried out the calculations for several examples of characters. Some illustrative examples are cited below. We restricted the tables below for $n\leq 40$. However, one can find the other values of $n>40$ represented in the graphs 1-4.\
[**Remark.**]{} In fact, by the summation formula (\[eq.22\]), we could compute more coefficients $\lambda_{\chi}(n)$ with less time consuming way than by the arithmetic formula (\[eq.3\]), where computation of the first 50 coefficients lasted more than a week.\
Based on the tables below, we conjecture the following result.\
[**Conjecture. The coefficients $\lambda_{\chi}(n)$ are positive and increasing in $n$.**]{}\
This conjecture was partially numerical verified for the case of the Riemann zeta function (see [@2 Appendix D] and [@8]) and by the authors in a work in progress for the Hecke $L$-functions [@13].
-------------- ----------------------------------- ------------------------- ---------- ---------------------------------- ------------------------- -- --
$\chi$(mod3)
$n$ $\tilde{\lambda}_{\chi}(n,M)\ $ $\lambda_{\chi}(n,N)$ $n$ $\tilde{\lambda}_{\chi}(n,M)$ $\lambda_{\chi}(n,N)$
1 0.05316 0.056442 19 17.18050 17.16170
2 0.22763 0.22542 20 18.58480 18.69100
3 0.14844 0.50592 21 20.01400 20.24310
4 0.89344 0.89624 22 21.46700 21.81300
5 1.35725 1.39404 23 22.94280 23.39600
6 2.12951 1.99635 24 24.44030 24.98820
7 2.98573 2.69962 25 25.95870 26.58590
8 3.91334 3.49978 26 27.49700 28.18600
9 4.40970 4.39225 27 29.05460 29.78580
10 5.94841 5.37202 28 30.63070 31.38330
11 7.04344 6.43371 29 32.22460 32.97700
12 8.18382 7.57163 30 33.83580 34.56580
13 9.36580 8.77987 31 35.46370 36.14940
14 10.58620 10.05230 32 37.10770 37.72780
15 11.84230 11.38280 33 38.76730 39.3014
16 12.81150 12.76510 34 40.44210 40.87120
17 14.45250 14.19300 35 42.13150 42.43870
18 15.80260 15.66050 36 43.83530 44.00550
-------------- ----------------------------------- ------------------------- ---------- ---------------------------------- ------------------------- -- --
![ Case of $\chi$ (mod3)[]{data-label="fig:1"}](graphmod3){height="8cm"}
-------------- ----------------------------------- ------------------------- ---------- ---------------------------------- ------------------------- -- --
$\chi$(mod5)
$n$ $\tilde{\lambda}_{\chi}(n,M)\ $ $\lambda_{\chi}(n,N)$ $n$ $\tilde{\lambda}_{\chi}(n,M)$ $\lambda_{\chi}(n,N)$
1 0.13183 0.08562 21 25.37770 26.21450
2 0.29872 0.34152 22 27.08610 27.92160
3 0.91468 0.76482 23 28.81730 29.60960
4 1.58476 1.35081 24 30.57020 31.27780
5 2.63432 2.09300 25 32.34400 32.92720
6 3.66199 2.98332 26 34.13770 34.56020
7 4.77362 4.01225 27 35.95070 36.18030
8 5.95664 5.16902 28 37.78220 37.79200
9 7.06010 6.44188 29 39.63160 39.40090
10 8.50254 7.81828 30 41.49820 41.01320
11 9.85298 9.28519 31 43.38150 42.63540
12 11.24880 10.82930 32 45.28090 44.2746
13 12.68620 12.43740 33 47.19590 45.93760
14 14.16200 14.09650 34 49.12610 47.63100
15 15.67350 15.79410 35 51.07100 49.36130
16 17.68370 17.51860 36 53.03020 51.13410
17 14.45250 19.25930 37 55.00320 52.95430
18 20.40000 21.00670 38 56.98980 54.82600
19 22.03340 22.75260 39 58.98950 56.75210
20 23.69300 24.49030 40 61.00210 58.73450
-------------- ----------------------------------- ------------------------- ---------- ---------------------------------- ------------------------- -- --
![Case of $\chi$ (mod5)[]{data-label="fig:1"}](graphmod5){height="7cm"}
--------------- --------------------------------- ------------------------- ---------- ---------------------------------- ------------------------- -- --
$\chi$(mod20)
$n$ $\tilde{\lambda}_{\chi}(n,M)$ $\lambda_{\chi}(n,N)$ $n$ $\tilde{\lambda}_{\chi}(n,M)$ $\lambda_{\chi}(n,N)$
1 0.695021 0.319128 21 39.93370 41.70260
2 1.68502 1.24419 22 42.33530 44.31350
3 2.99412 2.68343 23 44.75970 46.59570
4 4.48123 4.50032 24 47.20580 48.55720
5 6.10005 6.53527 25 49.67270 50.26430
6 7.82087 8.63067 26 52.15960 51.83150
7 9.62565 10.65500 27 54.66570 53.403100
8 11.50180 12.52230 28 57.19040 55.12930
9 13.32220 14.20280 29 59.73290 57.14130
10 15.43400 15.72450 30 62.29260 59.52940
11 17.47760 17.16450 31 64.86910 62.32740
12 19.56650 18.63130 32 67.46160 65.50710
13 21.69710 20.24300 33 70.06980 68.98220
14 23.86610 22.10320 34 72.69310 72.62260
15 26.07070 24.28030 35 75.33110 76.27560
16 28.54690 26.79300 36 77.98350 79.79060
17 30.57800 29.60520 37 80.64970 83.04340
18 32.87670 32.63050 38 83.32940 85.95580
19 35.20320 35.74610 39 86.02230 88.50750
20 37.55600 38.81360 40 88.72800 90.73760
--------------- --------------------------------- ------------------------- ---------- ---------------------------------- ------------------------- -- --
![Case of $\chi$ (mod20)[]{data-label="fig:1"}](graphmod20){height="7cm"}
--------------- ----------------------------------- ------------------------- ---------- ---------------------------------- ------------------------- -- --
$\chi$(mod60)
$n$ $\tilde{\lambda}_{\chi}(n,M)\ $ $\lambda_{\chi}(n,N)$ $n$ $\tilde{\lambda}_{\chi}(n,M)$ $\lambda_{\chi}(n,N)$
1 1.12226 0.48626 21 51.46920 50.88960
2 2.78363 1.86950 22 54.42010 52.52830
3 4.64204 3.94169 23 57.39370 54.44350
4 6.83662 6.41363 24 60.38910 56.86290
5 8.84658 8.98530 25 63.40530 59.89590
6 11.11670 11.41720 26 66.44150 63.50750
7 13.47080 13.58380 27 69.49700 67.53000
8 15.89630 15.49640 28 72.57090 71.70750
9 18.06830 17.28820 29 75.6628 75.7637
10 20.92710 19.16770 30 78.77180 79.47310
11 23.52000 21.35250 31 81.89750 82.71770
12 26.15820 24.00100 32 85.03940 85.51520
13 28.83810 27.16160 33 88.19690 88.00960
14 31.55630 30.75170 34 91.36950 90.42920
15 34.31030 34.57380 35 94.55690 93.02160
16 37.56690 38.36300 36 97.75850 95.98430
17 39.91620 41.85530 37 100.97400 99.40850
18 42.76420 44.85610 38 104.20300 103.25300
19 45.64000 47.29300 39 107.44500 107.35200
20 48.54210 49.23760 40 110.70000 111.46000
--------------- ----------------------------------- ------------------------- ---------- ---------------------------------- ------------------------- -- --
![Case of $\chi$ (mod60)[]{data-label="fig:1"}](graphmod60){height="7cm"}
[**Acknowledgements**]{}. The authors would like to thank Maciej Radziejewski and Mark Coffey for their many valuable comments about the published paper.
[99]{}
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[^1]: 2000 Mathematics Subject Classification: 11M06,11M26, 11M36.\
Keywords and Phrases: Dirichlet $L$-functions, Li’s criterion, Riemann hypothesis.
|
---
abstract: 'We compute the analytic expression of the probability distributions $F_{FTSE100,+}$ and $F_{FTSE100,-}$ of the normalized positive and negative FTSE100 (UK) index daily returns $r(t)$. Furthermore, we define the $\alpha$ re-scaled FTSE100 daily index positive returns $r(t)^\alpha$ and negative returns $(-r(t))^\alpha$ that we call, after normalization, the $\alpha$ positive fluctuations and $\alpha$ negative fluctuations. We use the Kolmogorov-Smirnov statistical test, as a method, to find the values of $ \alpha$ that optimize the data collapse of the histogram of the $ \alpha$ fluctuations with the Bramwell-Holdsworth-Pinton (BHP) probability density function. The optimal parameters that we found are $\alpha^{+}= 0.55$ and $\alpha^{-}= 0.55$. Since the BHP probability density function appears in several other dissimilar phenomena, our results reveal an universal feature of the stock exchange markets.'
author:
- |
Rui Gonçalves $^{\rm a}$, Helena Ferreira $^{\rm b}$ and Alberto Pinto $^{\rm c}$\
$^{\rm a}$[*[LIAAD-INESC Porto LA and Faculty of Engineering, University of Porto, R. Dr. Roberto Frias s/n, 4200-465 Porto, Portugal]{}*]{}\
$^{\rm b}$ [*LIAAD-INESC Porto LA, Portugal*]{}\
$^{\rm c}$ [*LIAAD-INESC Porto LA and Department of Mathematics, Faculty of Sciences, University of Porto. Rua do Campo Alegre, 687, 4169-007, Portugal*]{}
title: Universal Fluctuations of the FTSE100
---
Introduction
============
The modeling of the time series of stock prices is a main issue in economics and finance and it is of a vital importance in the management of large portfolios of stocks [@Gabaixetal03; @LilloMan01; @ManStan95]. Here, we analyze the well known FTSE100 Index — also called FTSE100, FTSE, or, informally, the “footsie” that corresponds to a share index of the 100 most highly capitalised UK companies listed on the London Stock Exchange. It is the most widely used of the FTSE Group’s indices and is frequently reported as a measure of business prosperity. The FTSE100 companies represent about 81 $\%$ of the market capitalisation of the whole London Stock Exchange. The time series to investigate in our analysis is the *FTSE100 index* from April of 1984 to September of 2009. Let $Y(t)$ be the FTSE100 index adjusted close value at day $t$. We define the *FTSE100 index daily return* on day $t$ by $$r(t)=\frac{Y(t)-Y(t-1)}{Y(t-1)}.$$ We define the $\alpha$ *re-scaled FTSE100 daily index positive returns* $r(t)^\alpha$, for $r(t)>0$, that we call, after normalization, the $\alpha$ *positive fluctuations*. We define the $\alpha$ *re-scaled FTSE100 daily index negative returns* $(-r(t))^\alpha$, for $r(t)<0$, that we call, after normalization, the $\alpha$ *negative fluctuations*. We analyze, separately, the $\alpha$ positive and $\alpha$ negative daily fluctuations that can have different statistical and economic natures due, for instance, to the leverage effects (see, for example, [@Andersen; @Barnhartetal09; @Pinto; @Pinto1]). Our aim is to find the values of $\alpha$ that optimize the data collapse of the histogram of the $\alpha$ positive and $\alpha$ negative fluctuations to the universal, non-parametric, Bramwell-Holdsofworth-Pinton (BHP) probability density function. To do it, we apply the Kolmogorov-Smirnov statistic test to the null hypothesis claiming that the probability distribution of the $\alpha$ fluctuations is equal to the (BHP) distribution. We observe that the $P$ values of the Kolmogorov-Smirnov test vary continuously with $\alpha$. The highest $P$ values $P^{+}=0.19...$ and $P^{-}=0.14...$ of the Kolmogorov-Smirnov test are attained for the values $\alpha^{+}= 0.55...$ and $\alpha^{-}= 0.55...$, respectively, for the positive and negative fluctuations. Hence, the null hypothesis is not rejected for values of $\alpha$ in small neighborhoods of $\alpha^{+}= 0.55...$ and $\alpha^{-}= 0.55...$. Then, we show the data collapse of the histograms of the $\alpha^{+}$ positive fluctuations and $\alpha^{-}$ negative fluctuations to the BHP pdf. Using this data collapse, we do a change of variable that allow us to compute the analytic expressions of the probability density functions $f_{FTSE100,+}$ and $f_{FTSE100,-}$ of the normalized positive and negative FTSE100 index daily returns $$\begin{aligned}
f_{FTSE100,+}(x) &=& 8.73x^{-0.45}f_{BHP}(30.87x^{0.55}-1.95) \\
f_{FTSE100,-}(x) &=& 8.74x^{-0.45}f_{BHP}(28.88x^{0.55}-1.82)\end{aligned}$$ in terms of the BHP pdf $f_{BHP}$. We exhibit the data collapse of the histogram of the positive and negative returns to our proposed theoretical pdf´s $f_{FTSE100,+}$ and $f_{FTSE100,-}$. Similar results are observed for some other stock indexes, prices of stocks, exchange rates and commodity prices (see [@Gonc; @Gond]). Since the BHP probability density function appears in several other dissimilar phenomena (see, for instance, [@bramwellfennelleuphys2002; @DahlstedtJensen2001; @DahlstedtJensen2005; @Gona; @Gonb; @Gonf; @Gong; @Pinto]), our result reveals an universal feature of the stock exchange markets.
Positive FTSE100 index daily returns
====================================
Let $T^+$ be the set of all days $t$ with positive returns, i.e. $$T^+=\{t:r(t)>0\} .$$ Let $n^+=3367$ be the cardinal of the set $T^+$. The *$\alpha$ re-scaled FTSE100 daily index positive returns* are the returns $r(t)^\alpha$ with $t\in T^+$. Since the total number of observed days is $n=6442$, we obtain that $n^+/n=0.52$. The *mean* $\mu^+_{\alpha}=0.063...$ of the $\alpha$ re-scaled FTSE100 daily index positive returns is given by $$\mu^+_{\alpha}=\frac{1}{n^{+}}\sum_{t\in T^+}r(t)^\alpha
\label{eq2}$$ The *standard deviation* $\sigma^+_{\alpha}=0.032...$ of the $\alpha$ re-scaled FTSE100 daily index positive returns is given by $$\sigma^+_{\alpha}=\sqrt{\frac{1}{n^{+}}\sum_{t\in T^+} {r(t)^{2\alpha}} - (\mu^+_{\alpha})^2}
\label{eq3}$$ We define the $\alpha$ *positive fluctuations* by $$r^+_{\alpha}(t) = \frac{r(t)^\alpha - \mu^+_{\alpha}}{\sigma^+_{\alpha}}
\label{eq6}$$ for every $t\in T^+$. Hence, the $\alpha$ *positive fluctuations* are the normalized $\alpha$ re-scaled $FTSE100$ daily index positive returns. Let $L^+_{\alpha}=-1.88...$ be the *smallest* $\alpha$ positive fluctuation, i.e. $$L^+_{\alpha}=\min_{t\in T^+}\{r^+_{\alpha}(t)\}.$$ Let $R^+_{\alpha}=6.68...$ be the *largest* $\alpha$ positive fluctuation, i.e. $$R^+_{\alpha}=\max_{t\in T^+}\{r^+_{\alpha}(t)\}.$$ We denote by $F_{\alpha,+}$ the *probability distribution of the $\alpha$ positive fluctuations*. Let the *truncated BHP probability distribution* $F_{BHP,\alpha,+}$ be given by $$F_{BHP, \alpha,+}(x)=\frac{F_{BHP}(x)}{F_{BHP}(R^+_{\alpha})-F_{BHP}(L^+_{\alpha})}$$ where $F_{BHP}$ is the BHP probability distribution. We apply the Kolmogorov-Smirnov statistic test to the null hypothesis claiming that the probability distributions $F_{\alpha,+}$ and $F_{BHP,\alpha,+}$ are equal. The Kolmogorov-Smirnov $P$ *value* $P_{\alpha,+}$ is plotted in Figure \[fig1\]. Hence, we observe that $\alpha^+=0.55...$ is the point where the $P$ value $P_{\alpha,+} =0.19...$ attains its maximum.
![[]{data-label="fig1"}](fer_fig01.eps){width="8cm"}
It is well-known that the Kolmogorov-Smirnov $P$ value $P_{\alpha,+}$ decreases with the distance $\left\|F_{\alpha,+}-F_{BHP,\alpha,+}\right\|$ between $F_{\alpha,+}$ and $F_{BHP,\alpha,+}$. In Figure \[fig2\], we plot $D_{\alpha^+,+}(x)=\left|F_{\alpha^+,+}(x)-F_{BHP,\alpha^+,+}(x)\right|$ and we observe that $D_{\alpha^+,+}(x)$ attains its highest values for the $\alpha^+$ positive fluctuations above or close to the mean of the probability distribution.\
![[]{data-label="fig2"}](fer_fig02.eps){width="8cm"}
In Figures \[fig3\] and \[fig4\], we show the data collapse of the histogram $f_{\alpha^+,+}$ of the $\alpha^+$ positive fluctuations to the truncated BHP pdf $f_{BHP,\alpha^+,+}$.\
![[]{data-label="fig3"}](fer_fig03.eps){width="8cm"}
![[]{data-label="fig4"}](fer_fig04.eps){width="8cm"}
Assume that the probability distribution of the $\alpha^+$ positive fluctuations $r^+_{\alpha^+}(t)$ is given by $F_{BHP,\alpha^+,+}$ (see [@Gonb]). The pdf $f_{FTSE100,+}$ of the FTSE100 daily index positive returns $r(t)$ is given by $$f_{FTSE100,+}(x)= \frac{\alpha^+ x^{\alpha^+-1}f_{BHP}\left(\left(x^{\alpha^+}-\mu^+_{\alpha^+}\right)/\sigma^+_{\alpha^+}\right)}{\sigma^+_{\alpha^+}\left(F_{BHP}\left(R^+_{\alpha^+}\right)-F_{BHP}\left(L^+_{\alpha^+}\right)\right)}.$$
Hence, taking $\alpha^+=0.55...$, we get $$f_{FTSE100,+}(x)=8.73... x^{-0.45...}f_{BHP}(30.87...x^{0.55...}-1.95...).$$ In Figures \[fig5\] and \[fig6\], we show the data collapse of the histogram $f_{1,+}$ of the positive returns to our proposed theoretical pdf $f_{FTSE100,+}$.
![[]{data-label="fig5"}](fer_fig05.eps){width="8cm"}
![[]{data-label="fig6"}](fer_fig06.eps){width="8cm"}
Negative FTSE100 index daily returns
====================================
Let $T^-$ be the set of all days $t$ with negative returns, i.e. $$T^-=\{t:r(t)<0\} .$$ Let $n^-=3074$ be the cardinal of the set $T^-$. Since the total number of observed days is $n=6442$, we obtain that $n^-/n=0.48$. The *$\alpha$ re-scaled FTSE100 daily index negative returns* are the returns $(-r(t))^\alpha$ with $t\in T^-$. We note that $-r(t)$ is positive. The *mean* $\mu^-_{\alpha}=0.063...$ of the $\alpha$ re-scaled FTSE100 daily index negative returns is given by $$\mu^-_{\alpha}=\frac{1}{n^{-}}\sum_{t\in T^-}(-r(t))^\alpha
\label{eq2}$$ The *standard deviation* $\sigma^-_{\alpha}=0.035...$ of the $\alpha$ re-scaled FTSE100 daily index negative returns is given by $$\sigma^-_{\alpha}=\sqrt{\frac{1}{n^{-}}\sum_{t\in T^-} {(-r(t))^{2\alpha}} - (\mu^-_{\alpha})^2}
\label{eq3}$$ We define the $\alpha$ *negative fluctuations* by $$r^-_{\alpha}(t) = \frac{(-r(t))^\alpha - \mu^-_{\alpha}}{\sigma^-_{\alpha}}
\label{eq6}$$ for every $t\in T^-$. Hence, the $\alpha$ *negative fluctuations* are the normalized $\alpha$ re-scaled $FTSE100$ daily index negative returns. Let $L^-_{\alpha}=-1.74...$ be the *smallest* $\alpha$ negative fluctuation, i.e. $$L^-_{\alpha}=\min_{t\in T^-}\{r^-_{\alpha}(t)\}.$$ Let $R^-_{\alpha}=7.27...$ be the *largest* $\alpha$ negative fluctuation, i.e. $$R^-_{\alpha}=\max_{t\in T^-}\{r^-_{\alpha}(t)\}.$$ We denote by $F_{\alpha,-}$ the *probability distribution of the $\alpha$ negative fluctuations*. Let the *truncated BHP probability distribution* $F_{BHP,\alpha,-}$ be given by $$F_{BHP, \alpha,-}(x)=\frac{F_{BHP}(x)}{F_{BHP}(R^-_{\alpha})-F_{BHP}(L^-_{\alpha})}$$ where $F_{BHP}$ is the BHP probability distribution. We apply the Kolmogorov-Smirnov statistic test to the null hypothesis claiming that the probability distributions $F_{\alpha,-}$ and $F_{BHP,\alpha,-}$ are equal. The Kolmogorov-Smirnov $P$ *value* $P_{\alpha,-}$ is plotted in Figure \[fig1x\]. Hence, we observe that $\alpha^-=0.55...$ is the point where the $P$ value $P_{\alpha^-} =0.68...$ attains its maximum.
![[]{data-label="fig1x"}](fer_fig07.eps){width="8cm"}
The Kolmogorov-Smirnov $P$ value $P_{\alpha,-}$ decreases with the distance $\left\|F_{\alpha,-}-F_{BHP,\alpha,-}\right\|$ between $F_{\alpha,-}$ and $F_{BHP,\alpha,-}$. In Figure \[fig2x\], we plot $D_{\alpha^-,-}(x)=\left|F_{\alpha^-,-}(x)-F_{BHP,\alpha^-,-}(x)\right|$ and we observe that $D_{\alpha^-,-}(x)$ attains its highest values for the $\alpha^-$ negative fluctuations below the mean of the probability distribution.\
![[]{data-label="fig2x"}](fer_fig08.eps){width="8cm"}
In Figures \[fig3x\] and \[fig4x\], we show the data collapse of the histogram $f_{\alpha^-,-}$ of the $\alpha^-$ negative fluctuations to the truncated BHP pdf $f_{BHP,\alpha^-,-}$.\
![[]{data-label="fig3x"}](fer_fig09.eps){width="8cm"}
![[]{data-label="fig4x"}](fer_fig10.eps){width="8cm"}
Assume that the probability distribution of the $\alpha^-$ negative fluctuations $r^-_{\alpha^-}(t)$ is given by $F_{BHP,\alpha^-,-}$, (see [@Gonb]). The pdf $f_{FTSE100,-}$ of the FTSE100 daily index (symmetric) negative returns $-r(t)$, with $T \in T^-$, is given by $$f_{FTSE100,-}(x)= \frac{\alpha^- x^{\alpha^-1}f_{BHP}\left(\left(x^{\alpha^-}-\mu^-_{\alpha^-}\right)/\sigma^-_{\alpha^-}\right)}{\sigma^-_{\alpha^-}\left(F_{BHP}\left(R^-_{\alpha^-}\right)-F_{BHP}\left(L^-_{\alpha^-}\right)\right)}.$$
Hence, taking $\alpha^-=0.55...$, we get $$f_{FTSE100,-}(x)=8.74...x^{-0.45...}f_{BHP}(28.88...x^{0.55...}-1.82...)$$
In Figures \[fig5x\] and \[fig6x\], we show the data collapse of the histogram $f_{1,-}$ of the negative returns to our proposed theoretical pdf $f_{FTSE100,-}$.
![[]{data-label="fig5x"}](fer_fig11.eps){width="8cm"}
![[]{data-label="fig6x"}](fer_fig12.eps){width="8cm"}
Conclusions
===========
We used the Kolmogorov-Smirnov statistical test to compare the histogram of the $\alpha$ positive fluctuations and $\alpha$ negative fluctuations with the universal, non-parametric, Bramwell-Holdsworth-Pinton (BHP) probability distribution. We found that the parameters $\alpha^{+}= 0.55...$ and $\alpha^{-}= 0.55...$ for the positive and negative fluctuations, respectively, optimize the $P$ value of the Kolmogorov-Smirnov test. We obtained that the respective $P$ values of the Kolmogorov-Smirnov statistical test are $P^{+}=0.19...$ and $P^{-}=0.14...$. Hence, the null hypothesis was not rejected. The fact that $\alpha^+$ is different from $\alpha^-$ can be do to leverage effects. We presented the data collapse of the corresponding fluctuations histograms to the BHP pdf. Furthermore, we computed the analytic expression of the probability distributions $F_{FTSE100,+}$ and $F_{FTSE100,-}$ of the normalized FTSE100 index daily positive and negative returns in terms of the BHP pdf. We showed the data collapse of the histogram of the positive and negative returns to our proposed theoretical pdfs $f_{FTSE100,+}$ and $f_{FTSE100,-}$. The results obtained in daily returns also apply to other periodicities, such as weekly and monthly returns as well as intraday values.
In [@Gonc; @science], it is found the data collapses of the histograms of some other stock indexes, prices of stocks, exchange rates, commodity prices and energy sources [@s6] to the BHP pdf.
Bramwell, Holdsworth and Pinton [@BHP1998] found the probability distribution of the fluctuations of the total magnetization, in the strong coupling (low temperature) regime, for a two-dimensional spin model (2dXY) using the spin wave approximation. From a statistical physics point of view, one can think that the stock prices form a non-equilibrium system [@Chowdhury; @Gopikrishnanetal98; @LilloMan01; @Plerouetal99]. Hence, the results presented here lead to a construction of a new qualitative and quantitative econophysics model for the stock market based in the two-dimensional spin model (2dXY) at criticality (see [@Gond]).
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Henrik Jensen, Peter Holdsworth and Nico Stollenwerk for showing us the relevance of the Bramwell-Holdsworth-Pinton distribution. This work was presented in PODE09, EURO XXIII, Encontro Ciência 2009 and ICDEA2009. We thank LIAAD-INESC Porto LA, Calouste Gulbenkian Foundation, PRODYN-ESF, POCTI and POSI by FCT and Ministério da Ciência e da Tecnologia, and the FCT Pluriannual Funding Program of the LIAAD-INESC Porto LA. Part of this research was developed during a visit by the authors to the IHES, CUNY, IMPA, MSRI, SUNY, Isaac Newton Institute and University of Warwick. We thank them for their hospitality.
Definition of the Bramwell-Holdsworth-Pinton probability distribution
=====================================================================
The universal nonparametric BHP pdf was discovered by Bramwell, Holdsworth and Pinton [@BHP1998]. The *BHP probability density function (pdf)* is given by $$\begin{aligned}
&\!&f_{BHP}(\mu)=\int_{-\infty}^{\infty}\frac{dx}{2\pi}
\sqrt{\frac{1}{2N^2}\sum_{k=1}^{N-1}\frac{1}{\lambda_k^2}}
e^{ix\mu\sqrt{\frac{1}{2N^2}\sum_{k=1}^{N-1}\frac{1}{\lambda_k^2}}}\nonumber\\\!
&\!&
.e^{-\sum_{k=1}^{N-1}\left[\frac{ix}{2N}\frac{1}{\lambda_k}-\frac{i}{2}
\mbox{arctan}\left(\frac{x}{N\lambda_k}\right)\right]}.e^{-\sum_{k=1}^{N-1}\left[\frac{1}{4}\mbox{ln}{\left(1+\frac{x^2}{N^2\lambda_k^2}\right)}\right]}\nonumber\\
\label{eq1}\end{aligned}$$ where the $\{\lambda_k\}_{k=1}^L$ are the eigenvalues, as determined in [@Bramwelletal2001], of the adjacency matrix. It follows, from the formula of the BHP pdf, that the asymptotic values for large deviations, below and above the mean, are exponential and double exponential, respectively (in this article, we use the approximation of the BHP pdf obtained by taking $L=10$ and $N=L^2$ in equation (\[eq1\])). As we can see, the BHP distribution does not have any parameter (except the mean that is normalize to 0 and the standard deviation that is normalized to 1) and it is universal, in the sense that appears in several physical phenomena. For instance, the universal nonparametric BHP distribution is a good model to explain the fluctuations of order parameters in theoretical examples such as, models of self-organized criticality, equilibrium critical behavior, percolation phenomena (see [@BHP1998]), the Sneppen model (see [@BHP1998] and [@DahlstedtJensen2001]), and auto-ignition fire models (see [@SinharayBordaJensen2001]). The universal nonparametric BHP distribution is, also, an explanatory model for fluctuations of several phenomenon such as, width power in steady state systems (see [@BHP1998]), fluctuations in river heights and flow (see [@Bramwelletal2001; @DahlstedtJensen2005; @Gona; @Gonb; @Gonf]), for the plasma density fluctuations and electrostatic turbulent fluxes measured at the scrape-off layer of the Alcator C-mod Tokamaks (see [@VanMilligen05]) and for Wolf’s sunspot numbers fluctuations (see [@Gong]).
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---
abstract: 'Gene drives have the potential to rapidly replace a harmful wild-type allele with a gene drive allele engineered to have desired functionalities. However, an accidental or premature release of a gene drive construct to the natural environment could damage an ecosystem irreversibly. Thus, it is important to understand the spatiotemporal consequences of the super-Mendelian population genetics prior to potential applications. Here, we employ a reaction-diffusion model for sexually reproducing diploid organisms to study how a locally introduced gene drive allele spreads to replace the wild-type allele, even though it possesses a selective disadvantage $s>0$. Using methods developed by N. Barton and collaborators, we show that socially responsible gene drives require $0.5<s<0.697$, a rather narrow range. In this “pushed wave” regime, the spatial spreading of gene drives will be initiated only when the initial frequency distribution is above a threshold profile called “critical propagule”, which acts as a safeguard against accidental release. We also study how the spatial spread of the pushed wave can be stopped by making gene drives uniquely vulnerable (“sensitizing drive”) in a way that is harmless for a wild-type allele. Finally, we show that appropriately sensitized drives in two dimensions can be stopped even by imperfect barriers perforated by a series of gaps.'
author:
- Hidenori Tanaka
- 'Howard A. Stone'
- 'David R. Nelson'
bibliography:
- 'reference.bib'
title: Spatial gene drives and pushed genetic waves
---
The development of the CRISPR/Cas9 system [@cong2013multiplex; @jinek2013rna; @mali2013rna; @wright2016biology], derived from an adaptive immune system in prokaryotes [@marraffini2015crispr], has received much recent attention, in part due to its exceptional versatility as a gene editor in sexually-reproducing organisms, compared to similar exploitations of homologous recombination such as zinc-finger nucleases (ZFNs) and the TALENS system [@jiang2015crispr; @wright2016biology]. Part of the appeal is the potential for introducing a novel gene into a population, allowing control of highly pesticide-resistant crop pests and disease vectors such as mosquitoes [@alphey2014genetic; @burt2014heritable; @esvelt2014concerning; @gantz2016dawn]. Although the genetic modifications typically introduce a fitness cost or a “selective disadvantage”, the enhanced inheritance rate embodied in CRISPR/Cas9 gene drives nevertheless allows edited genes to spread, even when the fitness cost of the inserted gene is large. The idea of using constructs that bias gene transmission rates to rapidly introduce novel genes into ecosystems has been discussed for many decades [@curtis1968possible; @foster1972chromosome; @burt2003site; @sinkins2006gene; @gould2008broadening; @deredec2008population]. Similar “homing endonuclease genes” (in the case of CRISPR/Cas9, the homing ability is provided by a guide RNA) were considered earlier by ecologists in the context of control of malaria in Africa [@north2013modelling; @Eckhoff:2017:10.1073/pnas.1611064114].
![ Schematics of the gene drive machinery with a perfect conversion efficiency $c=1$. (A) Every time an individual homozygous for the drive construct and a wild-type mate, heterozygotes in the embryo are converted to homozygotes by the mutagenic chain reaction (MCR). (B) Gene drives enhance their inheritance rate beyond that of the conventional Mendelian population genetics and can spread even with a selective disadvantage. []{data-label="Fig_1"}](./Figure_1.pdf){width="0.9\columnwidth"}
As a hypothetical example of a gene drive applied to a pathogen vector requiring both a vertebrate and insect host, consider plasmodium, carried by mosquitoes and injected with its saliva into humans (Fig. \[Fig\_1\]). Female mosquitoes typically hatch from eggs in small standing pools of water and, after mating, search for a human to feed on. They then lay their eggs and repeat the process, thus spreading the infection over a few gonotrophic cycles. A gene drive could alter the function of a protein manufactured in the salivary gland of female mosquitoes from, say, type $a$, anesthetizing nerve cells when it bites humans, to instead type $A$, clogging up essential chemoreceptors in plasmodium and thus killing these eukaryotes. In the absence of a gene drive, there would be a selective disadvantage or fitness cost $s$ to losing this protein. Even if the fitness cost $s$ were zero, it is unlikely that this new trait would be able to escape genetic drift in large populations. However, as we describe below, the trait could spread easily if linked to a gene drive that converts heterozygotes to homozygotes with efficiency $c$ close to 1 (Fig. \[Fig\_1\]A). Remarkably, high conversion rates have already been achieved with the mutagenic chain reaction (“MCR”) realized by the CRISPR/Cas9 system [@cong2013multiplex; @jinek2013rna; @mali2013rna] for yeast ($c_{\rm{yeast}}>0.995$) [@dicarlo2015safeguarding], fruit flies ($c_{\rm{flies}}=0.97$) [@gantz2015mutagenic] and malaria vector mosquito, *Anopheles stephensi* with engineered malaria resistance ($c_{\rm{mosquito}}\geq 0.98$) [@gantz2015highly].
However, the gene drives’ intrinsic nature of irreversibly altering wild-type populations raises biosafety concerns [@esvelt2014concerning], and calls for confinement strategies to prevent unintentional escape and spread of the gene drive constructs [@akbari2015safeguarding]. While various genetic design or containment strategies have been discussed [@chan2011insect; @henkel2012monitoring; @esvelt2014concerning; @gantz2015mutagenic], and a few computational simulations were conducted [@huang2011gene; @north2013modelling; @Eckhoff:2017:10.1073/pnas.1611064114], the *spatial* spreading of the gene drive alleles has received less attention.
To understand such phenomena in a spatial context, we will exploit a methodology developed by N. Barton and collaborators, originally in an effort to understand adaptation and speciation of diploid sexually reproducing organisms in genetic hybrid zones [@barton1979dynamics; @barton1989adaptation; @barton2011spatial]. We apply these techniques to a spatial generalization of a model of diploid CRISPR/Cas9 population genetics proposed by Unckless *et al.* [@unckless2015modeling], and highlight two distinct ways in which gene drive alleles can spread spatially. The non-Mendelian (or “super-Mendelian” [@Noble057281]) population genetics of gene drives are remarkable because individuals homozygous for a gene drive can in fact spread into wild-type populations even if they carry a positive selective disadvantage $s$ (Fig. \[Fig\_1\]B). First, for small selective disadvantages ($0 < s < 0.5$ in our case), the spatial spreading proceeds via a well-known Fisher-Kolmogorov-Petrovsky-Piskunov wave [@fisher1937wave; @kolmogorov]. Such pulled genetic waves [@stokes1976two; @lewis2016finding; @gandhi2016range] are driven by growth and diffusive dispersal at the leading edge, and are difficult to slow down and stop.
However, for somewhat larger selective disadvantages ($0.5 < s < 0.697$) we find that propagation proceeds instead via a pushed genetic wave [@stokes1976two; @lewis2016finding; @gandhi2016range], where the genetic wave advances via accentuated growth from populations somewhat behind the front that spill over the leading edge. These waves, characterized by a strong Allee effect [@lewis1993allee; @taylor2005allee], are more socially responsible than the pulled Fisher waves because: (i) only inoculations whose spatial size and density exceed a critical nucleus, or “critical propagule”[@barton2011spatial] are able to spread spatially, thus providing protection against a premature or accidental release of a gene drive, (ii) the gene drive pushed waves can be stopped by making them uniquely vulnerable to a specific compound (“sensitizing drive” [@esvelt2014concerning]), which is harmless for a wild-type allele, and (iii) appropriately sensitized gene drives can be stopped even by barriers punctuated by defects, analogous to regularly spaced fire breaks used to contain forest fires. Similar pushed or “excitable” waves also arise, for example, in neuroscience, in simplified versions of the Hodgkin-Huxley model of action potentials [@nelson2004biological]. When the selective disadvantage associated with the gene drive is too large ($s > 0.697$ in our model) the excitable wave reverses direction and the region occupied by the gene drive homozygotes collapses to zero.
![ Schematic phase diagram of the spatial evolutionary games in one dimension [@frey2010evolutionary; @korolev2011competition; @lavrentovich2014asymmetric]. The parameters $\alpha$ and $\beta$ control interactions between red and green haploid organisms. Positive $\alpha$ means the presence of the green allele favors the red allele, positive $\beta$ enhances the green allele when red is present, etc. (see SI Appendix for a detailed description of the model.) Pulled Fisher wave regimes (controlling, for example, the dynamics of selective dominance in the second and four quadrants) and the pushed excitable wave regimes (third quadrant, competitive exclusion dynamics) are bounded by the black dashed spinodal lines $\alpha=0, \beta < 0$ and $\alpha<0, \beta = 0$. These two bistable regimes are separated by the first-order phase transition (PT) line $\alpha=\beta<0$, drawn as a black solid line. []{data-label="Fig_2"}](./Figure_2.pdf){width="0.8\columnwidth"}
The same mathematical analyses applies to spatial evolutionary games of two competing species in one dimension, which are governed by a class of reaction-diffusion equations that resemble the gene drive system. The fitnesses of the two interacting red and green species ($w_R$, $w_G$) are related to their frequencies ($f(x,t)$, $1-f(x,t)$) by $w_R (x,t) = g + \alpha (1-f(x,t)), w_G (x,t) = g + \beta f(x,t)$, where $g$ is a background fitness, assumed identical for the two alleles for simplicity. The mutualistic regime $\alpha>0, \beta>0$ in the first quadrant of Fig. \[Fig\_2\] has been studied already [@korolev2011competition], including the effect of genetic drift, with two lines of directed “percolation” transitions out of a mutualistic phase. Here, we apply the methods of [@barton2011spatial] to study the evolutionary dynamics near the line of first-order transitions that characterize the competitive exclusion regime in the third quadrant of Fig. \[Fig\_2\]. Because the mathematics parallels the analysis inspired by gene drive systems in the main text, we relegate discussion of this topic to the SI Appendix, which also discusses conversion efficiencies $c<1$, an analogy with nucleation theory, laboratory tests and other matters.\
Mathematical model of the CRISPR gene drives {#mathematical-model-of-the-crispr-gene-drives .unnumbered}
--------------------------------------------
We start with a Hardy-Weinberg model [@hartl1997principles] and incorporate a mutagenic chain reaction (“MCR”) with $100\%$ conversion rate to construct a model for a well-mixed system. This model is the limiting case of “$c=1$” in the work of Unckless *et al* [@unckless2015modeling]. Conversion efficiencies $c<1$ can be handled by similar techniques. First, we consider a well-mixed diploid system with a wild-type allele $a$ and a gene drive allele $A$ with frequencies $p=p(t)$ and $q=q(t)$ respectively at time $t$, with $p(t)+q(t)=1$. Within a random mating model, the allele frequencies after one generation time $\tau_g$ are given by $$(pa+qA)^2 = p^2 (a,a) + 2pq(a,A)+q^2(A,A),$$ and the ratios of fertilized eggs with diploid types $(a,a)$, $(a,A)$ and $(A,A)$ are $p^2:2pq:q^2$. In a heterozygous $(a,A)$ egg, the CRISPR/Cas9 machinery encoded on a gene drive allele $A$ converts the wild-type allele $a$ into a gene drive allele $A$. Here, we assume a perfect conversion rate $(a,A)\xrightarrow[\rm{MCR}]{c=1}(A,A)$ in the embryo, as has been approximated already for yeast [@dicarlo2015safeguarding] and fruit flies [@gantz2015mutagenic]. Genetic engineering will typically reduce the fitness of individuals carrying the gene drive alleles compared to wild-type organisms, which have already gone through natural evolution and may be near a fitness maximum.
The selective disadvantage of a gene drive allele $s$ is defined by the ratio of the fitness $w_{\rm{wild}}$ of wild-type organisms $(a,a)$ to the fitness $w_{\rm{drive}}$ of $(A,A)$ individuals carrying the gene drive, $$\frac{w_{\rm{drive}}}{w_{\rm{wild}}}\equiv1-s,~0\leq s.$$ (In the limit $c\rightarrow 1$ no heterozygous $(a,A)$ individuals are born [@unckless2015modeling].) Taking the fitness into account, the allele frequencies after one generation time $\tau_g$ are $$p' : q' = w_{\rm{wild}} p^2 : w_{\rm{wild}} (1-s)(q^2 + 2pq),$$ where $p'\equiv p(t+\tau_g)$ and $q'\equiv q(t+\tau_g)$. Upon approximating $q'-q=q(t+\tau_g)-q(t)$ by $\tau_g \frac{dq}{dt}$, we obtain a differential equation $$\begin{split}
\tau_g \frac{dq}{dt} &= \frac{(1-s)(q^2 + 2pq)}{p^2+(1-s)(q^2 + 2pq)} - q\\
&=\frac{sq(1-q)(q-q^*)}{1-sq(2-q)},\textnormal{ where}~q^* = \frac{2s-1}{s},
\end{split}
\label{eq4}$$ which governs population dynamics of the mutagenic chain reaction with $100\%$ conversion efficiency in a well-mixed system. To take spatial dynamics into account, we add a diffusion term [@barton2011spatial] and obtain a deterministic reaction-diffusion equation for the MCR model, namely $$\label{rdMCR}
\tau_g \frac{\partial q}{\partial t} = \tau_g D \frac{\partial^2 q}{\partial x^2}+ \frac{sq(1-q)(q-q^*)}{1-sq(2-q)},$$ which will be the main focus of this article. For later discussions, we name the reaction term of the reaction-diffusion equation, $$f_{\rm{MCR}}(q,s) = \frac{sq(1-q)(q-q^*)}{1-sq(2-q)}.
\label{FMCR}$$ The reaction term reduces to a simpler cubic expression $$f_{\rm{cubic}}(q,s)=sq(1-q)(q-q^*)
\label{Fcubic}$$ by ignoring $-sq(2-q)$ in the denominator, which is a reasonable approximation if the selective disadvantage $s$ is small. This form of the reaction-diffusion equation has been well studied, as reviewed in [@barton2011spatial].\
Although population genetics is often studied in the limit of small $s$, $s$ is in fact fairly large in the regime of pushed excitable waves of most interest to us here, $0.5 < s < 1.0$. Hence, we will keep the denominator of the reaction term, as was also done in [@barton2011spatial] with a different reaction term. Comparison of results for the full nonlinear reaction term with those for the cubic approximation will give us a sense of the robustness of the cubic approximation. Although it might also be of interest to study corrections to the continuous time approximation arising from higher order time derivatives in $(q'-q)/\tau_g = \frac{\partial q}{\partial t} + \frac{1}{2} \tau_g \frac{\partial^2 q}{\partial t^2}+...$ (contributions from $\tau_g \frac{\partial^2 q}{\partial t^2}$ are formally of order $s^2$ ), this complicated problem will be neglected here; see, however, [@turellibartonpre] for a study of the robustness of the continuous time approximation, motivated by a model of dengue-suppressing Wolbachia in mosquitoes.
Initiation of the pushed waves {#initiation-of-the-pushed-waves .unnumbered}
------------------------------
![ (A) Spatial dynamics of gene drives can be determined by both the selective disadvantage $s$ and (when $0.5<s<0.697$), the size and intensity of the initial condition. (B) The energy landscapes $U(q)$ with various selective disadvantages $s$. i) Pulled Fisher wave regime: When $s$ is small, $s\leq s_{\rm{min}}=0.5$ (lowermost red and yellow curves), fixation of the gene drive allele ($q=1$) is the unique stable state and there is no energy barrier between $q=0$ and $1$. Any finite introduction of a gene drive allele is sufficient to initiate a pulled Fisher population wave that spreads through space to saturate the system. ii) Pushed excitable wave regime: When $s$ is slightly larger (green curve), and satisfies $s_{\rm{min}}=0.5<s<s_{\rm{max}}=0.697$, $q=1$ is still the preferred stable state, but an energy barrier at $q=q^*$ appears between $q=0$ and $1$. In this regime, the introduction of the gene drive allele at sufficient concentration and over a sufficiently large spatial extent is required for a pushed wave to spread to global fixation. iii) Wave reverses direction: When $s$ is large, $s>s_{\rm{max}}= 0.697$ (topmost blue and purple curves), $q=0$ is the unique ground state and the gene drive species cannot establish a traveling population wave and so dies out. []{data-label="Fig_3"}](./Figure_3.pdf){width="0.8\columnwidth"}
The reaction terms $f_{\rm{MCR}}(q,s)$ and $f_{\rm{cubic}}(q,s)$ have three identical fixed points, $q=0,~1$ and $~q^*\big(=\frac{2s-1}{s}\big)$. As discussed in the SI Appendix in connection to classical nucleation theory in physics, and following [@barton1979dynamics], we can define the potential energy function $$U(q) = -\frac{1}{\tau_g} \int^{q}_{0} \frac{sq'(1-q')(q'-q^*)}{1-sq'(2-q')} dq'$$ to identify qualitatively different parameter regimes. In a well-mixed system, without spatial structure, the gene drive frequency $q(t)$ obeys Eq. \[eq4\], and evolves in time so that it arrives at a local minimum of $U(q)$. For the spatial model of interest here, $q(x,t)$ shows qualitatively distinct behaviors in three parameter regimes depending on the selective disadvantage $s$ (see Fig. \[Fig\_3\]*A*). We plot the potential energy functions $U(q)$ in these parameter regimes in Fig. \[Fig\_3\]*B*.\
i) First, when $s<s_{\rm{min}}=0.5$, fixation of a gene drive allele $q(x)=1$ for all $x$ is the unique stable state and there is no energy barrier to reach the ground state starting from $q\approx0$. In this regime, any finite frequency of gene drive allele locally introduced in space (provided it overcomes genetic drift) will spread and replace the wild-type allele. The frequency profile will evolve as a pulled traveling wave $q(x,t)=Q(x-vt)$ with wave velocity $v$. Such a wave was first found by Fisher [@fisher1937wave] and by Kolmogorov, Petrovsky and Piskunov [@kolmogorov] in the 1930s, in studies of how locally introduced organisms with advantageous genes spatially spread and replace inferior genes. However, the threshold-less initiation of population waves of engineered gene drives with relatively small selective disadvantages seems highly undesirable, since the accidental escape of a single gene drive construct can establish a population wave that spreads freely into the extended environment.
ii) There is a second regime for $0.5<s<0.697$ in which the potential energy function $U(q)$ exhibits an energy barrier between $q=0$ and $q=1$. In this regime, a pushed traveling wave can be excited only when a threshold gene drive allele frequency is introduced over a sufficiently broad region of space that exceeds the size of a critical nucleus, which we investigate in the next section. The existence of this threshold acts as a safeguard against accidental release. In addition, such excitable waves are easier to stop as we will discuss later. It appears that gene drives in this relatively narrow intermediate regime are the most desirable from a biosafety perspective.
iii) When $s>s_{\rm{max}}=0.697$, the fixation of a gene drive allele throughout space is no longer absolutely stable (Fig. \[Fig\_3\]*B*), and a gene drive population wave cannot be established. Indeed, the excitable wave reverses direction for $s>s_{\rm{max}}$. An implicit equation for $s_{\rm{max}}$ results from equating $U(0)=U(1)=0$, which yields $$\begin{split}
0 &= \int^{1}_{0} \frac{sq(1-q)(q-q^*)}{1-sq(2-q)}dq,\\
\textnormal{or } 0 &=\frac{-2+s_{\rm{max}}+2\sqrt{-1+\frac{1}{s_{\rm{max}}}}\arcsin(\sqrt{s_{\rm{max}}})}{2s_{\rm{max}}}\\
&\Rightarrow s_{\rm{max}}\approx 0.697,
\end{split}$$ where we used $q^*=(2s-1)/s$. When $s>s_{\rm{max}}$, the locally introduced gene drive allele contracts rather than expands relative to the wild-type allele and simply dies out. See SI Appendix for the analogous results with an arbitrary conversion rate ($0<c<1$).
Critical nucleus in the pushed wave regime {#critical-nucleus-in-the-pushed-wave-regime .unnumbered}
------------------------------------------
![ The excitable population wave carrying a gene drive can be established only when the initial concentration is above a threshold distribution and over a region of sufficient spatial extent (the critical nucleus or “critical propagule” [@barton2011spatial]). Numerical solutions of $\tau_g \frac{\partial q}{\partial t}=\tau_g D \frac{\partial^2 q}{\partial x^2} + \frac{sq(1-q)(q-q^*)}{1-sq(2-q)}$ with $q^*=\frac{2s-1}{s}$ are plotted with time increment $\Delta t = 2.5 \tau_g$. The early time response is shown in red with later times in blue. Selective disadvantage of the gene drive allele relative to the wild-type allele is set to $s=0.58$. In the case illustrated here, the gene drive allele can either die out or saturate the entire system, depending on the width of initial Gaussian population profile of $q(x,0)=ae^{-(x/B)^2}$. (A) With a narrow distribution of the initially introduced gene drive species $(a=0.5, B=3.0\sqrt{\tau_g D})$, the population quickly fizzles out. (B) With a broader distribution of the initial gene drive allele $(a=0.5, B=6.0\sqrt{\tau_g D})$, the gene drive allele successfully establishes a pushed population wave leading to $q(x)=1$ over the entire system. []{data-label="Fig_4"}](./Figure_4.pdf){width="0.8\columnwidth"}
When the selective disadvantage $s$ is in the intermediate regime, $s_{\rm{min}}=1/2<s<s_{\rm{max}}= 0.697$, we can control initiation of the pushed excitable wave by the initial frequency profile of the gene drive allele $q(x,0)$ as shown in Fig. \[Fig\_4\]. For example, in Fig. \[Fig\_4\]*A*, an initially introduced gene drive allele (in the form of a Gaussian) diminishes and dies out since the width of the initial frequency distribution $q(x,0)$ is not sufficient to excite the population wave. In contrast, the results in Fig. \[Fig\_4\]*B* show the successful establishment of the excitable wave starting from a sufficiently broad (Gaussian) initial distribution of a gene drive allele. Roughly speaking (provided $\frac{1}{2}<s<s_{\rm{max}}$), two conditions must be satisfied to obtain a critical propagule: (1) The initial condition $q(0,0)$ at the center of the inoculant must exceed $q^* = \frac{2s-1}{s}$, the local maximum of the function $U(q)$ plotted in Fig. \[Fig\_3\]; and (2) The spatial spread $\Delta x$ of the inoculant $q(x,t=0)$ must satisfy $\Delta x \gtrsim \rm{const}\sqrt{D \tau_g}$ where the dimensionless constant depends on $s$. Thus, the initial width should exceed the width of the pushed wave that is being launched.
![ Initial critical frequency profiles of the mutagenic chain reaction (MCR) allele $q_{\rm{c}}(x)$ just sufficient to excite a pushed genetic wave in 1D (critical propagule). Numerically calculated critical propagules for the MCR model of Eq. \[rdMCR\] (solid lines) are compared with analytical results available for the cubic model Eq. \[Fcubic\] (dashed lines) [@barton2011spatial]. When $s=0.51$, the two equations gives almost identical results, but as $s$ increases the critical propagule shape of the MCR model deviates significantly from that of the cubic model. The critical propagule of the cubic equation consistently overestimates the height of the $q_{\rm{c}}(x)$, since the $sq(2-q)>0$ term in the denominator of the MCR model always increases the growth rate. []{data-label="Fig_5"}](./Figure_5.pdf){width="0.8\columnwidth"}
We show the spatial concentration profile $q_{\rm{c}}(x)$ that constitutes that (Gaussian) critical nucleus just sufficient to initiate an excitable wave in Fig. \[Fig\_5\]. The solid lines represent numerically obtained critical nuclei of the MCR model. Note the consistency for $s=0.58$ with the pushed excitable waves shown in Fig. \[Fig\_4\]. The dashed lines represent analytically derived critical propagules of the cubic model as a reference (see SI Appendix for details). Fig. \[Fig\_5\] shows that the cubic model overestimates the height of critical propagule, particularly for larger $s$. The difference between the reaction terms of the MCR model $f_{\rm{MCR}}(q)$ (see Eq. \[FMCR\]) and that of its cubic approximation $f_{\rm{cubic}}(q)$ (see Eq. \[Fcubic\]), arises from the term $-sq(2-q)$ in the denominator of Eq. \[rdMCR\]. In the biologically relevant regime $(0<s<1,~0<q<1)$, $sq(2-q)$ is always positive and $f_{\rm{MCR}}(q)>f_{\rm{cubic}}(q)$ is satisfied, which explains why there is a larger critical propagule in the cubic approximation, and the discrepancy is larger for larger $s$. The critical nucleus with a step-function-like circular boundary is studied both numerically and analytically in two dimensions in the SI Appendix.
Stopping of pushed, excitable waves by a selective disadvantage barrier {#stopping-of-pushed-excitable-waves-by-a-selective-disadvantage-barrier .unnumbered}
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Thus far, we have found that (i) we can control initiation of the spatial spread of a gene drive provided $s_{\rm{min}}=0.5<s<s_{\rm{max}}=0.697$, and (ii) the pushed population waves in this regime slow down and eventually stop (and reverse direction) when $s>s_{\rm{max}}$, see SI Appendix. In this section, we examine alternative ways to confine an excitable gene drive wave to attain greater control over its spread in this regime.
Imagine exploiting the CRISPR/Cas9 system to encode multiple functionalities into the gene drive machinery [@cong2013multiplex; @jinek2013rna; @mali2013rna; @gantz2015mutagenic]. For example, one could produce genetically engineered mosquitoes that are not only resistant to malaria, but also specifically vulnerable to an insecticide that is harmless for the wild-type alleles. Such a gene drive, which is uniquely vulnerable to an otherwise harmless compound, is a sensitizing drive [@esvelt2014concerning]. The effect of laying down insecticide in a prescribed spatial pattern on a sensitizing drive can be incorporated in our model by increasing the selective disadvantage to a value $s_b(>s)$ within a “selective disadvantage barrier” region.
![ Numerical simulations of pushed, excitable waves generated by Eq. \[rdMCR\] with barriers in one dimension, with time increments $\Delta t = 5.0 \tau_g$. As the waves advance from left to right, the early time response is shown in red with later times in blue. The fitness disadvantage inside the barrier is set to $s_b=0.958$ within a region $25\sqrt{\tau_g D}<x<27\sqrt{\tau_g D}$ (shown as a purple bar). The initial conditions are step-function-like, $q(x,0)=q_0/(1+e^{10(x-x_0)/\sqrt{\tau_g D}})$, with $q_0 = 1.0$ and $x_0=5.0\sqrt{\tau_gD}$, similar to the initial condition Eq. S30 we used in two dimensions (see SI Appendix). (A) In the case of a Fisher wave with $s =0.479 <s_{\rm{min}}=0.5$, a small number of individuals diffuse through the barrier, which is sufficient to reestablish a robust traveling wave. (B) In the case of the excitable wave $s =0.542>s_{\rm{min}}=0.5$, a small number of individuals also diffuse through the barrier. However, since the tail of the penetrating wave front is insufficient to create a critical nucleus, the barrier causes the excitable wave to die out. []{data-label="Fig_6"}](./Figure_6.pdf){width="0.8\columnwidth"}
In Fig. \[Fig\_6\], we numerically simulate the mutagenic chain reaction model defined by Eq. \[rdMCR\] in one dimension with a barrier of strength $s_b=0.958$ placed in a region $25\sqrt{\tau_g D}<x<27\sqrt{\tau_g D}$. When the selective disadvantage outside the barrier is small $(s<0.5)$ and the population wave travels as the pulled Fisher wave, even a tiny fraction of MCR allele diffusing through the insecticide region can easily reestablish the population wave, as shown in Fig. \[Fig\_6\]*A*. However, when the system is in the pushed wave regime $0.5<s<0.697$, the wave can be stopped provided the spatial profile of the gene drive allele that leaks through does not constitute a critical nucleus, as illustrated in Fig. \[Fig\_6\]*B*. See the SI Appendix for numerically calculated plots of the critical width and barrier selective disadvantage needed to stop pushed waves for various values of $s$.
Excitable Wave Dynamics with Gapped Barriers in Two Dimensions {#excitable-wave-dynamics-with-gapped-barriers-in-two-dimensions .unnumbered}
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![ Population waves impeded by a selective disadvantage barrier of strength $s_b = 1.0$ (colored purple) with a gap. This imperfect barrier has a region without insecticide in the middle of width $6\sqrt{\tau_g D}$. (A) The pulled Fisher wave with $s=0.48<0.5$ always leaks through the gap and reestablishes the gene drive wave (colored red and yellow). (B) The pushed wave that arises when $s=0.62>0.5$ is deexcited by a gapped barrier, provided the gap width is comparable to or smaller than the width of the gene drive wave. []{data-label="Fig_7"}](./Figure_7.pdf){width="0.9\columnwidth"}
In the previous section, we showed that pushed excitable waves can be stopped by a selective disadvantage barrier in one dimension. However, in two dimensions, it may be difficult to make barriers without defects. Hence, we have also studied the effect of a gap in a two-dimensional selective disadvantage barrier. We find that while the gene drive population wave in the Fisher wave regime $s<0.5$ always leaks through the gaps, the excitable wave with $0.5<s<0.697$ can be stopped, provided the gap is comparable or smaller than the width of the traveling wave front. In Fig. \[Fig\_7\], we illustrate the gene drive dynamics for two different parameter choices. Both in Fig. \[Fig\_7\]*A* and *B*, the strength of the selective disadvantage barrier is set to be $s_b=1.0$ and the width of the gap in the barrier is set to be $6\sqrt{\tau_g D}$. The engineered selective disadvantage in the non-barrier region $s$ differs in the two plots. In Fig. \[Fig\_7\]*A* $s=0.48<0.5$, so the gene drive wave propagates as a pulled Fisher wave and the wave easily leaks through the gap. If genetic drift can be neglected, we expect that Fisher wave excitations will leak through any gap however small. However, when the selective disadvantage barrier is in the pushed wave regime $0.5<s<0.697$, the population wave can be stopped by a gapped selective disadvantage barrier as shown in Fig. \[Fig\_7\]*B*. To stop a pushed excitable wave, the gap dimensions must be smaller than the front width; alternatively, we can say that the gap must be smaller than size of the critical nucleus.
Discussion {#discussion .unnumbered}
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The CRISPR/Cas9 system has greatly expanded the design space for genome editing and construction of mutagenic chain reactions with non-Mendelian inheritance. We analyzed the spatial spreading of gene drive constructs, applying reaction-diffusion formulations that have been developed to understand spatial genetic waves with bistable dynamics [@barton1979dynamics; @barton1989adaptation; @barton2011spatial]. For a continuous time and space version of the model of Unckless *et al* [@unckless2015modeling], in the limit of $100\%$ conversion efficiency, we found that a critical nucleus or propagule is required to establish a gene drive population wave when the selective disadvantage satisfies $0.5<s<0.697$. Our model led us to study termination of pushed gene drive waves using a barrier that acts only on gene drive homozygotes, corresponding to an insecticide in the case of mosquitoes. In this parameter regime, the properties of pushed waves allow safeguards against the accidental release and spreading of the gene drives. One can, in effect, construct switches that initiate and terminate the gene drive wave. In the future, it would be interesting to study the stochasticity due to finite population size (genetic drift), which is known to play a role in the first quadrant of Fig. \[Fig\_2\] [@korolev2011competition; @lavrentovich2014asymmetric]. We expect that genetic drift can be neglected provided $N_{\rm{eff}} \gg 1$, where $N_{\rm{eff}}$ is an effective population size, say, the number of organisms in a well-mixed critical propagule. See the SI Appendix for a brief discussions on genetic drift. It could also be important to study the effect of additional mutations on an excitable gene drive wave, particularly those that move the organism outside the preferred range $0.5<s<0.697$. Finally we address possible experimental tests of the theoretical predictions. Since it seems inadvisable to conduct field tests without thorough understanding of the system, laboratory experiments with microbes would be a good starting point. Recently, the transition from pulled to pushed waves was qualitatively investigated with haploid microbial populations [@gandhi2016range]. Because the mutagenic chain reaction has already been realized in *S. Cerevisiae* [@dicarlo2015safeguarding], it may also be possible to test the theory in the context of range expansions on a Petri dish, as has already been done for haploid mutualistic yeast strains in [@muller2014genetic]. Here, the frontier approximates a one dimensional stepping stone model, and jostling of daughter cells at the frontier leads to an effective diffusion constant in one dimension [@RevModPhys.82.1691; @hallatschek2007genetic]. Finally, as illustrated in Fig. S2, the mathematics of the spatial evolutionary games in one dimension parallels the dynamics of diploid gene drives in the pushed wave regime, providing another arena for experimental tests, including the effects of genetic drift.
Numerical Simulations {#numerical-simulations .unnumbered}
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To simulate the dynamics governed by Eq. \[FMCR\] in Figs. \[Fig\_4\],\[Fig\_6\],\[Fig\_7\] and S6, we used the method of lines and discretized spatial variables to map the partial differential equation to a system of coupled ordinary equations (“ODE”). Then we solved the coupled ODEs with a standard ODE solver. The width of the spatial grids were varied from $\frac{1}{200}\sqrt{\tau_g D}$ to $\frac{1}{20}\sqrt{\tau_g D}$ always making sure that the mesh size was much smaller than the width of the fronts of the pushed and pulled genetic waves we studied.
Acknowledgement {#acknowledgement .unnumbered}
---------------
We thank N. Barton, S. Block, S. Sawyer, T. Stearns, and M. Turelli for helpful discussions and two anonymous reviewers for useful suggestions. N. Barton also provided a critical reading of our manuscript. Work by HT and DRN was supported by the National Science Foundation, through grants DMR1608501 and via the Harvard Materials Science Research and Engineering Center via grant DMR1435999. HAS acknowledges support from NSF grants MCB1344191 and DMS1614907.
Supporting Information (SI) {#supporting-information-si .unnumbered}
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Nucleation theory of the gene drive population waves
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Here we identify different parameter regimes of various types of gene drive waves by establishing an analogy between zero temperature nucleation theory and the reaction-diffusion equation of the prescribed mutagenic chain reaction, $$\label{rdMCR2}
\frac{\partial q}{\partial t} = D \frac{\partial^2 q}{\partial x^2}+ \frac{1}{\tau_g} \frac{sq(1-q)(q-q^*)}{1-sq(2-q)},$$ using the methods reviewed in [@barton2011spatial]. First, we introduce a potential energy function $U(q)$ $$U(q) = -\frac{1}{\tau_g} \int^{q}_{0} \frac{sq'(1-q')(q'-q^*)}{1-sq'(2-q')} dq', q^* = \frac{2s-1}{s},$$ and rewrite Eq. \[rdMCR2\] as $$\frac{\partial q}{\partial t} = D \frac{\partial^2 q}{\partial x^2} - \frac{dU(q)}{dq}.$$ It is useful to recast the reaction-diffusion dynamics in terms of a functional derivative $$\frac{\partial q(x,t)}{\partial t} = -\frac{\delta \mathcal{F}[q(y,t)] }{\delta q(x,t)},$$ where the functional $\mathcal{F}[q(y,t)]$ is given by $$\mathcal{F}[q(y,t)] = \int^{\infty}_{-\infty} \bigg \{ \frac{1}{2} D \Big( \frac{\partial q(y,t)}{ \partial y} \Big)^2 +U[q(y,t)] \bigg \} dy,$$
and we have $$\begin{split}
&-\frac{\delta \mathcal{F}[q(y,t)] }{\delta q(x,t)}
= - \lim_{\epsilon \rightarrow 0} \frac{ \mathcal{F}[q(y,t) + \epsilon \delta(y-x)] - \mathcal{F}[q(y,t)] }{\epsilon}\\
&= - \int_{-\infty}^{\infty}
\Big\{ D \frac{\partial q(y,t)}{\partial y} \frac{\partial \delta (y-x)}{\partial y} + \frac{dU[q(y,t)]}{dq} \delta(y-x) \Big\} dy\\
&= D \frac{\partial^2 q(x,t)}{\partial x^2} - \frac{dU[q(x,t)]}{dq}.
\end{split}$$
Since $\mathcal{F}(t)$ always decreases in time, $$\begin{split}
\frac{d \mathcal{F}(t)}{dt}
&=\int_{-\infty}^{\infty} \frac{\partial q(x,t)}{\partial t} \frac{\delta \mathcal{F}[q(y,t)]}{\delta q(x,t)}dx\\
&= - \int_{-\infty}^{\infty}\Big( \frac{\partial q(x,t)}{\partial t} \Big)^2 dx \leq 0,
\end{split}$$ $\mathcal{F}[q(y,t)]$ plays the role of the free energy in a thermodynamic system.
The potential energy function $U(q)$ with various selective disadvantages $s$ is plotted in Fig. 3. $U(1)$ becomes the absolute minimum when $0.5<s$ and population waves behave as pushed waves, because both $U(0)$ and $U(1)$ are locally stable [@barton1979dynamics; @barton1989adaptation; @barton2011spatial]. The pushed gene drive wave stalls out when the two stable points have the same potential energy (blue curve in Fig. 3). The maximum value of the selective disadvantage $s_{\rm{max}}$ supporting the pushed wave of the gene drive allele can be derived by equating $U(0)=U(1)$, which leads to $$\begin{split}
0 &=\int^{1}_{0} \frac{sq(1-q)(q-q^*)}{1-sq(2-q)}dq\\
&=\frac{-2+s_{\rm{max}}+2\sqrt{-1+\frac{1}{s_{\rm{max}}}}\arcsin(\sqrt{s_{\rm{max}}})}{2s_{\rm{max}}}.\\
&\Rightarrow s_{\rm{max}} \approx 0.697
\end{split}$$ The excitable gene drive wave of primary interest to us thus arises when the selective disadvantage satisfies $$0.5 < s < 0.697.$$
The range of the pushed wave regime with an arbitrary conversion rate
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![ $s_{\rm{min}}$ and $s_{\rm{max}}$ as a function of the conversion rate $c$ when the fitness of heterozygotes individuals is (A) recessive ($h=0$), (B) additive ($h=0.5$) and (C) dominant ($h=1.0$) of gene drives, where the fitness of heterozygotes is $1-hs$. The socially responsible pushed wave regime ($s_{\rm{min}}<s<s_{\rm{max}}$) is always widest when $c=1$, i.e., for $100\%$ conversion efficiency. Note that the results become independent of $h$ when $c=1$. The gene drive wave reverses direction and dies out in the white regions of this diagram. []{data-label="Fig_S1"}](./Figure_S1.pdf){width="0.8\columnwidth"}
In the main text, we assumed perfect conversion efficiency ($c=1$) of the mutagenic chain reaction. However, in reality, some fraction of the reactions can be unsuccessful and the conversion rate $c$ will be $0<c<1$. As a result there will be heterozygous individuals with fitness $1-hs$, where $h$ controls dominance of the gene drive allele. When $h=1$, the gene drive allele is dominant and the fitness of the heterozygous genotype is $1-s$. The choices $h=0, 0.5$ correspond to the recessive and additive cases respectively. As derived by Unckless *et al.* [@unckless2015modeling], the reaction term in Eq. 5 is now given by
$$\begin{split}
&q(t+\tau_g) - q(t)
=\bar{f}(q)\\
&=\frac{q^2(1-s)+q(1-q)\big[ (1-c) (1-hs) + 2c(1-s) \big]}{q^2 (1-s) +2q(1-q)(1-c)(1-hs)+2q(1-q)c(1-s)+(1-q)^2} - q
\end{split}$$
There are again three fixed points $q=0,1,q^*$ where the third fixed point is $$q^*=\frac{c+cs(h-2)-hs}{s(1-2c-2h+2ch)}.$$ Following [@unckless2015modeling], we find that $q^*$ first becomes positive for $s>s_{\rm{min}}$, where $$s_{\rm{min}} = \frac{c}{2c - (c-1)h}.$$ For $0\leq s \leq s_{\rm{min}}$, $q^*<0$ and the spatial dynamics is again controlled by pulled waves. We can also calculate $s_{\rm{max}}$ by recalculating the potential function analogy discussed in SI, Sec **A** and in the main text, $$\bar{U}(q)= -\frac{1}{\tau_g} \int_0^q \bar{f}(q') dq',$$ and numerically solving for $\bar{U}(q=0,c,h,s_{\rm{max}})=\bar{U}(q=1,c,h,s_{\rm{max}})$ to obtain $s_{\rm{max}}(c)$ given $h$, with the results shown in Fig. S1. The gene drive spreads spatially as a pushed excitable wave for $s_{\rm{min}} < s < s_{\rm{max}}$. Note that the relevant range of $s$ when $c < 1$ shrinks compared to $c=1$.
Spatial evolutionary games in one dimension
-------------------------------------------
![ A schematic phase diagram of the spatial evolutionary games in one dimension ignoring genetic drift. The parameters $\alpha$ and $\beta$ describe interactions between red and green genetic variants, with growth rates written as $w_R (x,t) = g + \alpha (1-f(x,t))$ and $w_G (x,t) = g + \beta f(x,t)$ respectively. (The parameter $g>0$ is a background growth rate.) Inserted graphs show schematically the potential energy function $U(f)$, where each of the green and red dot corresponds to $f=0$ and $f=1$ respectively ($0 \leq f \leq 1$). By searching for barriers in $U(f)$ as a function of $\alpha$ and $\beta$, we identify the bistable regimes that require a critical nucleus and pushed excitable waves to reach a stable dynamical state and the pulled Fisher wave regimes which do not require the nucleation process. The two regimes are separated by two solid black lines $\alpha=0$, $\beta<0$, and $\alpha<0$, $\beta=0$, which correspond limits of metastability. The solid line along $\alpha=\beta<0$ between the two bistable states is analogous to a first-order phase transition line (equal depth minima in $U(q)$), along which the excitable genetic wave separating red and green stalls out. []{data-label="Fig_S2"}](./Figure_S2.pdf){width="0.5\columnwidth"}
In this SI section, we show that genetic waves mathematically quite similar to the pushed gene drive waves studied here arise in spatial evolutionary games of two interacting asexual species that are colored red (“$R$”) and green (“$G$”) using the analogy with nucleation theory introduced in the previous SI section. We start from the continuum description of the one dimensional stepping stone model (following [@RevModPhys.82.1691; @korolev2011competition]), $$\frac{\partial f(x,t)}{\partial t} = D \frac{\partial^2 f(x,t)}{\partial x^2} + s[f]f(1-f) + \sqrt{D_g f(1-f)} \Gamma (x,t),
\label{FullRG}$$ where $f(x,t)$ is the frequency of red species and $D$ is the spatial diffusion constant representing migration. The last term, where $\Gamma(x,t)$ is an Ito correlated Gaussian white noise source and $D_g$, proportional to an inverse effective population size, represents genetic drift. We henceforth neglect genetic drift and set this term to zero. The function $s[f]$ represents the difference in relative reproduction rates between the two species, and is given by [@RevModPhys.82.1691] $$s[f]=w_{\rm{eff}} = \frac{w_R - w_G}{\frac{1}{2} (w_R + w_G)},$$ where $w_R$ and $w_G$ are fitnesses of alleles $R$ and $G$. If $g$ is a background reproduction rate, we have $$\begin{split}
w_R (x,t) &= g + \alpha (1-f(x,t)),\\
w_G (x,t) &= g + \beta f(x,t),
\end{split}$$ where the interactions between the two competing variants are characterized by constants $\alpha$ and $\beta$. With the definitions above, we have $$s[f]=-\frac{(\alpha + \beta)(f-\frac{\alpha}{\alpha+\beta})}{g+\frac{1}{2}\alpha (1-f) + \frac{1}{2} \beta f},$$ which leads to a reaction term similar to that in Eq. 5 and introduces an additional fixed point into the dynamics of Eq. \[FullRG\] at $f^* = \frac{\alpha}{\alpha + \beta}$ in addition to $f=0,1$. A diagram summarizing the dynamics of this model is shown in Fig. 2. This “phase diagram” was worked out including genetic drift in Eq. \[FullRG\] which affects the shape and location of the phase transition lines in the first quadrant of Fig. 1. [@korolev2011competition]. If the genetic drift term in Eq. \[FullRG\] is neglected, the lines labelled “DP” in Fig. 2 would coincide with the positive $\alpha$ and $\beta$ axes and would merge at the origin. Upon setting $D_g = 0$ in Eq. \[FullRG\], we employ the argument presented above and define a potential energy function, $$U_b (f)= -\int^{f}_0 s[f']f'(1-f') df'.
\label{Ub}$$
The schematic picture of $U_b (f)$ in different parameter regimes is drawn in Fig. \[Fig\_S2\]. The mutualistic regime ($\alpha>0, \beta>0$) has already been studied in detail, including effects of genetic drift [@korolev2011competition]. By studying shapes of the potential energy function $U[f]$ we identify two important parameter regimes. In the bistable regime (dark green), there is a finite energy barrier between the two locally stable states and a nucleation process is required to establish an excitable wave.
However, in the Fisher wave regimes (light green and light red), there is no energy barrier to reach the unique stable configuration and thus nucleation is not required. The two regimes are separated by the two black solid lines $\alpha=0, \beta<0$ or $\alpha<0, \beta=0$, which are limits of metastability. We also draw a solid black line between the two bistable states along $\alpha=\beta<0$, where the pushed waves stall out. This line is analogous to a line of first-order transitions. When $\alpha \neq \beta$, the integral in Eq. \[Ub\] for the effective thermodynamic potential is given by
$$\begin{split}
U[f] &=
\frac{1}{3(\alpha-\beta)^4}\Bigg( (\alpha-\beta) f \bigg\{ \alpha^3 f (2f-3)
+ \alpha^2 f (9\beta - 2\beta f +6g)\\
&+ \alpha \Big(\beta^2 \big(12-f(3+2f) \big) + 36\beta g+24g^2 \Big)
+\beta \big( \beta^2 f(-3+2f) - 6\beta(-2+f)g+24g^2 \big) \bigg\}\\
&+ 12(\alpha+2g)(\beta+2g) \big( \alpha \beta + (\alpha + \beta)g \big) \log \Big[1-\frac{\alpha-\beta}{\alpha+2g} f \Big]
\Bigg).
\label{long}
\end{split}$$
When $\alpha=\beta$, we can simplify $s[f]$ $$s[f] = -\frac{2 \alpha (f-\frac{1}{2})}{g+\frac{1}{2}\alpha},$$ and the integral gives $$U[f] = \frac{2 \alpha }{g+\frac{1}{2}\alpha} \int^{f}_0 f'(1-f') \Big( f'-\frac{1}{2} \Big) df' =-\frac{\alpha}{2g+\alpha} {f}^2 ({f}-1)^2.$$
When $\alpha=-\beta$, $\alpha \ll g$ and $1 \ll \big| \frac{g}{\alpha}+\frac{1}{2} \big|$, we have $$s[f] = \frac{\alpha}{g+\frac{1}{2}\alpha (1-2f)}$$ and $$\begin{split}
&U[f] = -\int^{f}_0 \frac{f'^2 -f'}{f' - \big( \frac{g}{\alpha}+\frac{1}{2} \big)} df'\\
&= -\frac{1}{2}f \Big(f+ \frac{2g}{\alpha} -1 \Big) - \bigg(\frac{g}{\alpha} + \frac{1}{2} \bigg) \bigg(\frac{g}{\alpha} - \frac{1}{2} \bigg) \log \bigg[
1-\frac{2\alpha f}{\alpha + 2g}
\bigg]
\label{amib}
\end{split}$$ The last term diverges at $f=\frac{g}{\alpha}+\frac{1}{2}$, but we focus on the weak interaction limit $1 \ll \big| \frac{g}{\alpha}+\frac{1}{2} \big|$, where the biologically relevant regime $0 \leq f \leq 1$ will not be affected. If we substitute $\alpha=-\beta$ into Eq. \[long\], we recover Eq. \[amib\], as expected.
Calculation of the critical propagules in one dimension
-------------------------------------------------------
In this SI section, we describe details of the calculation of the critical propagules shown in Fig. 5. Reaction-diffusion equations in one dimension with a general reaction term $R[q(x,t)]$ can be written as $$\label{general}
\tau_g \frac{\partial q(x, t)}{\partial t} = \tau_g D \frac{\partial^2 q(x,t)}{\partial x^2} + R[q(x,t)].$$ The critical propagule profile $q_{\rm{c}}(x)$ can be defined as a stationary solution of Eq. \[general\], i.e., $$0 = \tau_g D \frac{\partial^2 q_c}{\partial x^2} + R[q_c].
\label{ODE}$$ Upon multiplying both sides by $\frac{d q_{\rm{c}}}{d x}$ and integrating we obtain, $$\tau_g D \Big ( \frac{d q_{\rm{c}}}{d x} \Big)^2 =2 \int^0_{q} R[\tilde{q}] d\tilde{q}.$$ If we assume a symmetric critical propagule about $x=0$, so that $\frac{d q_{\rm{c}}}{d x} = 0$ at $x=0$, we can obtain $q_{\rm{m}} \equiv q_{\rm{c}}(0) $ from
$$\int^0_{q_{\rm{m}}} R[\tilde{q}] d\tilde{q}=0.$$
Since the slope $\frac{dx_{\rm{c}} (q)}{dq}$ is given by $$\frac{d x_{\rm{c}}(q)}{d q} = \frac{\sqrt{\frac{\tau_g D}{2}}}{\sqrt{ \int^0_{q} R[\tilde{q}] d\tilde{q}}},$$ we obtain the critical propagule profile $x_{\rm{c}}(q)$ by integrating both sides from $q_{\rm{m}}$ to $q$. The calculations described above can be carried out analytically for the cubic reaction term Eq. 7 and critical propagules for $s=0.66,0.58,0.51$ are plotted in Fig. 5 with dashed lines. For the full MCR equation, the corresponding numerical results are plotted with solid lines.
Critical radius and allele concentration in two dimensions
----------------------------------------------------------
![ In two dimensions the gene drive allele is introduced uniformly over a disk-shaped region with radius $r_0$ with uniform frequency $q_0$ inside as illustrated in the inset image. We numerically determined the critical frequency $q_0$ and radius $r_0$ just sufficient to initiate an excitable wave in two dimensions.[]{data-label="Fig_S3"}](./Figure_S3.pdf){width="0.5\columnwidth"}
In practice, it is important to model the distribution of MCR alleles to be released locally to initiate its traveling genetic wave in a two-dimensional space. Upon assuming circular symmetry of the traveling wave solution, the reaction-diffusion equation governing the radial frequency profile of the MCR allele $q(r,t)$ reads in radial coordinates, $$\tau_g \frac{\partial q}{\partial t}= \tau_g D \Big( \frac{\partial^2 q}{\partial r^2}+\frac{1}{r} \frac{\partial q}{\partial r} \Big) + \frac{sq(1-q)(q-q^*)}{1-sq(2-q)}.$$ The only correction to the one dimensional case is the derivative term $\frac{1}{r} \frac{\partial q}{\partial r}$, which can be neglected relative to $\frac{\partial^2 q}{\partial r^2}$ in the limit of $r\rightarrow \infty$. However, we keep this term in the calculation of the critical nucleus as this term is not negligible where $r$ is comparable to or smaller than the width of the excitable wave being launched. In our numerical calculations, instead of a Gaussian initial condition, it is convenient to introduce the gene drive allele with a uniform frequency $q_0$ over a circular region with radius $r_0$. Indeed, in an actual release of a gene drive organism, it is plausible that the release would be implemented by creating a gene drive concentration $q_0$ in a circular region of radius $r_0$ with a sharp boundary. To model the radial frequency profiles, we used a circularly symmetric steep logistic function as an initial condition, $$\label{logistic}
q(r,t=0)=\frac{q_0}{1+e^{10(r-r_0)/\sqrt{\tau_g D}}},$$ instead of a step function to insure numerical stability. Fig. \[Fig\_S3\] shows the parameter regimes where a pushed wave is excited for various selective disadvantages $s$. The pushed waves successfully launched for initial conditions whose parameters are above the curves $q_0(r_0)$, shown for a variety of selective disadvantages $s$ in the pushed wave regime.
Line tension, energy difference and analogy with nucleation theory in two dimensions
------------------------------------------------------------------------------------
The scenario studied in the previous section (sharp boundary, adjustable initial drive concentration $q_0$ and inoculation radius $r_0$) seems appropriate for many engineered releases of gene drives, at least in situations with large effective population sizes $N_{\rm{eff}}$, so that genetic drift can be neglected. (See the discussion of genetic drift in SI Sec. **J**.)
However, when genetic drift is important, stochastic contributions like the term $\sqrt{D_g f(1-f) \eta(x,t)}$ in, e.g., Eq. \[FullRG\], can act on spatial gradients at the interfaces of pushed and pulled waves [@polechova2011genetic; @polechova2015limits] in a manner somewhat reminiscent of thermal fluctuations near a first-order phase transition. Provided strong genetic drift is able to produce something analogous to local thermal equilibrium after a gene drive release, it is interesting to explore an analogy with classical nucleation theory. Nucleation leads to a pushed wave when $s_{\rm{min}}<s<s_{\rm{max}}$. One might then expect the two-dimensional analog of the total energy function discussed in SI Sec. **A** for an equilibrated circular droplet with $q_0=1$ and radius $r_0$ in two dimensions to take the form $$\begin{split}
\mathcal{F}[q(\bm{r})] &= \int d \bm{r} \bigg\{ \frac{1}{2} D \big( \bm{\nabla} q(\bm{r}) \big)^2 + U[q(\bm{r})] \bigg\}\\
&= 2\pi \int_0^{\infty} dr \frac{rD}{2} \Big( \frac{dq}{dr} \Big)^2 + 2 \pi \int_0^{\infty} dr r U[q(r)]\\
&\approx 2\pi r_0 \int_0^{\infty} dr \frac{D}{2} \Big( \frac{dq}{dr} \Big)^2 + \pi r_0^2 \big(U(1) - U (0) \big)\\
& \equiv 2\pi r_0 \gamma - \pi r_0^2 | \Delta U |
\end{split}$$ where we have assumed a sharp interface between saturated gene drive and wild-type states. Here, $\Delta U$, the “energy” difference between the gene drive and wild type, causes the droplet to expand, and the role of an energy barrier to nucleation is played by the line tension term $\gamma$ [@barton1985analysis]. This is indeed the case. For simplicity, we illustrate the nucleation approach with the cubic reaction term given by Eq. 7 in the main text.
First, we assume the logistic form of the spatial profile derived in the 1d limit by Barton and Turelli [@barton2011spatial] $$q(r) = \frac{1}{1+e^{\sqrt{s/2 \tau_g D}(r-r_0)}},$$ and the line tension term is $$\gamma = \int_0^{\infty} dr \frac{D}{2} \Big( \frac{dq}{dr} \Big)^2
= \frac{\sqrt{sD/2\tau_g } (e^{3 r_0 \sqrt{s/2 \tau_g D}} + 3 e^{2 r_0 \sqrt{s/2 \tau_g D}} ) }{12(e^{r_0 \sqrt{s/2 \tau_g D}}+1)^3}
\approx \frac{\sqrt{sD /2 \tau_g }}{12},$$ in the limit of $1 \ll r_0 \sqrt{s/2\tau_g D}$. The energy difference is given by $$\Delta U = U(1) - U(0) = \frac{3s-2}{12 \tau_g},$$ and the critical radius of the nucleus $r_{\rm{c}}$ which corresponds to the saddle point barrier of the free energy landscape is $$r_{\rm{c}} = \frac{\sqrt{s \tau_g D/2}}{2-3s}$$ as plotted in Fig. \[Fig\_S4\]. This result shows the divergence of $r_{\rm{c}}$ in the limit of $s\rightarrow s_{\rm{max}} (=2/3)$ and the above approximation ($r_0 \sqrt{s/ 2 \tau_g D} \gg 1$) becomes exact in this limit. The diverging $r_{\rm{c}} (s)$ shown in Fig. \[Fig\_S4\] is qualitatively consistent with the behavior found for the simplified gene drive initial condition in two dimensions shown in Fig. \[Fig\_S3\] in the limit $q_0 \rightarrow 1$
![ The critical radius of the nuclei $r_{\rm{c}}$ as a function of the selective disadvantage $s$. []{data-label="Fig_S4"}](./Figure_S4.pdf){width="0.5\columnwidth"}
Wave velocities of the excitable waves
--------------------------------------
![ Asymptotic wave velocities $v$ of the excitable waves are plotted as a function of selective disadvantage $s$. The pink circular dots are numerically calculated wave velocities for the MCR model. The blue curve is an analytically derived result for the simple cubic approximation, $ v(s)=(2-3s) \sqrt{D/2\tau_g s} $ [@barton2011spatial] and the blue squares are from numerical calculations, which confirm good agreement with the analytical result.[]{data-label="Fig_S5"}](./Figure_S5.pdf){width="0.5\columnwidth"}
The reaction-diffusion equation admits traveling wave solutions with a continuous family of velocities. It selects the slowest speed asymptotically in the large time limit [@van2003front]. The pink circular dots in Fig. \[Fig\_S5\] are numerically calculated asymptotic wave velocities for the MCR model in the pushed wave regime. We also plot the known wave velocity for the cubic approximation $ v(s)=(2-3s) \sqrt{D/2\tau_g s} $ [@barton1979dynamics; @barton1989adaptation; @barton2011spatial] for comparison. Due to the larger reaction term $f_{\rm{MCR}}(q)>f_{\rm{cubic}}(q)$ (see discussion in Fig. 5), the wave velocity for the MCR model is always faster than the cubic approximation given the same selective disadvantage $s$. In both cases, a larger selective disadvantage $s$ decreases the wave velocity, which eventually becomes zero at $s_{\rm{max}} = 0.697$ for the MCR model and the slightly smaller value $s_{\rm{max}}=2/3$ within the cubic approximation.\
Calculation of the speed of the excitable waves
-----------------------------------------------
In this section, we review the numerical method for calculating the speed of the excitable waves, following [@barton2011spatial]. First, we assume a traveling waveform of the solution $$q(x,t)=Q(x-vt)=Q(z),~z\equiv x-vt,$$ with boundary conditions $$\begin{split}
Q(z)\rightarrow 1~(z \rightarrow - \infty),~Q(z)\rightarrow 0~(z \rightarrow + \infty),\\
\frac{dQ}{dz} \rightarrow0~(z \rightarrow \pm \infty).
\end{split}$$ By substituting $Q(z)$ into $$\tau_g \frac{\partial q}{\partial t} = \tau_g D \frac{\partial^2 q}{\partial x^2} + R[q],$$ we obtain $$0=\tau_g D \frac{d^2 Q}{d z^2} + v \tau_g \frac{d Q}{d z} + R[Q].$$ If we define the gradient $G$ as a function of $Q$, $G[Q] \equiv \frac{dQ}{dz}$ we arrive an ordinary differential equation $$0=\tau_g D G\frac{dG}{dQ} + v \tau_g G + R[Q],$$ with boundary conditions $$G[0]=G[1]=0.$$ It is known that there exists a unique velocity of the excitable wave $v$ that has solution $G[Q]$ of the above differential equation with the boundary condition [@keener1998mathematical]. We used a shooting method to determine such $v$ and plotted the results in Fig. \[Fig\_S5\].
Critical barrier strength
-------------------------
![ Stopping power of a selective advantage barrier in one dimension. Numerical solutions of Eq. 5 are shown with time increment $\Delta t = 10.0 \tau_g$. The early time response is shown in red with later times in blue. The selective disadvantage of the barrier is $s_b$ within the purple bar of width $L=5$ occupying the spatial region $25\sqrt{\tau_g D}<x<30\sqrt{\tau_g D}$ (shaded in blue) and $s=0.625$ otherwise. (A) The excitable wave propagates with constant speed when the barrier vanishes for $s_b = 0.625$. (B) With $s_b =0.688>s=0.625$, the wave significantly slows down at the barrier, but recovers and propagates onwards. (C) The excitable wave is stopped when $s_b =0.708$.[]{data-label="Fig_S6"}](./Figure_S6.pdf){width="0.5\columnwidth"}
Fig. \[Fig\_S6\] shows how the excitable wave can be slowed down and finally stopped by increasing the strength of a selective disadvantage barrier $s_b > s$. As a reference, we first show dynamics of the excitable wave without a barrier ($s_b = 0.625$ matches the selective disadvantage $s=0.625$ outside) in Fig. \[Fig\_S6\]*A*. When a small barrier is erected ($s_b=0.688<0.697$), the excitable wave significantly slows down within the barrier as expected from the results shown in Fig. \[Fig\_S5\]. However, the wave recovers and propagates through the barrier as in Fig. \[Fig\_S6\]*B*. When the barrier strength exceeds a critical value (in Fig. \[Fig\_S6\]*C* we plot the case $s_b = 0.708$) the excitable wave is stopped.
![ Critical width $L$ and the selective disadvantage $s_b$ of a barrier that is just sufficient to stop a pushed gene drive wave in one dimension. The values are numerically obtained by placing the barrier in a region $25\sqrt{\tau_g D}<x<(25+L)\sqrt{\tau_g D}$. Results are plotted for a variety of selective disadvantages $s$ outside the barrier region. Given $s$, the excitable population wave can be stopped by a barrier whose parameters $(s_b, L / \sqrt{\tau_g D})$ lie above the curves. []{data-label="Fig_S7"}](./Figure_S7.pdf){width="0.5\columnwidth"}
In Fig. \[Fig\_S7\], we plot the critical width $L$ and selective disadvantage within the one dimensional barrier region $s_b$ just sufficient to stop the excitable population wave of the gene drive species. The values are numerically obtained by placing the barrier in a region $25\sqrt{\tau_g D}<x<(25+L)\sqrt{\tau_g D}$. For example, when the selective disadvantage outside the barrier region is set to be $s \approx 0.65$, the excitable gene drive wave can be stopped by increasing $s$ by $\sim 20\%$ within the barrier region of thickness $\sim \sqrt{\tau_g D/s}$.
Fluctuations due to finite population size
------------------------------------------
In this section, we estimate effects of fluctuations due to a finite population size using mosquitos as an example. First, we define the effective spatial population size $N_{\rm{eff}}$ to be the number of mosquitos with which an individual might conceivably mate during its generation time $\tau_g$ [@hartl1997principles]. Given a diffusion constant $D$, the two dimensional area an individual can explore during its life time $\tau_g$ is $\pi (\sqrt{4D\tau_g})^2$ and the effective population size in two dimensions is $$N_{\rm{eff}} \equiv 4 \pi D \tau_g n,$$ where $n$ is the area density of organisms. Here, we estimate $N_{\rm{eff}}$ using parameters appropriate to mosquitos: $\tau_g\sim 10[\text{days}]$ [@deredec2011requirements], $D\sim 0.1 [\text{km}^2/\text{day}]$ and $n\sim 1 [\rm{m}^{-2}] = 10^6 [\rm{km}^{-2}]$ to get $N_{\rm{eff}}\sim 10^5 - 10^6$. With such a large effective population size, we believe that the dynamics can be well described by the deterministic limit explored here. Fluctuations *can* play a role for systems with smaller populations and such effects have been thoroughly investigated in the physics literatures [@brunet1997shift; @van2003front; @brunet2015exactly; @cohen2005fluctuation; @hallatschek2009fisher]. Pulled waves are more sensitive to fluctuations, with a Fisher wave velocity that changes according to $$v=v_F[ 1 - O(1/\ln^2 N_{\rm{eff}})],$$ where $v_F$ is the velocity of the pulled wave in the deterministic limit [@brunet1997shift].
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abstract: 'A quantum network consists of independent sources distributing entangled states to distant nodes which can then perform entangled measurements, thus establishing correlations across the entire network. But how strong can these correlations be? Here we address this question, by deriving bounds on possible quantum correlations in a given network. These bounds are nonlinear inequalities that depend only on the topology of the network. We discuss in detail the notably challenging case of the triangle network. Moreover, we conjecture that our bounds hold in general no-signaling theories. In particular, we prove that our inequalities for the triangle network hold when the sources are arbitrary no-signaling boxes which can be wired together. Finally, we discuss an application of our results for the device-independent characterization of the topology of a quantum network.'
author:
- 'Marc-Olivier Renou'
- Yuyi Wang
- Sadra Boreiri
- Salman Beigi
- Nicolas Gisin
- Nicolas Brunner
date: '[ JanuaryFebruaryMarchAprilMayJune JulyAugustSeptemberOctoberNovemberDecember ]{}'
title: 'Limits on correlations in networks for quantum and no-signaling resources'
---
Introduction
============
Quantum nonlocality, i.e., the fact that distant observers performing local measurements on a shared entangled quantum state can violate a Bell inequality, is a key feature of quantum theory [@bell]. In recent years, considerable efforts have been devoted, both theoretically and experimentally, to deepen our understanding of this phenomenon [@review]. Of particular interest is the investigation of quantum nonlocality in the context of general networks [@branciard; @branciard2; @fritz]. Here, a set of distant observers share entanglement distributed by several sources which are assumed to be independent from each other. As each source distributes entanglement to only certain subsets of observers, new limits on possible correlations arise. Moreover, observers can correlate particles coming from different independent sources (e.g. via entangled quantum measurements, as in quantum teleportation [@bennett]), and thus generate strong correlations across the entire network. Notably, this leads to astonishing new effects, such as the possibility of violating a Bell inequality without the need for inputs [@fritz; @branciard2]. Beyond the fundamental interest, these ideas are also directly relevant to the development of real-world quantum networks [@kimble; @simon].
It is fair to say, however, that our understanding of quantum nonlocality in networks is still very limited [@NG2018]. A first challenge is to characterize classical correlations in networks, i.e. when all sources distribute only classical variables. Due to the independence condition of the sources, the set of classical correlations is no longer convex (contrary to the standard Bell scenario, featuring a single common source, see e.g. [@review]). Therefore, relevant Bell inequalities must be nonlinear. Examples of such inequalities have been derived (see e.g. [@branciard; @branciard2; @chaves2; @armin; @rosset; @chaves; @wolfe; @luo]), but the general structure of this problem is still not understood.
Another challenge, which represents the starting point of this work, is to understand the limits of quantum correlations in networks. Specifically, given a certain network, we aim at determining fundamental constraints on achievable correlations when using quantum resources. Hence we consider any possible quantum strategy compatible with the network topology. This involves sources producing arbitrary quantum states (of any Hilbert space dimension), and nodes performing arbitrary joint quantum measurements. Interestingly it turns out that fundamental limits arise here even without involving any inputs, in contrast to the standard Bell scenario. This means that the network topology imposes fundamental limitations on achievable correlations.
We start our investigation with the case of networks featuring three observers, each of them providing an output (but receiving no input). We discuss in detail the notably challenging case of the “triangle network” [@branciard2; @fritz; @gisin; @wolfe], for which we identify a nonlinear inequality capturing partly the set of quantum correlations. More generally we derive a family of inequalities satisfied by quantum correlations considering an arbitrary network with bipartite sources. Interestingly these inequalities can be viewed as quantum versions of the Finner inequalities [@finner], introduced in a completely different context, namely graph theory.
An interesting application of our inequalities is that they allow one to test the topology of an unknown quantum network in a device-independent manner. That is, by simply considering the observed correlations, one can tell whether a certain network topology is compatible or not. If the observed data violates one of our inequalities, then the corresponding network topology can be ruled out immediately.
Finally, we go beyond quantum correlations, and consider more general no-signaling resources [@PR; @barrettPR]. That is, each source now distributes a no-signaling (NS) box, and each node performs a joint operations on these resources [@barrett]. Notably, we show that, for the triangle network, the nonlinear inequality we obtained for quantum correlations also holds in a general no-signaling theory, where each source can produce an arbitrary number of bipartite NS boxes, and each node performs an arbitrary wiring on the received resources. This leads us to the conjecture that Finner inequalities captures in fact the limit of correlations in networks for any possible no-signaling theory. It thus represents a general limit of achievable correlations in networks, independently of the underlying physical model, given the latter does not allow for instantaneous communication (in other words, is compatible with special relativity).
Three-observer networks
=======================
![All inequivalent three-party networks.[]{data-label="threeparties"}](threeparties_allscenarios.JPG)
Consider a quantum network with three observers $A$, $B$ and $C$, featuring one (or more) sources, distributing quantum states to subset of the parties. Each party then performs a measurement on the received quantum systems, leading to outputs denoted $a$, $b$ and $c$. Here for simplicity we assume that $a, b, c\in \{0,1\}$ are binary, but later consider larger output sets.
One can consider three inequivalent networks here. The first, depicted in Fig. \[threeparties\](a), features a single common source distributing a quantum state to all three observers. This corresponds to the situation considered in the standard Bell scenario (see e.g. [@review]), except that parties receive no inputs in our case. It is straightforward to see that any possible distribution $P(abc)$ can be achieved. In fact, it is enough to restrict to classical sources here. The source samples from the distribution $P(abc)$, and then distributes the obtained outputs to each observer. Geometrically, the set of possible attainable distributions $P(abc)$ is nothing but the whole probability simplex, which is a 7-dimensional simplex in $\mathbb R^8$ due to the normalization constraint $\sum_{a,b,c} P(abc) =1$.
A more interesting scenario is when the network features two independent sources, as in Fig. \[threeparties\](b). The first source distributes a common state to $A$ and $B$, and the second independent source to $B$ and $C$. This scenario is known as bilocality [@branciard], and corresponds to the setup of entanglement swapping. In this case, the parties $A$ and $C$ are initially independent, and can only be correlated via $B$. Hence, if one traces out $B$, the marginal statistics of $A$ and $C$ must factorize. We have the causality condition: $$\label{biloc}
\sum_b P(abc) = P_{AC}(ac) = P_A(a) P_C(c) \,.$$ Hence, contrary to the first network discussed above, not all correlations are possible in the bilocality network. It turns out however that the constraint is enough to characterize achievable correlations: any $P(abc)$ satisfying can be achieved. It is again enough to consider only classical variables. Specifically, let the first (resp. second) source sample from $P_A(a)$ (resp. $P_C(c)$) and distribute the output to $A$ and $B$ (resp. $B$ and $C$) and $B$ use local randomness to sample $P(b|ac)$. Geometrically, the set of achievable distributions $P(abc)$ forms a 6-dimensional curved manifold in $\mathbb{R}^8$.
Next we move to the third—and arguably the most interesting and challenging—configuration, i.e. the triangle network (see Fig. \[threeparties\](c)). Consider the bilocality network again, and add a source connecting $A$ and $C$. Due to this additional source, the independence condition does no longer hold. In fact, one can show that, besides the normalization constraint, there is no other equality constraint for this network (which would reduce the dimension of the set). This follows from the fact that the maximally mixed (uniform) distribution $P_u(abc) = 1/8$ $\forall a,b,c$ is surrounded by a ball of achievable distributions; see Appendix \[cube\_representation\].
It turns out, however, that not all distributions $P(abc)$ are achievable in the triangle scenario, as shown in Appendix \[cube\_representation\] and Ref. [@wolfe] via specific examples. Thus, the set of possible distributions forms a strict subset of the probability simplex, yet its characterization is a challenging problem. Here we derive a relevant nonlinear inequality that necessarily holds in quantum theory.
\[Finner\_triangle-1\] In the triangle network (Fig. \[threeparties\](c)), quantum correlations necessarily satisfy $$\label{Finner_triangle}
P(abc) \leq \sqrt{P_A(a)P_B(b)P_C(c)} \,.$$
As quantum correlations are stronger than classical ones, inequality also holds for the case where the sources emit classical variables: in this case Theorem \[Finner\_triangle-1\] can be derived by applying two Cauchy-Schwarz inequalities on $\mathbb E[f_A g_B h_C]$ where $f_A, g_B$ and $h_C$ are the characteristic functions of the sets $\{a\}, \{b\}$ and $\{c\}$ respectively. Back to quantum sources, Theorem \[Finner\_triangle-1\] can be proven by essentially the same ideas, but as we will later prove a generalization of this theorem, we skip the proof here.
In the classical case, the set of all possible strategies can be understood intuitively in geometrical terms, as a 3-dimensional cube. In this case, the inequality follows from the Loomis-Whitney inequality, capturing the fact that the volume of a 3-dimensional object is upper bounded by the product of the areas of the object’s projections in three orthogonal directions (see Appendix \[cube\_representation\]).
The inequality allows us to prove that a large range of distributions cannot be achieved in the quantum triangle network. Consider for instance the family of distributions
$$\label{pq}
P_{p,q} = p \delta_{000} + q \delta_{111} + (1-p-q) P_{\text{diff}}$$
where $\delta_{abc}$ represents the distribution that always outputs $a$, $b$ and $c$ deterministically, and $P_{\text{diff}}$ is the uniform distribution over $\{0,1\}^3\setminus\{000, 111\}$, i.e., $P_{\text{diff}}= (\delta_{001} + \delta_{010}+ \delta_{100}+ \delta_{011}+ \delta_{101}+ \delta_{110})/6$. From inequality it follows that $P_{p,q} $ is not realizable in the triangle network when $q> 1+p -2p^{2/3}$; see Fig. \[pq\_region\]. This also shows that the set of quantum distributions achievable in the triangle network is not star convex [^1] since distributions of the form $r \delta_{111}+ (1-r) P_u$ violate inequality when $7/8 <r<1$; $P_u$ denotes the uniform distribution over $\{0,1\}^3$.
The above example also illustrates how our results can be used to test the topology of an initially unknown network. Suppose Alice, Bob and Charlie observe a distribution $P(abc)$ that violates inequality . Then, they can certify that the underlying network is not of the triangle type (neither bilocal indeed), but must feature a common source distributing information to all three parties. Note that this test is device-independent, as it is based only on the observed data $P(abc)$.
![Geometrical representation of the set of distributions $P_{p,q}$ of Eq. . Distributions in the shaded area are not achievable in the triangle network in quantum theory, as they do not satisfy inequality . Distributions below the red curve satisfy the inequality, and are thus potentially achievable in quantum mechanics. The blue circle represents the maximally mixed distribution $P_u$, while the blue square is the so-called GHZ distribution $p=q=1/2$. Distributions of the form $r \delta_{111}+ (1-r) P_u$ (dashed line) violate inequality for $7/8<r<1$ (see inset), showing that the quantum set is not star convex.[]{data-label="pq_region"}](FinnerFig.JPG)
In the remainder of the paper, we will generalize Theorem \[Finner\_triangle-1\] in two different directions. First, we will show how to derive similar nonlinear inequalities for a larger class of networks. Second, we will prove that inequality holds also in the triangle network for a generalized probabilistic theory.
General networks
================
We now consider networks with an arbitrary number of parties (outputs) yet we mostly restrict to bipartite sources. That is, we assume that our networks $\mathcal N$ consist of $n$ parties $A_1, \dots, A_n$, and an arbitrary number of sources each of which is connected to a pair of parties. Thus a network $\mathcal N$ can be thought of as a graph over $n$ vertices whose edges represent sources (e.g., the triangle network is represented by the triangle graph). The following theorem presents a generalization of Theorem \[Finner\_triangle-1\] for arbitrary graphs whose proof is given in Appendix \[finner\_quantum\].
\[thm:quantum-finner\] (Quantum Finner inequality) Consider a network $\mathcal{N}$ with $n$ observers $A_1, \dots, A_n$ and some bipartite sources. Let $\eta=\{\eta_1, ..., \eta_n\}$ be a fractional independent set of $\mathcal{N}$, i.e. weights $\eta_j$ attributed to $A_j$’s are such that, for each source the sum of weights of parties connected to it is smaller than or equal to 1. [^2]Let $f_{j}$ be any real positive local post-processing (function) of the classical output of party $A_{j}$. Then, any distribution $P$ achievable in quantum network $\mathcal N$ satisfies $$\label{NQFinner}
{\mathds{E}\left[\prod_j f_j\right]} \leq \prod_j {\left\lVertf_j\right\rVert}_{1/\eta_j},$$ where ${\left\lVertf\right\rVert}_{1/\eta}=({\mathds{E}\left[ f^{1/\eta}\right]})^{\eta}$ and the expectations are with respect to $P$. In particular, letting $f_{j}$ being the indicator function of the output of $A_j$ being $a_j$, we have $$\label{NQFinnerProba}
P(a_1...a_n)\leq \prod_{j=1}^n \left(P_{A_{j}}(a_j)\right)^{\eta_j}.$$
Inequality has been derived by Finner [@finner] in the context of graph theory, as a generalization of Hölder’s inequality. In our setting, the proof of Finner directly applies to arbitrary networks with multipartite classical sources. Our theorem here generalizes Finner’s inequality for arbitrary networks with bipartite sources that are quantum.
Note that although Finner’s inequality is presented as a continuous family of inequalities depending on the choice of weights $(\eta_1, \dots, \eta_n)$, it can be reduced to a finite set of inequalities for a given network (see Appendix \[app:HR\]). In particular, for the triangle network, the only nontrivial fractional independent set corresponds to $ \boldsymbol\eta =(1/2, 1/2, 1/2)$, in which case reduces to .
Triangle network with no-signaling boxes
========================================
We now consider the correlations achievable in the triangle network in a generalized no-signaling theory [@barrett]. In this model, the resources are not quantum states, but general NS boxes. In the triangle network, each source can thus distribute some NS boxes to the two parties connected to it. We emphasis that in general, each source can distribute several NS boxes that are not necessarily identical and can have arbitrary number of inputs and outputs. These NS boxes thus serve as a resource for the parties to generate correlated outputs; each party having an arbitrary number of NS boxes shared with others, can locally “wire” these boxes in the most general way to determine an output[^3]. That is, the inputs of certain boxes can be chosen by the party, while others can be determined by wirings, the output of one box being used as in the input for another one. We can prove the following result.
\[Finner\_boxworld\] Suppose that a distribution $P$ is achievable in the triangle network when the sources distribute arbitrary NS boxes and the parties perform arbitrary local wirings. Then $P$ satisfies .
Clearly, this result does not follow from Theorem \[Finner\_triangle-1\], as here the sources can distribute stronger nonlocal resources than what is possible in quantum theory [@PR]. However, we also point out that Theorem \[Finner\_boxworld\] does not imply Theorem \[Finner\_triangle-1\], as squantum theory allows for joint entangled measurements which cannot be described as wirings and admit no equivalent in general no-signaling theories [@barrett; @short1].
Here we give the proof ingredients while all details are left for Appendix \[finner\_boxworld\_proof\]. A key point in the proof is the notion of the Hypercontractivity Ribbon (HR) studied in [@beigi] as a monotone measure of non-local correlations.
\[hypercontractivity\_ribbon\] The HR ${\mathfrak{R}}(P)$ of a tripartite distribution $P(abc)$ is the set of non-negative triplets $(\alpha, \beta, \gamma)$ such that for any real functions $f(a)$, $g(b)$ and $h(c)$ of the outputs $a$, $b$, $c$, we have $$\label{hypercontractivity_ribbon_2}
{\mathds{E}\left[fgh\right]}\leq {\mathds{E}\left[|f|^{1/\alpha}\right]}^\alpha {\mathds{E}\left[|g|^{1/\beta}\right]}^\beta {\mathds{E}\left[|h|^{1/\gamma}\right]}^\gamma \,.$$
By Hölder’s inequality, the HR of the maximally mixed distribution $P_u$ (the weakest resource for establishing correlations) is the entire unit cube. Also it is not hard to verify that the HR of the GHZ distribution $(\delta_{000}+\delta_{111})/2$ which is the best possible resource, is the half cube given by the vertices $(0,0,0), (1,0,0), (0,1,0), (0,0,1)$.
The main feature of HR is its monotonicity under local operations. That is, if a tripartite distribution $Q(a'b'c')$ can be obtain by local post-processing of outcomes of another distribution $P(abc)$, then ${\mathfrak{R}}(P)\subseteq {\mathfrak{R}}(Q)$. In particular, with the GHZ distribution we can simulate $P_u$, but no the other way around.
Now with the definition of HR in hand, equation essentially says that $(1/2,1/2,1/2)\in{\mathfrak{R}}(P)$. In Appendix \[app:HR\], we give an alternative characterization of HR in terms of mutual information which allows us to prove Theorem \[Finner\_boxworld\].
Discussion
==========
We have presented fundamental constraints on quantum correlations achievable in networks. The constraints take the form of nonlinear inequalities, that can be viewed as the quantum version of the Finner inequalities. In particular, we have discussed in detail the case of the triangle network, as well as the problem of device-independently testing the topology of an unknown quantum network.
A natural question is indeed whether the quantum Finner inequalities fully capture the set of quantum correlations in networks. This appears not to be the case in general. Indeed, for the triangle network, there exist correlations that are provably not achievable in quantum theory that do not violate our inequality [^4]. It would be interesting to derive other forms of constraints. A possibility in this direction would be to exploit the “reverse Holder” inequality, the quantum version of which can be straightforwardly derived. Whether this new inequality will turn out to be stronger than the ones we presented is not clear. More generally, one should generalise the quantum Finner inequalities to the case of sources producing multipartite quantum states.
Finally, we also discussed the limits of correlations in networks when considering theories beyond quantum mechanics. In particular, we could show that inequality also holds for the triangle network with generalized no-signaling resources. More generally, we conjecture that the inequality holds for any no-signaling theory, for more general networks. Note that for the triangle network, this does not follow from our results, as each generalized probabilistic theory features its own set of allowed no-signaling correlations and the set of allowed joint measurements; the two sets being dual to each other [@barrett; @short_barrett]. If our conjecture is correct, this means that the Finner inequalities capture the limits of achievable correlations in a network, that must hold in any no-signaling theory; the Finner inequalities could thus be viewed as a generalisation of the standard no-signaling condition to networks.
*Acknowledgements.—*We thank Denis Rosset, Armin Tavakoli, Amin Gohari and Elie Wolfe for discussions. We acknowledge financial support from the Swiss national science foundation (Starting grant DIAQ, NCCR-QSIT).
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R.J.Evans, Annals of Statistics, Vol. 46, No. 6A, 2623-2656 (2018)
A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency, Springer, 2003.
Alternative description of networks {#network_formalism}
===================================
A network consists of a pair $(\mathcal A, \mathcal S)$ where $\mathcal A=\{A_1, \dots, A_n\}$ is the set of parties and $\mathcal S=\{S_1, \dots, S_m\}$ is the set of sources each of which is shared among a specific subset of parties. Each party $A_j$ produces an output after a post-processing her available sources.
Here we do not limit the amount of information provided by the sources, but restrict ourselves to distributions in which the output of each party $A_j$ has a finite alphabet set. We also restrict ourselves to minimal networks, where no source is connected to a subgroup of parties already connected by another source. However, as the amount of information provided by the source is not restricted, we can always (for convenience) add additional sources shared by parties already connected by another source: in Appendix \[cube\_representation\], we sometimes assume that the parties have there own source of randomness.
Since here we are mostly interested in bipartite sources we may think of the network as a graph. We may think of $\mathcal N$ as a graph whose vertices are labeled by $A_j$’s and whose edges are labeled by $S_i$’s. That is, for any $i$ there is an edge $e_i$ of the graph that connects the two parties that share the source $S_i$. For later use, we adopt the notation $S_i\rightarrow A_j$ (or simply $i\rightarrow j$) to represent that $A_j$ receives a share from the source $S_i$, i.e., $A_j$ is connected to the edge $e_i$.
Classical variable models {#formalism_HVM}
-------------------------
In the classical case we assume that the sources $S_i$ share randomness among the parties, and each party $A_j$ applies a function on the receives randomnesses to determine her output. As proven in [@rosset] in this case, we can assume that each source $S_i$ only takes a finite number of values. Alternatively, we can suppose that $S_i$ is associated to a uniform random variable $s_i\in [0,1]$ that is discretized by a single step function over $[0,1]$ by the parties. Here we adopt the latter notation. Then the party $A_j$ applies a function on the sources she receives, i.e., on $\{s_i: i\rightarrow j\}$, and outputs $a_j$. Let us denote by $r^j_a$ the indicator function that $a_j$ equals $a$, i.e., for a given realization of the sources $\{s_i\}$ we define $r^j_a(\{s_i: i\to j\})$ to be equal to $1$ if the output of $A_j$ given inputs $\{s_i: i\to j\}$ equals $a$, and $0$ otherwise. Then the joint output distribution can be written as $$P(a_1, \dots, a_n)=\int \Big( \prod_j r^{j}_{a_j} \big(\{s_i: i\to j\}\big)\Big)\prod_i {\mathrm{d}}s_i,$$ and the marginals are given by $$P_{A_{j}}(a_j)=\int r^{j}_{a_j} \big(\{s_i: i\to j\}\big)\prod_{i: i\rightarrow j} {\mathrm{d}}s_i .$$ We denote the set of all distributions achievable in the classical variable model for a given network $\mathcal N$ by $\mathcal N_{\mathcal L}$.
Quantum models
--------------
In the quantum model we assume that sources $S_i$ share quantum states, and the parties determine their outputs by applying measurements. As we have no dimension restriction, we can assume that the shared states are pure and the measurements are projective. We denote by $\rho_i$ the quantum state distributed by source $S_i$ and write $\rho=\bigotimes_i\rho_i$. Moreover, we denote the measurement operators of $A_j$ by $\big\{M^{(j)}_{a_j}\big\}$. Then the resulting output distribution equals $$P(a_1, \dots, a_n) = \text{Tr}\bigg( \rho \cdot \bigotimes_j M^{(j)}_{a_j} \bigg).$$ We denote the set of all distributions achievable in the quantum model for a given network $\mathcal N$ by $\mathcal N_{\mathcal Q}$.
Boxworld model
--------------
In this model, each source distributes an arbitrary number of NS boxes to the parties to which it connects. These NS boxes serve as a resource for the parties to generate correlated outputs. Each party can locally “wire” her boxes in hand to determine her output. That is, each party successively choose a box and its input, and receives an output. She is free to choose the order in which she uses her boxes and the input of it with some stochastic post-processing of her transcript at that time, i.e., previous choices of boxes, their inputs and their outputs. The final output of each party is a stochastic post-processing of her final transcript. We will later formalize this definition in a more precise way. We denote the set of all distributions achievable in the boxworld model for a given network $\mathcal N$ by $\mathcal N_{\mathcal B}$.
Fractional independent sets
---------------------------
Later we will use the following definition.
A fractional independent set of a network $\mathcal{N}$ is a vector of non-negative $\boldsymbol\eta=(\eta_1, ..., \eta_n)$ which corresponds a weight each party such that for each source the summation of the weights of parties connected to it at most 1. Formally, $\eta_j\geq 0$ for all $j$ and $$\label{fractionnal_matchings}
\sum_{j:\, i\rightarrow j} \eta_j \leq 1, \qquad \forall i.$$
A vector $\boldsymbol \eta$ satisfying the above conditions is called a fractional independent set since assuming that $\eta_j$’s are either $0$ or $1$, the subset $\{A_j:\, \eta_j=1\}$ forms an independent set of the associated graph, i.e., a subset of vertices to two of which are adjacent.
Basic properties in the triangle scenario {#cube_representation}
=========================================
In the following, we present some basic results about correlations achievable in the triangle network.
Cube representation of strategies in the triangle scenario
----------------------------------------------------------
![Any local strategy in the triangle scenario can be mapped to a unit cube. Three orthogonal axis of the cube are labeled by the hidden variables $\alpha,\beta,\gamma$. Alice’s response for a given ($\beta, \gamma$) is written on the face orthogonal to the $\alpha$ direction (and similarly for Bob and Charlie). In this representation, $P(000)$ is the volume of points which project to $0$ on all the three faces, and $P_A(0)$ is the area of points one the face orthogonal to the $\alpha$ direction which are labeled $0$.[]{data-label="strat_triangle_appendix"}](cube_strategy.JPG)
Any classical strategy for generating a given tripartite distribution in the triangle scenario can geometrically be represented by a cube with labels on it sides. Consider a unit cube in three dimensions. We label three mutually orthogonal edges of the cube by the three sources $\alpha, \beta, \gamma$ and label $A$, $B$, $C$ the faces respectively orthogonal to the edges $\alpha, \beta, \gamma$. Recall that we assume that the sources $\alpha, \beta$ and $\gamma$ take values in $[0,1]$. Thus any values of $\beta, \gamma\in [0,1]$ correspond to a point on face $A$, and to an answer of Alice when she receives $(\beta,\gamma)$ from the sources, i.e., $a(\beta,\gamma)$. That is, points of face $A$ are labeled by $a$’s. On the other hand, as we mentioned before, for each source the interval $[0, 1]$ is divided in a finite number of subintervals, and the parties are ignorant of the exact value of the source, but its subinterval index. Therefore, face $A$ is indeed partitioned in some aligned rectangles which are labeled by $a$’s. The same applies to faces $B$ and $C$. See Fig. \[strat\_triangle\_appendix\].
Basic properties of ${\mathcal{N}_\mathcal{L}}$ in the triangle scenario
------------------------------------------------------------------------
In the following, we recall and give basic properties of the set of correlation in ${\mathcal{N}_\mathcal{L}}$ for the triangle scenario. We choose to illustrate some proofs with the cube representation of strategies discussed above. For simplicity of presentation, we limit ourselves to the case where the outputs are all binary; generalization of these proofs to larger output alphabet sizes is straightforward.
Let $\mathcal N$ denote the triangle network. Then the followings hold:
1. The GHZ distribution and the W distribution (in which exactly one of the parties, chosen uniformly at random, outputs $1$ and the others output $0$) are not in ${\mathcal{N}_\mathcal{L}}$ (see also [@wolfe]).
2. ${\mathcal{N}_\mathcal{L}}$, as well as ${\mathcal{N}_\mathcal{Q}}$, ${\mathcal{N}_\mathcal{B}}$, are contractible (even though they are not star convex), hence do not contain holes.
3. ${\mathcal{N}_\mathcal{L}}$ contains an open ball in the probability simplex around $P_u$ the maximally mixed distribution (see also [@evans]). Thus the dimension of ${\mathcal{N}_\mathcal{L}}$ equals the dimension of the probability simplex, which is $7$ when outputs are all binary. The same holds for ${\mathcal{N}_\mathcal{Q}}$ and ${\mathcal{N}_\mathcal{B}}$ as they contain ${\mathcal{N}_\mathcal{L}}$.
4. The Finner inequality is valid for any distribution in ${\mathcal{N}_\mathcal{L}}$.
($i$) can be proven based on the cube representation of strategies. The proof for the W distribution is given in Fig. \[NoStratW\]; the proof for GHZ is left for the reader.
![Suppose that a cube gives the W distribution. As Alice sometimes answer 1, by relabeling the hidden variables, we can suppose that there is a 1 at the location given in $(a)$. Then, there are two possibilities: either there is no 0 on the left of that 1 (case $(a1)$), or there is one (case $(a2)$). The first case is not possible: as Alice and Charlie never answer 1 together, there must be 0 everywhere on Charlie’s face, i.e., Charlie always answers 0. In the second one, as when Alice says 1 Bob and Charlie cannot say 1, we end up with the cube $(a2)$. As when Alice and Bob both say 0, Charlie must say 1, we obtain the cube $(a22)$. However, in that cube Bob should not say 1 when Charlie already says it; he must answer 0 all the time, which is absurd.[]{data-label="NoStratW"}](cube_proof.JPG)
($ii$) Recall that a set ${\mathcal{S}}$ is said to be contractible if it can be continuously shrunk to a point within ${\mathcal{S}}$. More precisely, there is a *continuous* map $\Phi: (t, P)\in [0,1]\times {\mathcal{S}}\mapsto \Phi_t(P)\in{\mathcal{S}}$ such that $\Phi_1(P)=P$ and $\Phi_0(P)=P_u$ for some fixed $P_u$. For the set ${\mathcal{N}_\mathcal{L}}$ such a map $\Phi_t(P)$ for an arbitrary $P\in {\mathcal{N}_\mathcal{L}}$ is constructed as follows, and can similarly be defined for ${\mathcal{N}_\mathcal{Q}}$ and ${\mathcal{N}_\mathcal{B}}$.
In the cubic representation of strategies, since the parties can also have access to local randomness independent of common sources, we may add a question mark symbol ‘$?$’ telling the party to choose her output uniformly at random. Thus, the maximally mixed (uniform) distribution $P_u$ corresponds to a cube all of whose three orthogonal faces are labeled by the question mark. Now consider a strategy for generating a distribution $P$ and construct a cube whose corner $t\times t\times t$ sub-cube, for some $0\leq t\leq 1$, is filled according to the *renormalized* strategy for $P$, and the rest of it is filled by the question mark (see Fig. \[contractile\]). Call the resulting distribution $P_t=\Phi_P(t)$. When the outputs are binary, a simple computation verify that $$\begin{aligned}
\label{eq:P_t-abc}
P_t(abc) =& \frac{1}{8}\Big(1-3t^2+2t^3 + 8 t^3 P(abc) \nonumber\\
&\qquad + 2t^2(1-t) \big(P_A(a)+P_B(b)+P_C(c)\big) \Big).\end{aligned}$$ Clearly, $P_t=\Phi_P(t)$ is continuous in $(P,t)$.
\[contractile\]
($iii$) Let $P^{(j)}$, $j=1, \dots, k$, be some distributions in ${\mathcal{N}_\mathcal{L}}$. Pick arbitrary $\epsilon_j\geq 0$ with $\sum_j \epsilon_j\leq 1$ and on each of the three orthogonal sides of the cube pick disjoint intervals of sizes $\epsilon_j$ for any $j$. Then in the cube one finds $k$ (disjoint) sub-cubes of sizes $\epsilon_j\times \epsilon_j\times \epsilon_j$ for any $j$ (see Fig. \[ballP0strategies\]). Now similar to the proof of part ($ii$) fill the $j$-th sub-cube according to the scaled strategy associated to $P^{(j)}$. This gives a distribution $Q\in {\mathcal{N}_\mathcal{L}}$ which can be derived following similar computation as that of : $$\begin{aligned}
Q(abc) = &\frac{1}{8}\Big( 1+ \sum_j \Big[-3\epsilon_j^2+2\epsilon_j^3 + 8 \epsilon_j^3 P^{(j)}(abc) \nonumber\\
&\quad + 2\epsilon_j^2(1-\epsilon_j) \big(P^{(j)}_A(a)+P^{(j)}_B(b)+P^{(j)}_C(c)\big)\Big] \Big),\end{aligned}$$ where 1 has to be interpreted as a vector full of ones (similarly for $\frac{1}{8}$ and $\frac{1}{2}$ in the following).
To continue the proof it is instructive two write down the distributions $P^{(j)}$ as $$P^{(j)} = \frac{1}{8}+ R^{(j)},$$ with marginals $P^{(j)}_A = \frac{1}{2}+ R_A^{(j)}$ etc. Then letting $Q=\frac{1}{8} + S$ the above equation can be rewritten as $$\begin{aligned}
\label{eq:S-abc}
S(abc) = &\frac{1}{8} \sum_j \Gamma_{P^{(j)}, \epsilon_j}(abc),\end{aligned}$$ where $$\begin{aligned}
\Gamma_{P^{(j)}, \epsilon_j}(abc)=&8 \epsilon_j^3 R^{(j)}(abc)\\
& + 2\epsilon_j^2(1-\epsilon_j) \big(R^{(j)}_A(a)+R^{(j)}_B(b)+R^{(j)}_C(c)\big).\end{aligned}$$
Now to finish the proof we need to show that any $S$ satisfying $\sum_{a, b, c} S(abc)=0$ and with sufficiently small coordinates can be written as for some $P^{(j)}\in {\mathcal{N}_\mathcal{L}}$ and some choices of $\epsilon_j$’s.
Let $P^{(xyz)}$, for $(x, y, z)\in \{0,1 , ?\}^3$ be the distribution coming from the cube whose Alice’s face is labeled $x$, Bob’s face is labeled $y$, and Charlie’s face is labeled $z$. For instance we have $$P^{(0??)} (abc) = \frac{1}{4}\delta_{a=0},$$ with $R^{(0??)}_A(a) = \frac 12 (-1)^a$ and $R^{(0??)}_B=R^{(0??)}_C=0$. Then we have $$\Gamma_{P^{(0??)}, \epsilon}(abc) = \epsilon^2 (-1)^a,$$ and also $\Gamma_{P^{(1??)}, \epsilon}(abc) = - \Gamma_{P^{(0??)}, \epsilon}(abc)$. We similarly can compute $$\Gamma_{P^{(00?)}, \epsilon}(abc) = \epsilon^2\big( (-1)^a+(-1)^b \big) + \epsilon^3 (-1)^{a+b},$$ and $$\begin{aligned}
\Gamma_{P^{(000)}, \epsilon}(abc) = &\epsilon^2\big( (-1)^a+(-1)^b +(-1)^c\big) \\
&+ \epsilon^3 \big( (-1)^{a+b} +(-1)^{a+c} +(-1)^{b+c} \big)\\
& + \epsilon^3(-1)^{a+b+c}.\end{aligned}$$ Comparing the above equations, we find that by considering the summations of these $\Gamma$ terms for different choices of $(x, y, z)\in \{0, 1, ?\}^3$, we can write the functions $$\begin{aligned}
&\pm\delta (-1)^{a}, \pm\delta (-1)^{b}, \pm\delta (-1)^{ c},\\
&\pm\delta (-1)^{ a +b} , \pm\delta (-1)^{a+c}, \pm\delta (-1)^{a+ c}\\
&\pm\delta (-1)^{+ a+ b+ c},\end{aligned}$$ in the form of when $\delta$ is sufficiently small. Observing that these functions, which also include their negations, form a basis for the space of functions $S$ with $\sum_{abc} S(abc)=0$, the proof is concluded.
Reference [@evans] is related to the same question and exploits a totally different framework.
![ Suppose that two disjoint sub-cubes of size $\epsilon\times \epsilon\times \epsilon$ (with disjoint projections on the three orthogonal directions) are filled with constant $0$ and constant $1$, and the rest of the cube with totally random choices. The resulting tripartite distribution would be equal to $P=P_u+\frac{\epsilon^3}{4}V$ where $P_u$ is the maximally mixed (uniform) distribution and $V$ is given by $V(000)=V(111)=3$ and $V(abc)=-1$ if $(a, b,c)\in \{0,1\}^3\setminus\{000, 111\}$. []{data-label="ballP0strategies"}](ballP0strategies1.jpg)
($i\nu$) This is a direct consequence of the Loomis-Whitney inequality, asserting that in $\mathbb R^3$, the square of the volume of any measurable subset is bounded by the product of the areas of its projections in the three orthogonal directions.
The Finner inequality in term of the Hypercontractivity Ribbon {#app:HR}
==============================================================
The Hypercontractivity Ribbon (HR) is a measure of correlation that can be defined in terms of parameters for which the Finner inequality is satisfied. In the tripartite case on which we focus, the HR of a distribution $P_{ABC}$ is the set of $(\alpha, \beta, \gamma)\in [0,1]^3$ for which $$\label{hypercontractivity_ribbon_2}
{\mathds{E}\left[f_Ag_Bh_C\right]}\leq \|f_A\|_{1/\alpha}\cdot \|g_B\|_{1/\beta}\cdot \|h_C\|_{1/\gamma},$$ for all choices of functions $f_A, g_B$ and $h_C$. We denote the HR of $P_{ABC}$ by ${\mathfrak{R}}(A, B, C)$. An important property of HR is that it expands under local post-processing. That is, if $A', B', C'$ are obtained by local post-processing of $A, B, C$ respectively, then we have $$\begin{aligned}
\label{eq:HR-monotone}
{\mathfrak{R}}(A, B, C)\subseteq {\mathfrak{R}}(A', B', C').\end{aligned}$$ More interesting is the tensorization property of HR saying that ${\mathfrak{R}}(A^n, B^n, C^n) = {\mathfrak{R}}(A, B, C)$ where the former is computed with respect to the iid distribution $P_{ABC}^{\otimes n}$. See [@beigi] and references therein for more details.
Finner’s inequality can be stated in terms of HR. In the triangle scenario, for instance, Finner’s inequality says that for every $P_{ABC}\in {\mathcal{N}_\mathcal{L}}$ we have $(1/2, 1/2, 1/2)\in {\mathfrak{R}}(A, B, C)$. In general, Finner’s inequality says that any fractional independent set of a network belongs to the HR of any distribution achievable in that network.
A crucial property of ${\mathfrak{R}}(A,B,C)$ is that it can be expressed in terms of the mutual information function as follows. ${\mathfrak{R}}(A, B, C)$ consists of the set of non-negative triples $(\alpha, \beta, \gamma)$ such that for any *auxiliary* random variable $U$ given by $P_{U|ABC}$ we have $$\label{hypercontractivity_ribbon_1}
I(U;ABC)\geq \alpha I(U;A) + \beta I(U;B) + \gamma I(U;C)$$ From this characterization of HR it is clear that ${\mathfrak{R}}(A, B, C)$ is a convex set. This is a property that will be used in the proof of Theorem \[thm:quantum-finner\].
The Finner inequality holds in ${\mathcal{N}_\mathcal{Q}}$ {#finner_quantum}
==========================================================
We now present a proof of Theorem \[thm:quantum-finner\], that the Finner inequality holds for any $P\in {\mathcal{N}_\mathcal{Q}}$ when the sources in the network $\mathcal N$ are all bipartite.
First of all, as mentioned in Appendix \[app:HR\], given a distribution $P$, the set of weights $\boldsymbol\eta = (\eta_1, \dots, \eta_n)$ for which the Finner inequality holds for all choices of $f_j$’s, is a convex set. That is, if holds for $\boldsymbol \eta$ and $\boldsymbol{\eta'}$, then it holds for any convex combination of them. Therefore, in order to show that the Finner inequality is satisfied for all weights $\boldsymbol \eta$ that form a fractional independent set (which itself is a convex set), it suffices to prove it for the extreme points of the set of fractional independent set. That is, in the proof we may assume that $\boldsymbol \eta =(\eta_1, \dots, \eta_n)$ is an extreme point of the set of fractional independent sets.
Second, we use the assumption that all sources in the network $\mathcal N$ are bipartite. It is well-known that in any graph the extreme points of the set of fractional independent sets are half-integers [@Schrijver Theorem 64.7]. In other words, if $\boldsymbol \eta =(\eta_1, \dots, \eta_n)$ is an extreme fractional independent set, for all $j$ we have $\eta_j\in \{0, 1/2, 1\}$. On the other hand, if one of $\eta_j$’s, say $\eta_n$, equals $0$, then we have $$\|f_n\|_{1/\eta_n} = \|f_n\|_\infty = \max_{a_n: P_{A_n}(a_n)\neq 0} f(a_n),$$ and $$\mathbb E\Big[\prod f_j\Big] \leq \mathbb E\bigg[\,\prod_{j=1}^{n-1} f_j\,\bigg] \cdot \|f_n\|_\infty.$$ This means that if holds ignoring the $n$-th party (for the marginal distribution $P_{A_1, \dots, A_{n-1}}$), it also holds including her and putting $\eta_n=0$. We conclude that we may restrict ourselves to weights $\boldsymbol \eta =(\eta_1, \dots, \eta_n)$ such that $\eta_j\in \{1/2, 1\}$ for all $j$. Even more, for such weights if there is $j$, say $j=n$, with $\eta_j=1$, then the $n$-th party cannot share any source with others. This is because if $A_{j'}$ shares a source with $A_n$ then we must have $\eta_n+\eta_{j'}\leq 1$ that is a contradiction since $\eta_{j'}$ is assumed to be in $\{1/2, 1\}$. This means such a party $A_n$ with $\eta_n=1$ is isolated and shares nothing with others. In this case we have $$\mathbb E\Big[\prod f_j\Big] = \mathbb E\bigg[\,\prod_{j=1}^{n-1} f_j\,\bigg] \cdot \mathbb E[f_n],$$ and of course $\|f_n\|_{1/\eta_n} = \|f_n\|_1= \mathbb E[f_n]$. As a result, parties whose weights are equal to $1$ can be ignored. Putting all these together we may assume that all the weights are equal to $\eta_j=1/2$ and we need to prove $$\begin{aligned}
\label{eq:Q-Finner-2}
{\mathds{E}\left[\prod_j f_j\right]} \leq \prod_j {\left\lVertf_j\right\rVert}_{2},\end{aligned}$$
Third, recall that $f_j$ is an arbitrary function applied on the output of the $j$-th party $A_j$. Composing the measurement operators of $A_j$ with this classical post-processing, we may assume that $f_j$ is the outcome of some quantum observable $X_j$ that the $j$-th party applies on quantum systems in her hand. Indeed, we may put $$X_j = \sum_{a_j} f_j(a_j) M_{a_j}^{(j)},$$ where $\big\{M_{a_j}^{(j)}\big\}$ is the projective measurement applied by $A_j$. Thus we have $$\begin{aligned}
\label{eq:mean-f-j-X}
\mathbb E\Big[ \prod_j f_j \Big] = \text{Tr} \Big[ \rho\cdot \bigotimes_j X_j \Big],\end{aligned}$$ where as before $\rho=\bigotimes_i \rho_i$ and $\rho_i$ is the pure state associated to the $i$-th source. Recall that for each source $i$, $\rho_i$ is a pure bipartite state, so we may consider its Schmidt decomposition. For simplicity of notation we may assume that the dimension of all subsystems in $\rho_i$’s are equal (by taking the maximum of all these local dimensions). Moreover, by applying appropriate local rotations we may assume that the Schmidt basis of all $\rho_i$’s are the same. Thus we may write $\rho_i = {\ensuremath{\left|\psi_i\right\rangle}}{\ensuremath{\left\langle\psi_i\right|}}$ with $${\ensuremath{\left|\psi_i\right\rangle}} = \sum_{\ell=1}^d \lambda_\ell^{(i)} {\ensuremath{\left|\ell\right\rangle}}\otimes {\ensuremath{\left|\ell\right\rangle}},$$ where $\lambda_\ell^{(i)}\geq 0$ denote Schmidt coefficients of $\rho_i$. Then reduces to $$\mathbb E\Big[ \prod_j f_j \Big] = \sum_{\stackrel{\ell_1, \dots, \ell_m} {\ell'_1, \dots, \ell'_m}} \prod_i \lambda^{(i)}_{\ell_i} \lambda^{(i)}_{\ell'_i} \cdot \prod_j \text{Tr}\Big[ X_j\cdot \bigotimes_{i: i\to j} {\ensuremath{\left|\ell_i\right\rangle}}{\ensuremath{\left\langle\ell'_i\right|}} \Big].$$
For any $j$ define $g_j: \prod_{i: i\to j} \{1, \dots, d\}^2\to \mathbb R$ by $$g_j\Big( (\ell_i, \ell'_i )_{i: i\to j} \Big) = \prod_{i:i\to j} \sqrt{\lambda^{(i)}_{\ell_i} \lambda^{(i)}_{\ell'_i}} \cdot \text{Tr} \Big[ X_j\cdot \bigotimes_{i: i\to j} {\ensuremath{\left|\ell_i\right\rangle}}{\ensuremath{\left\langle\ell'_i\right|}}\Big].$$ Also, let $R_i$ be the uniform random variable taking values in $\{1, \dots, d\}^2$, i.e., $R_i$ equals $(\ell, \ell')$ with probability $d^{-2}$. Then the previous equation can be written as $$\mathbb E\Big[ \prod_j f_j \Big] = d^{2m} \mathbb E\Big[ \prod_j g_j\big( (R_i)_{i\to j} \big) \Big]$$ Now we may think of $R_i$’s as sources of randomnesses that are shared to the parties who apply local functions $g_j$ on them. Then by the (classical) Finner inequality we have $$\mathbb E\Big[ \prod_j f_j \Big]\leq d^{2m} \prod_j \|g_j\|_{2}.$$ Let us compute the factors on the right hand side: $$\begin{aligned}
&d^{2\cdot|\{i:\, i\to j\}|}\cdot\|g_j\|^2_{2} = d^{2\cdot|\{i:\, i\to j\}|}\cdot\mathbb E\big[ g_j^2 \big]\\
& = \sum_{(\ell_i, \ell'_i)_{i: i\to j}} \prod_{i:i\to j} \lambda^{(i)}_{\ell_i} \lambda^{(i)}_{\ell'_i} \cdot \text{Tr} \Big[ X_j\cdot \bigotimes_{i: i\to j} {\ensuremath{\left|\ell_i\right\rangle}}{\ensuremath{\left\langle\ell'_i\right|}}\Big]^2\\
& = \text{Tr} \big[ \sqrt{\sigma_j} X_j\sqrt{\sigma_j} X_j\big],\end{aligned}$$ where $$\sigma_j= \sum_{(\ell_i)_{i:i\to j}} \Big(\prod_{i:i\to j}\lambda^{(i)}_{\ell_i}\Big)^2 \bigotimes_{i:i\to j} {\ensuremath{\left|\ell_i\right\rangle}}{\ensuremath{\left\langle\ell_i\right|}}.$$ We continue $$\begin{aligned}
d^{2\cdot|\{i:\, i\to j\}|}\cdot\|g_j\|^2_{2} & = \text{Tr} \Big[ \big(\sigma_j^{1/4} X_j\sigma_j^{1/4}\big)^2\Big]\\
& \leq \text{Tr} \Big[ \sigma_j^{1/2}X_j^2 \sigma_j^{1/2} \Big]\\
&= \text{Tr} \Big[ \sigma_j X_j^2 \Big]\\
& = \|f_j\|_2^2,\end{aligned}$$ where the inequality follows from the Araki-Lieb-Thirring inequality, and the last equality is verified by an easy computation. We conclude that $$\begin{aligned}
\mathbb E\Big[ \prod_j f_j \Big]&\leq d^{2m} \prod_j \|g_j\|_{2}\\
&= \prod_j d^{|\{i:\, i\to j\}|} \cdot \|g_j\|_2\\
&\leq \prod_j \|f_j\|_2,\end{aligned}$$ where the equality follows from the fact that the sources are bipartite and for each $i$ there are exactly two $j$’s for which $i\to j$.
Finner inequality holds for triangle scenario in the Boxworld {#finner_boxworld_proof}
=============================================================
In this section we show that the Finner inequality is satisfied for the triangle scenario in the Boxworld, where bipartite no-signaling boxes are wired by the parties to produce an output. As explained in Appendix \[finner\_quantum\], we can restrict ourselves to coefficients 1/2.
\[Finner inequality in Boxworld\]\[finner\_bw\] Letting $\mathcal N$ be the triangle network, for any $P_{ABC}\in {\mathcal{N}_\mathcal{B}}$ we have $${\mathds{E}\left[f_Ag_Bh_C\right]}\leq \|f_A\|_{2}\cdot \|g_B\|_{2}\cdot \|h_C\|_{2}.$$
Before getting into the details of the proof, let us briefly explain the proof ideas. First, as mentioned in Appendix \[app:HR\], the above theorem says that for any $P_{ABC}\in {\mathcal{N}_\mathcal{B}}$ we have $(1/2, 1/2, 1/2)\in {\mathfrak{R}}(A, B, C)$. On the other hand, since HR is monotone under local post-processing (equation ), it suffices to prove that $(1/2, 1/2, 1/2)\in {\mathfrak{R}}(T, R, S)$ where $T, R$ and $S$ denote all information available to Alice, Bob and Charlie respectively, at the end of the wirings. This is because, $A, B, C$ are functions of $T, R, S$ respectively. Next we can use the second equivalent characterization of HR, and in order to prove $(1/2, 1/2, 1/2)\in {\mathfrak{R}}(T, R, S)$ show that $$\begin{aligned}
\chi = I(U; TRS) - \frac{1}{2}I(U;T) - \frac{1}{2}I(U;R) - \frac{1}{2}I(U;S)\ge 0. \end{aligned}$$ The proof of this inequality is based on the chain rule of mutual information. In the wiring, each party uses her boxed in hand one by one: for each time-step new information is added to her transcript. Therefore, we expand each mutual information term in the above equation as a summation over time-steps using the chain rule. We further expand each time step into the choice of box and input, and the creation of the output. As a result we obtain $\chi = \chi_I + \chi_O$, where $\chi_I$ (respectively, $\chi_O$) is a summation of terms corresponding to all the choices of boxes and their inputs (respectively, outputs) by the parties.
As the choice of boxes and their inputs are done locally and independently of the sources, $\chi_I$ can easily be bounded. As the order in which Alice, Bob and Charlie choose there boxes is a priori not the same, bounding $\chi_O$ is more tricky. One has to first reorder the summation in $\chi_O$ to put terms associated to a given box together, and then use properties of mutual information to bound $\chi_O$. We use the no-signaling assumption in this last step.
This proof follows the ideas introduced in [@beigi], in which the author prove that PR boxes cannot be purified using wirings.
Notations
---------
Here we introduce the notations we need to prove Theorem \[finner\_bw\]. We define notations for Alice and then recap them for Bob and Charlie.
Let $J_{AB}$ (respectively $J_{AC}$) be the set of index of all boxes shared between Alice and Bob (respectively Alice and Charlie). Let $N_A$ be the number of boxes available to Alice, i.e., $N_A = |J_{AB}|+|J_{AC}|$.
For $j \in J_{AB}$, let $X_j$, $Y_j$ (respectively $A_j$, $B_j$) be the inputs (respectively, the outputs) of the box $j$. This box is described by a no-signaling conditional distribution $P(A_j B_j|X_j Y_j)$.
In each time-step Alice chooses which box to use next and its input as a random function of whatever she has so far, i.e. previous choices of boxes, their inputs and their outputs. We denote by $\Pi_j$ the time-step at which Alice uses box $j$. Thus $\Pi$ is a permutation of the boxes available to Alice. We denote the inverse of this distribution by $\tilde{\Pi}$. That is, $\tilde{\Pi}_i$ is the index of the box used by Alice at time-step $i$.
Let $T_j$ be the *transcript* of Alice before using the $j$-th box, i.e., whatever she has seen before using the $j$-th box. We also denote Alice’s *extended transcript* by ${T}_j^e$ that is $T_j$ together with $\Pi_j$ and $X_j$: $$T_j^e=( T_j,\Pi_j,X_j).$$ We also use the notations $\tilde{X}_i=X_{\tilde{\Pi}_i}$, $\tilde{A}_i=A_{\tilde{\Pi}_i}$, $\tilde{T}_i=T_{\tilde{\Pi}_i}$ and $\tilde{T}^e_i=T^e_{\tilde{\Pi}_i}$. Observe that, for instance, $\tilde X_i$ is the box that Alice uses in time-step $i$. With these notations we have $$T_j := \big ( \tilde{\Pi}_1, \dots, \tilde{\Pi}_{\Pi_{j-1}}, \tilde{X}_1, \dots, \tilde{X}_{\Pi_{j-1}}, \tilde{A}_1, \dots, \tilde{A}_{\Pi_{j-1}} \big ).$$
Here is a summary of notations for later use:
- $\Pi_j$ : Alice uses the $j$-th box in her $\Pi_j$-th action.
- $\tilde{\Pi}_i$: Alice uses the $\tilde{\Pi}_i$-th box in her $i$-th action.
- $X_j$ : Alice’s input of the $j$-th box.
- $\tilde{X}_i$ : Alice’s input in her $i$-th action.
- $A_j$ : Alice’s output of the $j$-th box.
- $\tilde{A}_i$ : Alice’s output in her $i$-th action.
- $T_j$ : Alice’s transcript before using the $j$-th box.
- $\tilde{T}_i$ :Alice’s transcript before her $i$-th action.
- $T_j^e=\{ T_j,\Pi_j,X_j\}$.
- $\tilde{T}_i^e=\{ \tilde{T}_i,\tilde{\Pi}_i,\tilde{X}_i\}$.
We use superscript $N_A$ to denote the full set of variables at the end, e.g., $A^{N_A}= (A_1, \ldots, A_{N_A})$. At the end, Alice determines her final output by applying a stochastic map on all information available to her, i.e., on $\Pi^{N_A}, X^{N_A}, A^{N_A}$ which we denote by $$\begin{aligned}
\label{eq:T-transcript}
T=\big(\Pi^{N_A}, X^{N_A}, A^{N_A}\big).\end{aligned}$$
At the end Alice, Bob, and Charlie apply their stochastic maps to determine their final output.
The corresponding variables in the above list for Bob are $\Omega_j, \tilde{\Omega}_i, Y_j, \tilde{Y}_i, B_j, \tilde{B}_i, S_j, \tilde{S}_i, S_j^e, \tilde S_i^e$ respectively. The corresponding variables in the above list for Charlie are $\Gamma_j, \tilde{\Gamma}_i, Z_j, \tilde{Z}_i, C_j, \tilde{C}_i, R_j, \tilde{R}_i, R_j^e, \tilde R_i^e$ respectively
Auxiliary lemmas
----------------
In this section, we introduce some lemmas deduced from the no-signaling condition.
\[lemma:input\_part\] For every $1\leq i\leq N_B$ we have $$I(\tilde{Y}_i \tilde{\Omega}_i; T | \tilde{S}_i ) = 0,$$ and for every $1\leq i\leq N_C$ we have $$I(\tilde{Z}_i \tilde{\Gamma}_i; T S | \tilde{R}_i ) = 0.$$
These expressions are simple consequences of the fact that each party at time-step $i$ chooses a box and its input locally as a (stochastic) function of the transcript at step $i$, and other parties cannot signal using the boxes.
\[B\_given\_A\] For boxes available to Bob we have
1. For $j \in J_{AB}$: $$I(B_j; T|T_j^e A_j S_j^e) = 0$$
2. For $j \in J_{BC}$: $$I(B_j;T |S_j^e) = 0$$
$(i)$ states the independence of Bob’s output of the $j$-th box and the future information in Alice’s side, given all the information of Alice and Bob, except $B_j$, up to just after using this box. To prove this, it is enough to show that $H(B_j|T_j^e A_j S_j^e)=H(B_j|T S_j^e)$, for which we compute $$\begin{aligned}
H(B_j|T S_j^e) & = H(A_jB_j|T\setminus\{A_j\} S_j^e) - H(A_j| T\setminus\{A_j\} S_j^e) \\
& = H(A_jB_j|T_j^e S_j^e) - H(A_j| T_j^e S_j^e)\\
& = H(B_j| T_j^e A_j S_j^e).\end{aligned}$$ Here the second line follows from the fact that $A_j$ and $B_j$ are determined independently of the other variables once the inputs of the $j$-th box are fixed.\
$(ii)$ states that given the input of Bob for a box shared between Bob and Charlie, Bob’s output is independent of Alice’s transcript. To prove this we compute $$\begin{aligned}
I(T;B_j|S_j^e) & \leq I(TR_j^e; B_j| S_j^e) \\
& = I(T;B_j|S_j^e R_j^e)\\
& \leq I(T;B_j C_j|S_j^e R_j^e).\end{aligned}$$ Here the inequalities follow from the data processing inequality, and the equality follows from the chain rule and $I(B_j;R_j^e|S_j^e) = 0$, the no-signaling condition. Then the desired result follows once we note that $I(T;B_j C_j|S_j^e R_j^e)=0$ since the output of the $j$-th box are determined independently of other variables once its inputs are fixed.
The following lemma presents similar statements as above for boxes available for Charlie. We skip its proof as it follows from similar ideas as above.
\[C\_given\_AB\] For boxes available to Charlie we have
1. For $j \in J_{BC}$: $$I(C_j;T S|S_j^e B_j R_j^e) = 0$$
2. For $j \in J_{AC}$: $$I(C_j;S T|T_j^e A_j R_j^e) = 0$$
In the following for four random variables $X, Y, Z, W$ we use the notation $$\begin{aligned}
I(X; Y; Z|W) =&H(X|W)+H(Y|W)+H(Z|W) \\
&- H(XY|W)-H(XZ|W)-H(YZ|W)\\
&+ H(XYZ|W).\end{aligned}$$ We will frequently use the following expression for $I(X; Y; Z|W)$ which can easily be verified: $$\begin{aligned}
\label{important0}
I(X; Y; Z) = I(X; Y|W) - I(X; Y|WZ).\end{aligned}$$ We indeed use the symmetry in the definition of $I(X; Y; Z|W)$ which gives $$\begin{aligned}
\label{important}
I(X; Y|W) - I(X; Y|WZ) = I(Y; Z|W) - I(Y; Z| WX).\end{aligned}$$
Proof of Theorem \[finner\_bw\]
-------------------------------
As mentioned before, we need to show that $(1/2, 1/2, 1/2)\in {\mathfrak{R}}(A, B, C)$. Moreover, since HR satisfies the monotonicity property , and $A, B, C$ are determined by post-processing of $T, S, R$ respectively, it suffices to prove that $(1/2, 1/2, 1/2)\in {\mathfrak{R}}(T, S, R)$. That is, we need to show that for any $P_{U|TSR}$ we have $$\begin{aligned}
\chi = I(U; TRS) - \frac{1}{2}I(U;T) - \frac{1}{2}I(U;R) - \frac{1}{2}I(U;S)\ge 0. \end{aligned}$$
We first write $$\begin{aligned}
\label{eq:chain-order}
I(U; TRS)=I(U;T) + I(U;S|T) +I(U;R|T S).\end{aligned}$$ Then noting that, say, $T$ itself consists of several random variables as in , we apply chain rule once again to each of the above terms. This decomposes $\chi$ into two terms $\chi = \chi_I + \chi_O$ associated to the *input parts* and the *output parts* given by $$\begin{aligned}
\chi_I = &\sum_{i=1}^{N_A} \Big[ I(U; \tilde{X}_i \tilde{\Pi}_i|\tilde{T}_i) - \frac{1}{2} I(U; \tilde{X}_i \tilde{\Pi}_i | \tilde{T}_i) \Big] \\
&+ \sum_{i=1}^{N_B} \Big[ I(U;\tilde{Y}_i\tilde{\Omega}_i|T \tilde{S}_i) - \frac{1}{2} I(U; \tilde{Y}_i \tilde{\Omega}_i | \tilde{S}_i) \Big]\\
&+\sum_{i=1}^{N_C} \Big[ I(U;\tilde{Z}_i \tilde{\Gamma}_i| T S \tilde{R}_i)-\frac{1}{2}I(U;\tilde{Z}_i\tilde{\Gamma}_i|\tilde{R}_i)\Big]\end{aligned}$$ and $$\begin{aligned}
\chi_O = &\sum_{i=1}^{N_A} \Big[ I(U; \tilde{A}_i |\tilde{T}_i^e) - \frac{1}{2} I(U; \tilde{A}_i | \tilde{T}_i^e) \Big] \\
&+ \sum_{i=1}^{N_B} \Big[ I(U;\tilde{B}_i|T \tilde{S}_i^e) - \frac{1}{2} I(U; \tilde{B}_i | \tilde{S}_i^e) \Big]\\
&+ \sum_{i=1}^{N_C} \Big[ I(U;\tilde{C}_i| T S \tilde{R}_i^e)-\frac{1}{2}I(U;\tilde{C}_i|\tilde{R}_i^e)\Big]\end{aligned}$$ We will show separately that both $\chi_I$ and $\chi_O$ are non-negative.
Let us first start with $\chi_I\geq 0$ that is easy (as each party chooses its box and input independently, a stronger inequality holds with the terms $1/2$ replaced by $1$). The first summand in $\chi_I$ is non-negative since $I(U; \tilde{X}_i \tilde{\Pi}_i|\tilde{T}_i) - \frac{1}{2} I(U; \tilde{X}_i \tilde{\Pi}_i | \tilde{T}_i)= \frac{1}{2} I(U; \tilde{X}_i \tilde{\Pi}_i | \tilde{T}_i)\geq 0$. For the second summand we compute $$\begin{aligned}
&I(U;\tilde{Y}_i\tilde{\Omega}_i|T \tilde{S}_i) - \frac{1}{2} I(U; \tilde{Y}_i \tilde{\Omega}_i | \tilde{S}_i) \\
&\qquad\geq I(U;\tilde{Y}_i\tilde{\Omega}_i|T \tilde{S}_i) - I(U; \tilde{Y}_i \tilde{\Omega}_i | \tilde{S}_i)\\
&\qquad = - I(U; \tilde{Y}_i\tilde{\Omega}_i; T| \tilde S_i)\\
&\qquad = I(T ;\tilde{Y}_i\tilde{\Omega}_i| U\tilde{S}_i) - I(T ;\tilde{Y}_i\tilde{\Omega}_i| \tilde{S}_i)\\
&\qquad = I(T ;\tilde{Y}_i\tilde{\Omega}_i| U\tilde{S}_i)\\
&\qquad \geq 0,\end{aligned}$$ where the third equality follows from Lemma \[lemma:input\_part\]. The proof that the third summand is non-negative is similar. Therefore, $\chi_I \geq 0$.
We now show that $\chi_O \geq 0$. Observe that by definitions $\tilde A_i= A_{\tilde \Pi_i}$ etc. Then splitting the summands in $\chi_O$ in terms of boxes shared between different pairs of parties, we obtain: $$\begin{aligned}
\chi_O =& \sum_{i:\,\tilde{\Pi}_i \in J_{AB}} \Big[ I(U; A_{\tilde{\Pi}_i}|T_{\tilde{\Pi}_i}^e) - \frac{1}{2} I(U; A_{\tilde{\Pi}_i} | T_{\tilde{\Pi}_i}^e) \Big] \\
&+\sum_{i:\,\tilde{\Pi}_i \in J_{AC}} \Big[ I(U; A_{\tilde{\Pi}_i}|T_{\tilde{\Pi}_i}^e) - \frac{1}{2} I(U; A_{\tilde{\Pi}_i} | T_{\tilde{\Pi}_i}^e) \Big] \\
&+ \sum_{i:\,\tilde{\Omega}_i \in J_{AB}} \Big[ I(U; B_{\tilde{\Omega}_i}|T S_{\tilde{\Omega}_i}^e) - \frac{1}{2} I(U; B_{\tilde{\Omega}_i} | S_{\tilde{\Omega}_i}^e) \Big] \\
&+ \sum_{i:\,\tilde{\Omega}_i \in J_{BC}} \Big[ I(U; B_{\tilde{\Omega}_i}|T S_{\tilde{\Omega}_i}^e) - \frac{1}{2} I(U; B_{\tilde{\Omega}_i} | S_{\tilde{\Omega}_i}^e) \Big]\\
&+ \sum_{i:\,\tilde{\Omega}_i \in J_{BC}} \Big[ I(U; C_{\tilde{\Gamma}_i}|T S R_{\tilde{\Gamma}_i}^e) - \frac{1}{2} I(U; C_{\tilde{\Gamma}_i} | R_{\tilde{\Gamma}_i}^e) \Big]\\
&+ \sum_{i:\,\tilde{\Gamma}_i \in J_{AC}} \Big[ I(U; C_{\tilde{\Gamma}_i}|T S R_{\tilde{\Gamma}_i}^e) - \frac{1}{2} I(U; C_{\tilde{\Gamma}_i} |R_{\tilde{\Gamma}_i}^e) \Big].\end{aligned}$$ Next, we rewrite $\chi_O$ by reordering the summands in terms of the indices of boxes and not time-steps: $$\begin{aligned}
\chi_O = &\sum_{j \in J_{AB}} \Big[ I(U; A_j|T_j^e) - \frac{1}{2} I(U; A_j | T_j^e) \Big] \\
&+\sum_{j \in J_{AC}} \Big[ I(U; A_j|T_j^e) - \frac{1}{2} I(U; A_j | T_j^e) \Big] \\
&+ \sum_{j \in J_{AB}} \Big[ I(U; B_j|T S_j^e) - \frac{1}{2} I(U; B_j | S_j^e) \Big] \\
&+ \sum_{j \in J_{BC}} \Big[ I(U; B_j|T S_j^e) - \frac{1}{2} I(U; B_j | S_j^e) \Big]\\
&+ \sum_{j \in J_{BC}} \Big[ I(U; C_j|T S R_j^e) - \frac{1}{2} I(U; C_j |R_j^e) \Big] \\
&+ \sum_{j \in J_{AC}} \Big[ I(U; C_j|T S R_j^e) - \frac{1}{2} I(U; C_j |R_j^e) \Big].\end{aligned}$$ Using and Lemma \[B\_given\_A\] (i), for $j \in J_{AB}$ we have $$\begin{aligned}
\label{g1}
I(U;& B_j|T S_j^e) - I(U; B_j|T_j^e A_j S_j^e) \nonumber \\
&=I(B_j; T|T_j^e A_j S_j^e U) \geq 0.\end{aligned}$$ Moreover, by and Lemma \[B\_given\_A\] (ii), for $j \in J_{BC}$ we have $$\begin{aligned}
\label{g2}
I(U; B_j|T S_j^e) - I(U; B_j| S_j^e) = I(B_j;T |S_j^e U) \geq 0\end{aligned}$$ We similarly for $j \in J_{BC}$ have $$\begin{aligned}
\label{g3}
I(U; C_j|T S R_j^e) - I(U; C_j|S_j^e B_j R_j^e)\nonumber \\
= I(C_j;T S|S_j^e B_j R_j^e U) \geq 0\end{aligned}$$ and for $j \in J_{AC}$ have $$\begin{aligned}
\label{g4}
I(U; C_j|T S R_j^e) - I(U; C_j|T_j^e A_j R_j^e)\nonumber \\ = I(C_j;S T|T_j^e A_j R_j^e U) \geq 0\end{aligned}$$ Putting these together we find that $$\chi_O \geq \chi_{O_1} + \chi_{O_2} + \chi_{O_3},$$ where, $$\begin{aligned}
\chi_{O_1} =& \sum_{j \in J_{AB}} \Big[ I(U; A_j|T_j^e) + I(U; B_j|T_j^e A_j S_j^e) \\
&- \frac{1}{2} I(U; A_j | T_j^e) - \frac{1}{2} I(U; B_j | S_j^e) \Big], \\ \\
\chi_{O_2} =& \sum_{j \in J_{AC}} \Big[I(U; A_j|T_j^e) + I(U; C_j|T_j^e A_j R_j^e) \\
&- \frac{1}{2} I(U; A_j | T_j^e) - \frac{1}{2} I(U; C_j |R_j^e) \Big],\\ \\
\chi_{O_3} =& \sum_{j \in J_{BC}} \Big[I(U; B_j| S_j^e) + I(U; C_j|S_j^e B_j R_j^e) \\
&- \frac{1}{2} I(U; B_j | S_j^e) -\frac{1}{2} I(U; C_j |R_j^e) \Big].\\\end{aligned}$$ By adding and subtracting $I(U; A_j|T_j^e S_j^e)$ and using and the chain rule we have $$\begin{aligned}
\chi_{O_1} &= \sum_{j \in J_{AB}} \Big[ I(U; A_j; S_j^e|T_j^e) + I(U; A_jB_j|T_j^e S_j^e) \\
&\qquad - \frac{1}{2} I(U; A_j | T_j^e) - \frac{1}{2} I(U; B_j | S_j^e) \Big].\end{aligned}$$ Next by the data processing inequality we have $$\begin{aligned}
\chi_{O_1} &\geq \sum_{j \in J_{AB}} \Big[ I(U; A_j; S_j^e|T_j^e) \\
&\qquad +\frac12 I(U; A_j|T_j^e S_j^e)+\frac12 I(U; B_j|T_j^e S_j^e) \\
&\qquad - \frac{1}{2} I(U; A_j | T_j^e) - \frac{1}{2} I(U; B_j | S_j^e) \Big].\end{aligned}$$ On the other hand, by the no-signaling condition $I(S_j^e; A_j|T_j^e)=0$ we have $$\begin{aligned}
I(U; A_j|T_j^e S_j^e) &= I(US_j^e; A_j| T_j^e) - I(S_j^e; A_j|T_j^e)\\
&=I(US_j^e; A_j| T_j^e)\\
& = I(U; A_j| T_j^e) + I(S_j^e; A_j| T_j^e U)\end{aligned}$$ We similarly have $$I(U; B_j|T_j^e S_j^e) = I(U; B_j| S_j^e) + I(T_j^e; B_j| S_j^e U).$$ Therefore, $$\begin{aligned}
\chi_{O_1} \geq & \sum_{j \in J_{AB}} \Big[ I(U; A_j; S_j^e |T_j^e ) \\
& \qquad +\frac 12I(S_j^e; A_j| T_j^e U) + \frac12 I(T_j^e; B_j| S_j^e U) \Big]. \end{aligned}$$ Next, using and $I(S_j^e; A_j|T_j^e)=0$ we find that $I(U; A_j; S_j^e |T_j^e ) = -I(A_j; S_j^e| T_j^eU)$. Putting these together we arrive at $$2\chi_{O_1} \geq \sum_{j \in J_{AB}} \Big[ I(B_j;T_j^e|S_j^e U)- I(A_j; S_j^e|T_j^e U) \Big] := L_{A\to B}.$$ Following similar computations we also obtain $$2\chi_{O_2} \geq \sum_{j \in J_{AC}} \Big[ I(C_j;T_j^e|R_j^e U) - I(A_j; R_j^e|T_j^e U)\Big] := L_{A\to C},$$ and $$2\chi_{O_3} \geq \sum_{j \in J_{BC}} \Big[ I(C_j;S_j^e|R_j^e U)- I(B_j; R_j^e|S_j^e U) \Big] := L_{B\to C}.$$ Hence, $$2\chi_O \geq L_{A\to B} + L_{A\to C} + L_{B\to C}.$$ If $ L_{A\to B} + L_{A\to C} + L_{B\to C} \geq 0$ the proof is complete. Otherwise, from the beginning we could change the order in which the chain rule in is expanded and repeat the same computations. If instead of the order Alice, Bob and Charlie in we expand $I(U; TSR)$ in the reverse order Charlie, Bob and Alice we obtain the inequality $$2\chi_O \geq L_{B\to A} + L_{C\to A} + L_{C\to B}.$$ Now the proof completes once we note that $L_{B\to A} =-L_{A\to B}$ etc.
$\hfill \square$
Tightness {#Appendix_Tightness}
=========
In this appendix we show that the Finner inequalities that we derive in the paper are tight in the following sense.
For any network $\cal{N}$ with parties $\{A_1, \dots, A_n\}$ and sources $\{S_1, \dots, S_m\} $, and arbitrary numbers $0\leq p_j\leq 1$, there exists a fractional independent set $(\eta_1, \dots, \eta_n)$ of $\mathcal N$ and a binary distribution $P_{A_1\dots A_n}\in {\mathcal{N}_\mathcal{L}}$ such that $P_{A_j}(1) =p_j$ for all $j$ and $$P(1, \dots, 1) = \prod_{j=1}^n \left(P_{A_{j}}(1)\right)^{\eta_j}.$$
Let $(\eta_1^*, \dots, \eta_n^*)$ be a fractional independent set of $\mathcal N$ that optimizes the following linear program: $$\begin{aligned}
\max & \quad -\sum_{j=1}^n \eta_j\log p_j \\
\hbox{s.t.} & \quad \sum_{j: i\to j} \eta_j\le 1 \quad \forall i \qquad \\
& \quad \eta_j\ge 0 \quad\quad \forall j.\end{aligned}$$ Consider the dual of this linear program: $$\begin{aligned}
\min & \quad \sum_{i=1}^m c_i \\
\hbox{s.t.} & \quad \sum_{i:i\to j} c_i\ge -\log p_j \quad \forall j\\
& \quad c_i\ge 0 \quad \forall i. \end{aligned}$$ Let $(c^*_1, \dots, c^*_m)$ be an optimal solution of this dual program is $c^*$. Then by the strong duality of linear programs we have $$\begin{aligned}
\label{eq:SDLP}
\sum_i c_i^* = -\sum_j \eta_j^* \log p_j.\end{aligned}$$ Let us split the set of parties in terms of the constraints of the dual linear program: $$\begin{aligned}
E&= \Big\{j\,\Big|\, \sum_{i: i\to j} c_i^{*}>-\log p_j\Big\},\\
F&= \Big\{j\,\Big|\, \sum_{i: i\to j} c_i^{*} = -\log p_j\Big\}.\end{aligned}$$ For any $j\in E$ and any source $i$ connected to it (i.e., with $i\to j$) pick some $0\leq d^{(j)}_i\leq c_i$ such that $$\sum_{i: i\to j} d^{(j)}_i = -\log p_j.$$ Note that since for $j\in E$ we have $ \sum_{i: i\to j} c_i^{*}>-\log p_j$, such $d^{(j)}_i$’s exist.
Define independent Bernoulli random variables $(X_1, \dots, X_m)$ by $$P(X_i=1) = 2^{-c^*_i}.$$ Also for any $j\in E$ and source $i$ connected to it define the binary random variable $Y^{(j)}_{i}$ by $$\begin{aligned}
P\big(Y_{i}^{(j)}=1\big| X_{i}=1\big)=1,\\
P\big(Y_{i}^{(j)}=1\big)= 2^{-d_i^{(j)}}.\end{aligned}$$ Observe that such a random variable $Y_{i}^{(j)}$ exists since $d_i^{(j)}\leq c_i$, and that $Y_i^{(j)}$ can be computed given $X_i$ and independent of the rest of random variables.
Now suppose that the $i$-th source distributes $X_i$. Then party $j\in F$ outputs $$A_j = \prod_{i: i\to j} X_j,$$ and party $j\in E$ outputs $$A_j = \prod_{i:i\to j} Y_i^{(j)}.$$ We emphasis once again that $Y_i^{(j)}$ can be computed locally by the $j$-th party having access to $X_i$. Then by definition the resulting joint distribution $P_{A_1\dots A_n}$ belongs to ${\mathcal{N}_\mathcal{L}}$. Also, for every $j\in E$ we have $$\begin{aligned}
P(A_j=1) = \prod_{i:i\to j} P\big(Y_i^{(j)}=1\big) =\prod_{i:i\to j} 2^{-d_i^{(j)}}
= p_j. \end{aligned}$$ Similarly for every $j\in F$ we have $P(A_j=1)= p_j$. Next we compute $P(1, \dots, 1)$. Note that for every $i$ there exists some $j_i\in F$ with $i\to j$ since otherwise we can decrease $c_i^*$ and improve the objective value of the dual linear program. Then $(A_1, \dots, A_n) = (1, \dots, 1)$, and in particular $A_{j_i}=1$ only if $X_i=1$. On the other hand, by definitions if $X_i=1$ then $Y_i^{(j)}=1$ for all $j\in E$. We conclude that $(A_1, \dots, A_n) = (1, \dots, 1)$ is equivalent to $X_i=1$ for all $i$. Therefore, $$\begin{aligned}
P(1, \dots, 1) & = \prod _i P(X_i=1) = \prod_i 2^{-c_i^*} = \prod_j p_j^{\eta_j^*},\end{aligned}$$ where we used .
[^1]: Note that here we show that the set is not star convex with respect to the identity, which implies that the set is not star convex in general (invoking symmetry arguments and the fact that if a set is star convex with respect to two points, then it is star convex with respect to any points between those two).
[^2]: Note that here we do not use the classical definition of (fractional) independent sets in hypergraph theory.
[^3]: Note that this model corresponds essentially to the generalized probabilistic theory of “boxworld” [@barrett], except for the fact that boxworld allows for certain multipartite effects (i.e., measurements) that are not wirings [@short_barrett].
[^4]: The so-called “W” distribution, $P_W =( \delta_{001}+\delta_{010}+\delta_{100})/3$, cannot be done with quantum resources, which can be proven using the inflation technique of [@wolfe].
|
---
abstract: 'Liquid droplets sliding along solid surfaces are a frequently observed phenomenon in nature, e.g., raindrops on a leaf, and in everyday situations, e.g., drops of water in a drinking glass. To model this situation, we use a phase field approach. The bulk model is given by the thermodynamically consistent Cahn–Hilliard Navier–Stokes model from \[Abels et al., Math. Mod. Meth. Appl. Sc., 22(3), 2012\]. To model the contact line dynamics we apply the generalized Navier boundary condition for the fluid and the dynamically advected boundary contact angle condition for the phase field as derived in \[Qian et al., J. Fluid Mech., 564, 2006\]. In recent years several schemes were proposed to solve this model numerically. While they widely differ in terms of complexity, they all fulfill certain basic properties when it comes to thermodynamic consistency. However, an accurate comparison of the influence of the schemes on the moving contact line is rarely found. Therefore, we thoughtfully compare the quality of the numerical results obtained with three different schemes and two different bulk energy potentials. Especially, we discuss the influence of the different schemes on the apparent contact angles of a sliding droplet.'
address:
- |
Process Dynamics and Operations Group, Technische Universität Berlin,\
10623 Berlin, Germany
- |
Center for Mathematical Sciences, Technische Universität München,\
85748 Garching bei München, Germany
author:
- Henning Bonart
- Christian Kahle
- 'Jens-Uwe Repke'
title: 'Comparison of Energy Stable Simulation of Moving Contact Line Problems using a Thermodynamically Consistent Cahn–Hilliard Navier–Stokes Model '
---
Multiphase flows ,Drop phenomena ,Contact line dynamics ,Phase field modeling 35Q30 ,35Q35 ,76D05 ,76M10 ,76T99
Introduction {#sec:I}
============
Liquid droplets sliding along solid surfaces are a frequently observed phenomenon in nature, e.g., raindrops on a leaf, and in everyday situations, e.g., drops of water in a drinking glass. Furthermore, sliding droplets (and consequently the suppression of those) are crucial in many industrial applications such as coating or painting and separation or reaction processes involving multiple phases and thin liquid films. The position where the interface between the sliding droplet and the surrounding fluid intersects the solid surface is the moving contact line (or contact point if a two dimensional problem is observed). For details about liquids on surfaces and moving contact lines see the reviews [@Bonn2009; @Snoeijer2013] and the references therein. In a continuum approach, applying the common no-slip boundary condition at the solid surface close to the contact line leads to a non-physical, logarithmically diverging energy dissipation. One possibility to circumvent this difficulty is the coupling of the incompressible Navier–Stokes equations with the Cahn–Hilliard equation [@Jacqmin2000]. This phase field method models the interface between the fluids with a diffuse interface of positive thickness and describes the distribution of the different fluids by a smooth indicator function. Especially, the Cahn–Hilliard equation allows the contact line to move naturally on the solid surface due to a diffusive flux across the interface, which is driven by the gradient of the chemical potential. Furthermore, the phase field method is able to calculate topological changes like breakup of droplets or merging interfaces [@Anderson1998]. For example in experiments by [@Carlson2012; @Eddi2013], it is found that during the rapid spreading of a droplet the contact angle can differ from the equilibrium angle given by Young’s equation. To allow for nonequilibrium contact angles, [@Jacqmin2000] proposes a relaxation of the static contact angle boundary condition, see , and [@2006_QianWangShen_Variational_MovingContactLine__BoundaryConditons] extends this approach to include the slip at the contact line stemming from the uncompensated Young stress.
In [@AbelsGarckeGruen_CHNSmodell] a thermodynamically consistent Cahn–Hilliard Navier–Stokes phase field model is proposed to describe the dynamics of the two phases in the bulk domain. It is valid also for different densities of the involved fluids, but specific contact line dynamics are not included. Recently, several numerical schemes for solving this system have been proposed, see for example, [@GarckeHinzeKahle_CHNS_AGG_linearStableTimeDisc; @Gruen_Klingbeil_CHNS_AGG_numeric; @GruenMetzger__CHNS_decoupled; @Aland__time_integration_for_diffuse_interface; @Tierra_Splitting_CHNS]. All these schemes are thermodynamically consistent in the sense, that they mimic the energy law from [@AbelsGarckeGruen_CHNSmodell] in the time discrete or even in the fully discrete setting. They range from fully coupled and nonlinear to decoupled and linear, where decoupled means, that the Navier–Stokes and the Cahn–Hilliard equations are solved sequentially.
These schemes are extended to the Cahn–Hilliard Navier–Stokes system with moving contact lines in various papers. Here the concepts from the aforementioned papers for the discretization of the bulk equations are straightforwardly applied. For the case of equal densities, schemes are proposed, e.g., in [@2011-HeGlowinskiWang-LeastSquaresCHNSMCL; @2012-GaoWang-gradientStableSchemePhaseFieldMCL; @2015-ShenYangYu-EnergyStableSchemesForCHMCL-Stabilization; @AlandChen__MovingContactLine] and for the case of different densities in [@YuYang_MovingContactLine_diffDensities; @GruenMetzger__CHNS_decoupled]. The model from [@AbelsGarckeGruen_CHNSmodell] contains an additional flux term in the momentum equation, that renders the model thermodynamically consistent. This term is often neglected, see e.g., [@ShenYang_CHNS_DingSpelt_Consistent]. For the resulting model several discretization schemes are proposed and we refer to the references in [@YuYang_MovingContactLine_diffDensities] for details. In all these simulations involving moving contact lines a polynomial bulk energy potential is applied. In contrast, we include a double obstacle potential, which is subsequently relaxed, see [@HintermuellerHinzeTber]. In [@GarckeHinzeKahle_CHNS_AGG_linearStableTimeDisc; @2017_SPP1506_book_AlaHKN__CoparativeSurfactants] in a numerical benchmark setting the results with this kind of energy are typically closer to sharp interface numeric than with the polynomial potential.
To prepare future research on the passive control of droplets sliding on structured or chemically patterned surfaces, we extend the work of [@GarckeHinzeKahle_CHNS_AGG_linearStableTimeDisc] in this paper to the case of moving contact line dynamics and compare the numerical results with the corresponding decoupled scheme from, e.g., [@GruenMetzger__CHNS_decoupled] and a fully linear scheme, that both uses decoupling and stabilization as in [@2015-ShenYangYu-EnergyStableSchemesForCHMCL-Stabilization]. We test both the polynomially and the relaxed double-obstacle bulk energy potential, so that in total we compare six different combinations of bulk energy potentials and solution schemes.
The remainder of the paper is organized as follows. In the second part of the introduction, , we introduce the continuous model as well as the bulk energy potentials and the contact line energies. Afterwards, we derive a weak formulation in and the numerical schemes in . In we compare the different combinations at first in the bulk without any contact line. Finally, we compare simulation results of sliding droplets on inclined surfaces to investigate the accuracy and efficiency of the linearization and decoupling strategies as well as the bulk energy potentials for moving contact line problems in . We conclude our work in .
Model {#ssec:model}
-----
In the fluid domain we consider the thermodynamically consistent model for the simulation of two-phase flow presented in [@AbelsGarckeGruen_CHNSmodell], in the variant for nonlinear density functions proposed in [@AbelsBreit_weakSolution_nonNewtonian_DifferentDensities Eq. 1.10]. To model the contact line dynamics we use generalized Navier boundary conditions for the velocity field together with dynamically advected boundary conditions for the phase field as proposed in [@2006_QianWangShen_Variational_MovingContactLine__BoundaryConditons].
In strong form the model reads as follows. Let $\Omega \subset \mathbb{R}^{d}$ with $d \in\{2,3\}$ denote an open, polygonally/polyhedrally bounded Lipschitz domain and $I = (0,T]$ with $0<T<\infty$ denote a time interval. The outer unit normal on $\partial\Omega$ is $\nu_\Omega$. At time $t \in I$ the primal variables are given by the velocity field $v$, the pressure field $p$, the phase field $\varphi$ and the chemical potential $\mu$. They satisfy the following system of equations $$\begin{aligned}
\rho\partial_t v + ((\rho v + J)\cdot\nabla) v + R\frac{v}{2}
-\mbox{div}\left(2\eta Dv\right) + \nabla p &= -\varphi\nabla \mu + \rho g
&& \mbox{ in } \Omega, \label{eq:M:1_NS1}\\
-\mbox{div}(v) &= 0
&& \mbox{ in } \Omega,\label{eq:M:2_NS2}\\
\partial_t \varphi + v \cdot\nabla \varphi - b\Delta \mu &= 0
&& \mbox{ in } \Omega,\label{eq:M:3_CH1}\\
-\sigma\epsilon\Delta \varphi + \frac{\sigma}{\epsilon}W^\prime(\varphi) &= \mu
&& \mbox{ in } \Omega,\label{eq:M:4_CH2}\\
v \cdot \nu_\Omega &= 0
&& \mbox{ on } \partial\Omega, \label{eq:M:5_NS_BC_1}\\
[2\eta Dv \nu_\Omega + l(\varphi)v_{tan} - L(\varphi)\nabla \varphi]
\times \nu_\Omega &= 0
&& \mbox{ on } \partial\Omega, \label{eq:M:6_NS_BC}\\
rB + L(\varphi)&=0
&& \mbox{ on } \partial\Omega,\label{eq:M:7_CH_BC}\\
\nabla \mu \cdot\nu_\Omega &= 0
&&\mbox{ on } \partial\Omega,
\label{eq:M:8_mu_neumann}\end{aligned}$$ where we abbreviate $J := -b\frac{\partial\rho}{\partial\varphi}\nabla \mu$, $R:= -b\nabla\frac{\partial\rho}{\partial\varphi}\cdot\nabla \mu$, $B := \partial_t\varphi + v \cdot \nabla \varphi$, $ L:= \sigma\epsilon \nabla\varphi\cdot\nu_\Omega + \gamma^\prime(\varphi)$. The gravitational acceleration is denoted by $g$ and we abbreviate $2Dv := \nabla v + (\nabla v)^t$. The function $W(\varphi)$ denotes a dimensionless potential of double-well type, with two strict minima at $\pm 1$. We refer to for a discussion of possible choices for $W$. We formulate \[eq:M:1\_NS1\] with a shifted pressure variable $p = p^{phys} - \mu\varphi$, where $p^{phys}$ denotes the physical pressure.
The contact line energy is denoted by $\gamma$, see . The strictly positive, constant parameters for the equations in $\Omega$ are given by the mobility $b >0$, the scaled surface tension $\sigma$, see , and the interfacial thickness parameter $\epsilon$. The constant mobility is used for simplicity but the following is also valid for mobilities that depend on $\varphi$.
The (nonlinear) density function is denoted by $\rho \equiv \rho(\varphi) > 0$ and satisfies $\rho(-1) = \rho_1$ and $\rho(1) = \rho_2$, with $\rho_2 > \rho_1 > 0$ denoting the constant densities of the two involved fluids. The (nonlinear) viscosity function is $\eta \equiv \eta(\varphi) > 0$ and satisfies $\eta(-1) =
\eta_1$ and $\eta(1) = \eta_2$, with $\eta_1,\eta_2$ denoting the viscosities of the involved fluids.
In general there is no quantitative upper bound available for $\varphi$ and thus in particular $|\varphi| > 1$ is possible. Thus a linear relation between $\varphi$ and $\rho$ can lead to negative densities in practice. This might appear for especially large density ratios, compare for example [@GruenMetzger__CHNS_decoupled Rem. 4.1]. Note, that $\varphi$ can be proven to be bounded in $L^\infty$ if $W^{\prime\prime}$ is uniformly bounded, see e.g. [@1995_CaffarelliMulder_LinftyForCahnHilliard].
It is a common approach to cut $\varphi$ when inserting it into the linear function for $\rho$, see e.g. [@YuYang_MovingContactLine_diffDensities]. This leads to a nonsmooth relation between $\varphi$ and $\rho$. However, as we require differentiability of $\rho$ to define $R$ and $J$ this is not admissible here. A second approach is to clip $\rho$ of at some positive value, see e.g. [@GruenMetzger__CHNS_decoupled; @Gruen_convergence_stable_scheme_CHNS_AGG]. This leads to a uniform bound on $\rho$ based on the Atwood number ${\operatorname{\mbox{\textit{At}}}}= \frac{\rho_2 - \rho_1}{\rho_2 + \rho_1}$. Here we use the latter approach and define $\rho$ as the following smooth, monotone and strictly positive function $$\begin{aligned}
\rho(\varphi) =
\begin{cases}
\frac{1}{4}\rho_1 & \mbox{ if } \varphi \leq -{\operatorname{\mbox{\textit{At}}}}^{-1},\\
\frac{1}{\rho_1}\left( \frac{\rho_2-\rho_1}{2}\varphi + \frac{\rho_2+\rho_1}{2} \right)^2 + \frac{1}{4}\rho_1
& \mbox{ if } -{\operatorname{\mbox{\textit{At}}}}^{-1} < \varphi < -1-\frac{ \rho_1}{\rho_2-\rho_1}, \\
\frac{ \rho_2- \rho_1}{2}\varphi + \frac{\rho_2+\rho_1}{2}
& \mbox{ if } -1-\frac{ \rho_1}{\rho_2-\rho_1}
\leq \varphi \leq
1+\frac{\rho_1}{\rho_2-\rho_1},\\ -\frac{1}{\rho_1}\left( \frac{\rho_2-\rho_1}{2}\varphi - \frac{\rho_2+\rho_1}{2} \right)^2 + \rho_2 + \frac{3}{4}\rho_1
& \mbox{ if } 1+\frac{\rho_1}{\rho_2-\rho_1} < \varphi < {\operatorname{\mbox{\textit{At}}}}^{-1},\\
\rho_2 + \frac{3}{4}\rho_1 & \mbox{ if } {\operatorname{\mbox{\textit{At}}}}^{-1} \leq \varphi.
\end{cases}\label{eqn:cutoff_phi}\end{aligned}$$ For a discussion we refer to [@Gruen_convergence_stable_scheme_CHNS_AGG Rem. 2.1]. The nonlinear viscosity $\eta(\varphi)$ can be defined analogously.
We note, that the total mass $\int_\Omega \rho(\varphi){\,dx}$ is only conserved if $\rho(\varphi)$ is a linear function on the (a-priori unknown) image of $\varphi$, see e.g., [@GarckeHinzeKahle_CHNS_AGG_linearStableTimeDisc Rem. 1], while $\int_\Omega \varphi{\,dx}$ is a conserved quantity,
As boundary data we use generalized Navier boundary conditions for the velocity field and dynamically advected contact angle boundary conditions for the two-phase equation, see [@2006_QianWangShen_Variational_MovingContactLine__BoundaryConditons Eq. 4.4, Eq. 4.5]. Here $\gamma$ denotes the fluid-solid interfacial free energy, see [@2006_QianWangShen_Variational_MovingContactLine__BoundaryConditons Sec. 4], $l(\varphi)$ is a slip coefficient for the generalized Navier boundary condition applied to the tangential part of the velocity $v_{tan} := v - (v\cdot \nu_\Omega)\nu_\Omega$, while $L(\varphi)\nabla \varphi \times \nu_\Omega$ is the uncompensated Young stress and $L$ is the chemical potential at the solid surface. The static contact angle is denoted by $\theta_s$ and $r \geq 0$ is a phenological parameter allowing for nonequilibrium at the contact line. For $r\equiv 0$ \[eq:M:7\_CH\_BC\] reduces to $\sigma\epsilon \nabla\varphi\cdot\nu_\Omega = - \gamma^\prime(\varphi)$, which means, that a static contact angle at the interface is assumed. Furthermore, for $\gamma^\prime(\varphi) \equiv 0$ (or rather $\theta_s \equiv 90\degree$, see ), \[eq:M:7\_CH\_BC\] further simplifies to $\nabla\varphi\cdot\nu_\Omega = 0$, which is a no-flux condition for $\varphi$ at the solid surface. The no-slip condition for $v$ is obtained from \[eq:M:6\_NS\_BC\] by $L\equiv 0$ and $l \rightarrow \infty$ (or rather the slip length $l_s \equiv 0$, see ).
Concerning the existence of solutions to \[eq:M:1\_NS1,eq:M:2\_NS2,eq:M:3\_CH1,eq:M:4\_CH2,eq:M:5\_NS\_BC\_1,eq:M:8\_mu\_neumann\] together with no-slip for $v$ and a homogeneous Neumann (or no-flux) boundary condition for $\varphi$ as well as with different assumptions on $b$ and $W$, we refer to [@AbelsDepnerGarcke_CHNS_AGG_exSol; @AbelsDepnerGarcke_CHNS_AGG_exSol_degMob; @AbelsBreit_weakSolution_nonNewtonian_DifferentDensities; @Gruen_convergence_stable_scheme_CHNS_AGG]. For the boundary conditions considered here we are not aware of such results, but refer to [@2016-GalGrasselliMiranville-CHNSMCL-EqualDensityExSol] for the Cahn–Hilliard Navier–Stokes system with equal densities, to [@2017_ColliGilardiSprekels_CH_with_dynamicBoundary] for analytical results for the Cahn–Hilliard system with dynamic boundary conditions, and to [@GruenMetzger__CHNS_decoupled] for a Cahn–Hilliard Navier–Stokes model with dynamical contact angle condition, but no-slip condition for the Navier–Stokes equation. Concerning sharp interface limits, we refer to [@AbelsGarckeGruen_CHNSmodell] for the bulk model with homogeneous boundary conditions. Sharp interface analysis for the model with equal densities including contact line dynamics is available in [@2018-XuDiHu-SharpInterfaceLimit-NavierSlipBoundary].
For \[eq:M:1\_NS1,eq:M:2\_NS2,eq:M:3\_CH1,eq:M:4\_CH2,eq:M:5\_NS\_BC\_1,eq:M:8\_mu\_neumann\] together with no-slip for $v$ and no-flux for $\varphi$, several thermodynamically consistent discretization schemes were proposed in the last years. Here, we refer to [@Tierra_Splitting_CHNS; @Gruen_Klingbeil_CHNS_AGG_numeric; @GarckeHinzeKahle_CHNS_AGG_linearStableTimeDisc]. Especially in [@GarckeHinzeKahle_CHNS_AGG_linearStableTimeDisc] the influence of spatial adaptivity on the fully discrete energy law is discussed. We further refer to [@Aland__time_integration_for_diffuse_interface], where the benefit of using fully coupled schemes is shown numerically, and to [@GonzalesTierra_linearSchemes_CH] for an extensive discussion of several discretization schemes for the bulk energy potential $W^{poly}$. For the full model \[eq:M:1\_NS1\]–\[eq:M:8\_mu\_neumann\] thermodynamically consistent schemes are for example proposed in [@AlandChen__MovingContactLine] for the case of constant density, and in [@YuYang_MovingContactLine_diffDensities] for the general case. The case with no-slip boundary condition for $v$ and dynamically advected boundary condition for $\varphi$ is numerically and analytically considered in [@GruenMetzger__CHNS_decoupled].
\[rm:M:freeEnergies\] Throughout this work we consider polynomially bounded potentials for $W$. To state the precise assumptions we split $W = W_+ + W_-$ with $W_+$ denoting the convex part of $W$ and $W_-$ denoting the concave part. We assume that $W:\mathbb R \to \mathbb R$ is continuously differentiable and that $W$ and its derivatives $W_+^\prime$ and $W_-^\prime$ are polynomially bounded, i.e., there exists $C>0 $ such that $$\begin{aligned}
|W(\varphi)| \leq C(1+|\varphi|^4),\quad
|W_+^\prime(\varphi)| \leq C(1+|\varphi|^3), \quad
|W_-^\prime(\varphi)| \leq C(1+|\varphi|^3).\end{aligned}$$ Note, that these bounds on the polynomial degree might be relaxed, see [@GarckeHinzeKahle_CHNS_AGG_linearStableTimeDisc (A3)], and that these assumptions are used to show the existence of discrete solutions.
These assumptions are for example fulfilled by the commonly used polynomial potential $$\begin{aligned}
W^{poly}(\varphi) := \frac{1}{4}(1-\varphi^2)^2,
\quad
W^{poly_2}(\varphi) :=
\begin{cases}
\frac{1}{4}(1-\varphi^2)^2 & \mbox{if } |\varphi|\leq 1,\\
(|\varphi|-1)^2 & \mbox{else,}
\end{cases} \end{aligned}$$ where $W^{poly_2}$ is a modification of $W^{poly}$ that guarantees an $L^\infty$ bound on $\varphi$, see [@1995_CaffarelliMulder_LinftyForCahnHilliard].
Another potential that fulfills the assumptions is $$\begin{aligned}
W^{{s}}(\varphi) := \frac{1}{2}\left( 1 - (\xi\varphi)^2
+ {s}\lambda(\xi \varphi)^2 \right) + \theta,\end{aligned}$$ where $\lambda(x) := \max(0,x-1) + \min(0,x+1) $, $\theta := \frac{1}{2({s}-1)}$ and $\xi := \frac{{s}}{{s}-1}$ are chosen such that $W(\pm1) \equiv 0$ are the two minima of $W^{{s}}$. Here ${s}\gg1$ is a penalization parameter. It appears as Moreau–Yosida relaxation of the double obstacle potential $W^{\infty}$, see [@BloweyElliott_I; @HintermuellerHinzeTber]. In a synthetic rising bubble benchmark, [@Hysing_Turek_quantitative_benchmark_computations_of_two_dimensional_bubble_dynamics], our results with this potential are typically closer to the results from sharp interface methods than with the potential $W^{poly}$, see [@GarckeHinzeKahle_CHNS_AGG_linearStableTimeDisc Tab. 1].
In the following, whenever we use the letter $W$, we mean any of the three mentioned bulk energy potentials.
In preparation of later results, we state the splittings of the potentials $W$ into $W(\varphi)= W_+(\varphi) + W_-(\varphi)$. These are $$\begin{aligned}
W^{poly}_+(\varphi) &= \frac{1}{4}\varphi^4 - \frac{1}{4},
& W^{poly}_-(\varphi) &= \frac{1}{2}(1- \varphi^2),\\
W^{poly_2}_+(\varphi) &=
\begin{cases}
\frac{1}{4}\varphi^4 - \frac{1}{4} & \mbox{if } |\varphi| \leq 1,\\
(|\varphi|-1)^2-\frac{1}{2}(1-\varphi^2) & \mbox{if } |\varphi| > 1,
\end{cases}
&
W^{poly_2}_-(\varphi) &= \frac{1}{2}(1-\varphi^2),\\
W^{{s}}_+(\varphi) &= \frac{{s}}{2}\lambda(\xi \varphi)^2 + \theta,
& W^{{s}}_-(\varphi) &= \frac{1}{2}(1-(\xi\varphi)^2).
\end{aligned}$$ These splittings are not unique, and we refer for example to [@2014-WuZwietenZee-StabilizedSecondOrderConvecSplittingCHmodels] for an alternative splitting of $W^{poly_2}$. We further refer to [@GonzalesTierra_linearSchemes_CH] for a discussion on the dissipation that is introduced by the convex-concave splitting and also for an elaborated discussion on the dissipation that in general is introduced by splitting $W$. In our numerical tests, splittings that have a quadratic convex part and thus give linear systems, typically lead to broader interfaces during the simulation and require smaller time steps to prevent this effect. Thus it is favorable to use non-linear systems as obtained by the proposed splittings above.
To define the scaled surface tension $\sigma$ we introduce the constant $c_W$ as $c_W^{-1} = \int_{-\infty}^\infty 2W(\Phi_0(z))\,dz
= \int_{-\infty}^\infty (\partial_z\Phi_0(z))^2\,dz$, where $\Phi_0$ denotes the first order approximation of $\varphi$ depending on $W$. It satisfies $\Phi_0(z)_{zz} = W^\prime(\Phi_0(z))$, see [@AbelsGarckeGruen_CHNSmodell Sec. 4.3.3]. Then $\sigma = c_W \sigma_{12}$, where $\sigma_{12}$ denotes the physical value of the surface tension between phase $1$ and phase $2$. As the dynamics of the diffuse model depend on the particular form of $W$, this scaling is necessary to guarantee that the same sharp interface dynamic is approximated independently of $W$. Using $W^{poly}$ and $W^{poly_2}$ it holds $\Phi_0(z) = \tanh(z/\sqrt{2})$ and $c_W = \frac{3}{2\sqrt 2}$. For $W^{s}$ one obtains by elementary calculation $$\begin{aligned}
\Phi_0(z) =
\begin{cases}
-\Phi_0(-z) & \mbox{if } z<0,\\
\sqrt{\xi}^{-1}\sin(\xi z) & \mbox{if } 0\leq z \leq z_0 := \xi^{-1}\arctan(\sqrt{s-1}),\\
1-s^{-1}\exp(-\xi\sqrt{s-1}(z-z_0)) & \mbox{if } z > z_0,
\end{cases}\end{aligned}$$ and $$\begin{aligned}
c_W^{-1} =
(1-{s}^{-2})\arctan(\sqrt{{s}-1})
+{s}^{-2}({s}+2)\sqrt{{s}-1}.\end{aligned}$$ For ${s}\to\infty$ we recover the well-known scaling $c_W = \frac{2}{\pi}$ for the double-obstacle potential.
\[rm:M:ContactEnergy\] The basic formula to derive the contact line energy is given by Young’s law, namely $$\begin{aligned}
\sigma_{s1}-\sigma_{s2} = \sigma_{12}\cos\theta_s.
\end{aligned}$$ Here $\sigma_{s1}$ and $\sigma_{s2}$ denote the physical surface tensions between phase 1 ($\varphi=-1$) and the solid ($\sigma_{s1}$) and phase 2 ($\varphi=1$) and the solid ($\sigma_{s2}$). Further $\sigma_{12}$ denotes the surface tension between phase 1 and phase 2 and $\theta_s$ denotes the static equilibrium contact angle between the solid and the interface and is measured in phase 2.
We use the ansatz $$\begin{aligned}
\gamma(\varphi) :=
\frac{\sigma_{s1}+\sigma_{s2}}{2}
-\sigma_{12}\cos\theta_s\vartheta(\varphi)
\end{aligned}$$ and choose $\vartheta(\varphi)$ to fulfill $$\begin{aligned}
\gamma(-1) = \sigma_{s1},
\quad \gamma(0) = \frac{\sigma_{s1}+\sigma_{s2}}{2},
\quad \gamma(1) = \sigma_{s2},
\quad \gamma^\prime(\pm1) = 0.
\end{aligned}$$ In particular it holds, that $\vartheta(-1) = -\frac{1}{2}$ and $\vartheta(1)=\frac{1}{2}$. Here, the unscaled value of the surface tension appears as can be shown by matched asymptotic expansions, see [@AbelsGarckeGruen_CHNSmodell Sec. 4.3.4].
Common choices for $\vartheta$ contain the sine function $\vartheta^{\sin}(\varphi):=
\frac{1}{2}\sin(\frac{\pi}{2}\varphi)$, for example proposed in [@2006_QianWangShen_Variational_MovingContactLine__BoundaryConditons Sec. 4], or a cubic polynomial $\vartheta^{poly}(\varphi) = \frac{1}{4}(3\varphi-\varphi^3)$, for example proposed in [@2018-XuDiHu-SharpInterfaceLimit-NavierSlipBoundary]. An alternative is given in [@2007_DingSpelt_WettingCondition_inDiffuseInterface_ContactLineMotion]. Here the assumption of equipartition of energy, i.e., $\frac{\epsilon}{2}|\nabla \varphi|^2 \approx \frac{1}{\epsilon}W(\varphi)$, is used to derive $(\vartheta^{W})^\prime(\varphi) = c_W\sqrt{2W(\varphi)}$. Finally, we state a contact line energy, that is the sum of a convex and a concave function namely $\vartheta^{cc}(\varphi) = \vartheta^{cc}_+(\varphi) + \vartheta^{cc}_-(\varphi)$ with $$\begin{aligned}
\vartheta^{cc}_+(\varphi) &=
\begin{cases}
-\frac{1}{2} & \mbox{if } \varphi\leq-1,\\
\frac{1}{2}(\varphi+1)^2 -\frac{1}{2} & \mbox{if } \varphi \in (-1,0),\\
\varphi& \mbox{if } \varphi \geq 0,
\end{cases}
&
\vartheta^{cc}_-(\varphi) &=
\begin{cases}
0 & \mbox{if } \varphi \leq 0,\\
-\frac{1}{2}\varphi^2 & \mbox{if } \varphi \in (0,1),\\
\frac{1}{2}-\varphi & \mbox{if } \varphi \geq 1.
\end{cases}
\end{aligned}$$ Here $\vartheta^{cc}_+$ is convex and $\vartheta^{cc}_-$ is concave and $\vartheta^{cc} \in C^{1,1}(\mathbb R)$ with $\vartheta^{\prime\prime} \in
L^{\infty}(\mathbb R)$.
Note, that for any $\vartheta$ that has a bounded second derivative, we can define a convex-concave splitting via $$\begin{aligned}
\vartheta_+(\varphi) &= \vartheta(\varphi) + \frac{1}{2}\max_{\phi\in\mathbb R}(\vartheta^{\prime\prime}(\phi))\varphi^2,
&
\vartheta_-(\varphi) &= - \frac{1}{2}\max_{\phi\in\mathbb R}(\vartheta^{\prime\prime}(\phi))\varphi^2,
\end{aligned}$$ compare [@2018_BackofenVoigt_ConvexitySplittingPhaseField]. This is very similar to the stabilization approach, proposed for example in [@2015-ShenYangYu-EnergyStableSchemesForCHMCL-Stabilization], that essentially resembles one of Eyre’s linear schemes [@GonzalesTierra_linearSchemes_CH]. In the following we always assume a convex-concave splitting of $\gamma$. This approach can also be used for the potential $W$.
To the best of our knowledge, there is no consent yet which combinations of bulk energy potential and contact line energy are most appropriate from both a physical and numerical point of view. From an analytical point of view, all combinations are reasonable that lead to the correct sharp interface limit, see [@2018-XuDiHu-SharpInterfaceLimit-NavierSlipBoundary] for results on formal sharp interface asymptotics. Here, the authors use the combination of $W^{poly}$ and $\vartheta^{poly}$. However, this topic is subject to future work. Further note, that using the notation from [@2018-XuDiHu-SharpInterfaceLimit-NavierSlipBoundary] we are in the setting $L_d = \mathcal O(\epsilon)$, and $V_s = \mathcal O(1)$.
The weak formulation {#sec:F}
====================
We next derive the weak formulation that is the basis for our numerical scheme proposed in \[sec:S\]. We assume sufficient regularity of all appearing functions. Multiplying \[eq:M:3\_CH1\] with $\frac{\partial\rho}{\partial\varphi}$ we observe $$\begin{aligned}
\partial_t \rho + \mbox{div}(\rho v+J) = R.
\label{eq:M:MassConservationLaw}\end{aligned}$$ Note that if $\rho$ is a nonlinear function $R \neq 0$ holds and thus mass conservation can be violated as soon as a nonlinear function for $\rho$ is used to guarantee $\rho>0$. Note that the conservation of $\varphi$ is not affected. Using \[eq:M:MassConservationLaw\] the momentum equation \[eq:M:1\_NS1\] can equivalently be written as $$\begin{aligned}
\partial_t(\rho v)
+ \mbox{div}\left(v \otimes (\rho v + J)\right)
- R\frac{v}{2}
-\mbox{div}\left(2\eta Dv\right)
+\nabla p &=- \varphi\nabla\mu + \rho g,
\label{eq:M:1.2_NS1.2}\end{aligned}$$ see [@AbelsBreit_weakSolution_nonNewtonian_DifferentDensities Eq. (1.12)]. We stress that this reformulation is independent of the actual boundary condition.
To define the weak formulation we multiply both \[eq:M:1\_NS1\] and \[eq:M:1.2\_NS1.2\] by a solenoidal test function $\frac{1}{2}w$ that satisfies $w|_{\partial\Omega}\cdot \nu_\Omega = 0$ and sum up the equations to achieve $$\begin{aligned}
\frac{1}{2}\int_\Omega(\rho \partial_t v + \partial_t(\rho v)) \cdot w{\,dx}- \int_\Omega \mbox{div}\left(2\eta Dv\right) \cdot w{\,dx}+\int_\Omega (\varphi\nabla \mu-\rho g)\cdot w{\,dx}&
\nonumber\\
+\frac{1}{2} \int_\Omega ((\rho v + J) \cdot \nabla) v \cdot w{\,dx}+\frac{1}{2} \int_\Omega \mbox{div}\left(v \otimes (\rho v + J)\right) \cdot w{\,dx}= 0. &\end{aligned}$$ Using integration by parts together with the boundary conditions $v\cdot \nu_\Omega = 0$ and $\nabla \mu \cdot \nu_\Omega = 0$ we observe $$\begin{aligned}
&\frac{1}{2} \int_\Omega ((\rho v + J)\cdot \nabla) v \cdot w{\,dx}+\frac{1}{2} \int_\Omega \mbox{div}\left(v \otimes (\rho v + J)\right)\cdot w{\,dx}\\
=& \frac{1}{2}\int_\Omega ((\rho v + J)\cdot \nabla) v \cdot w - ((\rho v + J)\nabla) w \cdot v {\,dx}\\
=:& a(\rho v + J,v,w).\end{aligned}$$ Note that $a(\cdot,v,v) = 0$ holds. Using integration by parts for the viscous stress we observe $$\begin{aligned}
- \int_\Omega \mbox{div}\left(2\eta Dv\right)\cdot w{\,dx}=&
\int_\Omega 2\eta Dv : Dw{\,dx}-\int_{\partial\Omega} 2\eta Dv \nu_\Omega\cdot w {\,ds},\\
=&\int_\Omega 2\eta Dv : Dw{\,dx}+\int_{\partial\Omega} (l(\varphi)v_{tan} + rB\nabla
\varphi) \cdot w {\,ds}\end{aligned}$$ where $Dv:Dw := \sum_{ij=1}^n (Dv)_{ij}(Dw)_{ij}$ and we use the boundary conditions \[eq:M:6\_NS\_BC\] and \[eq:M:7\_CH\_BC\].
The weak form of \[eq:M:3\_CH1\]–\[eq:M:4\_CH2\] is derived by the standard procedure. Summarizing the equations, we obtain the following weak form of \[eq:M:1\_NS1\]–\[eq:M:8\_mu\_neumann\]:
Find sufficiently smooth $v,\mu,\varphi$, with $v$ solenoidal, $v\cdot \nu_\Omega = 0$, such that for all $w$, $\psi$, $\phi$, with $w$ solenoidal, the following equations are satisfied: $$\begin{aligned}
\frac{1}{2}\int_\Omega(\rho \partial_t v + \partial_t(\rho v))\cdot w{\,dx}+a(\rho v + J,v,w)
+\int_\Omega 2\eta Dv : Dw{\,dx}&\nonumber\\
+\int_{\partial\Omega} (l(\varphi)v_{tan} + r B(\varphi_t,\varphi,v)\nabla \varphi) \cdot w {\,ds}-\int_\Omega(-\varphi\nabla \mu +\rho g )\cdot w{\,dx}&= 0,
\label{eq:M:weak_1}
\\
\int_\Omega \varphi_t \psi{\,dx}- \int_\Omega \varphi v\cdot \nabla \psi{\,dx}+\int_\Omega b\nabla \mu\cdot\nabla \psi{\,dx}&= 0,
\label{eq:M:weak_2}\\
\int_\Omega
\sigma\epsilon\nabla \varphi\cdot\nabla \phi
+ \frac{\sigma}{\epsilon}W^\prime(\varphi)\phi {\,dx}- \int_\Omega\mu \phi {\,dx}&\nonumber\\
+\int_{\partial\Omega} \left( r B(\varphi_t,\varphi,v) + \gamma^\prime(\varphi)\right)
\phi {\,ds}&= 0.
\label{eq:M:weak_3}\end{aligned}$$
The weak form \[eq:M:weak\_1\]–\[eq:M:weak\_3\] allows us to derive the following energy identity.
Assume there exists a sufficiently smooth solution to \[eq:M:weak\_1\]–\[eq:M:weak\_3\]. Then the following energy identity holds $$\begin{aligned}
\frac{d}{dt}\left(\int_\Omega \frac{1}{2}\rho |v|^2{\,dx}+ \sigma\int_\Omega\frac{\epsilon}{2}|\nabla \varphi|^2 +
\frac{1}{\epsilon}W(\varphi){\,dx}+ \int_{\partial\Omega}\gamma{\,ds}\right)\\
+ \int_\Omega 2\eta |Dv|^2{\,dx}+ \int_\Omega b|\nabla \mu|^2{\,dx}\\
+\int_{\partial\Omega}l(\varphi)|v_{tan}|^2{\,ds}+ r \int_{\partial\Omega} |B(\varphi_t,\varphi,v)|^2{\,ds}= \int_\Omega \rho g \cdot v{\,dx}.
\label{eq:M:EnergyIdent}
\end{aligned}$$ Note that the energy in the system can only increase by the gravitational acceleration.
Use $w \equiv v$, $\Psi \equiv \mu$, and $\Phi \equiv \partial_t \varphi$ as test functions in \[eq:M:weak\_1\]–\[eq:M:weak\_3\] and sum up the resulting equations.
The numerical schemes {#sec:S}
=====================
For a practical implementation in a finite element scheme we introduce a time grid $0 = t_0 < t_1 < \ldots<t_{m-1} < t_m< \ldots <t_M = T$ on $I = [0,T]$. For the sake of notational simplicity let the time grid be equidistant with step size $\tau>0$. We further introduce a triangulation $\mathcal T_h$ of $\overline \Omega$ into cells $T_i$, such that $\mathcal T_h = \bigcup_{i=1}^{N}T_i$ covers $\overline \Omega$ exactly.
On $\mathcal T_h$ we introduce the finite element spaces $$\begin{aligned}
V_1 &:= \{v \in C(\overline\Omega) \,|\, v|_{T_i} \in \mathcal P_1\},\\
V_2 &:= \{v \in C(\overline\Omega)^d \,|\, v|_{T_i} \in (\mathcal P_2)^2,\,
v \cdot \nu_\Omega = 0\},\end{aligned}$$ where $\mathcal P_k$ denotes the space of polynomials of order up to $k$. We use $V_1$ to define discrete approximations $\varphi_h$, $\mu_h$, and $p_h$ of the corresponding continuous variables, and $V_2$ to define the discrete approximation $v_h$ of $v$. This means that we use standard Taylor–Hood elements for the Navier–Stokes part and explicitly denote the pressure variable in the following.
The scheme reads as follows:\
Given $\varphi^{m-1} \in V_1$, $\mu^{m-1} \in V_1$, and $v^{m-1} \in V_2$, find $\varphi^m_h \in V_1$, $\mu^m_h \in V_1$, $p^m_h \in V_1$ and $v^m_h \in V_2$, such that for all $w \in V_2$, $q \in V_1$, $\Phi \in V_1$, and $\Psi \in V_1$ the following equations hold $$\begin{aligned}
\frac{1}{\tau}\left( \frac{\rho^m+\rho^{m-1}}{2} v^m_h -\rho^{m-1}v^{m-1},w\right)\nonumber\\
+ a(\rho^{m-1}v^{m-1} + J^{m-1},v^m_h,w)
+ (2\eta^{m-1}Dv^m_h,Dw) - (\mbox{div} w,p^m_h)\nonumber\\
+ (l(\varphi^{m-1})v^m_{h,tan} + r B^m_h\nabla \varphi^{m-1},w)_{\partial\Omega}\nonumber\\
+( \varphi^{m-1}\nabla \mu^m_h,w) -(g\rho^{m-1},w) &= 0,
\label{eq:S:1_NS_1}\\
-(\mbox{div} v^m_h,q) &= 0, \label{eq:S:2_NS_2}\\
\frac{1}{\tau}(\varphi_h^{m} - \varphi^{m-1},\Psi)
-(\varphi^{m-1}v^m_h,\nabla \Psi)
+(b\nabla \mu^m_h,\nabla \Psi) &= 0 \label{eq:S:3_CH1},\\
\sigma \epsilon(\nabla \varphi^m_h,\nabla \Phi)
+\frac{\sigma}{\epsilon}(W_+^\prime(\varphi^m_h) + W_-^\prime(\varphi^{m-1}),\Phi)
- (\mu^m_h,\Phi) \nonumber\\
+\left( r B^m_h,
\Phi\right)_{\partial\Omega}
+\left( \gamma^\prime_+(\varphi^{m}_h) + \gamma^\prime_-(\varphi^{m-1}),
\Phi\right)_{\partial\Omega}
&= 0,
\label{eq:S:4_CH2}\end{aligned}$$ with $J^{m-1} := -b\frac{\partial \rho}{\partial\varphi}(\varphi^{m-1})\nabla \mu^{m-1}$, $B^m_h:= \left(
\frac{\varphi^m_h-\varphi^{m-1}}{\tau} + v^m_h\cdot \nabla \varphi^{m-1}
\right)$, $\rho^{m-1} := \rho(\varphi^{m-1})$, and $\eta^{m-1} := \eta(\varphi^{m-1})$.
Using Brouwer’s fixed-point theorem one can show the existence of at least one solution following [@GarckeHinzeKahle_CHNS_AGG_linearStableTimeDisc Thm. 2]. The uniqueness stays unclear due to the nonlinearity $\rho^mv^m_h$ in \[eq:S:1\_NS\_1\]. The scheme fulfills a fully discrete variant of the formal energy identity \[eq:M:EnergyIdent\].
\[thm:M:enerInequDisc\] Let $\varphi^m_h \in V^m_1$, $\mu_h^m \in V^m_1$, and $v_h^m \in V_2^m$ denote a solution to \[eq:S:1\_NS\_1\]–\[eq:S:4\_CH2\]. Then the following energy inequality holds $$\begin{aligned}
\frac{1}{\tau}\left(\frac{1}{2}\int_\Omega \rho^m|v^m_h|^2
+\sigma\int_\Omega\frac{\epsilon}{2}|\nabla \varphi^m_h|^2
+ \frac{1}{\epsilon}W(\varphi^m_h){\,dx}+ \int_{\partial\Omega} \gamma(\varphi^m_h){\,ds}\right)\\
+\int_\Omega 2\eta^{m-1}|Dv^m_h|^2 {\,dx}+ b \int_\Omega |\nabla \mu^m_h|^2{\,dx}+\int_{\partial\Omega}l(\varphi^{m-1})|v^m_{h,tan}|^2{\,ds}+ r \int_{\partial\Omega} |B^m_h|^2{\,ds}\\
+ \frac{1}{\tau}\left(
\frac{1}{2}\int_\Omega \rho^{m-1}|v^m_h-v^{m-1}|^2{\,dx}+ \frac{\sigma\epsilon}{2}\int_\Omega |\nabla \varphi^m_h-\nabla \varphi^{m-1}|^2{\,dx}\right)\\
\leq \frac{1}{\tau}\left(\frac{1}{2}\int_\Omega \rho^{m-1}|v^{m-1}|^2
+\sigma\int_\Omega\frac{\epsilon}{2}|\nabla \varphi^{m-1}|^2
+ \frac{1}{\epsilon}W(\varphi^{m-1}){\,dx}+ \int_{\partial\Omega} \gamma(\varphi^{m-1}){\,ds}\right)\\
+\int_\Omega \rho^{m-1} g\cdot v^m_h{\,dx}.
\end{aligned}$$
We use $w \equiv v^m_h$, $q = p^m_h$, $\Psi \equiv \mu^m_h$ and $\Phi \equiv \frac{\varphi^m_h-\varphi^{m-1}}{\tau}$ as test functions in \[eq:S:1\_NS\_1\]–\[eq:S:4\_CH2\] and sum up to obtain $$\begin{aligned}
\frac{1}{\tau}\left(\frac{1}{2}\int_\Omega \rho^m|v^m_h|^2 -
\frac{1}{2}\int_\Omega \rho^{m-1}|v^{m-1}|^2+
\frac{1}{2}\int_\Omega \rho^{m-1}|v^{m}_h - v^{m-1}|^2 {\,dx}\right)\\
+\int_\Omega 2\eta^{m-1}|Dv^m_h|^2 {\,dx}-\int_\Omega \rho^{m-1} g\cdot v^m_h{\,dx}\\
+\int_{\partial\Omega}l(\varphi^{m-1})v^m_{h,tan}\cdot v^m_h{\,ds}+r\int_{\partial\Omega}B^m_h\nabla \varphi^{m-1}\cdot v^m_h{\,ds}\\
+ b \int_\Omega |\nabla \mu^m_h|^2{\,dx}\\
+\frac{\sigma\epsilon}{2\tau}\left(
\int_\Omega |\nabla \varphi^m_h|^2
- |\nabla \varphi^{m-1}|^2
+|\nabla \varphi^m_h-\nabla \varphi^{m-1}|^2{\,dx}\right)\\
+\frac{\sigma}{\epsilon}\int_\Omega (W^\prime_+(\varphi^{m}_h)+W^\prime_-(\varphi^{m-1}))
\frac{\varphi^m_h-\varphi^{m-1}}{\tau}{\,dx}\\
+r\int_{\partial\Omega} B^m_h\frac{\varphi^m_h-\varphi^{m-1}}{\tau}{\,ds}+ \int_{\partial\Omega}
(\gamma_+^\prime(\varphi_h^{m}) +\gamma_-^\prime(\varphi^{m-1}))
\frac{\varphi^m_h-\varphi^{m-1}}{\tau}{\,ds}= 0.\end{aligned}$$ Using convexity and concavity of $W_+$ and $W_-$, and $\gamma_+$ and $\gamma_-$ it holds $$\begin{aligned}
\int_\Omega(W^\prime_+(\varphi^{m}_h)+W^\prime_-(\varphi^{m-1}))
\frac{\varphi^m_h-\varphi^{m-1}}{\tau} {\,dx}\geq \frac{1}{\tau}\int_\Omega W(\varphi^m_h)-W(\varphi^{m-1}){\,dx},\\
\int_\Omega(\gamma^\prime_+(\varphi^{m}_h)+\gamma^\prime_-(\varphi^{m-1}))
\frac{\varphi^m_h-\varphi^{m-1}}{\tau} {\,ds}\geq \frac{1}{\tau}\int_\Omega \gamma(\varphi^m_h)-\gamma(\varphi^{m-1}){\,ds}.\end{aligned}$$ Summing up and using $v\cdot \nu_\Omega = 0$, we obtain the desired result.
In general, in diffuse interface simulations it is advantageous to use adaptive meshes to resolve the interfacial region. Then in every time step additional prolongation operators between subsequent meshes are required. As a consequence, in this case the energy inequality from only holds with the prolongated data for the energy from the old time instance. We further note that special care has to be taken for prolongating the velocity field, as the prolongated velocity field typically is not solenoidal with respect to the new mesh. We refer to [@GarckeHinzeKahle_CHNS_AGG_linearStableTimeDisc; @2011-BesierWollner-PressureApproximationIncompressibleFlowDynamicallySpatialMesh] for further discussion of this topic.
Variants {#ssec:S:V}
--------
Let us state variants of the above discretization scheme \[eq:S:1\_NS\_1\]–\[eq:S:4\_CH2\] for numerical comparison. We note, that \[eq:S:1\_NS\_1\]–\[eq:S:4\_CH2\] is a fully coupled and non-linear scheme.
### A stable decoupled scheme {#ssec:S:V:decoupled}
If $r\equiv 0$ the scheme is only coupled by the transport term $(\varphi^{m-1} v^m_h,\nabla \Psi) $ in \[eq:S:3\_CH1\]. The same holds for $l \to \infty$, which results in the commonly used no-slip condition for the Navier–Stokes equation. In the case of no-slip conditions $B$ is independent of $v$ and thus again the only coupling is the transport term in \[eq:S:3\_CH1\].
In both cases we can decouple the Navier–Stokes equation and the Cahn–Hilliard equation by using an augmented velocity field in \[eq:S:3\_CH1\], see for example [@Minjeaud_decoupling_CHNS; @Tierra_Splitting_CHNS; @GruenMetzger__CHNS_decoupled; @YuYang_MovingContactLine_diffDensities]. Here we substitute $-\int_\Omega \varphi^{m-1}v^m_h \cdot\nabla \Psi{\,dx}$ in \[eq:S:3\_CH1\] by $$\begin{aligned}
-\int_\Omega \varphi^{m-1}v^{m-1}\cdot \nabla \Psi{\,dx}+ \tau\int_\Omega (\rho^{m-1})^{-1}|\varphi^{m-1}|^2
\nabla \mu^m_h\cdot\nabla \Psi{\,dx}.
\label{eq:S:V:transport}\end{aligned}$$ The resulting scheme is decoupled; we can first solve \[eq:S:3\_CH1,eq:S:4\_CH2\] and thereafter \[eq:S:1\_NS\_1,eq:S:2\_NS\_2\]. This scheme is also energy stable, as the additional integral compensates terms arising from Hölder’s and Young’s inequality to balance the first integral with the numerical dissipation $\frac{1}{2}\int_\Omega
\rho^{m-1}|v^m_h-v^{m-1}|^2{\,dx}$. This scheme with no-slip conditions for Navier–Stokes and $r\equiv 0$ is analyzed in [@GruenMetzger__CHNS_decoupled] for different treatments of $W^\prime$. We also refer to [@KayStylesWelford] for an alternative decoupling in the case of constant density. Here the systems are decoupled by using $v^{m-1}$ in \[eq:S:3\_CH1\], and the energy stability is obtained by introducing a step size restriction for the temporal discretization.
If $r >0$, we use $v^{m-1}$ in the definition of $B^m_h$ in \[eq:S:4\_CH2\] and $v^m_h$ in the corresponding term in \[eq:S:1\_NS\_1\] and can still derive an energy inequality containing an error of order $r \int_{\partial\Omega} (v^m_h-v^{m-1}) \cdot \nabla \varphi^{m-1}{\,ds}$. In [@GruenMetzger__CHNS_decoupled] a no-slip condition is assumed for $v$ to decouple the boundary conditions. Then the decoupling proposed in \[eq:S:V:transport\] is sufficient to decouple the Navier–Stokes and the Cahn–Hilliard equation.
We note that this scheme can be applied for any bulk energy potential that admits a convex-concave splitting.
### A stable decoupled and linear scheme {#ssec:S:V:stablelinear}
Using the decoupling proposed in \[ssec:S:V:decoupled\], the only nonlinearity in the scheme arises from $W_+^\prime$. In [@2015-ShenYangYu-EnergyStableSchemesForCHMCL-Stabilization; @AlandChen__MovingContactLine; @YuYang_MovingContactLine_diffDensities], a stabilized linear scheme is used and the term $W_+^\prime(\varphi^m_h) + W_-^\prime(\varphi^{m-1})$ is substituted by $W^\prime(\varphi^{m-1}) + S_W(\varphi^{m}_h - \varphi^{m-1})$, where $S_W$ is a suitable stabilization parameter. For smooth $W$ it satisfies $S_W \geq \frac{1}{2}\max_{t} |W^{\prime\prime}(t)|$. It can be derived by Taylor expansion of $W$ at $\varphi^{m-1}$, see for example [@2015-ShenYangYu-EnergyStableSchemesForCHMCL-Stabilization]. As $W^{s}$ is of class $C^{1,1}$ only, $({W^{s}})^{\prime\prime}$ jumps at $\xi^{-1}$ from $-\xi^2$ to $({s}-1)\xi^2$. In this case we use $S_W \geq {s}/2$. For large values of ${s}$ we expect that this stabilization will prevent changes in $\varphi$ and thus might have a deep impact on the allover dynamics. This is investigated in and especially discussed in Remark \[rm:relaxation\]. To linearize $\gamma$ we substitute $\gamma^\prime_+(\varphi^m_h) + \gamma^\prime_-(\varphi^{m-1})$ by $\gamma^\prime(\varphi^{m-1}) + S_\gamma(\varphi^m_h - \varphi^{m-1})$ with $S_\gamma \geq \frac{1}{2}\max_{t}|\gamma^{\prime\prime}(t)|$ and especially $S_\gamma \geq \frac{1}{2}\sigma_{12}|\cos(\theta_s)|$ in the case of $\gamma^{cc}$. Here, again $S_\gamma$ is obtained by Taylor expansion of $\gamma$ at $\varphi^{m-1}$.
For further discretization schemes of the bulk energy density $W$ we refer for example to [@GruenMetzger__CHNS_decoupled; @GonzalesTierra_linearSchemes_CH; @2014-WuZwietenZee-StabilizedSecondOrderConvecSplittingCHmodels]. Second order schemes for the Cahn–Hilliard equation are for example proposed and analyzed in [@GonzalesTierra_linearSchemes_CH; @2014-Tierra_SecondOrder_and_Adaptivity; @2014-WuZwietenZee-StabilizedSecondOrderConvecSplittingCHmodels; @2016-DiegelWangWise-StabilityConvergenceSecondOrderSchemeCH; @2018-WangYu-SecondOrderCahnHilliardStabilized]. Recently the Invariant Energy Quadratization approach for $W\equiv W^{poly}$ was proposed in [@2017-YangJu-InvariantEnergyQuadratization]. It is used in [@2018-YangYu-SecondOrderCHMCL_IEQ] for the Cahn–Hilliard moving contact line model together with a Crank–Nicolson and a BDF2 scheme in time. However, typically for these schemes either higher regularity than $W^s$ provides is required for $W$, or the particular $W^{poly}$ is assumed and necessary. Moreover, unconditional energy stability is typically not proven yet.
\[rm:S:energyStable\] Considering the energy inequality from , the terms in the first line correspond to the discrete energy of the system, while the second line corresponds to the energy dissipation of the system, and the third line corresponds to numerical dissipation of the scheme. Based on this we can define four different values to define the energies in our system. These are the energy $E^m$ at time instance $m$, the physical dissipation $\Delta^m_p$ at time instance $m$, the energy $E_g^m$ introduced from gravity at time instance $m$, and the numerical dissipation $\Delta^m_n$ at time instance $m$. They are defined by $$\begin{aligned}
E^m &:= \frac{1}{2}\int_\Omega \rho^m|v^m_h|^2{\,dx}+\sigma\int_\Omega\frac{\epsilon}{2}|\nabla \varphi^m_h|^2
+ \frac{1}{\epsilon}W(\varphi^m_h){\,dx}+ \int_{\partial\Omega} \gamma(\varphi^m_h){\,ds},\\
\Delta^m_p &:= \tau\int_\Omega 2\eta^{m-1}|Dv^m_h|^2 {\,dx}+ \tau \int_\Omega b |\nabla \mu^m_h|^2{\,dx}\nonumber\\
&\phantom{:= }+\tau\int_{\partial\Omega}l(\varphi^{m-1})|v^m_{h,tan}|^2{\,ds}+ \tau \int_{\partial\Omega} r |B^m_h|^2{\,ds},\label{eq:S:phyDiss} \\
E_g^m & :=\tau\int_\Omega \rho^{m-1}g\cdot v_h^m{\,dx},\\
\Delta^m_n &:= E^{m-1} + E_g^m - E^m - \Delta^m_p. \end{aligned}$$ We call a scheme thermodynamically consistent if is fulfilled without the explicit form of the numerical dissipation, thus if $$E^m + \Delta^m_p \leq E^{m-1} + E_g^m
\label{eq:S:EnergyInequ}$$ holds, i.e., $\Delta_n^m \geq 0$. We investigate this energy inequality numerically in .
Numerics {#sec:N}
========
In this section we numerically investigate the three schemes under consideration. In we briefly give results from the well-known second benchmark in [@Hysing_Turek_quantitative_benchmark_computations_of_two_dimensional_bubble_dynamics], where no contact line motion is included, to estimate the difference of the schemes in the bulk. In we thereafter investigate the behavior of the contact line for a gravity-driven droplet sliding on an inclined surface in a two-dimensional setting.
We implement the schemes in Python3 using FEniCS 2018.1.0 [@fenics1; @fenics_book]. For the solution of the arising nonlinear and linear systems and subsystems the software suite PETSc 3.8.4 [@petsc_webpage; @petsc-user-ref; @petsc-efficient] together with the direct linear solver MUMPS 5.1.1 [@mumps_1; @mumps_2] are utilized. Note, that we do not apply any preconditioning or subiterations except for the Newton iterations.
Rising Bubble {#ssec:N:rising_bubble}
-------------
At first, we discuss the accuracy of the proposed schemes without moving contact lines. Later on, this allows for an evaluation of the influence of the schemes on the moving contact line. We employ the quantitative benchmark case proposed in [@Hysing_Turek_quantitative_benchmark_computations_of_two_dimensional_bubble_dynamics]. In [@Aland_Voigt_bubble_benchmark] it is found, that three different diffuse interface approximations together with the polynomial potential $W^{poly}$ agree well with the sharp interface results from [@Hysing_Turek_quantitative_benchmark_computations_of_two_dimensional_bubble_dynamics]. In [@GarckeHinzeKahle_CHNS_AGG_linearStableTimeDisc] the benchmark is used to compare to a phase field model with a relaxed double obstacle potential.
### Setup
lists the properties of our simulations, which correspond to the second benchmark case in [@Hysing_Turek_quantitative_benchmark_computations_of_two_dimensional_bubble_dynamics]. For details on the setup we refer to the references above. Note, that $\sigma_{12}$ denotes the physical surface tension, yielding $\sigma\approx1.24$ for $W^{{s}=100}$, $\sigma\approx1.22$ for $W^{{s}=10}$ and $\sigma\approx 2.07$ for $W^{poly_2}$. Following [@Hysing_Turek_quantitative_benchmark_computations_of_two_dimensional_bubble_dynamics], we introduce a characteristic length scale $L=2r_0$, where $r_0$ equals the initial radius of the bubble, and a characteristic velocity scale $U=\sqrt{Lg}$. To classify our simulations we indicate in the dimensionless numbers Reynolds ${\operatorname{\mbox{\textit{Re}}}}= \frac{\rho_l U L}{\eta_l}$, Eötvös (or Bond) ${\operatorname{\mbox{\textit{Eo}}}}=\frac{\rho_l g L^2}{\sigma}$, Capillary ${\operatorname{\mbox{\textit{Ca}}}}= \frac{\eta_l U}{\sigma}$, Atwood ${\operatorname{\mbox{\textit{At}}}}= \frac{\rho_l - \rho_g}{\rho_l + \rho_g}$, Cahn ${\operatorname{\mbox{\textit{Cn}}}}= \frac{\epsilon}{L}$ and Péclet ${\operatorname{\mbox{\textit{Pe}}}}= \frac{L U \epsilon}{b \sigma}$, see [@Khatavkar2006].
We apply no-slip boundary conditions for the velocity on the top and bottom walls, free-slip on the left and symmetry at the centerline through the bubble at $x=0.5$. Similar to [@GarckeHinzeKahle_CHNS_AGG_linearStableTimeDisc], we set $b=10^{-3}\epsilon$ and $\epsilon=0.02$. The time discretization step is set to different values and the final time is $t=3$. We initialize the simulations by solving the Cahn–Hilliard equations without convection until a steady state is reached. In total, we perform 7 distinct simulations using the three schemes from with $W^{poly2}$ and $W^{{s}}$ with ${s}=100$, and one additional simulation with ${s}=10$ for the fully linear and stabilized scheme with $W^{{s}}$, see the first three columns in . To get an impression of the influence of the discretization parameters, we use different values for $\tau$ and $h_{min}$, see columns four and five in .
In [@Hysing_Turek_quantitative_benchmark_computations_of_two_dimensional_bubble_dynamics] a set of benchmark parameters is used, that we define in the phase field setting as follows.
The center of mass is calculated using $$\begin{aligned}
(x_c, y_c) &= \frac{\int_\Omega (x, y)\frac{1+\varphi}{2}{\,dx}}{\int_\Omega \frac{1+\varphi}{2}{\,dx}}\;,
\label{eq:S:BM-y}\end{aligned}$$ where $\frac{1+\varphi}{2} = 1$ indicates the droplet.
We define the mean velocity in unit direction $a\in \mathbb R^2$ as $$\begin{aligned}
v_a = \frac{\int_\Omega v \cdot a \frac{1+\varphi}{2}{\,dx}}{\int_\Omega \frac{1+\varphi}{2}{\,dx}}\;.
\label{eq:S:BM-v}\end{aligned}$$ If $a$ denotes the unit vector in rising direction, this is called rising velocity $v_r$, while if $a$ points in sliding direction, we call this value sliding velocity $v_s$.
Finally we define the stretching of the interface as $$\begin{aligned}
c = \frac{c_W\int_\Omega (\frac{\epsilon}{2}|\nabla \varphi_0|^2 + \frac{1}{\epsilon}W(\varphi_0)){\,dx}}{c_W\int_\Omega (\frac{\epsilon}{2}|\nabla \varphi|^2 + \frac{1}{\epsilon}W(\varphi)){\,dx}}\;.
\label{eq:S:BM-c}\end{aligned}$$ Here the denominator denotes an approximation to the length of the interface represented by $\varphi$, and the numerator denotes the same for the initial phase field $\varphi_0$. If $\varphi_0$ denotes a sphere, this is equivalent to the circularity as defined in [@Hysing_Turek_quantitative_benchmark_computations_of_two_dimensional_bubble_dynamics] as the volume of the bubble is constant over time.
\[rm:relaxation\] For the choice of the relaxation parameter ${s}$ in $W^{s}$, see , several points must be considered. To reduce the inter-mixing between the phases and increase the rate at which the equilibrium profile of $\varphi$ is reestablished after a deformation, it is desirable to exhibit a large spinodal region and subsequently a small metastable region [@Donaldson2011]. The metastable region of the bulk energy potential $W^{{s}}$ is located between $1 > |\varphi|> \xi^{-1} = 1-\frac{1}{{s}}$, while the metastable region for $W^{poly}$ is located between $1> |\varphi| > \sqrt{3^{-1}} \approx 0.577$. Thus already for small values of ${s}$, say ${s}= 10$, the metastable region of $W^{{s}}$ is significantly smaller than the metastable region of $W^{poly}$. Furthermore, referring to [@GarckeHinzeKahle_CHNS_AGG_linearStableTimeDisc; @Kahle_Linfty_bound], the value of ${s}$ controls the deviation of the $L^\infty$ norm of $\varphi$ from 1. Since $\rho$ and $\eta$ directly depend on $\varphi$ a small deviation is desirable, which is achieved by a large value of ${s}$.
On the other hand, the stable decoupled and linear scheme, , includes a stabilization parameter $S_W$ which has to be chosen like $S_W>{s}/2$ for $W^{s}$. In this case a large value of ${s}$ has a severe impact on the overall dynamics as the stabilization can be interpreted as adding the quadratic potential $\frac{S_W}{2}\|\varphi-\varphi^{m-1}\|^2$ to $W$ for given $\varphi^{m-1}$. For large values of $S_W$ thus $\varphi \equiv \varphi^{m-1}$ is preferred. To show the influence of $S_W$ in the case $W\equiv W^{s}$ we test the linear and decoupled scheme with two values of ${s}$.
---------------------------------------------------------------- ------------------------------------------ ---------- ---------- ---------- ------- --------------------------------------- --------------------------------------- --------------------------------------- ---------------------------------------
$\sigma_{12}$ $\rho_l$ $\rho_g$ $\eta_l$ $\eta_g$ $g_y$ ${\operatorname{\mbox{\textit{Re}}}}$ ${\operatorname{\mbox{\textit{Eo}}}}$ ${\operatorname{\mbox{\textit{Ca}}}}$ ${\operatorname{\mbox{\textit{At}}}}$
(r)[1-6]{}(l)[7-10]{} 1.96 1000 1 10 0.1 -0.98 35 125 3.5 0.99
(r)[1-6]{}(l)[7-10]{} $\epsilon$ $b$ ${\operatorname{\mbox{\textit{Cn}}}}$ ${\operatorname{\mbox{\textit{Pe}}}}$
(r)[1-6]{}(l)[7-10]{} $\num[scientific-notation = true]{2e-2}$ $\num[scientific-notation = true]{2e-5}$ 0.04 178
---------------------------------------------------------------- ------------------------------------------ ---------- ---------- ---------- ------- --------------------------------------- --------------------------------------- --------------------------------------- ---------------------------------------
: Parameters used in the rising bubble simulations.[]{data-label="tab:rb_setup"}
### Results
The resulting benchmark values are listed in . As it is not even clear in the sharp interface simulations whether or not topological changes develop, e.g. the separation of trailing gas filaments, we compare our results only up to time instance $t=2$, see [@Aland_Voigt_bubble_benchmark] . Our results show that all the schemes give very similar results compared to the sharp interface solution even for the significantly larger time step $\tau=0.001$ and on a coarse mesh with $h_{min}=0.0125$. In general, decoupling the two systems has a very small impact on the benchmark values. For even larger $\tau=0.008$ the coupled scheme is advantageous against the decoupled schemes. The latter might be explained by the fact, that the decoupling adds artificial diffusion of order $\tau$ to the Cahn–Hilliard system, see . Thus we expect a stronger influence of this decoupling for larger values of $\tau$. As expected, the stabilized linear scheme together with $W^{{s}=100}$ hinders the dynamics of the rising bubble. However, the results improve significantly with smaller ${s}$. All schemes together with $W^{{s}=100}$ give slightly better results compared to $W^{poly_2}$ except the decoupled/linear scheme. However, for very small $\tau$ and $h_{min}$ the results converge towards similar values.
Concerning the computational effort the difference in using $W^{{s}=100}$ or $W^{poly_2}$ is insignificant. The decoupled/nonlinear and decoupled/linear schemes are around 1.4 respectively 2.0 times faster than the coupled scheme. In the nonlinear schemes 2-3 Newton iterations are needed per time step. Note that the performance results strongly dependent on the solver and whether sophisticated preconditioning is applied. For an efficient preconditioner for the coupled/nonlinear system we refer to [@Bosch2016].
Sliding Droplet {#ssec:N:sliding_droplet}
---------------
To compare the influence of the numerical schemes from on the moving contact line, we perform simulations of single droplets sliding down an inclined surface. Besides the effect of gravity on the droplet movement, this test case allows to observe both an advancing and receding contact line.
### Setup
In the initial configuration is shown and lists the properties of our simulations. The fluid properties are chosen to be similar to the first rising bubble test case in [@Hysing_Turek_quantitative_benchmark_computations_of_two_dimensional_bubble_dynamics]. Note, that $\sigma_{12}$ denotes the physical surface tension, yielding $\sigma\approx15.58$ for $W^{{s}=100}$, $\sigma\approx15.34$ for $W^{{s}=10}$ and $\sigma\approx25.98$ for $W^{poly_2}$. A liquid droplet with radius $r_0=0.25$ is placed in a $0.5\times2.0$ rectangular domain at ($0$,$1.5$) on a smooth, solid surface with an initial contact angle of $90\degree$. The inclination angle of the plate is $45\degree$. The density of the droplet is greater than that of the surrounding fluid. We have no-slip boundary conditions for the velocity on the left and right side and free-slip on the top side. The conditions \[eq:M:6\_NS\_BC,eq:M:7\_CH\_BC\] are applied on the bottom solid surface, see . The influence of the boundary conditions \[eq:M:6\_NS\_BC,eq:M:7\_CH\_BC\] on the sliding droplets are examined by varying the static contact angle $\thetaori$, the relaxation factor $r$ and the slip coefficient $l$, see the fifth to seventh column in . We vary the contact angle from super-hydrophilic ($5\degree$) to super-hydrophobic ($150\degree$) [@Law2014]. We initialize the simulations by solving the Cahn–Hilliard equations without convection and a contact angle of $90\degree$ until a steady state is reached.
In a first step, we compare 21 distinct simulations obtained with the three schemes from with $W^{poly_2}$ and $W^{{s}}$ with ${s}=100$, and one additional simulation with ${s}=10$ for the fully linear and stabilized scheme with $W^{{s}}$, see the first two columns in . These simulation are performed with a relatively coarse mesh ($h_{min} = 0.0125)$ and large time step ($\tau=0.001$) to discuss the practical applicability of the solution schemes. Afterwards, we show the thermodynamic consistency of the schemes and compare the physical and numerical dissipation rates. To discuss the influence of the time step size on the results, we perform 14 additional simulations with $\tau$ between 0.008 and 0.00025. Finally, we perform 8 simulations with varying interfacial thicknesses $\epsilon$ on a very fine mesh ($h_{min}=0.0002$) to briefly discuss the convergence to the sharp-interface limit.
(0, 1.75) node(r1) arc(90:-90:0.25)node(r2); (0,0) node (a) – (0.5,0) node (b) – (0.5,2) node (c) – (0,2) node (d) – (0,0); (a) – ++(0, 2) – ++(-0.05,0) – ++(0,-2) – ++(0.05,0); ($(a)+(0,-0.1)$)–node\[rotate=45,below\] [$0.5$]{} ($(b)+(0,-0.1)$) ; ($(b)+(+0.1,0)$)–node\[rotate=-45,above\] [$2.0$]{} ($(c)+(+0.1,0)$) ; ($(a)+(-0.1,0)$)–node\[rotate=-45,below\] [$1.5$]{} ($(a)+(-0.1,1.5)$) ; ($(r1)+(-0.2,0)$)–node\[rotate=-45,below\] [$0.5$]{} ($(r2)+(-0.2,0)$) ; ($(d)+(0,+0.1)$)–node\[rotate=45,above\] [$0.25$]{} ($(d)+(0.25,+0.1)$) ; (0,0)–(-0.7,0); (-0.4,0) node\[above left\][$45\degree$]{} arc(180:135:0.4);
(-1.1,0.5)–(-0.85,0.5) node\[right\][$x$]{}; (-1.1,0.5)–(-1.1,0.25) node\[right\][$y$]{};
(0.1,1.3)–(0.1,1.1) node\[right\][Droplet]{}; (0,0.6)–(0.8,0.5) node\[right, align=left\][Wall with\
eqns. \[eq:M:6\_NS\_BC\],\[eq:M:7\_CH\_BC\]\
and $\theta_s$, $r$, $l$.]{}; (0.0,1.8)– node\[right\][$g$]{}(0.0,1.55);
---------------------------------------------------------------- ------------------------------------------ ---------- ---------- ---------- ------- --------------------------------------- --------------------------------------- --------------------------------------- ---------------------------------------
$\sigma_{12}$ $\rho_l$ $\rho_g$ $\eta_l$ $\eta_g$ $g$ ${\operatorname{\mbox{\textit{Re}}}}$ ${\operatorname{\mbox{\textit{Eo}}}}$ ${\operatorname{\mbox{\textit{Ca}}}}$ ${\operatorname{\mbox{\textit{At}}}}$
(r)[1-6]{}(l)[7-10]{} 24.5 1000 100 10 1 -0.98 35 10 0.28 0.81
(r)[1-6]{}(l)[7-10]{} $\epsilon$ $b$ ${\operatorname{\mbox{\textit{Cn}}}}$ ${\operatorname{\mbox{\textit{Pe}}}}$
(r)[1-6]{}(l)[7-10]{} $\num[scientific-notation = true]{2e-2}$ $\num[scientific-notation = true]{2e-5}$ 0.04 14
---------------------------------------------------------------- ------------------------------------------ ---------- ---------- ---------- ------- --------------------------------------- --------------------------------------- --------------------------------------- ---------------------------------------
: Parameters used in the sliding droplet simulations.[]{data-label="tab:sd_setup"}
\[rm:bc\_parameters\] For meaningful values of the relaxation parameter $r$ and the slip coefficient $l$, we write \[eq:M:6\_NS\_BC,eq:M:7\_CH\_BC\] in non-dimensionalized form, $$\begin{aligned}
\left[\frac{{\operatorname{\mbox{\textit{Ca}}}}}{{\operatorname{\mbox{\textit{Cn}}}}} 2\hat \eta D\hat v \nu_\Omega + \frac{{\operatorname{\mbox{\textit{Ca}}}}}{{\operatorname{\mbox{\textit{Cn}}}}} \frac{L}{l_s} - \hat L \hat \nabla \varphi\right] \times \nu_\Omega &= 0, \\
\frac{{\operatorname{\mbox{\textit{Ca}}}}}{{\operatorname{\mbox{\textit{Cn}}}}}\frac{r_s}{L}(\partial_{\hat t}\hat\varphi + \hat v \hat\nabla \varphi) + \hat L&=0,\end{aligned}$$ in which ${\operatorname{\mbox{\textit{Ca}}}}=\eta_l U/\sigma$ and ${\operatorname{\mbox{\textit{Cn}}}}=\epsilon/L$ are the Capillary number respectively the Cahn number, and $L$ and $U$ are some characteristic macroscopic length scale respectively velocity. We choose $r=r_s\eta_l$ and $l=\eta_l/l_s$ such that the dimensionless groups $\frac{{\operatorname{\mbox{\textit{Ca}}}}}{{\operatorname{\mbox{\textit{Cn}}}}} \frac{L}{l_s}$ and $\frac{{\operatorname{\mbox{\textit{Ca}}}}}{{\operatorname{\mbox{\textit{Cn}}}}}\frac{r_s}{L}$ are of $\mathcal{O}(1)$, see [@Sibley2013].
As benchmark values we again use the three values defined in with minor modifications. For the center of mass, we use a coordinate system that is aligned with the inclined plate, see , and for the now called sliding velocity, we use for $a$ the unit vector in tangential direction to the inclined plate. The stretching of the interface is defined as before.
Additionally, we evaluate two values that are specific for the moving contact line setup. For both the receding and advancing contact line the position of the contact points and a dynamic (or apparent) contact angle measured at some height above the contact points are evaluated. The position of a contact point is defined by $$\begin{aligned}
y_p = y \mbox{ on } \partial\Omega \mbox{ where }\varphi=0\;,\end{aligned}$$ and the dynamic contact angle $\thetaori_d$ is calculated by linear interpolation between $y_p$ and the intersection $y_{p+\Delta}\mbox{ where }\varphi=0$ and $\Delta=h_\text{min}$, see and [@Omori2017].
(0.25, 0) arc(180:0:0.25); (0,0) node (a) – (1,0) node (b); (0,0) – (1,0) – (1,-0.05) – (0,-0.05) – (0, 0); (0.75,0) circle (0.10); (0.85, 0) – (1.24,0.5) node(r1); (0.65, 0) – (0.26,0.5) node(r2); (0.75,1.0) circle (0.70); (0.75,1.0) circle (0.70); (0,0.6) – (2,0.6); (0,0.6) – (2,0.6) – (2,0.295) – (0,0.295) – (0, 0.6); (0.75, 0.6) to\[out=60,in=-40\] node\[above right\](bla)[$\mathclap{\varphi=0}$]{}(0.5, 1.8); (0.75, 0.6) node\[draw, circle, minimum size=3.0pt, solid\] – node\[draw, circle, minimum size=3.0pt, solid\](cross2) (1.0, 1.5); (1.1, 0.6)–node\[ below right, rotate=-0\][$h_\text{min}$]{}(1.1, 1.05); (0.45, 0.6) node\[rotate=0, above\][$\thetaori_d$]{} arc(180:75:0.3);
[background]{} (0.75,1.0) circle (0.70); (-1, 1.05) – (2, 1.05); (0,0.6) – (0.75, 0.6) to\[out=60,in=-40\] (0.5, 1.8) – (0, 1.8) – (0, 0.6);
### Results
#### Comparison of droplet shapes and characteristic values obtained on a coarse mesh and with a large time step
In dependence on $\thetaori_s$, $r$ and $l$ the droplets show characteristic developments. The calculated shapes for different combinations of $\thetaori_{s}$, $r$ and $l$ at $t=0.0; 0.5; 1.0; 1.5; 2.0$ are presented in . All the simulated droplets show the expected physical behavior: on the hydrophobic surface (third row) the droplet contracts, whereas the droplets spread on the hydrophilic surface (second row). In addition, the droplets slide down the surface due to the density difference and gravity. The different behavior at the advancing and receding contact points is visible and one can observe nonequilibrium contact angles in the second and third row. It is evident that there are virtually no differences between the coupled (solid black) and decoupled schemes (crosses) for all contact angles. In contrast, in the linearized scheme with ${s}=100$ (dashed black) the dynamics are greatly reduced. Similar as in the rising bubble case, a smaller ${s}$ (${s}=10$, dashed gray) leads to improvements. For comparison, we show the behavior of the droplet with the coupled scheme and $W^{poly_2}$ (dotted line). Here, a slightly different droplet shape is observed especially for later times and large contact angles.
The evolution of the slide velocity $v_s$, the position of the contact points $y_p$ along the surface and the dynamic contact angle $\thetaori_d$ are displayed in . Again, we show all the schemes together with $W^{{s}=100}$ and in addition the stabilized scheme together with $W^{{s}=10}$ and the coupled, nonlinear scheme with $W^{poly_2}$. To allow for a more quantitative comparison between the solution schemes, we list the characteristic values at $t=2$ in . As expected, in simulations without equilibrium contact angle relaxation ($r=0$) and slip ($l=1e6$) (first row) the apparent contact angles on both sides of the droplets stay near the equilibrium value $\thetaori_{s} = 90\degree$ the whole time. In contrast, applying the full boundary conditions \[eq:M:6\_NS\_BC,eq:M:7\_CH\_BC\] with $r=0.35$ and $l=140$ leads to clearly visible advancing and receding contact angles (third column). As before, no difference is visible between the coupled (solid black) and decoupled (black crosses) nonlinear schemes for all the characteristic quantities. The characteristic values at $t=2$ differ only very slightly. The results with the decoupled, stabilized scheme with ${s}=100$ (dashed black) are very far off and show very low sliding velocities (left column) and a different contact point behavior (middle column), especially for $\thetaori_{s}=150\degree$ (last row). The sliding velocities at $t=2$ differ greatly. In contrast to the comparison in the bulk only, see , the usage of the coupled scheme with $W^{poly_2}$ (dotted black) gives results which are noticeable different from the results with $W^{{s}=100}$. This is most obvious in the simulations with $\thetaori_{s}=5\degree$ (middle row): the sliding velocity (left column) is slower and the terminal velocity is reached later. In addition, the receding and advancing contact angles are both lower than in the simulations with $W^{{s}=100}$. For example, at $t=2$, the advancing contact angle for $W^{poly_2}$ is around $12\degree$ smaller than in the nonlinear simulations with $W^{{s}=100}$.
#### Thermodynamic consistency and comparison of dissipation rates
We reveal the thermodynamic consistency of the schemes by calculating the evolution of the energy inequality using \[eq:S:EnergyInequ\]. We use $\gamma^{cc}$ and set $\thetaori_{s}=150\degree$, $r=0.35$ and $l=140$. We observe, that $\Delta_n^m$ is positive for all times, which justifies that the schemes are thermodynamically consistent, see \[eq:S:EnergyInequ\]. Note that for $r>0$ we introduce an additional error as soon as we use the decoupling strategy. We observe, that the physical dissipation for the three nonlinear schemes are close together, while the physical dissipation for the linear model is strongly reduced. This corresponds to the reduced dynamics that are observed in for the linear schemes, especially for ${s}=100$. This influence can be reduced by using very small time steps and finer meshes, see the results for the rising bubble case in . Comparing the numerical and physical dissipation of the nonlinear schemes, $\Delta_n^m$ only accounts for around 25% of the total dissipation even for large time steps. Furthermore, by halving the time step $\tau$, the numerical dissipation $\Delta_n^m$ relative to the total dissipation $\Delta_n^m+\Delta_p^m$ is greatly reduced to around 12%, see the grey plots in the bottom figure of Figure \[fig:sd\_energy\].
table\[x=t,y=deltapwogravity\] [data/slidingdroplet/energies/ws100\_c\_nlin.dat]{}; table\[x=t,y=deltapwogravity\] [data/slidingdroplet/energies/ws100\_de\_nlin.dat]{}; table\[x=t,y=deltapwogravity\] [data/slidingdroplet/energies/ws100\_de\_lin.dat]{}; table\[x=t,y=deltapwogravity\] [data/slidingdroplet/energies/ws10\_de\_lin.dat]{}; table\[x=t,y=deltapwogravity\] [data/slidingdroplet/energies/poly2\_c\_nlin.dat]{}; table\[x=t,y=deltan\] [data/slidingdroplet/energies/ws100\_c\_nlin.dat]{}; table\[x=t,y=deltan\] [data/slidingdroplet/energies/ws100\_de\_nlin.dat]{}; table\[x=t,y=deltan\] [data/slidingdroplet/energies/ws100\_de\_lin.dat]{}; table\[x=t,y=deltan\] [data/slidingdroplet/energies/ws10\_de\_lin.dat]{}; table\[x=t,y=deltan\] [data/slidingdroplet/energies/poly2\_c\_nlin.dat]{};
table\[x=t,y expr=/(+)\] [data/slidingdroplet/energies/ws100\_de\_nlin.dat]{}; table\[x=t,y expr=/(+)\] [data/slidingdroplet/energies/ws100\_de\_nlin\_dt0.0005.dat]{}; (a) at (axis cs:1.3,0.21)[$\tau=0.001$]{}; (b) at (axis cs:1.3,0.15)[$\tau=0.0005$]{};
#### Comparison of characteristic values obtained with smaller time steps $\tau$
We show the behavior of the schemes for different time step sizes in Table \[tab:sd\_convergence\]. For small time steps, both the nonlinear schemes (coupled and decoupled) converge to the same characteristic values for the particular bulk energy potentials. However, by comparing the values between the different bulk energy potentials, we note, that the differences are still relatively large even for small time steps. Again, the linear scheme together with $W^{s}$ gives results far away from the solution obtained with the coupled schemes.
#### Convergence to sharp-interface limit
In we show solutions obtained with both bulk energy potentials and smaller $\epsilon$ on a very fine mesh ($h_{min}=0.0002$). To reduce the computational effort, the inclination angle of the plate, see , is set to zero and the simulation is already stopped at $t=0.2$. As expected for $b = \mathcal O(\epsilon)$ and $r = \mathcal O(1)$, see [@2018-XuDiHu-SharpInterfaceLimit-NavierSlipBoundary], the rate of convergence for both potentials is very slow and the sharp-interface limit is not reached yet. However, from our simulations we can conclude, that on the fine mesh both potentials give very similar results and exbibit the same behavior for smaller $\epsilon$. For larger $\epsilon$, solutions obtained with $W^{{s}=100}$ seem to diverge slightly faster from the sharp-interface solution than solutions obtained with $W^{poly_2}$.
Conclusion {#sec:conclusion}
==========
We compare the quality of the numerical results with three different schemes and two different bulk energy potentials. For simulations without a moving contact line (rising bubble case), we find very similar results in the bulk independent of the coupling and linearization for both potentials. However, the linearization of $W^{s}$ for large ${s}$ hinders the dynamics to a great extend but gets better for smaller ${s}$. For the simulations including moving contact lines (sliding droplet case), the differences between the polynomial potential $W^{poly_2}$ and the relaxed double-obstacle potential $W^{s}$ are more pronounced. Again, we observe a strong truncation of the allover dynamics using $W^{s}$ together with the linear scheme. In both cases, the influence of the decoupling of the Navier–Stokes and Cahn–Hilliard system slightly depends on the time step size. However, the decoupling has a negligible influence on the all-over dynamics even for larger time steps. Concerning the two tested bulk energy potentials, we observe, that both give in general physically sound results, but differences are still exists even for small time steps. The results and the behavior for smaller $\epsilon$ on a fine mesh are almost the same for both potentials. For larger $\epsilon$, solutions obtained with $W^{{s}=100}$ seem to diverge slightly faster from a sharp-interface solution than with $W^{poly_2}$.
Summarizing our results, we find that
- the decoupling strategy gives excellent results while the computational effort is significantly reduced compared to the fully coupled scheme,
- a further linearization of the Cahn–Hilliard system applying the stabilization is not recommended together with $W^{s}$ for large values of ${s}$,
- both bulk energy potentials produce sound and similar results in particular for a smaller interfacial thickness $\epsilon$.
To further judge whether one of the potentials lead to more accurate results, high fidelity sharp interface results on flows with moving contact lines (similar to the benchmark performed in [@Hysing_Turek_quantitative_benchmark_computations_of_two_dimensional_bubble_dynamics]) are critical. It is our hope, that the presented work sparks further comparisons of diffuse and sharp interface models especially for the frequently observed and relevant case of sliding droplets.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors thank Marion Dziwnik for helpful discussions on the scaling of the contact line surface tensions.
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abstract: 'Consider $n$ players whose “scores" are independent and identically distributed values $\{X_i\}_{i=1}^n$ from some discrete distribution $F$. We pay special attention to the cases where (i) $F$ is geometric with parameter $p\to0$ and (ii) $F$ is uniform on $\{1,2,\ldots,N\}$; the latter case clearly corresponds to the classical occupancy problem. The quantities of interest to us are, first, the $U$-statistic $W$ which counts the number of “ties" between pairs $i,j$; second, the univariate statistic $Y_r$, which counts the number of strict $r$-way ties between contestants, i.e., episodes of the form ${X_i}_1={X_i}_2=\ldots={X_i}_r$; $X_j\ne {X_i}_1;j\ne i_1,i_2,\ldots,i_r$; and, last but not least, the multivariate vector $Z_{AB}=(Y_A,Y_{A+1},\ldots,Y_B)$. We provide Poisson approximations for the distributions of $W$, $Y_r$ and $Z_{AB}$ under some general conditions. New results on the joint distribution of cell counts in the occupancy problem are derived as a corollary.'
author:
- |
Julia Eaton\
Department of Mathematics\
University of Washington\
Seattle, WA 98195, USA
- |
Anant P. Godbole\
Department of Mathematics\
East Tennessee State University\
Johnson City, TN 37614, USA
- |
Betsy Sinclair\
Department of Political Science\
University of Chicago\
Chicago, IL 60637, USA
nocite:
- '[@cmj]'
- '[@bhj]'
- '[@mori]'
- '[@bc]'
title: 'Competition between Discrete Random Variables, with Applications to Occupancy Problems'
---
¶[[Po]{}]{} V ł \[thm\][Proposition]{}
Introduction
============
In this paper we hope to shed new light on an old problem, studied extensively in, e.g., [@bhj], [@kolchin]. Consider $n$ players whose “scores" are independent and identically distributed values $\{X_i\}_{i=1}^n$ from some discrete distribution $F$. We consider the case of general distributions $F$ but pay special attention to the cases where (i) $F$ is geometric with parameter $p\to0$ and (ii) $F$ is uniform on $\{1,2,\ldots,N\}$; the latter case corresponds to the classical occupancy problem. The quantities of interest to us are
- the $U$-statistic $W$ which counts the number of “ties" between pairs $i,j$ (with $X_a=X_b=X_c=X_d$, for example, leading to a contribution of ${4\choose2}=6$ to the value of $W$);
- the univariate statistic $Y_r$ which counts the number of strict $r$-way ties between contestants, i.e., episodes of the form $X_i=x$ for some $x$ iff $i\in A$, $\vert A\vert=r$; and
- the multivariate vector $Z_{AB}=(Y_A,Y_{A+1},\ldots,Y_B)$.
We provide Poisson approximations for the distributions of $W$, $Y_r$ and $Z_{AB}$ under some general conditions. New results on the joint distribution of cell counts in the occupancy problem are derived as a corollary.
Consider the following elementary problem from [@cmj]: “Two players use a coin that lands heads with probability $p$ to play a game that consists of a sequence of rounds. In each round, the first player tosses the coin until a head appears. Then the second player tosses the coin until a head appears. If the players have the same number of flips in a round, the round is declared a tie and another round is played. If not, the player with the larger number of flips wins the game. Rounds are played successively until one of the two players wins the game." Readers are asked to find the expected number of rounds; the expected value of the total number of flips; and the probability distribution of the difference between the number of flips made by players 1 and 2 in a given round. We briefly mention the solution for the first two of these questions: The probability of a two person tie is clearly $$\sum\limits_{x=1}^\infty (1-p)^{2x-2} p^2 =
\frac{p}{2-p},$$ so that $\e(R)$, the expected number of rounds is given by $$\begin{aligned}
\e(R) &=&
\sum\limits_{x=1}^\infty x(\p({\rm tie}))^{x-1}(1-\p({\rm tie})) =
\sum\limits_{x=1}^\infty x\lr\frac{p}{2-p}\rr^{x-1}\lr1-\frac{p}{2-p}\rr\\ &=&
\frac{2-p}{2-2p},\end{aligned}$$ so that Wald’s lemma yields for $\e(F)$, the expected total number of flips, $$\e(F) = \e(
{F}/{R})\e(R)=\frac{2-p}{2-2p}\e(F/R)=\frac{2(2-p)}{p(2-2p)},$$ since the expected number $\e(F/R)$ of flips per round is clearly $2/p$. Computations for a three-person game, not mentioned in [@cmj], are similar, but we need to lay down some rules as follows: Three players each flip a $p$-coin until heads is flipped. The player with the highest number of flips wins unless there are ties between two or more players, in which case we repeat the process. That is, the value of each of the three geometric variables in question must be unique. We next compute the probability of a two- or three-way tie; the expected number of rounds; and the expected number of flips for $n=3$ – to convince the reader that the situation rapidly becomes quite complicated as $n$ increases. \[The authors had a lively discussion with Lloyd Douglas, NSF Program Officer, about the following “real-life" application of the $n$-person model with $p\to0$. We wish to rank $n$ of the greatest free-throw shooters (or slam dunkers, or,...) in the National Basketball Association. The players each shoot free throws until they miss – conditional on the fact that no two players miss on the same attempt. Rankings are then awarded in the obvious fashion.\]
With three players (A,B,C), there are 3!=6 ways to have a strict inequality and seven ways to tie, since there is one way for a three way tie (which we loosely write as “A = B = C") to occur; ${3\choose 1}=3$ ways for A $> $ B = C to occur; and ${3 \choose 2} = 3 $ ways for A = B $> $ C to occur. Note that $$\begin{aligned}
\p(A = B = C) &=&p^3 + p^3(1-p)^3 + p^3(1-p)^6
\ldots\\ &=& \sum\limits_{x=1}^\infty p^3(1-p)^{3x-3}\\ &=&
\frac{p^2}{3-3p+p^2}, \end{aligned}$$ while the table below
------------ -------------------------- ------ ------
Case A B C
\[.8ex\] 1 TH, TTH, $\ldots $ H H
2 TTH, TTTH, $\ldots $ TH TH
3 TTTH, TTTTH, $\ldots $ TTH TTH
4 TTTTH, TTTTTH, $\ldots $ TTTH TTTH
------------ -------------------------- ------ ------
reveals that $$\p(A>B=C)=
p^3 \sum\limits_{m=1}^\infty (1-p)^m \sum\limits_{i=0}^{m-1}
(1-p)^{2i} = \frac{p(1-p)}{3-3p+p^2}.$$ Finally, we observe from the table
------------ ------- ------- ------------------
Case A B C
\[.8ex\] 1 TH TH H
2 TTH TTH H, TH
3 TTTH TTTH H, TH, TTH
4 TTTTH TTTTH H, TH, TTH, TTTH
------------ ------- ------- ------------------
that $$\p(A = B>C) =p^3
\sum\limits_{m=1}^\infty(1-p)^{2m} \sum\limits_{i=0}^{m-1}(1-p)^i
= \frac{p(1-p)^2}{(2-p)(3-3p+p^2)},$$ which leads to $$\begin{aligned}
\p({\rm tie})&=&\p(A = B = C) + 3\p(A >B = C) + 3\p(A = B>C) \\
&=& \frac{5p^3 -
13p^2 + 9p}{(2-p)(3-3p+p^2)},\end{aligned}$$ and hence as before to $$\e(R) =
\frac{1}{1-{\frac{5p^3 -13p^2 +
9p}{(2-p)(3-3p+p^2)}}}$$ and $$\e(F)=\e(F/R)\e(R)=\frac{3}{p}\lr\frac{1}{1-\frac{5p^3 - 13p^2 +
9p}{(2-p)(3-3p+p^2)}}\rr.$$ Competitions of the kind discussed above are best formulated in the more general context of occupancy models as follows: $n$ balls are independently thrown into an infinite array of boxes so that any ball hits the $j$th box with probability $p_j$. Let $X_j$ be the number of balls in box $j$. Then, with $p_j=(1-p)^{j-1}p$, we have the game inspired by [@cmj] ending iff $X_j\le1\ \forall j$. Extremal versions of such questions have arisen in the literature before, often with surprising results. Motivated by a question, posed by Carl Pomerance and arising in an additive number theory context, Athreya and Fidkowski [@af] proved that the probability $\pi_n$ that the highest numbered non-empty box has exactly one ball in it converges to a constant (which is shown to be one) iff $\lim_{n\to\infty}p_n/\sum_{j=n}^\infty p_j=0$. This is a condition that is not satisfied by, e.g., the sequence $p_n=1/2^n$ for which, quite interestingly, the limit superior and the limit inferior of the sequence $\pi_n$ differ in the [*fourth*]{} decimal place. [These results had been independently obtained a few years earlier by Eisenberg et. al [@e2], [@e1], [@e3] and also by Bruss and O’Cinneide [@bc].]{} The comprehensive paper of Móri [@mori] is most relevant too: Here it is proven that given a double sequence of integer valued random variables, i.i.d. within rows, and letting $\mu(n)$ denote the multiplicity of the maximal value in the $n$th row, the limiting distribution of $\mu(n)$ does not exist in the ordinary sense – but that the intriguing empirical type a.s. limit result $$\lim_{t\to\infty}{{1}\over{\log t}}\sum_{n=1}^t{1\over n}I(\mu(n)=m)={{r^m}\over{m\log\lr{{1}\over{1-r}}\rr}},m=1,2,\ldots$$ holds, where $r$ is a parameter that depends on the distribution. The whole field appears to be extraordinarily rich with known facts and tantalizing possibilities.
Results of the kind described above indicate that the cell counts for the $n$ person (Geometric) coin game are unlikely to behave in an asymptotically smooth way if $p=p_n\not\to0$. This fact is borne out in Theorem 1, where we study the distribution of the number $W$ of pairs of equalities in the $n$ person game, with $W=0$ corresponding to the end of a “round" in the sense of [@cmj], and show that a good Poisson approximation is obtained if $np\to0$ (Geometric distribution) or $n/N\to0$ (Uniform distribution). Theorem 2 concerns itself with the distribution of the number $Y_r$ of strict $r$-way ties (=the number of boxes with exactly $r$ balls) and Theorem 3 with a multivariate generalization of Theorem 2. The approximating distribution is Poisson (Theorem 2) or a product of independent Poisson variates (Theorem 3). We note, moreover, that we were able to prove a result such as Theorem 3 relatively easily [*probably*]{} due to the approach taken – we use as a counter the event that $r$ [*specific*]{} balls go into the same urn, rather than the conventional approach (e.g., [@bhj], Section 6.2) of counting the number of urns with $r$ balls. See also [@agg], [@kolchin], [@abt1],[@abt2], and [@abt3].
Results
=======
Let $\{X_j\}_{j=1}^n$ be an integer valued sequence of i.i.d. random variables with $\p(X_1=i)=p_i$, and consider the U-statistic $$W=\sum_{K=1}^\nt I_K,$$ where, with $\ck$ denoting the $K$th 2-subset of $\{1,2,\ldots,n\}$, $I_K=1$ if $X_i=X_j; i,j\in\ck$ ($I_K=0$ otherwise). For any two discrete random variables $T$ and $U$, let $\tv(\cl(T),\cl(U))$ denote the usual total variation distance between their distributions $\cl(T)$ and $\cl(U)$, i.e., $$\tv(\cl(T),\cl(U))=\sup_{A\subseteq\z^+}\vert\p(T\in A)-\p(U\in A)\vert,$$ and let $\P(\l)$ be the Poisson r.v. with parameter $\l=\e(W)$. Then $$\tv(\cl(W),\P(\l))\le 2 n\pi+{{2 n\rho}\over{\pi}},$$ where $\pi=\p(X_1=X_2)$ and $\rho=\p(X_1=X_2=X_3)$.
[**Proof**]{} The proof is an elementary application of, e.g., Theorem 2.C.5 in [@bhj], which yields with $\l=\e(W)={n\choose2}\pi$ $$\begin{aligned}
&&\tv(\cl(W),\P(\l)\\
&&{}\le{{1-e^{-\l}}\over{\l}}\lr\sum_K\p^2(I_K=1)+\sum_K\sum_{\{L:\ck\cap\cl\ne\emptyset\}}\lc\e(I_KI_L)+\pi^2\rc\rr\\
&&{}\le\pi+{{2(n-2)\rho}\over{\pi}}+2(n-2)\pi\\
&&{}\le2n\pi+{{2n\rho}\over{\pi}},\end{aligned}$$ as asserted.
In Theorem 1, if the variables are uniform over $\{1,2,\ldots,N\}$, then $\pi=1/N; \rho=1/N^2$, so that $\tv(\cl(W),\P(\l))\le4n/N\to0$ if $N\gg n$, where, throughout this paper we write for $f_n,g_n\ge0$, $f_n\ll g_n$ (or $g_n\gg f_n$) if $f_n/g_n\to0$ as $n\to\infty$. If the variables are Geometric($p$), then the discussion in Section 1 yields $\pi=p/(2-p)$ and $\rho=p^2/(3-3p+p^2)$, so that we get $\tv(\cl(W),\P(\l))\le 2np/(2-p)+2np(2-p)/(3-3p+p^2)\le6np\to0$ if $p\ll1/n$. For the $n$ person game discussed in Section 1, we thus get $$\begin{aligned}
\p(W=0)&=&\p({\rm no\ ties})
=\exp\{-(n(n-1)p)/(2(2-p))\}\pm6np\\&=&\exp\{-\l\}\pm6np,\end{aligned}$$ $$\e(R)={{1}\over{\p(W=0)}}={{1}\over{e^{-\l}\pm6np}},$$ and $$\e(F)={n\over p}\e(R).$$
The random variable $W$, while providing us with some insight, does not yield the level of detail that we desire. For this reason, we turn our attention next to the variable $Y_r$ that counts the “number of strict $r$-way ties," or, in other words, the “number of boxes with exactly $r$ balls." The development that follows is alternative to that provided, say, in [@bhj], Theorems 6.C, 6.E, and particularly 6.F, though we do not make too many comparisons between our results and those of [@bhj], since our main focus will be on the multivariate Theorem 3; the strategy of looking at specific sets of $r$ players is what sets our method apart.
Letting as before $\{X_j\}_{j=1}^n$ be an integer valued sequence of i.i.d. random variables with $\p(X_1=i)=p_i$, we denote by $$\pi=\sum_{x=1}^\infty p_x^r(1-p_x)^{n-r}$$ the probability that a specific set of $r$ players are involved in a strict tie, and thus $$\l={n\choose r}\pi$$ is the expected number of boxes with exactly $r$ balls. Throughout this paper we will employ, as in the previous sentence, the dual analogies of “balls in boxes" and “ties between contestants." It may be readily verified that $\pi=(1/N^{r-1})(1-N^{-1})^{n-r}$ in the uniform case, and that in the geometric case $\pi=\sum_{x=1}^\infty(1-p)^{rx-r}p^r(1-(1-p)^{x-1}p)^{n-r}$ may be estimated as follows; the estimates may be seen to be tight provided that $p\to0$. First we have $$\begin{aligned}
\pi&=&\sum_{x=1}^\infty(1-p)^{rx-r}p^r(1-(1-p)^{x-1}p)^{n-r}\\
&\ge&\sum\limits_{x=1}^\infty(1-p)^{rx-r}p^r[1-(n-r)(1-p)^{x-1}p] \\
&=&
\sum\limits_{x=1}^\infty(1-p)^{rx-r}p^r -
(n-r)\sum\limits_{x=1}^\infty(1-p)^{rx-r}p^r(1-p)^{x-1}p \\
&\ge&
\frac{p^{r-1}}{r} - \frac{2(n-r)p^r}{(r+1)(2-rp)},\end{aligned}$$ where the above inequalities follow since $(1-p)^r\ge 1-rp$ and $(1-p)^{r+1}\le 1-(r+1)p+[(r+1)rp^2]/2$. Next note that $$\begin{aligned}
\pi&=&\sum_{x=1}^\infty(1-p)^{rx-r}p^r(1-(1-p)^{x-1}p)^{n-r}\\
&\le&\sum\limits_{x=1}^\infty(1-p)^{rx-r}p^r\cdot\\
{}&&\lr1-(n-r)(1-p)^{x-1}p +
\frac{(n-r)(n-r-1)}{2}(1-p)^{2x-2}p^2\rr\\
&=& \frac{p^r}{1-(1-p)^r} - \frac{(n-r)p^{r+1}}{1-(1-p)^{r+1}} +
\frac{(n-r)(n-r-1)}{2}\frac{p^{r+2}}{1-(1-p)^{r+2}}\\
&\le&
\frac{2p^{r-1}}{r(2-(r-1)p)} - \frac{(n-r)p^r}{r+1} +
\frac{(n-r)^2p^{r+1}}{(r+2)(2-(r+1)p)},\end{aligned}$$ so that in the geometric case, $\pi\sim p^{r-1}/r$ provided that $np^{(r+1)/2}\to0; rp\to0$.
We shall use the coupling approach as in [@bhj] to show that $\cl(Y_r)$ may be closely approximated by a Poisson distribution with the same mean. We need to first find, given a sum $\sum_{j=1}^nI_j$ of indicator variables, a sequence $\{J_{ij}\}$ of indicator variables, defined on the same probability space as the $I_j$s, so that for each $j$, $$\cl(J_{1j},J_{2j},\ldots,J_{nj})=\cl(I_1,I_2,\ldots,I_n\vert I_j=1).$$ Good error bounds on a Poisson approximation are obtained if the $J_{ji}$s are chosen in a fashion that makes them “not too far apart" from the $I_j$s. We proceed in a manner similar to that in Theorem 6.F in [@bhj], but the coupling we use is conditional, thus imparting a different flavor to the argument: We have $$Y_r=\sum_{j=1}^\nr I_j,$$ where $I_j=1$ if the $j$th $r$-set is engaged in a strict tie. Now we define the indicator variable $I_{jx}$ as being one if and only if $I_j$=1 and the members of the $j$th $r$-set all have “value" $x$. Now we proceed as follows: If $I_{jx}=1$, we “do nothing", setting $J_i=I_i$ for all $i$. If, however, $I_{jx}=0$, we move all members of the $j$th $r$ set into the $x$th box (some of these might of course have occupied the $x$th box to begin with), while ejecting all its “illegal" occupants and moving each these independently with probability $p_k/(1-p_x)$ to box $k;k\ne x$. Finally we set $J_i=J_{ijx}=1$ if the $i$th $r$ set is involved in a strict tie [*after*]{} this interchange. We need to verify that (1) holds in the modified form $$\cl(J_{1jx},J_{2jx},\ldots,J_{\nr jx})=\cl(I_1,I_2,\ldots,I_\nr\vert I_{jx}=1);$$ while this may be viewed as being “obvious," we provide a proof next. To show that (2) holds, it clearly suffices to show that any configuration (or sample point) corresponding to the members of the $j$th $r$-set being “the only occupants of the $x$th box" is equally likely under both the conditional and unconditional models in (2). This strategy will achieve more, in fact, since we will not have to verify a condition similar to (2) when we move on to the multivariate case.
We let $a_i$ denote the score of the $i$th player not in the $r$-clique in question ($a_i\ne x$), and $b_i$ the score of the $i$th player in the $r$-clique, so that $b_i = x$. Now
$$\begin{aligned}
\p({\rm configuration}\vert
I_{jx}=1)
&=& {{\p({\rm configuration})}\over{\p(I_{jx}=1)}}\\
&=&\frac{\p(a_1, a_2, \ldots, a_{n-r},
b_{n-r+1}, \ldots, b_n)}{p_x^r(1-p_x)^{n-r}}
\\
&=&\frac{\p(a_1)\p(a_2)\ldots\p(a_{n-r})}{(1-p_x)^{n-r}}.\end{aligned}$$
Note also that the probability of the configuration under the coupled model is given by $$\sum\limits_{l=0}^r {{r} \choose {l}}p_x^l[1-
p_x]^{r-l}\sum\limits_{S \subseteq {1, \ldots,n-r}} p_x^{\vert{S}\vert}\prod\limits_{j \in\{1,2,\ldots,n-r\}\setminus
S}p(a_j)\prod\limits_{j \in
S}\frac{p(a_j)}{1-p_x}.$$ Now $$\begin{aligned}
{}&&\sum\limits_{S \subseteq {1, \ldots,n-r}}p_x^{\vert{S}\vert}\prod\limits_{j \in\{1,2,\ldots,n-r\}\setminus
S}p(a_j)\prod\limits_{j \in
S}\frac{p(a_j)}{1-p_x}\\ \quad&=&
\sum\limits_{t=0}^{n-r}{{n-r} \choose
{t}}\lr\frac{p_x}{1-p_x}\rr^t\prod\limits_{j=1}^{n-r}p(a_j)\\
&=& \lr\frac{p_x}{1-p_x} +
1\rr^{n-r}\prod\limits_{j=1}^{n-r}p(a_j) \\
&=& \lr\frac{1}{1-p_x}\rr^{n-r}\prod_{j=1}^{n-r}p(a_j), \\\end{aligned}$$ which shows that (3) yields the same expression as before. This proves the claim.
Now Theorem 2.B in [@bhj] leads to the following inequality: $$\begin{aligned}
&&\tv(\cl(Y_r),\P(\l))\nonumber\\
&&{}\le\lr{{1-e^{-\l}}\over{\l}}\rr\sum_j\sum_x\p(I_{jx}=1)\lc\p(I_j=1)+\sum_{i\ne j}\p(I_i\ne J_{ijx})\rc\nonumber\\
&&{}\le\pi+\lr{{1-e^{-\l}}\over{\l}}\rr\sum_j\sum_x\p(I_{jx}=1)\sum_{i\ne j}\p(I_i\ne J_{ijx})\end{aligned}$$ where $\l=\e(Y_r)$, $\pi=\p(I_j=1)$, and the coupled sequence $\{J_i\}=\{J_{ijx}\}$ satisfies (2) for each $j$ and $x$. Consider first the case $\p(I_i=0,J_{ijx}=1)$, which is clearly impossible when $\vert i\cap j\vert\ge1$, and which we shall call Case I. We thus have for $\vert i\cap j\vert=0$, $$\sum_{i\ne j}\p(I_i=0,J_{ijx}=1)
={{n-r}\choose{r}}\sum_{y\ne x}\p(I_i=0,J_{ijx}=1,y),$$ where the summand $\p(I_i=0,J_{ijx}=1,y)$ represents the probability that the $i$th $r$-set is not engaged in a strict $r$-way tie before the coupling, but is part of such a tie with common value $y$ after the coupling. (5) thus yields $$\begin{aligned}
\sum_{i\ne j}\p(I_i=0,J_{ijx}=1)&=&\sum_{i\ne j}\sum_{y\ne x}\lc\p(J_{ijx}=1,y)-\p(I_i=1,J_{ijx}=1,y)\rc\\&=&{{n-r}\choose{r}}\sum_{y\ne x}\mu_y\end{aligned}$$ where $$\begin{aligned}
\mu_y&=&\sum_{q=0}^r{r\choose q}p_y^qp_x^{r-q}\sum_{s=0}^{n-2r}{{n-2r}\choose{s}}p_x^s(1-p_x-p_y)^{n-2r-s}\lr{{p_y}\over{1-p_x}}\rr^{r-q}\cdot\nonumber\\
&&{}\lr{{1-p_x-p_y}\over{1-p_x}}\rr^{s}\nonumber\\
&&{}-(1-p_y)^rp_y^r\sum_{s=0}^{n-2r}{{n-2r}\choose{s}}p_x^s(1-p_x-p_y)^{n-2r-s}\lr{{1-p_x-p_y}\over{1-p_x}}\rr^{s}\nonumber\\
&&{}=\sum_q{r\choose q}p_y^r\lr{{p_x}\over{1-p_x}}\rr^{r-q}\sum_s{{n-2r}\choose{s}}\lr{{p_x}\over{1-p_x}}\rr^{s}\cdot\nonumber\\&&{}(1-p_x-p_y)^{n-2r}\nonumber\\
&&{}-p_y^r(1-p_y)^r\sum_s{{n-2r}\choose{s}}\lr{{p_x}\over{1-p_x}}\rr^{s}(1-p_x-p_y)^{n-2r}\nonumber\\
&&{}=p_y^r\lr{{1}\over{1-p_x}}\rr^{n-2r}(1-p_x-p_y)^{n-2r}\lr{{1}\over{1-p_x}}\rr^{r}\nonumber\\
&&{}-p_y^r(1-p_y)^r(1-p_x-p_y)^{n-2r}\lr{{1}\over{1-p_x}}\rr^{n-2r}\nonumber\\
&&{}=p_y^r\lr1-{{p_y}\over{1-p_x}}\rr^{n-2r}\lc\lr{{1}\over{1-p_x}}\rr^{r}-(1-p_y)^r\rc\end{aligned}$$ We now check to see the nature of the bound (6) in the uniform and geometric cases: When the balls are distributed uniformly in $N$ boxes, we see that (6) leads to $$\begin{aligned}
%&&{{n-r}\choose{r}}(N-1){{1}\over{N^r}}\lr1-{1\over n}\rr^{n-2r}\lc{{1}\over{(1-{1\over n})^r}}-(1-{1\over n})^r\rc\nonumber\\&&{}\approx{{n-r}\choose{r}}{{2r}\over{N^r}}e^{-n/N}\nonumber\\
%&&{}\approx{{2r}\over{N}}\l,
&&{}\sum_{i\ne j}\p(I_i=0,J_{ijx}=1)\nonumber\\&&{}={{n-r}\choose{r}}\sum_{y\ne x}\mu_y\nonumber\\
&&{}\le\nr\cdot N\cdot {{1}\over{N^r}}\lr1-{1\over N}\rr^{n-2r}\lc\lr{{N}\over{N-1}}\rr^r-\lr{{N-1}\over{N}}\rr^r\rc\nonumber\\
&&{}=\l \lr1-{1\over N}\rr^{-2r}\lc1-\lr1-{1\over N}\rr^{2r}\rc\nonumber\\
&&{}\le{{2r}\over{N}}\exp\{2r/(N-1)\}\l,\end{aligned}$$ while in the geometric case we have $$\begin{aligned}
&&\sum_{i\ne j}\p(I_i=0,J_{ijx}=1)\nonumber\\
&&{}={{n-r}\choose{r}}\sum_{y\ne x}p_y^r\lr1-{{p_y}\over{1-p_x}}\rr^{n-2r}\lc{{1}\over{(1-p^x)^r}}-(1-p_y)^r\rc\nonumber\\
&&{}\le\nr\sum_{y=1}^\infty p_y^r(1-p_y)^{n-r}(1-p_y)^{-r}\lr(1-p_x)^{-r}-(1-p_y)^r\rr\nonumber\\
&&{}\le\l\max_{x,y}\lc(1-p_x)^{-r}(1-p_y)^{-r}-1\rc\nonumber\\
&&{}\le\l\lr{{1}\over{1-p}}\rr^{2r}\lc1-(1-p)^{2r}\rc\nonumber\\
&&{}\le\l\exp\{2rp/(1-p)\}2rp.\end{aligned}$$ We now consider the case where $I_i=1$ and $J_{ijx}=0$ (Case II). We clearly have $\p(I_i=1, J_{ijx}=0)=\p(I_i=1)$ for $\vert i\cap j\vert\ge1$ (Case II$'$), so we obtain $$\begin{aligned}
\sum_{\vert i\cap j\vert\ge1}\p(I_i=1,J_{ijx}=0)&\le&r{{n-1}\choose{r-1}}\pi\nonumber\\
&=&{{r^2}\over{n}}\l.\end{aligned}$$ Next assume (Case II$''$) that $\vert i\cap j\vert=0$, and we seek to estimate the probability $\p(I_i=1, J_{ijx}=0)$. If $y=x$, we bound $\p(I_i=1, J_{ijx}=0)$ by $\pi_x$, so that $$\sum_{\vert i\cap j\vert=0}\p(I_{ix}=1, J_{ijx}=0)\le{{n-r}\choose{r}}\pi_x.$$ If, on the other hand, $y\ne x$, then we have $$\begin{aligned}
\sum_{\vert i\cap j\vert=0}\p(I_{i}=1, J_{ijx}=0)&=&{{n-r}\choose{r}}\sum_{y\ne x}\p(I_{iy}=1, J_{ijx}=0)\nonumber\\
&\le&{{n-r}\choose{r}}\sum_{y\ne x}\sum_{q\ge 1}{{n-2r}\choose{q}}p_x^qp_y^r{{qp_y}\over{1-p_x}};\end{aligned}$$ the above equation follows since in order for $I_{iy}=1, J_{ijx}=0$ to occur, we must have at least one of the $q$ “bad" balls present in urn $x$ land in urn $y$ and thus “spoil" the fact that $I_{iy}=1$. We thus get $$\begin{aligned}
\sum_{\vert i\cap j\vert=0}\p(I_{i}=1, J_{ijx}=0)&\le&{{n-r}\choose{r}}\sum_{y\ne x}{{p_y^{r+1}}\over{1-p_x}}\sum_{q\ge1}{{n-2r}\choose{q}}qp_x^q\nonumber\\
&\le&{{n-r}\choose{r}}n{{p_x}\over{1-p_x}}(1+p_x)^{n-2r-1}\sum_{y\ne x}{p_y^{r+1}}\nonumber\\
&\le&{{n-r}\choose{r}}n{{p_x}\over{1-p_x}}e^{np_x}\sum_{y\ne x}p_y^{r+1}.\end{aligned}$$ Now (11) reduces in the uniform case to $$\begin{aligned}
{{n-r}\choose{r}}{n\over {N-1}}e^{n/N}{{N-1}\over{N^{r+1}}}&\le&{{n}\over{N^2}}\l{{e^{n/N}}\over{\lr1-{1\over N}\rr^{n-r}}}\nonumber\\
&\le&{{n}\over{N^2}}\l{e^{n/N}}\exp\{(n-r)/(N-1)\}\nonumber\\
&\le&{{n}\over{N^2}}\l e^{2n/(N-1)}\end{aligned}$$ and is bounded in the geometric case by $${{n-r}\choose{r}}{{np^{r+2}}\over{1-p}}\sum(1-p)^{(y-1)(r+1)}\le\l np^2e^{np}(1+o(1)),$$ provided that $np^{(r+1)/2}\to0, rp\to0$. Equations (4), (6), (9), (10) and (11) now yield $$\begin{aligned}
&&\tv(\cl(Y_r),\P(\l))\nonumber\\
&&{}\le\pi+\lr{{1-e^{-\l}}\over{\l}}\rr\sum_j\sum_x\p(I_{jx}=1)\sum_{i\ne j}\p(I_i\ne J_{ijx})\nonumber\\
&&{}\le\pi+\lr1\wedge{1\over\l}\rr\sum_j\sum_x\p(I_{jx}=1)\bullet\nonumber\\
&&{}\bigg[{{n-r}\choose{r}}\sum_{y\ne x}p_y^r\lr1-{{p_y}\over{1-p_x}}\rr^{n-2r}\lc\lr{{1}\over{1-p_x}}\rr^{r}-(1-p_y)^r\rc+{{r^2}\over{n}}\l\nonumber\\
&&{}+{{n-r}\choose{r}}\pi_x+{{n-r}\choose{r}}n{{p_x}\over{1-p_x}}e^{np_x}\sum_{y\ne x}p_y^{r+1}\bigg].\end{aligned}$$ We next evaluate (14) in the uniform case: Equations (4), (7), (9), (10) and (12) give $$\begin{aligned}
&&\tv(\cl(Y_r),\P(\l))\nonumber\\
&&{}\le\pi+\lr1\wedge{1\over\l}\rr\sum_j\sum_x\p(I_{jx}=1)\sum_{i\ne j}\p(I_i\ne J_{ijx})\nonumber\\
&&{}\le \pi+\lr1\wedge{1\over\l}\rr\l\cdot\nonumber\\
&&{}\lc{{2r}\over{N}}\l\exp\{2r/(N-1)\}+{{r^2}\over{n}}\l+{{n-r}\choose{r}}{\pi\over N}+{{n}\over{N^2}}\l\exp\{2n/(N-1)\}\rc\nonumber\\
&&{}=\pi+\lr\l\wedge{\l^2}\rr\cdot\nonumber\\
&&{}\lc{{2r}\over{N}}\exp\{2r/(N-1)\}+{{r^2}\over{n}}+{1\over N}+{{n}\over{N^2}}\exp\{2n/(N-1)\}\rc.\end{aligned}$$ We compare (15) with Equation 6.2.18 in [@bhj], which yields the upper bound $$\tv(\cl(Y_r),\P(\l))\le\lr\l\wedge{\l^2}\rr\lc{1\over N}+{{6n}\over{N^2}}+{{6r^2}\over{n}}\rc;$$ it is evident that (15) provides a better estimate if $${{2r}\over{N}}\exp\{2r/(N-1)\}\le{{5r^2}\over{n}}+(6-\exp\{2n/(N-1)\}){{n}\over{N^2}},$$ which is a condition that holds under a wide range of circumstances, and certainly if $n/N\to0$. Now in the geometric case, Equations (4), (8), (9), (10), and (13) reveal that (14) reduces as follows: $$\begin{aligned}
&&\tv(\cl(Y_r),\P(\l))\nonumber\\
&&{}\le\pi+\lr1\wedge{1\over\l}\rr\sum_j\sum_x\p(I_{jx}=1)\bullet\nonumber\\
&&{}\lr 2\l rp\exp\{2rp/(1-p)\}+{{r^2}\over{n}}\l+\nr\pi_x+\l np^2\rr\nonumber\\
&&{}\le \pi+(\l\wedge\l^2)\lc2rp\exp\{2rp/(1-p)\}+{{r^2}\over{n}}+rp+np^2e^{np}\rc.\end{aligned}$$ We have thus proved
Let $\{X_j\}_{j=1}^n$ be an integer valued sequence of i.i.d. random variables with $\p(X_1=i)=p_i$. Define $Y_r$ to be the number of strict $r$-way ties between these random variables. Then the total variation distance between $\cl(Y_r)$ and a Poisson distribution with the same mean is given by (14). This expression reduces to the one in Equation (15) when the distribution of the $X_i$s is uniform on $\{1,2,\ldots,N\}$ and to the expression in Equation (16) when $X_1\sim {\rm Geo}(p)$.
For the rest of the paper we will, for simplicity, restrict our attention to the classical occupancy problem of $n$ balls in $N$ boxes, assuming furthermore, that $n/N\to0$. The goal is to obtain a multivariate Poisson approximation for the vector $Z_{AB}=\{Y_A,Y_{A+1}\ldots,Y_B\}$, for suitably restricted $A$ and $B$, and where the approximating Poisson vector consists of independent components. First consider the quantities $\{\l_a:A\le a\le B\}.$ Since $$\begin{aligned}
\l_a&=&{n\choose a}\lr{1\over N}\rr^{a-1}\lr1-{1\over N}\rr^{n-a}\\&\sim&{{1}\over{\sqrt{2\pi a}}}N\lr{{ne}\over{Na}}\rr^a\exp\{(n-a)/N\}\\
&\sim&{{1}\over{\sqrt{2\pi a}}}N\lr{{ne}\over{Na}}\rr^a(1+o(1)),\end{aligned}$$ it follows, due to the fact that $n/N\to0$, that $\l_a$ is monotone decreasing in $a$. Suppose that $\l_A<\infty$ for some finite $A$. It then follows that the approximating Poisson distribution for $Y_{A+1}$ would have mean close to zero, making our agenda somewhat uninteresting. We shall assume therefore that $\l_A\to\infty$ as $n,N\to\infty$. Choices of the parameters that make this occur might be, e.g., $n=N^\alpha; \alpha<1$, when $\e(Y_a)\to0$ for all $a\ge A_0$, or, more interestingly, $n=N/\log N$ in which case the threshold $A_0$ would tend to infinity with $N$. We thus seek values of $A$ and $B$ for which we get an “interesting" multivariate Poisson approximation for the ensemble $(Y_A,\ldots,Y_B)$. Now Theorem 10.J in [@bhj] yields, using notation suggested by that used in the proof of Theorem 2, $$\begin{aligned}
&&{}\tv(\cl(Y_A,\ldots,Y_B),\prod_{a=A}^B\P(\l_a))\nonumber\\&&{}\le\sum_{a=A}^B\sum_{j=1}^{n\choose a}\sum_{x=1}^N\p(I_{ajx}=1)\lc\p(I_{aj}=1)+\sum_{biy\ne ajx}\p(I_{biy}\ne J_{biy})\rc,\end{aligned}$$ where the last sum does not include the case $bi=aj$. Correspondingly, we let $T_1,T_2,T_3$ denote the quantities $$\sum_{a=A}^B\sum_{j=1}^{n\choose a}\sum_{x=1}^N\p(I_{ajx}=1)\p(I_{aj}=1)$$ $$\sum_{a=A}^B\sum_{j=1}^{n\choose a}\sum_{x=1}^N\p(I_{ajx}=1)\sum_{iy\ne jx}\p(I_{aiy}\ne J_{aiy})$$ and $$\sum_{a=A}^B\sum_{j=1}^{n\choose a}\sum_{x=1}^N\p(I_{ajx}=1)\sum_{b\ne a}\sum_{i=1}^{n\choose b}\sum_{y=1}^N\p(I_{biy}\ne J_{biy})$$ respectively; we need to compute the sum $T_1+T_2+T_3.$ First, we see that $$\begin{aligned}
T_1&=&\sum_a\sum_j\p^2(I_{aj}=1)\nonumber\\
&=&\sum_a{n\choose a}\frac{1}{N^{2a-2}}\lr1-\frac{1}{N}\rr^{2n-2a}\nonumber\\
&\le&N^2\sum_{a=A}^B\lr\frac{ne}{aN^2}\rr^a\nonumber\\
&\le&N^2\sum_{a=2}^B\lr\frac{ne}{2N^2}\rr^a\nonumber\\
&=&\frac{e^2}{4}\frac{n^2}{N^2}(1+o(1))\to0\end{aligned}$$ for each $A,B$. The computation of $T_2$ follows as in the proof of Theorem 2. The first component, $T_{21}$ is, by (7), given by $$\begin{aligned}
T_{21}&=&\sum_{a=A}^B\sum_j\sum_x\p(I_{ajx}=1)\cdot\frac{2a}{N}\exp\{2a/(N-1)\}\l_a\nonumber\\
&\le&\sum_a\l_a^2\frac{2a}{N}(1+o(1)).\end{aligned}$$ Under what circumstances might the bound in (19) tend to zero? Let us pause to consider this question before continuing. If $n=\sqrt{N\log N}$ and $A=2$, then $\l_A\sim(\log N)/2$, the error bound of Theorem 2 is of magnitude $\sqrt{\log N/N}$, and the bound in (19) [*does*]{} approach zero. However in this case $\l_3\to0$ so we are able to derive little useful beyond a Poisson approximation for $Y_2$, the number of “days" with exactly two “birthdays". If $n=N^{0.9}$, then $\l_a\sim N^{1-0.1a}\to\infty$ for all $a=2,3,\ldots9$ but the summands in (19), asymptotically equal to $N^{1-0.2a}$, tend to zero only if $a=6,7,8,9$. We thus have a potential multivariate approximation for $(Y_6,Y_7,Y_8,Y_9)$. Finally, let $n=N/\log N$. In this case, $\l_a\sim(e/a\log N)^a\cdot N$, and, with $a=\log N/(2\log\log N)$, for example, we see that $$\begin{aligned}
\l_a&\sim&\lr\frac{e}{a\log N}\rr^a\cdot N\\
&=&\lr\frac{2e\log \log N}{\log^2 N}\rr^{\frac{\log N}{2\log\log N}}\cdot \lr e^{2\log\log N}\rr^{\frac{\log N}{2\log\log N}}\\
&=&\lr{2e\log \log N}\rr^{\frac{\log N}{2\log\log N}}\to\infty,\end{aligned}$$ while with $a=\log N/[(4-\varepsilon)\log\log N]$ (we use $a=\log N/(3\log\log N)$ below) we have $$\begin{aligned}
\frac{\l_a^2}{N}&\sim&\lr\frac{e}{a\log N}\rr^{2a}\cdot N\\
&=&\lr\frac{3e\log \log N}{\log^2 N}\rr^{\frac{2\log N}{3\log\log N}}\cdot \lr e^{\frac{3\log\log N}{2}}\rr^{\frac{2\log N}{3\log\log N}}\\
&=&\lr{\frac{3e\log \log N}{\log^{1/2}N}}\rr^{\frac{2\log N}{3\log\log N}},\end{aligned}$$ which leads, with $A=\log N/(3\log\log N)$ and $B=\log N/(2\log\log N)$, to $$T_{21}\le 2(B-A)\frac{\l_A^2}{N}\le\frac{\log N}{3\log\log N}\lr{\frac{3e\log \log N}{\log^{1/2}N}}\rr^{\frac{2\log N}{3\log\log N}}\to0;$$ we thus have a potential Poisson approximation for the vector $(Y_A\ldots Y_B)$. Next note that the term $T_{22}$ that corresponds to (9) is given by $$T_{22}=\sum_{a=A}^B\sum_{j=1}^{n\choose a}\sum_{x=1}^N\p(I_{ajx}=1)\frac{a^2}{n}\l_a=\sum_a\frac{a^2}{n}\l_a^2.$$ Finally we combine the two remaining terms (10) and (12), as reflected in (15), to get $$\begin{aligned}
T_{23}&=&\sum_{a=A}^B\sum_{j=1}^{n\choose a}\sum_{x=1}^N\p(I_{ajx}=1)\lr\frac{{n\choose a}\pi_a}{N}+\frac{n}{N^2}\l_a\exp\{2n/(N-1)\}\rr\nonumber\\
&=&\sum_a\frac{\l_a^2}{N}+\frac{n}{N^2}\l_a^2(1+o(1)).\end{aligned}$$ Turning to a computation of $T_3$, we first observe that for $a\ne b$ and $\vert i\cap j\vert\ge1$, it is impossible for $I_{biy}=0,J_{biy}=1$ to occur. Accordingly, as in the calculation leading up to (6) we see that $$\begin{aligned}
&&{}\sum_b\sum_i\sum_y\p(I_{biy}=0,J_{biy}=1)\nonumber\\
&&{}=\sum_b\sum_i\sum_y\{\p(J_{biy}=1)-\p(I_{biy}=1,J_{biy}=1)\}\nonumber\\
&&{}=\sum_b{{n-a}\choose{b}}\sum_y\Huge\{\sum_{q=0}^b{b\choose q}\lr\frac{1}{N}\rr^q\lr\frac{1}{N}\rr^{b-q}\cdot\nonumber\\
&&{}\sum_s{{n-a-b}\choose{s}}\lr\frac{1}{N}\rr^s\lr1-\frac{2}{N}\rr^{n-a-b-s}\lr\frac{1}{N-1}\rr^{b-q}\lr\frac{N-2}{N-1}\rr^s-\nonumber\\
&&{}\lr\frac{N-1}{N}\rr^a \lr\frac{1}{N}\rr^b\sum_s{{n-a-b}\choose{s}}\lr\frac{1}{N}\rr^s\lr1-\frac{2}{N}\rr^{n-a-b-s}\lr\frac{N-2}{N-1}\rr^s\Huge\}\nonumber\\
&&{}=\sum_b{{n-a}\choose{b}}N\cdot\nonumber\\
&&{}\lc\lr\frac{1}{N-1}\rr^b\lr\frac{N-2}{N-1}\rr^{n-a-b}-\lr\frac{N-1}{N}\rr^a\lr\frac{1}{N}\rr^b\lr\frac{N-2}{N-1}\rr^{n-a-b}\rc\nonumber\\
&&{}\le\sum_b\l_b\frac{(a+b)}{N}(1+o(1)).\end{aligned}$$ As in the univariate case, $\p(I_{biy}=1,J_{biy}=0)=\p(I_{biy}=1)$ if $\vert i\cap j\vert\ge1$. Hence $$\begin{aligned}
\sum_b\sum_{\vert i\cap j\vert\ge1}\sum_y\p(I_{biy}=1,J_{biy}=0)&\le&\sum_ba{{n-1}\choose{b-1}}\pi_b\nonumber\\
&=&\sum_b \frac{ab}{n}\l_b.\end{aligned}$$ If, however, $\vert i\cap j\vert=0$, then $$\sum_b\sum_{\vert i\cap j\vert=0}\p(I_{bix}=1,J_{bix}=0)\le\sum_b{{n-a}\choose{b}}\pi_{bx},$$ and, being rather crude with the final estimation $$\begin{aligned}
&&{}\sum_b\sum_{\vert i\cap j\vert=0}\sum_{y\ne x}\p(I_{biy}=1,J_{biy}=0)\nonumber\\
&&{}\le\sum_b{{n-a}\choose{b}}\frac{1}{N^b}\sum_y\sum_{q\ge1}{{n-a-b}\choose{q}}\frac{1}{N^q}\frac{q}{N-1}\nonumber\\
&&{}\le\sum_b{n\choose b}\frac{n}{N^{b+1}}\sum_{q\ge1}{{n-1}\choose{q-1}}\frac{1}{N^{q-1}}(1+o(1))\nonumber\\
&&{}\le\sum_b\frac{n}{N^2}\l_b(1+o(1)).\end{aligned}$$ Collecting equations (18) through (25), we see that the following holds:
When $n$ balls are randomly assigned to $N$ boxes, where $n\ll N$, the joint distribution of the multivariate vector $(Y_A,\ldots,Y_B)$ of exact box counts may be approximated by a Poisson vector with independent components. More specifically, $$\tv\lr\cl(Y_A,\ldots,Y_B),\prod_{a=A}^B\P(\l_a) \rr\le \varepsilon_{n,N,A,B}$$ where $\l_a=\e(Y_a)={n\choose a}\frac{1}{N^{a-1}}\lr1-\frac{1}{N}\rr^{n-a}$ and $\varepsilon_{n,N,A,B}$ is of magnitude $$\sum_{a=A}^B\l_a^2\lr\frac{2a}{N}+\frac{a^2}{n}+\frac{1}{N}+\frac{n}{N^2}\rr+\sum_a\l_a\lr\sum_{b\ne a}\l_b\lr \frac{(a+b)}{N}+\frac{ab}{n}+\frac{1}{N}+\frac{n}{N^2}\rr\rr.$$ In addition, an application, e.g., of Theorem 10.K in [@bhj] may provide slight improvements in the above, through a partial reinstatement of the so-called “magic factor".
[**Acknowledgment**]{} The research of all three authors was supported by NSF Grants DMS-0049015 and DMS-0552730, and was conducted at East Tennessee State University, when Eaton and Sinclair were undergraduate students at Rochester University and the University of the Redlands respectively. An earlier version of this work was presented at IWAP-Piraeus and a completed version was presented at Lattice Path Combinatorics-Johnson City.
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abstract: |
The coupled cluster method (CCM) is applied to the spin-one anisotropic Heisenberg antiferromagnet (HAF) on the square lattice at zero temperature using a new high-order CCM ground-state formalism for general quantum spin number ($s \ge 1/2$). The results presented constitute the first such application of this new formalism, and they are shown to be among the most accurate results for the ground-state energy and sublattice magnetisation of this model as yet determined. We “track” the solution to the CCM equations at a given level of approximation with respect to the anisotropy parameter, $\Delta$, from the trivial Ising limit ($\Delta
\rightarrow \infty$) down to a critical value $\Delta_c$, at which point this solution terminates. This behaviour is associated with a phase transition in the system, and hence a primary result of these high-order CCM calculations is that they provide an accurate and unbiased (i.e., [*ab initio*]{}) estimation of the position of the quantum phase transition point as a function of the anisotropy parameter. Our result is, namely, that this point occurs at (or slightly below) the isotropic Heisenberg point at $\Delta=1$, for this model.
PACS numbers: 75.10.Jm, 75.50.Ee, 03.65.Ca
author:
- 'D.J.J. Farnell, K.A. Gernoth, and R.F. Bishop'
title: 'Phase Transition Of The Spin-One Square-Lattice Anisotropic Heisenberg Model'
---
The area of quantum magnetic insulating systems at zero temperature has become increasingly well understood for one-dimensional (or quasi-one-dimensional) lattices via the existence of well-known exact solutions such as the Bethe ansatz (BA) [@ba1; @ba2] and also via more recent density matrix renormalisation group (DMRG) calculations [@DMRG1; @DMRG2; @DMRG3]. Similarly, zero-temperature quantum Monte Carlo (QMC) calculations [@qmc3; @qmc4] have been shown to yield very accurate results for spin-half, two-dimensional systems. QMC techniques, however, are severely limited in their range of application by the presence of the infamous “sign problem,” although we note that for non-frustrated spin systems it is often possible to determine a “sign rule” [@sign_rules1; @sign_rules2] which completely circumvents this minus-sign problem. By contrast, the technique of quantum many-body theory known as the coupled cluster method (CCM) is neither limited by the presence of frustration nor by the spatial dimensionality of the lattice [@ccm2; @ccm7; @ccm4; @ccm5; @ccmextra; @ccm6]. Indeed, a great strength of this method is that it is able to determine with great accuracy the positions of quantum phase transition points of quantum systems within an [*ab initio*]{}, and thus essentially unbiased, framework. These quantum phase transitions typically arise as some parameter within the Hamiltonian is varied, thus driving the system from one phase into another.
The behaviour of the quantum Heisenberg antiferromagnet (HAF) on the linear chain is highly dependent on whether the quantum spin number, $s$, has an integer or a half-odd-integer value. Indeed, Haldane [@haldane] first predicted that the spin-one isotropic HAF would contain an excitation energy gap, and this prediction has subsequently and conclusively been shown numerically to be correct by exact diagonalisations of short chains [@parkinson] and by more recent and extremely accurate DMRG calculations [@DMRG3]. We note that this is in stark contrast to the exact BA solution [@ba1; @ba2] of the spin-half isotropic HAF for the linear chain, which contains no such gap. However, such exact diagonalisations and DMRG calculations are considerably more difficult for systems of higher spatial dimensionality. In this article, we focus on the specific case of the spin-one anisotropic HAF on the square lattice at zero temperature using the CCM, especially in relation to its quantum phase transition, by applying a high-order CCM formalism for general quantum spin number for the first time.
The spin-one anisotropic HAF is given by, $$H = \sum_{\langle i , j \rangle} \biggl \{ s_i^x s_j^x +
s_i^y s_j^y + \Delta s_i^z s_j^z \biggr \} ~~ ,
\label{eq1}$$ where the symbol $\langle i , j \rangle$ indicates nearest-neighbour bonds on the square lattice, and where each bond is counted once and once only. For $\Delta \stackrel{>}{_{\sim}} 1$, the ground state of this model contains non-zero Néel-like order. The precise value of the phase transition point, at which this ordering breaks down is not exactly known although it is believed to be at (or near to) the Heisenberg point at $\Delta=1$. Very accurate results for the values of various ground-state properties of this model have been obtained via spin-wave theory (SWT) [@Hamer] and cumulant series expansions [@Zheng]. Both sets of results indicate that approximately $80\%$ of the classical ordering remains for the spin-one, square-lattice HAF model at $\Delta=1$. We note however that cumulant series expansions make an explicit assumption that the position of the phase transition point is exactly at $\Delta=1$ in order to perform Padé resummations of the otherwise divergent perturbation series. Also, conventional SWT essentially appears to “build in” a phase transition point at $\Delta=1$ for any lattice, at which point the excitation spectrum becomes soft. This is, of course, now known to be incorrect [@DMRG3; @haldane; @parkinson] for the spin-one linear-chain HAF model, as noted above.
We now briefly describe the general CCM formalism, and the interested reader is referred to Refs. [@ccm2; @ccm7; @ccm4; @ccm5; @ccmextra; @ccm6] for further details. The exact ket and bra ground-state energy eigenvectors, $|\Psi\rangle$ and $\langle\tilde{\Psi}|$, of a many-body system described by a Hamiltonian $H$ are parametrised within the single-reference CCM as follows: $$\begin{aligned}
|\Psi\rangle = {\rm e}^S |\Phi\rangle \; &;&
\;\;\; S=\sum_{I \neq 0} {\cal S}_I C_I^{+} \nonumber \; , \\
\langle\tilde{\Psi}| = \langle\Phi| \tilde{S} {\rm e}^{-S} \; &;&
\;\;\; \tilde{S} =1 + \sum_{I \neq 0} \tilde{{\cal S}}_I C_I^{-} \; .
\label{ccm_eq2} \end{aligned}$$ The single model or reference state $|\Phi\rangle$ is required to have the property of being a cyclic vector with respect to two well-defined Abelian subalgebras of [*multi-configurational*]{} creation operators $\{C_I^{+}\}$ and their Hermitian-adjoint destruction counterparts $\{ C_I^{-} \equiv
(C_I^{+})^\dagger \}$. Thus, $|\Phi\rangle$ plays the role of a vacuum state with respect to a suitable set of (mutually commuting) many-body creation operators $\{C_I^{+}\}$. Note that $C_I^{-} |\Phi\rangle = 0,
~ \forall ~ I \neq 0$, and that $C_0^{-} \equiv 1$, the identity operator.
=7.3cm
We note that the exponentiated form of the ground-state CCM parametrisation of Eq. (\[ccm\_eq2\]) ensures the correct counting of the [*independent*]{} and excited correlated many-body clusters with respect to $|\Phi\rangle$ which are present in the exact ground state $|\Psi\rangle$. It also ensures the exact incorporation of the Goldstone linked-cluster theorem, which itself guarantees the size-extensivity of all relevant extensive physical quantities.
The CCM equations are defined by the following coupled set of equations, $$\begin{aligned}
\langle\Phi|C_I^{-} {\rm e}^{-S} H {\rm e}^S|\Phi\rangle &=&
0 , ~ \forall \;\; I \neq 0 \;\; ; \label{ccm_eq7} \\
\langle\Phi|\tilde{S} {\rm e}^{-S} [H,C_I^{+}] {\rm e}^S|\Phi\rangle
&=& 0 , ~ \forall \;\; I \neq 0 \; . \label{ccm_eq8}\end{aligned}$$ We furthermore note that the ground-state energy at the stationary point has the simple form $$E_g = E_g ( \{{\cal S}_I\} ) = \langle\Phi| {\rm e}^{-S} H {\rm e}^S|\Phi\rangle
\;\; .
\label{ccm_eq9}$$ It is important to realize that this (bi-)variational formulation does [*not*]{} lead to an upper bound for $E_g$ when the summations for $S$ and $\tilde{S}$ in Eq. (\[ccm\_eq2\]) are truncated, due to the lack of exact Hermiticity when such approximations are made. However, it is clear that the important Hellmann-Feynman theorem [*is*]{} still preserved in all such approximations.
The CCM formalism is exact in the limit of inclusion of all possible multi-spin cluster correlations for $S$ and $\tilde S$, although in any real application this is usually impossible to achieve. It is therefore necessary to utilise various approximation schemes within $S$ and $\tilde{S}$. The three most commonly employed schemes previously utilised have been: (1) the SUB$n$ scheme, in which all correlations involving only $n$ or fewer spins are retained, but no further restriction is made concerning their spatial separation on the lattice; (2) the SUB$n$-$m$ sub-approximation, in which all SUB$n$ correlations spanning a range of no more than $m$ adjacent lattice sites are retained; and (3) the localised LSUB$m$ scheme, in which all multi-spin correlations over distinct locales on the lattice defined by $m$ or fewer contiguous sites are retained. We note that in order to carry out such LSUB$m$ or SUB$n$-$m$ calculations to high order in the truncation indexes $n$ and $m$ we must rely on computer-algebra techniques in order to generate the corresponding sets of coupled equations. These computational techniques are based on the fact that the similarity transformed Hamiltonian in Eqs. (\[ccm\_eq7\]-\[ccm\_eq9\]) may be written [*purely*]{} in terms of creation operators acting on the model state, so that evaluation of Eq. (\[ccm\_eq7\]) becomes purely a matter of pattern matching. The interested reader is referred to Refs. [@ccm4] and [@ccm6] for a full explanation of how this is achieved for spin-half models. We note that the generalisation of this procedure to the case of arbitrary quantum spin number, $s$, is relatively straightforward, although detailed, and will be described fully elsewhere [@ccm8].
$N_F$ $E_g/N$ $M$ $\Delta_c$
----------------------- ------- -------------- ----------- ------------
LSUB2 2 $-$2.295322 0.909747 0.3240
SUB2[^1] – $-$2.302148 0.8961 0.9109
LSUB4 30 $-$2.320278 0.869875 0.7867
LSUB6 1001 $-$2.325196 0.851007 0.8899
LSUB$\infty$ – $-$2.3292 0.8049 0.98
SWT[^2] – $-$2.3282 0.8043 –
Series Expansions[^3] – $-$2.3279(2) 0.8039(4) –
: CCM results for the ground state of the spin-one Heisenberg antiferromagnet at $\Delta =1$ on the square lattice using the LSUB$m$ approximation with $m=\{2,4,6\}$. Values for the CCM critical points, $\Delta_c$, of the anisotropic model as a function of the anisotropy, $\Delta$, are also presented. Note that $N_F$ indicates the number of fundamental clusters at each level of approximation.
\[tab1\]
We now apply the CCM formalism outlined above to the specific case of the anisotropic HAF, and we choose the Néel state, in which the spins lie along the $z$-axis, to be the model state. Furthermore, we perform a rotation of the local axes of the up-pointing spins by 180$^\circ $ about the $y$ axis, so that spins on both sublattices may be treated equivalently. The (canonical) transformation is described by, $$s^x \; \rightarrow \; -s^x, \; s^y \; \rightarrow \; s^y, \;
s^z \; \rightarrow \; -s^z \; .$$ The model state now appears $mathematically$ to consist of purely down-pointing spins in these rotated local axes. In terms of the spin raising and lowering operators $s_k^{\pm} \equiv s_k^x \pm {\rm i} s_k^y$ the Hamiltonian may be written in these local axes as, $$H = -\frac 14 \sum_{\langle i, j \rangle} \biggl \{
s_i^+ s_{j}^+ + s_i^-s_{j}^- + 2 \Delta s_i^z s_{j}^z
\biggr \} ~~ ,
\label{eq:newH}$$ where $i$ and $j$ runs over all nearest neighbours, although each nearest-neighbour bond is counted once and once only. Furthermore, the sublattice magnetisation, $M$, (after rotation of the local spin axes) is given by, $$M = -\frac 1{N} \sum_{i=1}^{N} s_{i}^z ~~ .
\label{ccm_j_j'_3}$$ In the limit $\Delta \rightarrow \infty$ we note that the model state is the exact ground eigenstate of the Hamiltonian of Eq. (\[eq:newH\]). Hence, all of the CCM correlation coefficients are zero at this point. We may track this solution for decreasing values of $\Delta$ until we reach a [*critical point*]{}, $\Delta_c$, at which point the real solution to our CCM equations for the LSUB$m$ and SUB$m$-$m$ approximation schemes terminates. This is associated with a phase transition in the real system [@ccm4; @ccm5; @ccmextra; @ccm6]. Similar behaviour was also seen for this model for the CCM SUB2 approximation for this model (see Ref. [@ccm7]), and once more this was associated with a phase transition of the real system.
$N_F$ $E_g/N$ $M$ $\Delta_c$
----------------------- ------- -------------- ----------- ------------
SUB2-2 1 $-$2.295041 0.910013 0.3499
SUB2[^4] – $-$2.302148 0.8961 0.9109
SUB4-4 15 $-$2.319755 0.871195 0.7843
SUB6-6 375 $-$2.324863 0.852559 0.8879
SUB$\infty$ – $-$2.3291 0.8067 0.98
SWT[^5] – $-$2.3282 0.8043 –
Series Expansions[^6] – $-$2.3279(2) 0.8039(4) –
: CCM results for the ground state of the spin-one Heisenberg antiferromagnet at $\Delta =1$ on the square lattice using the SUB$m$-$m$ approximation with $m=\{2,4,6\}$. Values for the CCM critical points, $\Delta_c$, of the anisotropic model as a function of the anisotropy, $\Delta$, are also presented. Note that $N_F$ indicates the number of fundamental clusters at each level of approximation.
\[tab2\]
We note that the “raw” results for the ground-state energy, sublattice magnetisation, and critical points, $\Delta_c$, have been obtained using both the LSUB$m$ and SUB$m$-$m$ approximation schemes. For both approximation schemes we may extrapolate these results as a function of $m$ in the limit $m \rightarrow \infty$ in order to obtain even better results. We note that SUB2-$m$ calculations and the full SUB2 calculation have also previously been performed [@ccm7] for the anisotropic square-lattice HAF with general quantum spin number. It was therefore possible to determine accurately the manner in which SUB2-$m$ results scale as a function of $m$, namely, that as $m \rightarrow \infty$ the ground-state energy and the critical points, $E_g(m)/N$ and $\Delta_c(m)$ respectively, scale linearly with $1/m^2$, and the sublattice magnetisation, $M(m)$, scales linearly with $1/m$. By analogy, we utilise a similar procedure here in order to extrapolate raw LSUB$m$ and SUB$m$-$m$ results. However, as we may utilise LSUB$m$ and SUB$m$-$m$ results for $m=\{2,4,6\}$ only, a quadratic function is fitted to these data in order to obtain the best possible results. A full and comprehensive explanation of the extrapolation process of CCM LSUB$m$ expectation values is given in Ref. [@ccm6] for the spin-half [*XXZ*]{} model for a variety of lattices, and the interested reader is referred to this article for further information.
CCM results for the position of the critical point using the LSUB$m$ and SUB$m$-$m$ approximation schemes are presented in Tables \[tab1\] and \[tab2\]. The extrapolated result of $\Delta_c = 0.98$ for both the LSUB$m$ and for the SUB$m$-$m$ approximation scheme indicates that the phase transition point is at (or perhaps slightly below) the isotropic Heisenberg point ($\Delta=1$). The strength of the CCM is that it can provide such an accurate value for the position of the quantum phase transition point as a function of the anisotropy within an [*ab initio*]{}, and thus fully unbiased, framework.
Figure \[fig1\] and Tables \[tab1\] and \[tab2\] indicate that the results for the ground-state energy of the spin-one HAF converge extremely quickly with $m$ for both the LSUB$m$ and SUB$m$-$m$ schemes. Indeed, even the “raw” unextrapolated results provide excellent estimates of the ground-state energy, although the extrapolated results of $E_g/N=-2.3292$ for the LSUB$m$ approximation scheme and $E_g/N=-2.3291$ for SUB$m$-$m$ approximate scheme are certainly even more accurate. By way of comparison, we note that third-order SWT [@Hamer] gives a result of $E_g/N=-2.3282$ and cumulant series expansions [@Zheng] a result of $E_g/N=-2.3279(2)$.
CCM results for the sublattice magnetisation are found to be similarly well converged as a function of the truncation index $m$ for both the LSUB$m$ and SUB$m$-$m$ schemes, as indicated in Fig. \[fig2\]. The extrapolated results of $M=0.8049$ and $M=0.8067$ for the LSUB$m$ and SUB$m$-$m$ schemes, respectively, are again in excellent agreement with the results of third-order SWT [@Hamer] and cumulant series expansions [@Zheng], which give values of $M=0.8043$ and $M=0.8039(4)$ respectively.
In this article it has been shown that the coupled cluster method (CCM) provides quantitatively accurate results for the ground-state properties of the spin-one square-lattice isotropic HAF by comparison with results of third-order spin-wave theory (SWT) and cumulant series expansions. Furthermore, we note that the results presented in this article constitute the first application of the CCM using a new high-order CCM ground-state formalism for general ($s \ge 1/2$) quantum spin number. (A fuller account of this new high-order formalism will be published elsewhere [@ccm8].) The best estimates of the ground-state energy of this model were found to be $E_g/N=-2.3292$ using the LSUB$m$ approximation scheme and $E_g/N=-2.3291$ using the SUB$m$-$m$ approximation scheme via heuristic extrapolation to the (exact) limit $m \rightarrow \infty$. The best estimates of the sublattice magnetisation are $M=0.8049$ using the LSUB$m$ approximation scheme and $M=0.8067$ using the SUB$m$-$m$ approximation scheme, again via extrapolation to the limit $m \rightarrow \infty$. The most important result of the CCM calculations presented here for this model is the prediction that the phase transition point of the spin-one square-lattice anisotropic HAF is at (or slightly below) the isotropic Heisenberg point. A strength of the method is that this prediction may be made using an unbiased (i.e., [*ab initio*]{}) treatment.
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[^1]: See Ref. [@ccm7]
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---
abstract: 'In this paper, a general methodology to study rigorously discontinuities in open waveguides is presented. It relies on a full vector description given by Maxwell’s equations in the framework of the finite element method. The discontinuities are not necessarily small perturbations of the initial waveguide and can be very general, such as plasmonic inclusions of arbitrary shapes. The leaky modes of the invariant structure [are first computed and then injected]{} as incident fields in the full structure with obstacles using a scattered field approach. The resulting scattered field is finally projected on the modes of the invariant structure making use of their bi-orthogonality. The energy balance is discussed. Finally, the modes of open waveguides periodically structured along the propagation direction are computed. The relevant complex propagation constants are compared to the transmission obtained for a finite number of identical cells. The relevance and complementarity of the two approaches are highlighted on a numerical example encountered in infrared sensing. Open source models allowing to retrieve most of the results of this paper are provided.'
author:
- 'Guillaume Demésy[^1] and Gilles Renversez\'
bibliography:
- 'biblio-guide\_louise.bib'
title: 'Discontinuities in photonic waveguides: Rigorous Maxwell-based 3D modeling with the finite element method'
---
Introduction {#sec:intro}
============
The study of discontinuities is an old research topic in waveguide studies due to its importance for practical applications in many areas of physics. One must cite the seminal contribution of Schwinger for the development of variational methods in the forties [@schinger68discontinuities-waveguides] and the results obtained by Lewin [@lewin75theory-waveguides].
These methods, often complex and specific, do not generally consider the exact solutions of Maxwell’s equations and rely on specific configurations, hypotheses, initial guesses for the solution forms. During the last two decades, the versatile Finite-Difference Time-Domain (FDTD) method allowed the study of waveguide discontinuities, including 3D ones, taking into account the full set of Maxwell’s equations [@taflove05; @taflove13]. Nevertheless, the computational resources both in terms of memory and time requirements are huge when realistic 3D photonic devices are considered with a uniform square grid, especially nanophotonic ones with high quality factors. As for harmonic methods, Fourier modal methods [@lalanne2000fourier; @lecamp2007theoretical] also allow to tackle discontinuities in waveguides but they are restricted to geometries with straight walls. In acoustics and optics, coupled modal-finite element techniques have been successfully used in varying cross-section waveguides [@pagneux1996study; @pelat2011coupled].
[Two types of non-Hermitian eigenvalue problems arise in open nanophotonic structures. When considering resonators in the general case, no particular ansatz can be guessed for the electromagnetic field and a natural eigenvalue is the (complex) frequency. Diffraction gratings are a special yet frequent case where the Bloch theorem applies ; the Bloch variable arises as a wavenumber and a corner stone of the dispersion relation of gratings consists in looking for *complex frequencies for a given real wavenumber*. Recent benchmarks of the numerical methods cited in the previous paragraph can be found in Refs. [@rosenkrantz2018benchmarking; @lalanne2019quasinormal] for this type of non-Hermitian eigenvalue problems. But when considering guiding structures as it is the case in this article, a more natural eigenvalue is *the propagation constant at a given real frequency*. When coupling a laser – indeed operating at a real frequency – into a waveguide, the relavant quantities are the light velocity and attenuation which are directly related to the real and imaginary parts of the eigenvalues (*i.e.* complex propagation constants) of the modes of the leaky waveguide.]{}
This article addresses the numerical characterization of open waveguides with discontinuities. We demonstrate that it can be carried out efficiently with adapted formulations of the finite element method (FEM) which has already proven its efficiency and versatility in many field of computational electrodynamics [@jin02FEM-electromag].
We can state four main advantages of the FEM-based method: i) curved geometries are naturally treated using high order mesh elements and corresponding shape functions, ii) conforming non-uniform meshing is now a standard for mesh generators which is particularly relevant when rapid and strong permittivity changes must be tackled, iii) the domain decomposition method, now available in several FEM solvers, allows the treatment of large scale 3D problem, iv) and the possibility to reuse the inverse matrix for several incident modes propagating in the invariant structure – a subtlety detailed later which is a key advantage for the optimization multi-mode guides. This is especially worthy when the simulations are performed within a topology optimization frame [@bendoe-sigmund04topo-opti].
The practical context motivating this theoretical and numerical study is the design of efficient plasmonic waveguides for infrared sensing [@gutierrez2016optical] since the mid-IR spectral domain is known to be the molecular fingerprint region, due to the fact that most molecule including pollutants have intense fundamental vibrational bands in this spectral range. The device configuration is fully integrated and based on a ridge waveguide upon which metallic scattering nano-objects will ensure the coupling between the guided modes and superstrate of the device. Chalcogenide glasses are chosen for the main layers due to their high transparencies for infrared wavelengths [@Desevedavy:09; @eggleton2011chalcogenide]. Ultimately, the metallic scatterers are planned to be functionalized in order to react to the targeted chemical species. The sensing property relies on the subsequent modification of the guidance of the full structure. [With this application in mind,]{} we present a general framework to study rigorously discontinuous waveguides using a full vector description given by Maxwell’s equations in the framework of the finite element method. The discontinuity can be very general and is not necessarily a small perturbation of the initial waveguide. The full structure under investigation is made of 3 segments: The input one is a uniform waveguide invariant along its main propagation axis, the intermediate one [(called “modified segment” in Fig. \[fig:scheme\])]{} contains the opto-geometrical modifications of the waveguide, the output one is again an invariant waveguide. In order to model the response of the resulting 3D guiding structure, we adopt a scattered field formulation consisting of three sequential steps, the output of first step being the input of the second one, the output of the first and second being the input of the third one.
First, we determine the leaky modes of the unperturbed 2D waveguide for a fixed frequency corresponding to the freespace wavelength of interest. [The example chosen to illustrate our method consists in]{} a ridge waveguide made of chalcogenide layers on a silicon substrate, assumed to be invariant along its propagation axis. We use our usual vector FEM method with the Galerkin approach to solve the relevant eigenvalue problem [@nedelec91; @jin02FEM-electromag; @renversez2012foundations]. This first step provides both the propagation constants (eigenvalues) and the associated modes profiles (eigenvectors).
Second, these guided modes are used as incident fields for the full 3D problem in the modified segment. The electromagnetic problem to solve for this second step is then a mere scattering problem [@demesy2010all]. It is possible to define a proper energy balance (transmission and reflection, absorption taking place into the obstacles, extra radiation losses) allowing to fully evaluate the impact of the modified segment on the energy propagation.
Third, outside of the modified region, the total field is [expanded]{} on a fixed number of leaky modes of the output segment of the full structure. [A special attention is paid to the coupling efficiency into the mode initially injected after crossing the modified segment.]{} Our method allows to compute all the required energy-related quantities to investigate quantitatively the behavior of the full structure, notably the impact of the modified segment, and to take into account the way it is excited by the selected input propagating mode.
Note that our approach differs from the one exposed in Ref. [@jin02FEM-electromag] where total field formulations making use of port boundary conditions are applied to closed discontinuous waveguides, whereas it is proposed here to use a general scattered field formulation to deal with open discontinuous waveguides.
Finally, we also compute the modes of waveguides infinitely periodically structured along the propagation axis and compare the relevant complex propagation constants to the transmission obtained with a finite number of identical cells. After deriving the formulation, the relevance and complementarity of the two approaches are highlighted on a numerical example.
Direct problem {#sec:3Ddirect}
==============
In this section, a direct – as opposed to modal – scattering approach is introduced. A typical and realistic structure is sketched in Fig. \[fig:scheme\]: A $z$-invariant dielectric rectangular waveguide (of width $w$ and thickness $h_g$ in blue) is deposited on a low index spacer (of thickness $h_l$, in green) lying on a semi-infinite substrate (in purple). The $z$-invariance of this guiding structure is locally broken, by adjunction of a finite number of obstacles. These obstacles can be in practice any bounded modification of permittivity: Ellipsoidal patches above the guiding layer labelled in Fig. \[fig:scheme\], holes in the guiding layer, obstacles or resonators next to the waveguide or even a combination of all …Note that the method applies irrespectively of the number of layers of the $z$-invariant structure and that the obstacles can be arbitrarily shaped and located in (or above) the structure. It is shown how the obstacles (more generally the modified waveguide segment) perturb a mode propagating in the $z$-invariant structure. A first step consists in the numerical computation of the modes of the invariant structure, which are used in a second step as incident fields for the full 3D structure.
![Scheme of the $z$-invariant structure (substrate in purple, low index layer in green and a rectangular waveguide in blue) with various discontinuities (or obstacles) breaking the $z$-invariance locally in a region called “modified segment”. Discontinuities can be ellipsoidal patches above the guiding layer labelled , holes in the guiding layer, obstacles or resonators next to the waveguide or even a combination of all . []{data-label="fig:scheme"}](scheme_chain_finite_cropped_ann2.pdf){width=".7\linewidth"}
Obtaining the incident fields {#sec:3Ddirect1}
-----------------------------
The classical guiding $z$-invariant structure is characterized by its permittivity function defined by parts as: $$\epsrdd(x,y)=
\left \{
\begin{array}{l}
{\ensuremath\varepsilon_{r,g}} \mbox{ in the guide,} \\
{\ensuremath\varepsilon_{r,l}} \mbox{ in the low index region,} \\
{\ensuremath\varepsilon_{r,s}} \mbox{ in the substrate,} \\
{\ensuremath\varepsilon_{r,t}} \mbox{ in the superstrate}
\end{array} \right. .$$
Between $z_{min}$ and $z_{max}$, one can now break the $z$-invariance by a local modification of the permittivity function which leads to a 3D scattering problem, which in turn can be characterized by its permittivity function defined by parts: $$\epsrddd(x,y,z)=
\left \{
\begin{array}{l}
{\ensuremath\varepsilon_{r,g}} \mbox{ in the guide,} \\
{\ensuremath\varepsilon_{r,l}} \mbox{ in the low index region,} \\
{\ensuremath\varepsilon_{r,s}} \mbox{ in the substrate,} \\
{\ensuremath\varepsilon_{r,t}} \mbox{ in the superstrate},\\
{\ensuremath\varepsilon_{r,d}}(x,y,z) {\crevme\mbox{ in the obstacles }}\\
\hspace{.5cm} {\crevme\mbox{ of the modified segment}}
\end{array} \right. .$$
The starting point consists in computing the modes of an annex problem formed by the $z$-invariant structure solely. This is a very classical problem [@snyder1983optical; @renversez2012foundations] where one introduces the ansatz $\bE(\bx)=\be(\x)e^{-i(\om_0 t-\beta z)}$ in the source-free Helmholtz equation: $$\label{eq:Hdd}
\curl\left(\tensmurdd^{-1}\,\curl\bE\right) = \tensepsrdd\left(\frac{\om_0}{c}\right)^2 \bE$$ [for a given real angular frequency $\om_0$]{}. Note that the relative permittivity and permeability in are tensors fields since cartesian PMLs adapted to each domain with infinite extension (superstrate, low index region and substrate) are used to damp the radially blowing leaky modes of this open structure [@renversez2012foundations]. The domains $\Omega$ involved in all the formulations of the paper are correspond to geometrical domains surrounded by appropriate PMLs of finite thicknesses.
It results in a quadratic non-Hermitian eigenvalue problem amounting to find non trivial pairs $(\beta_k,\bek)\in\mathbb{C}\times H^1(\Omega_{2D},\mathbf{curl})$ such that : $$\label{eq:devstrongform2}
\begin{split}
&\curl\left(\tensmurdd^{-1}\,\curl \bek\right)-k_0^2\,\tensepsrdd\,\bek \\
+i\betak&\left[\bzh\times \left(\tensmurdd^{-1}\curl \bek\right) + \curl\left(\tensmurdd^{-1}\,\bzh\times\bek \right)\right] \\
+(i\betak)^2\,& \bzh\times\left(\tensmurdd^{-1}\,\bzh\times\bek \right)=\mathbf{0}\,,
\end{split}$$ with $ k_0 := \om_0/c$.
(modeone) at (-.4,9.0) [![[The six modes with smallest attenuation supported by the $z$-invariant structure at $\lambda_0={\SI{7.7}{\micro\metre}}$. The edges of the cross-section are represented in colors matching the domains shown in Fig. \[fig:scheme\]. The eigenvalue corresponding to mode 1 in the inset (a) has the smallest imaginary part. The power attenuation (defined [@renversez2012foundations] in dB/cm as -$2000\,\mathrm{Im}\{\beta_k\}/\mathrm{ln}(10)$) is given at the top of each inset. The black and white maps (white is high) represent the norm of the electric eigenfields $|\bek|$ in the waveguide cross-section. The orange arrows indicate the real part of the electric eigenfields $\bek$.]{}[]{data-label="fig:modes2D"}](mode2D_0.png "fig:"){width="7.2cm"}]{}; (modetwo) at (7.5,9.0) [![[The six modes with smallest attenuation supported by the $z$-invariant structure at $\lambda_0={\SI{7.7}{\micro\metre}}$. The edges of the cross-section are represented in colors matching the domains shown in Fig. \[fig:scheme\]. The eigenvalue corresponding to mode 1 in the inset (a) has the smallest imaginary part. The power attenuation (defined [@renversez2012foundations] in dB/cm as -$2000\,\mathrm{Im}\{\beta_k\}/\mathrm{ln}(10)$) is given at the top of each inset. The black and white maps (white is high) represent the norm of the electric eigenfields $|\bek|$ in the waveguide cross-section. The orange arrows indicate the real part of the electric eigenfields $\bek$.]{}[]{data-label="fig:modes2D"}](mode2D_1.png "fig:"){width="7.2cm"}]{}; (modethr) at (-.4,4.5) [![[The six modes with smallest attenuation supported by the $z$-invariant structure at $\lambda_0={\SI{7.7}{\micro\metre}}$. The edges of the cross-section are represented in colors matching the domains shown in Fig. \[fig:scheme\]. The eigenvalue corresponding to mode 1 in the inset (a) has the smallest imaginary part. The power attenuation (defined [@renversez2012foundations] in dB/cm as -$2000\,\mathrm{Im}\{\beta_k\}/\mathrm{ln}(10)$) is given at the top of each inset. The black and white maps (white is high) represent the norm of the electric eigenfields $|\bek|$ in the waveguide cross-section. The orange arrows indicate the real part of the electric eigenfields $\bek$.]{}[]{data-label="fig:modes2D"}](mode2D_2.png "fig:"){width="7.2cm"}]{}; (modefou) at (7.5,4.5) [![[The six modes with smallest attenuation supported by the $z$-invariant structure at $\lambda_0={\SI{7.7}{\micro\metre}}$. The edges of the cross-section are represented in colors matching the domains shown in Fig. \[fig:scheme\]. The eigenvalue corresponding to mode 1 in the inset (a) has the smallest imaginary part. The power attenuation (defined [@renversez2012foundations] in dB/cm as -$2000\,\mathrm{Im}\{\beta_k\}/\mathrm{ln}(10)$) is given at the top of each inset. The black and white maps (white is high) represent the norm of the electric eigenfields $|\bek|$ in the waveguide cross-section. The orange arrows indicate the real part of the electric eigenfields $\bek$.]{}[]{data-label="fig:modes2D"}](mode2D_3.png "fig:"){width="7.2cm"}]{}; (modefiv) at (-.4,0.0) [![[The six modes with smallest attenuation supported by the $z$-invariant structure at $\lambda_0={\SI{7.7}{\micro\metre}}$. The edges of the cross-section are represented in colors matching the domains shown in Fig. \[fig:scheme\]. The eigenvalue corresponding to mode 1 in the inset (a) has the smallest imaginary part. The power attenuation (defined [@renversez2012foundations] in dB/cm as -$2000\,\mathrm{Im}\{\beta_k\}/\mathrm{ln}(10)$) is given at the top of each inset. The black and white maps (white is high) represent the norm of the electric eigenfields $|\bek|$ in the waveguide cross-section. The orange arrows indicate the real part of the electric eigenfields $\bek$.]{}[]{data-label="fig:modes2D"}](mode2D_4.png "fig:"){width="7.2cm"}]{}; (modesix) at (7.5,0.0) [![[The six modes with smallest attenuation supported by the $z$-invariant structure at $\lambda_0={\SI{7.7}{\micro\metre}}$. The edges of the cross-section are represented in colors matching the domains shown in Fig. \[fig:scheme\]. The eigenvalue corresponding to mode 1 in the inset (a) has the smallest imaginary part. The power attenuation (defined [@renversez2012foundations] in dB/cm as -$2000\,\mathrm{Im}\{\beta_k\}/\mathrm{ln}(10)$) is given at the top of each inset. The black and white maps (white is high) represent the norm of the electric eigenfields $|\bek|$ in the waveguide cross-section. The orange arrows indicate the real part of the electric eigenfields $\bek$.]{}[]{data-label="fig:modes2D"}](mode2D_5.png "fig:"){width="7.2cm"}]{}; at (modeone.north) [(a) Mode 1 ()]{}; at (modetwo.north) [(b) Mode 2 ()]{}; at (modethr.north) [(c) Mode 3 ()]{}; at (modefou.north) [(d) Mode 4 ()]{}; at (modefiv.north) [(e) Mode 5 ()]{}; at (modesix.north) [(f) Mode 6 ()]{};
This equation can be solved using a mixed finite element formulation involving edge elements for the discretization of the transverse component $(e_x,e_y)$ coupled to a nodal basis for the (continuous) longitudinal component $e_z$. [The rather lengthy details of the resulting weak formulation can be found in Refs. [@nicolet2004modelling; @renversez2012foundations]]{}.
Throughout the paper, the following numerical values are considered for the $z$-invariant waveguide [@kuriakose17Measurement-ultrafast-optical-Kerr-effect-chalco]: The operating freespace wavelength $\lambda_0={\SI{7.7}{\micro\metre}}$, ${\ensuremath\varepsilon_{r,g}}=7.1824$ ($\mbox{Se}_4$), ${\ensuremath\varepsilon_{r,l}}=6.2001$ ($\mbox{Se}_2$), ${\ensuremath\varepsilon_{r,s}}=11.69024481$ (silicon) [@chandler2005high], ${\ensuremath\varepsilon_{r,t}}=1$ (air), $w_g={\SI{14}{\micro\metre}}$, $h_g={\SI{2.2}{\micro\metre}}$ and $h_l={\SI{5.3}{\micro\metre}}$. All the materials are assumed to be non-magnetic: $\tensmurdd=\tensid$ (except in the PMLs where $\tensmurdd$ takes the appropriate value), where $\tensid$ is the $3\times3$ identity tensor.
[The modes of this structure associated with eigenvalues with lowest imaginary parts are depicted in Fig. \[fig:modes2D\](a-f), sorted in ascending attenuation ([*i*.e.]{} $\mbox{Im}\{\beta_1\}$ is the smallest). The black and white colormaps show the norm of the electric eigenfields $|\be_k|$ in Fig. \[fig:modes2D\](a-f) and the orange arrows represent the real part of $\be_k$, allowing to distinguish a TE-like mode from a TM-like one. These six modes are also those with electric eigenfield most confined into the core region of the structure (ridge).]{}
Finally, all geometries and conformal meshes have been obtained using the Gmsh software [@gmsh] and all the finite element formulations in this article are implemented thanks to the flexibility of the finite element software GetDP [@getdp]. Open source models allowing to retrieve most of the results of this article are provided [@code].
Computation of the scattered field {#sec:3Ddirect2}
----------------------------------
One can now use any of these 2D modes ${\ensuremath{\mathbf{E}_{k,2D}}}:=\be_k\,e^{i(\betak z-\omega_0 t)}$ as an incident field $\bEinc$ on the obstacles and look for $\bEtot$, the total field solution of the source-free Helmholtz equation: $$\label{eq:Hddd}
-\curl\left[\tensmurddd^{-1}\,\curl \bEtot \right] + k_0^2\,\tensepsrddd\,\bEtot = \mathbf{0}.$$ Let us define the scattered field as $\bEd \equiv \bEtot-\bEinc$ and from the linearity of Eqs. (\[eq:Hdd\],\[eq:Hddd\]), we obtain the following scattering problem: $$\label{eq:Hdddrad}
-\curl\left[\tensmurddd^{-1}\,\curl \bEd \right] + k_0^2\,\tensepsrddd\,\bEd = k_0^2\,(\tensepsrdd-\tensepsrddd)\bEinc.$$ Note that the support of the effective sources $(\tensepsrdd-\tensepsrddd)$ in this scattering problem has to be bounded to ensure a proper outgoing wave condition [@zolla1996method] to the scattered field $\bEd$, which is the case in our examples. Finally, 3D cartesian PMLs are used to bound the computational domain [@renversez2012foundations; @berenger94perfec-match-layer] [as shown in grey lines in Fig. \[fig:poynting\]]{}. Compared to a total field approach with a port condition [@jin02FEM-electromag], it is stressed that the electromagnetic sources of our equivalent radiation problem are located within the discontinuities. The PMLs of elongated structures are naturally built to damp fields radiating from the center of the computational box more efficiently than the total field radiating from a port located at one extremity of the elongated box, the resulting total field being more grazing than the scattered field when entering the PMLs.
Energy balance {#sec:3Ddirect3}
--------------
The Poynting vectors associated with the incident, diffracted and total fields are classically defined by respectively $\bSi={\ensuremath{\mathrm{Re}\{\bEinc\times\overline{{\bHinc}}\}}}/2$, $\bSd={\ensuremath{\mathrm{Re}\{\bEd\times\overline{\bHd}\}}}/2$ and $\bStot={\ensuremath{\mathrm{Re}\{\bEtot\times\overline{\bHtot}\}}}/2$, where the horizontal bar means complex conjugation. Then, the incoming, transmitted, reflected and absorbed powers can be defined as respectively : $$\label{eq:power}
\begin{split}
P_{in} &= \int_{\Gamma_{in}}\bSi\cdot \bn_\Gamma\,\mathrm{d}S,\\
P_{tr} &= \int_{\Gamma_{out}}\bStot\cdot \bn_\Gamma\,\mathrm{d}S,\\
P_{ref} &= \int_{\Gamma_{in}}\bS^{d}\cdot \bn_\Gamma\,\mathrm{d}S \mbox{ and }\\
P_{abs} &= \frac{\varepsilon_0\,\omega_0}{2}\int_{\Omega_{d}} {\crevme \mathrm{Im}\{\varepsilon_{r,d}\}}\,|\bEtot|^2\,\mathrm{d}\Omega,
\end{split}$$ where $\Gamma_{in}$ and $\Gamma_{out}$ are [the transverse plane surfaces before and after the obstacles depicted in transparent grey color in Fig. \[fig:poynting\]]{}, $ \bn_\Gamma$ is the unit vector normal to $\Gamma_{in}$ and $\Gamma_{out}$ and $\Omega_{d}$ is the support of the diffractive obstacles or of the localized region where the waveguide opto-geometrical parameters are modified. Finally one can define transmission (T), reflection (R) and absorption (A) coefficients as : $$\label{eq:RTA}
T=\frac{P_{tr}}{P_{in}} \mbox{ , } R=-\frac{P_{ref}}{P_{in}}\mbox{ and } A=\frac{P_{abs}}{P_{in}}.$$
![ Cuts of the [$z$-component of the total Poynting vector ($\bStot\cdot\bzh$ in purple/yellow colors) and norm of the total field ($|\bEtot|$ in blue/red colors) inside the four lossy obstacles (ellipsoidal patches) above the waveguide. The computational domain represents half of the structure due to the symmetry properties of both the geometry and the incident field. The colored edges represent the actual geometry of the structure with the same color code as in Fig. \[fig:scheme\] and the grey ones the cartesian PMLs adapted to each physical domain.]{} []{data-label="fig:poynting"}](fig_poynting_v2.pdf){width=".7\columnwidth"}
For clarity, Fig. \[fig:poynting\] illustrates the quantities at stake in the energy balance. This numerical set up is obtained for an incident field set to ${\ensuremath{\mathbf{E}_{1,2D}}}$ (cf Fig. \[fig:modes2D\](a)) with four ellipsoidal lossy patches placed above the same waveguide as in Section \[sec:3Ddirect1\]. This particular configuration will be discussed in detail in Section \[sec:disc\]. [The red/blue colormap represents the norm of the total electric field $|\bEtot|$ involved in the computation of the Joule losses, *i.e.* $P_{abs}$ in . The purple/yellow colormap represents three cuts of the $z$ component of the total Poynting vector $\bStot$ on three selected plane surfaces. The first cut is taken at $z=z_{min}$ (see left side of the figure), another one at $z=z_{max}$ (right side of the figure) and the last one along the symmetry plane $x=0$ of the structure. In this last cut, it is clear that the perturbation induced by the objects affects the $z$ component of the total Poynting vector since in absence of scattering objects above the ridge, this map would be constant along $z$]{}.
In this example, the transmission $T$ reaches 0.688, the reflection $R=0.007$ and the absorption $A=0.224$ ($T+R+A=0.904$). Note that $R$, $T$ and $A$ are defined here in order to match commonly measured quantities does but do not add up to unity. It is nonetheless expected since their sum does not correspond to a full Poynting balance. It can be easily completed by adding the flux contributions from the surfaces parallel to the $zOx$ and $zOy$ planes, which represent the extra radiative leakage induced by the modified segment ($9.6\%$ in the present case).
As shown in Sec. \[sec:3Ddirect4\], the scattering process can be further precised by expansion of the diffracted and total fields outside the modified segment on the modes of the 2D invariant structure.
Discretization and convergence
------------------------------
In the 2D eigenvalue problem of Sec. \[sec:3Ddirect1\], the longitudinal component of the electric field, which is a continuous scalar field, is discretized using classical $P_2$ nodal elements having one Degree Of Freedom (DOF) per node and one DOF per edge. The transverse components are discretized using edge elements of the second order (2 degrees of freedom per edge). For eigenvalue problems, the GetDP software relies on the high performance library SLEPc [@Hernandez2005-SSF] which implements advanced Krylov subspace methods for computing a small amount of eigenvalues of the large sparse matrices.
The 3D scattering problem uses high order Webb hierarchical edge elements [@geuzaine1999convergence; @webb1993hierarchal; @jin02FEM-electromag] with 26 DOFs per tetrahedron (3 DOFs per edge, 2 DOFs per face). The direct problem described in section \[sec:3Ddirect2\] is solved using the direct solver MUMPS [@mumps-userguide] interfaced in GetDP
.
[The convergence of the absorption $A$ as a function of the mesh refinement is shown in Fig. \[fig:conv\](a). The mesh size is parametrized by $n$ (in abscissa) and decreases as $\lambda_0/(n\,\mathrm{Re}\{\sqrt{\varepsilon_r}\})$ as $n$ increases. In other words, $n$ represents the average number of tetrahedrons per wavelength inside a dielectric material of relative permittivity $\varepsilon_r$. Note that in metals, the relevant physical length to consider for a proper spatial sampling of the field would be the skin depth rather that the wavelength. For $n$=1, that is roughly one tetrahedron per wavelength, the computational box in Fig. \[fig:poynting\] leads to 32000 DOFs solved in on a laptop equipped with 4 cores 16 Gb of RAM memory. In this case, the local values of the field are poorly approximated, but the order of magnitude of integral quantities such as the absorption is relevant, as can be noticed on the left side of the convergence plot shown in Fig. \[fig:conv\](a). For $n$=4, the number of DOFs is 650000, the model still runs on the same laptop within . For $n$=7, the number of DOFs becomes about 3 millions and a workstation equipped with 24 cores and 256 Gb RAM memory was used for a runtime of . Five significant digits are then obtained on energy-related quantities such as the absorption.]{}
Modal expansion of the scattered field {#sec:3Ddirect4}
--------------------------------------
The modes of the 2D invariant structure satisfy the following bi-orthogonality condition equivalent to the one given in Ref. [@sammut1976leaky; @snyder1983optical] which provides the normalization of each leaky modes: $$\label{eq:orth}
\int_S \be_j\times\bh_k\cdot\bzh\,\mathrm{d}S = \int_S \be_k\times\bh_j\cdot\bzh\,\mathrm{d}S=A_k\delta_{kj} \; \mbox{, where} $$ $S$ is an infinite cross-section of the open waveguide. In the case of leaky modes, it is suggested in [@sammut1976leaky] to perform a complex change of space variable as one moves far away from the waveguide to damp the exponential growth of the leaky mode. In our finite element approach that includes the PMLs which are an analytical continuation of the space variables, this integration simply corresponds to integrate over a full cross-section of computational domain including the PML regions. Therefore it is stressed that the cross-sections considered hereafter include the PMLs. Hence, away from the obstacles, it is possible to expand the scattered field as $\bEd=\sum_k r_k {\ensuremath{\mathbf{E}_{k,2D}}}$ and the total field as $\bEtot=\sum_k t_k {\ensuremath{\mathbf{E}_{k,2D}}}$ where the reflection and transmission coefficients are simply given as: $$\label{eq:rntn}
\left \{
\begin{array}{l}
t_k=\dint_{S_{out}} \bEtot\times\bh_k\cdot\bzh\,\mathrm{d}S / A_k \\[4mm]
r_k=\dint_{S_{in}} \bEd\times \bh_k\cdot\bzh\,\mathrm{d}S / A_k
\end{array} \right.$$ where $S_{in}$ can be any transverse section before the obstacles ($z<z_{min}$) and $S_{out}$ can be any transverse section after the obstacles ($z>z_{out}$). Note that this formalism can be extended to the computation of the scattering matrix of the waveguide, by considering sequentially all the modes of the invariant structure computed in Sec. \[sec:3Ddirect1\] as incident fields. In this example, the modulus of the mean value of the off-diagonal coefficients of the $12\times12$ bi-orthogonality matrix (see ) is less than $10^{-3}$ smaller than the modulus of the mean value of the diagonal coefficients $A_k$.
[The total field within a leaky guiding structure can be expanded [@snyder1983optical] as discrete sum over bounded modes plus an integral over the continuous spectrum. When using finite size PMLs, the continuous spectrum becomes discrete and the integral contribution turns into a discrete sum. Besides, the power orthogonality between the modes holding for self-adjoint eigenvalue problems such as the perfect metallic waveguide now fails in the our non-Hermitian case. This can be seen in the bi-orthogonality relations (see ) that involve the magnetic field rather than its complex conjugate. However, we are in a weakly leaky regime [@sammut1976leaky] were the weakly leaky modes decay rapidly as the radial distance $r=\sqrt{x^2+y^2}$ increases until experiencing an exponential blow at even larger radial distances. In this regime, it is interesting to see that the power exchange between the most highly confined modes (associated with eigenvalues with small imaginary part compared to their real part) can be neglected. In short, $\sum_k |t_k|^2\rightarrow T$ and $\sum_k |r_k|^2\rightarrow R$ hold to a very good approximation. In the particular configuration described in Fig. \[fig:poynting\], one obtains the values $\sum_{k=1}^{12}|t_k|^2=0.685$ (with a major contribution from $|t_1|^2=0.632$, to be compared to $T$=0.688 obtained in Sec. \[sec:3Ddirect3\]) and $\sum_{k=1}^{12}|r_k|^2=0.0007$. The comparison between $T$ and $\sum_{k=1}^M |t_k|^2$ as a function of the truncation order $M$ is shown in Fig. \[fig:conv\](b) for a fine mesh with $n=7$.]{}
![(a) [Convergence of the absorption $A$ of the lossy ellipsoidal patches as a function of the mesh size decreasing as $\lambda_0/(n\,\mathrm{Re}\{\sqrt{\varepsilon_r}\})$ as $n$ (in abscissa) increases. (b) The quantity $T-\sum_{k=1}^M |t_k|^2$ as a function of the truncation order $M$ for $n$=7.]{}[]{data-label="fig:conv"}](conv.pdf){width=".9\linewidth"}
Modes of the infinitely periodic 3D structure {#sec:3Dmodal}
=============================================
Variational formulation of the spectral problem {#sec:3Dmodal1}
-----------------------------------------------
In this section, we are now interested in a 3D spectral problem with one direction of periodicity defined by $$\label{eq:L3D}
\curl\left[\tensmurp(\bx)^{-1}\curl \bE\right]=\tensepsrp(\bx,\omega_0)\left(\frac{\omega_0}{c}\right)^2\,\bE\,.$$ where $ \tensepsrp(\bx,\omega_0)$ and $\tensmurp(\bx)$ are respectively the permittivity and permeability tensor fields at a fixed frequency $\om_0$ exhibiting a 1D $d$-periodicity along $Oz$. Bloch’s theorem states that, without loss of generality, one can look for solutions for the electric field $\bE$ under the form [@joannopoulos95photon-crist-modlin-flow-light; @renversez2012foundations] : $$\label{eq:solform}
\bE=\bEp(x,y,z)\,e^{-i(\omega_0 t - \gamma z)}\,,$$ where $\bE_\#$ is a $d$-periodic function in $z$ and $\gamma$ is the Bloch variable lying in the first reduced Brillouin zone \[0,$\pi/d$\].
One can choose to set $\gamma$ to a real value lying in the first Brillouin zone and to look for ($\omega_{\gamma,i}$,$\bE_{\gamma,i}$) eigenvalues and eigenvectors, by imposing Bloch conditions on the z-transverse surfaces of the cell and making the use of . An alternative option amounts to set $\omega_0$ to a real value, inject the ansatz in into and look for eigenvectors under the form of the periodic part $\bEp$ of the Bloch wave along with corresponding eigenvalue $\gamma$. In this latter case, two equations are to be fulfilled:
[align]{} \[eq:modal3Dcurl\]-+k\_0\^2()e\^[iz]{}&=0\
\[eq:modal3Ddiv\]&=0
.
It is stressed that the invariant 2D problem described in Sec. \[sec:3Ddirect1\] is a particular case of this 3D problem, an invariant structure along $z$ being trivially periodic in $z$ with arbitrary period. Unsurprisingly, expanding the $\mathbf{curl}$ term in in order to get rid of the $e^{i\gamma z}$ dependency, leads to an expression similar to the $z$-invariant counterpart of the problem (see ):
$$\label{eq:devstrongform3}
\begin{split}
-\curl\left[\tensmurp^{-1}\curl \bEp\right]&+k_0^2\,\tensepsrp(\bx)\,\bEp \\
-i\gamma\, & \bzh\times \left[\tensmurp^{-1}\curl \bEp\right] \\
-i\gamma\, & \curl\left(\tensmurp^{-1}\,\bzh\times\bEp \right) \\
-(i\gamma)^2\,& \bzh\times\left(\tensmurp^{-1}\,\bzh\times\bEp \right) =\mathbf{0} \,.\\
\end{split}$$
In a variational way, after classically integrating by part two curl operators, it holds that for any $\mathbf{W}\in H_\#^1(\Omega,\curl)$ : $$\label{eq:devstrongform3}
\begin{split}
-&\dint_\Omega \left[\tensmurp^{-1}\curl \bEp\right]\cdot\curl \bW \d\Omega \\
+&\dint_\Omega k_0^2\,\tensepsrp(\bx)\,\bEp \cdot\bW \d\Omega\\
-i\gamma\, &\dint_\Omega \bzh\times \left[\tensmurp^{-1}\curl \bE\right] \cdot\bW \d\Omega\\
-i\gamma\, &\dint_\Omega \left(\tensmurp^{-1}\,\bzh\times\bEp \right) \cdot\curl\bW \d\Omega \\
+(i\gamma)^2\,&\dint_\Omega \left(\tensmurp^{-1}\,\bzh\times\bEp \right)\cdot(\bzh\times\bW) \d\Omega\\
- &\dint_{\partial\Omega} \left[\bn_{|\partial\Omega}\times\left(\tensmurp^{-1}\curl \bEp\right)\right]\cdot \bW \d S\\
- i\gamma&\dint_{\partial\Omega} \left[\bn_{|\partial\Omega}\times\left(\tensmurp^{-1}\,\bzh\times\bEp \right) \right]\cdot \bW \d S \\
& =0
\end{split}$$ Note that the two boundary terms recombine into $-\int_{\partial\Omega} [\bn_{|\partial\Omega}\times(\tensmurp^{-1}(\curl \bEp+i\gamma\bzh\times\bEp) )]\cdot \bW \d S\propto\int_{\partial\Omega} [\bn_{|\partial\Omega}\times\bH]\cdot \bW \d S$ so that setting a Dirichlet or Neumann natural condition for $\bEp$ on non-periodic faces of the domain ([*i.e*]{}. the PML bounds) actually corresponds to a Dirichlet or Neumann natural condition for $\bH$.
The divergence condition in has to be handled carefully. Indeed, we are looking for divergence free solutions such that $\div(\tensepsrp\,\bE)=0$, that is: $$\label{eq:div}
\div\left(\tensepsrp\,\bEp e^{i\gamma z}\right)=0=\div(\tensepsrp\,\bEp)+i\gamma\bzh\cdot(\tensepsrp\,\bEp)\\$$ Consequently, $\tensepsrp\,\bEp$ is not divergence-free and, from the variational point of view, the following holds for any $\varphi\in H_\#^1(\Omega)$: $$\label{eq:div}
\begin{split}
\int_\Omega&\left[\div\left(\tensepsrp\,\bEp\right)+i\gamma\bzh\cdot\left(\tensepsrp\,\bEp\right)\right]\,\overline{\varphi}\,\d\Omega=0\\
=-\int_\Omega&\tensepsrp\,\bEp\cdot\,\overline{\grad\varphi}\,\d\Omega+i\gamma\int_\Omega\bzh\cdot\left(\tensepsrp\,\bEp\right)\,\overline{\varphi}\,\d\Omega\,,
\end{split}$$ [where the boundary term arising from the integration by part vanishes due to periodicity and homogeneous conditions at the back of the PMLs.]{} Finally, the proper way to ensure the divergence condition [@lackner2019] in a weak sense is to use $\varphi$ as a Lagrange multiplier. We are now in position to reformulate the eigenvalue problem at stake in this section. We are looking for non trivial pairs $\gamma_k,(\bEpk,\varphi_k)\in \mathbb{C}\times(H^1_\#(\Omega,\mathrm{curl})\times H_\#^1(\Omega))$ such that:
\[eq:weakmodal3D\]
[align]{} \[eq:weakmodal3Dcurl\]
-&\_\^[-1]{}\
+&\_k\_0\^2()\
-i&\_(\^[-1]{})\
-i&\_(\^[-1]{})\
+(i)\^2&\_(\^[-1]{})()\
+&\_\_k\
+i&\_\_k=0
\
\[eq:weakmodal3Ddiv\]
&\_\
-i&\_()=0
![[For validation of the 3D modal approach, the same structure as in Fig. \[fig:modes2D\] is extruded along $z$ with a distance $d=\SI{1}{\micro\meter}$. The spectrum obtained for the periodic 3D modal approach (, red pluses) is compared to the spectrum of a 2D invariant structure (blue crosses and orange circles, see the main text of Sec. \[sec:3Dmodal2\] for the details of the two approaches used for the genuine 2D problem). The insets represent the three periodic parts of the modes with smallest attenuation. For each inset, two cuts are performed in the 3D domain. The first one (black and white contour plot at $z$=-$d$/4) corresponds to the norm of the and the second one (yellow arrows at $z$=$d$/4) to the real part of the eigenvectors. Note that the modes profile in the insets correspond exactly to the modes shown in Fig. \[fig:modes2D\](a,b,c).]{}[]{data-label="fig:validation2D3D"}](fig_validation2D3D_final_v3_cropped.pdf){width=".65\columnwidth"}
Discretization
--------------
[The periodic vector unknown $\bEp$ is discretized using high order Webb hierarchical edge elements [@geuzaine1999convergence; @webb1993hierarchal] with 26 DOFs per tetrahedron (3 DOFs per edge, 2 DOFs per face). The scalar field $\varphi$ mapping the divergence is discretized using Lagrange $P_3$ elements, with 20 DOFs per tetrahedron (4 nodal DOFs, two DOFs per edge, one DOF per face). Periodic boundary conditions are imposed along the $z$ direction for both $\bEp$ and $\varphi$.]{}
Numerical validation {#sec:3Dmodal2}
--------------------
It is apropos to validate this 3D model numerically using an extruded 2D domain. The eigenvalue resulting from three finite element problems are shown in Fig. \[fig:validation2D3D\]. Two of them (orange circles and blue crosses) are variants of the 2D problem in . The problem is indeed quadratic and can be solved as is using the SLEPc library (orange circles) which implements its own internal numerical linearization. But as detailed in Ref. [@renversez2012foundations], it is possible to linearize the 2D problem by simply using for unknown $(e_x,e_y,i{\crevme\beta}\,e_z)$ instead of $(e_x,e_y,e_z)$. The resulting sparse systems resulting from the two methods are different and it is worth noting that the 30 eigenvalues are identical up to numerical precision. Now the third eigenproblem (red pluses in Fig. \[fig:validation2D3D\]) corresponds to Eqs. (\[eq:weakmodal3Dcurl\],\[eq:weakmodal3Ddiv\]) applied to the 3D problem obtained by simple extrusion along $z$ of the previous 2D problem by a period $d$= along the $z$ direction. The value of $d$ can be arbitrarily chosen because of the translational invariance. The small value of the period along $z$ corresponds to a large first Brillouin zone so that the eigenvalues computed do not belong to a folded dispersion branch and the periodic part of the Bloch vector field is constant along $z$ as depicted in Figs. \[fig:validation2D3D\](a-c). Up to the $\pi$/d folding of the dispersion curves expected from the application to the Bloch theorem, these 2D and 3D invariant problems are spectrally equivalent and the eigenvalues are indeed retrieved with excellent accuracy. The discrepancies obtained for Im$\{\gamma_k/k_0\}>0.2$ can be simply explained by the fact that the 3D mesh used in the simulation is comparatively coarser than the 2D mesh. For the modes with smallest attenuation labelled (a), (b) and (c), the relative error between the 2D and 3D eigenvalues is lower that $10^{-4}$ in modulus. Note that convergence tests have been performed and this discrepancy decreases with the mesh size at the expected rate given the type and order of the chosen finite elements spaces. The real parts of $\bE_{\#,k}$ are shown in the insets (a-c) of Fig. \[fig:validation2D3D\] (note that they cannot be directly compared to the yellow arrows of Fig. \[fig:modes2D\](a-c) since an eigenvector is defined up to an arbitrary complex number).
[Computing 30 eigenvalues of the periodic 3D structure takes on a laptop with low order elements (twelve DOFs per tetrahedron for the vector unknown, 10 DOFs per tetrahedron for the scalar one, about 100000 DOFs in total) with the mesh paramater $n$=4. The same computation with $n$=8 and higher order edge elements (26 DOFs per tetrahedron for the vector unknown, twenty DOFs per tetrahedron for the scalar one, about 800000 DOFs in total) takes on the 24 cores workstation.]{}
We are finally in position to compare the results derived from the modes of the infinitely structured waveguide (3D modal problem defined in Sec. \[sec:3Dmodal\]) to the transmission properties of scattering problems with a finite number of periods (3D direct problems defined in Sec. \[sec:3Ddirect\]).
![[The various transmission definitions (see the main text in Sec. \[sec:disc\]) introduced as a function of the number of obstacles. The black curve represents the transmission $T$ defined in obtained using the direct problem (each black bullet corresponds to a direct Finite Element run). The green curve corresponds to the sum of square modulus of the expansion coefficients $|t_k|$ defined in . The yellow curve shows the sole contribution of $|t_1|^2$. The red curve is obtained by solving one single 3D modal problem and shows the spatial exponential decay of the power associated with the 3D mode with lowest attenuation. The norm of the corresponding electric eigenfield is shown in at the top right corner.]{}[]{data-label="fig:compa_modal_direct3D"}](compare_modal_direct_9p2i_v4.pdf){width=".65\textwidth"}
Discussion {#sec:disc}
==========
The direct 3D approach detailed in Sec. \[sec:3Ddirect\] and the 3D modal one based on $z$-periodicity presented in the previous section are now compared. In both cases, the unit cell of the waveguide guide contains one lossy ellipsoidal patch defined by its relative permittivity ${\ensuremath\varepsilon_{r,d}}=9+2i$, height $h={\SI{1}{\micro\metre}}$, transverse and longitudinal radii $r_t={\SI{5}{\micro\metre}}$ and $r_l={\SI{1.5}{\micro\metre}}$, and period $d={\SI{4}{\micro\metre}}$ (see in Fig. \[fig:scheme\]). In the finite size problem, the number of patches is $N$ and the incident field is the fundamental mode labelled 1 in Fig. \[fig:modes2D\](a).
Figure \[fig:compa\_modal\_direct3D\] shows the transmission $T_N$ (black curve, cf ) as a function of the number $N$ unit cells (or scatterers). The orange curve represents the coefficient $|t_1|^2$ (cf ), [which is a good approximation of]{} the fraction of incident energy carried into mode 1 and remaining in this channel after crossing the modified waveguide segment containing the scatterers. This last curve lies below the green curve that represents the sum of the amplitudes transmitted into all 6 channels (or modes) of the $z$-invariant waveguide shown in Fig. \[fig:modes2D\]. This numerical set up corresponds exactly to the configuration depicted in Fig. \[fig:poynting\] with $N=4$.
Finally, one can correlate these results to the infinitely periodic structure and superimpose the last red curve that represents [$T_1\,e^{-2\gamma''(N-1)d}$]{} (where $\gamma''=\mathrm{Im}\{\gamma\}$), the spatial damping of power associated with the mode with smallest propagation losses found using the modal approach detailed in Sec. \[sec:3Dmodal\]. Note that the normalization factor $T_1$ (transmission obtained for the direct 3D problem for one single obstacle) accounts for the fact the damping of the 3D mode computed with one obstacle does not make any sens in absence of obstacle ([*i*.e.]{} for $N=0$) and represents the input impedance of the structured waveguide. [The norm of this 3D mode is represented in purple/yellow colors at the top right of the Fig. \[fig:compa\_modal\_direct3D\]. The direction of the eigenfield is not represented for clarity, but it is globally polarized along $x$. This 3D mode is very similar to mode 1 in Fig. \[fig:modes2D\](a), which is the one injected in the direct problem to obtain the three other black, orange and green curves.]{} The consistency between these four quantities is remarkable.
Conclusion
==========
In conclusion, we present in this paper a general finite element frame for the study of discontinuous waveguides, from isolated discontinuities to fully periodic ones. A first method, adapted to a finite set of discontinuities allows to compute, given the modes of the invariant structure, the field scattered by the local discontinuity, all relevant energy related quantities, and the projection of the scattered field on the modes of the invariant structure (that is the elements of the transition scattering matrix).
When the modified region extends to infinity with periodic discontinuities, the relevant quantity is the dispersion relation of the so formed structured waveguide. An adequate weak treatment of divergence condition allows to determine these modes with accuracy.
The two numerical models presented in this paper show great interest for the design of structured waveguides. Note that the methodology adopted for the direct problem is very general and can readily be applied to a large variety of guiding structure and geometry of objects located in the modified waveguide segment.
[The two methods are in fact complementary. To give a concrete example in sensing applications, the adjunction of well chosen periodic scatterers [@fevrier2012giant] above a waveguide allows for instance to strengthen the interaction of the light flowing in the superstrate near the ridge where *e.g.* the molecules to detect lie. In this case, it is enough to study the modes profiles of the infinitely periodic 3D structure as detailed in Sec. \[sec:3Dmodal\]. It is indeed much faster than optimizing the scatterers properties using the direct problem introduced in Sec. \[sec:3Ddirect\] applied to a very long finite chain of scatterers. However, once the properties of the scatterers optimal for the targeted application, the practical device consists indeed of a finite chain. Then, the direct problem introduced in Sec. \[sec:3Ddirect\] is the ideal tool to study and optimize the coupling of an incident mode into the modified segment.]{}
Finally, open source models allowing to retrieve most of the results of this paper are provided [@code]. They can be tuned to handle different geometries and material properties.
This work will later be extended to the case where the input and output invariant structures mismatch using a coupled mode-FE approach [@pelat2011coupled] to compute the relevant incident fields.
Acknowledgement {#acknowledgement .unnumbered}
===============
This research was supported by ANR Louise project, grant ANR-15-CE04-000164 of the French Agence Nationale de la Recherche. The authors would like to thank the developers of MUMPS [@mumps-userguide], PETSc [@petsc-user-ref], SLEPc [@Hernandez2005-SSF], Gmsh [@gmsh] and GetDP [@getdp] for maintaining and making their respective libraries freely available. Finally, the authors acknowledge Sonia Fliss (INRIA POEMS) for fruitful discussions.
Disclosures {#disclosures .unnumbered}
===========
The authors declare no conflicts of interest.
[^1]: Corresponding author : `guillaume.demesy@fresnel.fr`
|
---
abstract: |
Photometry of galaxies has typically focused on small, faint systems due to their interest for cosmological studies. Large angular size galaxies, on the other hand, offer a more detailed view into the properties of galaxies, but bring a series of computational and technical difficulties that inhibit the general astronomer from extracting all the information found in a detailed galaxy image. To this end, a new galaxy photometry system has been developed (mostly building on tools and techniques that have existed in the community for decades) that combines ease of usage with a mixture of pre-built scripts. The audience for this system is a new user (graduate student or non-optical astronomer) with a fast, built-in learning curve to offer any astronomer, with imaging data, a suite of tools to quickly extract meaningful parameters from decent data. The tools are available either by a client/server web site or by tarball for personal installation. The tools also provide simple scripts to interface with various on-line datasets (e.g. 2MASS, Sloan, DSS) for data mining capability of imaged data.
As a proof of concept, we preform a re-analysis of the 2MASS Large Galaxy Atlas to demonstrate the differences in an automated pipeline, with its emphasis on speed, versus this package with an emphasis on accuracy. This comparison finds the structural parameters extracted from the 2MASS pipeline is seriously flawed with scale lengths that are too small by 50% and central surface brightness that are, on average, 1 to 0.5 mags too bright. A cautionary tale on how to reduce information-rich data such as surface brightness profiles. This document and software can be found at http://abyss.uoregon.edu/$\sim$js/archangel.
author:
- James Schombert
title: ARCHANGEL Galaxy Photometry System
---
Introduction
============
The photometric analysis of large galaxies is a double edged sword. While increased resolution, compared to distant galaxies, provides avenues for more detailed analysis of galaxy properties (to name a few: examination of star formation regions, examination of core parameters to search for massive blackholes, spiral arm analysis, isophotal irregularities that may signal rings, bars or other secular evolution processes), it is a data reduction fact that the larger number of pixels complicates the extraction of simple parameters, such as total magnitude, mean surface brightness and isophotal radius. For example, the fact that the galaxy is spread over a larger area of the sky means that the outer pixels have more sky luminosity than galaxy luminosity, increasing the error in any galaxy value. In addition, increased angular size has frequently prevented a fair comparison between distant and nearby galaxy samples simply because the techniques used to extract parameters from nearby galaxies differ from those used on small galaxies.
In general, the analysis of large galaxies (i.e. ones with many pixels) requires a full surface photometry study of the isophotes and their shape. For most extragalactic systems, the shape of choice is an ellipse. The astrophysics behind this assumption is that galaxy light traces stellar mass, and stellar mass follows elliptical orbits as given by Kepler’s 1st law. Certainly, this is true the case of early-type galaxies (elliptical and S0’s) as demonstrated by studies that examined the residuals from elliptical isophotes (Jedrzejewski 1987). This is also mostly true for disk galaxies, although the lumpiness of their luminosity distribution due to recent star formation increases the noise around each ellipse (see the discussion in §2.4). For dwarf irregular systems, any regular contours are poor describers of their shape, thus an ellipse is used because it is the simplest shape (aside from a circle) to describe an irregular isophote.
Therefore, the analysis of large galaxies begins with the reduction of their 2D images into 1D surface photometry as described by elliptical isophotes. In turn, the 1D profiles can be fit to various functions in order to extract characteristic luminosities (stellar mass), scale lengths and standard surface brightnesses (luminosity density). When combined with kinematic information, these three parameters form the Fundamental Plane for galaxies, a key relationship for understanding the formation and evolution of galaxies.
Aside from direct relevance to the Fundamental Plane, the need for better galaxy photometric tools has also increased with the influx of quality HST imaging. Before HST, distant galaxies were mere point sources, but now with WFPC2, ACS and NICMOS data, there is the need to perform full surface photometric studies on a much larger volume of the Universe. The sizes of our database on the photometric structure of galaxies has increased a thousandfold in the last 10 years, but most of the tools used to reduce this new data are up to 20 years out-of-date. Thus, the analysis of high resolution space imaging data is far behind spectroscopic and high energy data, not due to lack of interest, but due to the inadequacy of our 2D analysis tools.
The goal of this software project (called the ARCHANGEL project for obtuse historical reasons) has been to produce a series of proto-NVO type tools, related to surface photometry, and develop a computing environment which will extend the capability of individual observers (or entire data centers) to perform virtual observatory science. There is no attempt herein to replace existing analysis packages (i.e. PyIRAF), but rather our goal is to supplement existing tools and provide new avenues for data reduction as it relates to galaxy photometry. We hope that the fundamental components in this package will provide the community with new methods to which they can add their own ideas and techniques as well as provide learning environment for new researchers. In addition, there is growing amount of data by non-optical astronomers as new space missions provide imaging data in wavelength regions previously unexplored. Thus, there is a new and growing community of non-optical astronomers with 2D analysis needs that we hope to serve.
Package Philosophy
==================
The tools described herein are not intended to be a complete data reduction package per say, but rather a set of basic modules that allows the user to 1) learn the procedures of galaxy photometry, 2) tailor the tools to their particular needs, 3) begin an advanced learning curve of combining basic modules to produce new and more sophisticated tools. Turning raw data (level 1 or 2 data) from the telescope (space or ground) into calibrated, flattened images is the job of other, more powerful packages such as PyRAF. The tools presented herein bridge the intermediate step between calibrated data and astrophysically meaningful values. Specifically, we are concerned with the analysis of 2D images into 1D values, such as a surface brightness profile, and further tabulation into final values, such as total luminosity or scale length.
With respect to galaxy images, the numbers most often valued are luminosity, scale length and luminosity density. Unfortunately, due to the extended nature of galaxies, the quality and accuracy of these values can varying depending on the type of value desired. For example, luminosities can be extracted in metric form (luminosity within 16 kpc) or isophotal (luminosity inside the 26.5 mag arcsecs$^{-2}$ isophote or the total luminosity, an extrapolation to infinite radius. Scale length can be expressed as the radius of half-light or a formula fit to the luminosity distribution (e.g. Seric function). Luminosity density can be described through a detailed surface photometry profile, or integrated as a mean surface brightness within an isophote, or again a fitted curve such as an exponential disk. The tools provided by this project allow an inexperienced user the capability to explore their dataset and extract meaningful results while outlining the limitations to that data.
For the experienced researcher, these tools enhance their previous background in data reduction and provide new, and hopefully, faster avenues of analysis. To this end, the tools provided by this package provide a user with most basic of descriptions of a galaxy’s light, then allowing the option to select any meaningful parameter by toggling a switch. For most parameters, such as aperture magnitudes, the switch is simple and automatic. For more complicated parameters, such as a profile fit or an asymptotic magnitude, the switch is understandably more sophisticated and needing more explanation to the user for accurate use.
Basic Steps
-----------
This paper is divided into five sections describing the major components of the reduction package: 1) sky determination, 2) 2D profile fitting, 3) aperture photometry, 4) extraction of 1D parameters from surface brightness profiles and 5) extracting total magnitudes. Each section contains examples of the reduction of galaxy images from the 2MASS archive.
Quick Start
-----------
The fastest way to introduce the techniques and tools used in our package is to walk through the analysis of a several different types of galaxy images. A more non-linear reader can refer to the Appendix for a listing of the major tools. A script titled $profile$ is included which outlines the usage of the routines described below. For a majority of galaxy images, this script will produce a usable surface brightness profile, and this script forms the core of the client/server version of this package (see §5). But, a sharper understanding of the data requires more interaction with the techniques, the user is encouraged to run through the examples given in the package.
To illustrate our tools, we have selected 2MASS $J$ images of several galaxies found in the Revised Shapley-Ames catalog with the characteristics of smooth elliptical shape (NGC 3193), disk shape (IC 5271), spiral features (NGC 157) and low in surface brightness/low contrast to sky (NGC 2082). The analysis procedure for each galaxy is divided into five basic parts; 1) sky determination, 2) cleaning, 3) ellipse fitting, 4) aperture photometry and 5) profile fitting.
Before starting it is assumed that the data frame has be initially processed for flatfielding, dark subtraction and masking of chip defects. Small defects, such as cosmic rays, are cleaned by the analysis routines. But the errors are always reduced if known features are removed before analysis. The following routines work on poorly flattened data (e.g. gradients or large-scale features), and will signal the poorness by the resulting errors, but the removal of large-scale flattening problems requires more interaction then acceptable for this package and remains the responsibility of the user.
Sky Determination
-----------------
Any galaxy photometry analysis process begins with an estimate of the image’s sky value. While this is not critical for isophote fitting, it is key for actually finding targets, cleaning the frame of stars and smaller galaxies, plus determination of the photometry zeropoints. Accurate sky determination will, in the end, serve as the final limit to the quality of surface photometry data since a majority of a galaxy’s luminosity distribution is near the sky value. For this reason, sky determination has probably received as much attention in astronomical data literature as any other comparable reduction or analysis problem.
The difficulty in sky determination ranges from too few photons to know the behavior of the instrumental response (e.g., high energy data) to a high temporal varying flux of sky photons that overwhelms the galaxy signal (e.g., near-IR data). Surface fitting, drift scans, sky flats and super flats are all procedures used to minimize the sky contribution to the noise levels of the final data. Several clever, but not technically challenging, algorithms were included in the NOAO IRAF system to handle time averaged flats and data, median co-adding and cosmic ray subtraction. In the end, improved CCD quality lowered the demands of sky subtraction as the production of linear, good charge transfer and uniform sensitivity chips replaced the earlier generations and their wildly inaccurate backgrounds.
For a cosmetically smooth image, an efficient, but crude, sky fit is one that simply examines the border of the frame and does an iterated average, clipping pixels more than 4$\sigma$ from the mean. A border sky fit is often sufficient to find the starting center of the galaxy (for the ellipse fitting routines), clean the frame of stars/galaxies external to the object of interest (the ellipse fitting routines will clean along the isophotes, see below) and provide a preliminary error estimate to the photometry. This error estimate is preliminary in that the true limiting error in the surface (and aperture) photometry of large galaxies is not the RMS of an isophote, but how well the sky value is know. Once the number of pixels involved in a calculation (be it an isophote or an aperture) becomes large (greater than 50 for typical readout noises), then the error is dominated by the precision of the sky value.
The disadvantage to a border sky fit is the occasional inconvenient occurrence of stars or bright galaxies on the edge of the frame. An iterated mean calculation will remove small objects. And large objects will be signaled with large $\sigma$’s in an iterative mean search. In an automated procedure, more than likely, the task will have to halt and request human intervention to find a starting sky value.
After years of experimentation, the method of choice for accurate sky determination for extended galaxies is to evaluate sky boxes. This is a procedure where boxes of a set sized are placed semi-randomly (semi in the sense of avoiding stars and other galaxies) in the frame. An algorithm calculates an iterative mean and $\sigma$ for each box. These means (and $\sigma$’s) are then super-summed to find the value of the sky as the mean of the means (and likewise, the error on the sky value is the $\sigma$ on this mean).
From an analysis point of view, there are several advantages to this technique. One is that each box exists as a measurement independent of the other boxes. Thus, the statistical comparison of the boxes is a real measure of the quality of the sky determination for the frame in terms of its accuracy and any gross trends with the frame. Another advantage is that contaminating objects are relatively easy to avoid (visual choice of sky boxes) or to sort by the higher $\sigma$ per box. Lastly, sky boxes are the easiest method of finding regions for sky determination outside the galaxy itself, particularly where an irregular object may fill a majority of the data frame.
The most difficult decision in sky determination by boxes is, of course, where to place the boxes. When done visually, the user selects region (usually with a cursor) that are free of stars and sufficiently far away from the target galaxy to be clear of its envelope light. For an automated process, the procedure returns for a final sky estimate after the ellipse fitting process is completed and when all the stars/galaxies are cleaned (set to values of not-a-number, NaN). Then, the outer edge of the large galaxy is determined and an iterative analysis of sky boxes outside this radius is used to determine the true sky and, most importantly, the variation on the mean of those boxes as a measure of how well the sky is known. This procedure is the role of [*sky\_box*]{}, see the Appendix for a more detailed description of its options.
Ellipse Fitting
---------------
Reduction of a 2D image into a 1D run of intensity versus radius in a galaxy assumes some shape to the isophote. Very early work on galaxies used circles since the data was obtained through circular apertures in photoelectric photometers. For early type galaxies, the ellipse is the shape that most closely follows the shape of the isophotes. This would confirm that the luminosity being traced by an isophote is due to stellar mass, which follow elliptical orbits (Kepler’s 1st law). As one moves to along the Hubble sequence to later type galaxies, the approximation of an ellipse to the isophotes begins to break down due to recent star formation producing enhancements in luminosity density at semi-random positions. However, no consistent shape describes the isophotes of irregular galaxies, so an ellipse is the best shape, to first order, and provides a common baseline for comparison to more regular galaxies.
Fitting a best ellipse to a set intensity values in a 2D image is a relatively straight forward technique that has been pioneered by Cawson (1987) and refined by Jedrzejewski (1987) (see also an excellent review by Jensen & Jorgensen 1999). The core routine from these techniques (PROF) was eventually adopted by STSDAS IRAF (i.e. ELLIPSE). The primary fitting routine in this package follows the same techniques (in fact, uses much of the identical FORTRAN code from the original GASP package of Cawson) with some notable additions.
These codes start with an estimated x-y center, position angle and eccentricity to sample the pixel data around the given ellipse. The variation in intensity values around the ellipse can be expressed as a Fourier series with small second order terms. Then, an iterative least-squares procedure adjusts the ellipse parameters searching for a best fit, i.e. minimized coefficients. There are several halting factors, such as maximum number of iterations or minimal change in the coefficients, which then moves the ellipse outward for another round of iterations. Once a stopping condition is met (edge of the frame or sufficiently small change in the isophote intensity), the routine ends. A side benefit to above procedure is that the cos(4$\theta$) components to each isophote fit are easily extracted, which provides a direct measure of the geometry of the isophote (i.e. boxy versus disk-like, Jedrzejewski 1987).
One new addition, from the original routines, is the ability to clean (i.e. mask) pixels along an isophote. Basically, this routine first allows a few iterations to determine a mean intensity and RMS around the ellipse. Any pixels above (or below) a multiple of the RMS (i.e. 3$\sigma$) are set to not-a-number (NaN) and ignored by further processing. Due to the fact that all objects, stars and galaxies, have faint wings, a growth factor is applied to the masked regions. While this process is efficient in early-type galaxies with well defined isophotes, it may be incorrect in late-type galaxies with bumpy spiral arms and HII regions. The fitting will be smoother, but the resulting photometry will be underestimated. This process can be controlled early in the analysis pipeline by the user with an initial guess of the galaxy’s Hubble type. Also, the erased pixels are only temporary stored until an adequate fit is found. Once a satisfactory ellipse is encountered, only then are the pixels masked for later ellipse fitting. The masked data is written to disk at the end of the routine as a record of the cleaning. The ellipse fitting is the function of [*efit*]{} as described in the Appendix.
For early-type galaxies, lacking any irregular features, the cleaning process is highly efficient. The pipeline first identifies the galaxy and its approximate size by moment analysis. It then cleans off stars/galaxies outside the primary galaxy by moment identification and radius growth for masking. Stars/galaxies inside the primary galaxy are removed by the ellipse fitting routine. The resulting ellipses are inspected for crossover (isophotes that crossover are assumed to be due to errors or embedded stars/galaxies and removed by averaging nearby ellipse isophotes, this is not true for disk galaxies). The smoothed ellipses are used by a more robust cleaning algorithm and the whole ellipse fitting process is repeated on the cleaned frame.
An example of the analysis of an elliptical is found in Figure 1, a 2MASS $J$ image of NGC 3193. The top panel is the raw 2MASS image, the bottom panel is the resulting cleaned image output at the end of the reduction process. The cleaning process efficiently removed all the stars on the frame, including the brighter object on the northern edge of the frame and its diffraction spikes. The star closest to the galaxy core is a problem in two arenas. The first is in the calculation of ellipse, as the inner star would drag the calculated moments off center. The isophote erasing routine has handled this as can be seen in Figure 2, where the fitting ellipses are shown and are not deflected by the erased star. Second, is that calculated total magnitudes would either be over estimated (if the star is not masked) or under estimated (if the star is masked and the galaxy light from those pixels is not replaced). This problem will be discussed in §2.6.
As one goes towards later type galaxies, there is an increase in the non-elliptical nature to their isophotes and an increase in luminosity density enhancements (HII regions, stellar clusters, spiral features) which are legitimate components to the galaxy’s light distribution and should not be cleaned. The user can specify the galaxy type and the cleaning restrictions will be tightened (only to stellar objects and at a higher cleaning threshold) plus the restrictions on overlapping ellipses is loosened (e.g. the transition from a round bulge to a flat disk). Most importantly, while some galaxy features are cleaned for the sake of a harmonious ellipse fit, those pixels need to be filled for later aperture photometry.
An example of this behavior can be found in Figure 3, the $J$ image of IC 5271. The red ellipses indicate isophote fits that crossover. While flagged as an error, this is in fact the real behavior of the isophotes as one transitions from bulge to disk. The resulting intensities are probably overestimated due to the crossover effect, but this error will be minor compared to the errors that would result from an off-center or overly round ellipse.
The quality of the fitting procedure can be judged by the behavior of the ellipse parameters such as eccentricity, position angle and center. If there are large jumps in any of the parameters that determine shape, then this may signal a feature in the galaxy that needs to be cleaned (a buried star for example). Slightly less abrupt changes may signal an astrophysically interesting features, such as a bar or lens morphology. Under the assumption that the isophotes of a typical galaxy are a smooth function with radius, the ellipse fitting algorithm checks for ellipse parameters that indicate a crossing of the isophotal lines. These ellipses are smoothed and flagged (the mean of the inner and outer ellipse parameters is used). In certain scenarios, crossing isophotes are to be expected, for example the transition region from a bulge to a disk (see Figure 2), and the smoothing criteria is relaxed. This is the function of $prf\_smooth$ as described in the Appendix.
An example of $prf\_smooth$’s corrections can be seen in Figure 4, the $J$ image of NGC 157. Several interior ellipses display erratic behavior, but $prf\_smooth$ took the mean average of nearby ellipses (in green) to produce a more rational fit. The resulting intensities were also more stable, although the RMS is going to be highly than the typical isophotes found in an elliptical.
An example of a LSB galaxy fit is found in Figure 5. The ellipse fitting routine, recognizing that the target is low in contrast with respect to sky, widened the annulus for collecting pixel values. This increases the S/N at some loss of spatial resolution. Since resolution is usually not important in a galaxy’s halo region, this is an acceptable trade off. LSB galaxies are susceptible to fitting instability, the fitting routines are tightened against rapid changes in eccentricity and centering to prevent this behavior.
LSB galaxies also demonstrate a key point in determining errors from surface photometry. There are two sources of error per isophote, the RMS around the ellipse and the error in the sky value. The RMS value is a simple calculation using the difference between the mean and the individual pixel values. This RMS then reflects into an observable error as the $\sqrt{N}$. However, as the isophote intensity approaches the sky value, the number of pixels increases and the error due to RMS becomes an artificially low value. In fact, at low intensities, the knowledge of the sky value dominates and the error in the isophote is reflected by the sky error (preferably as given by the $\sigma$ on the means of a large number of sky boxes).
Surface Photometry
------------------
With a file of isophotal intensities versus radius in hand, it is a simple step to producing a surface brightness profile for the galaxy. There are a few tools are in the package to examine the quality of the ellipse fitting (e.g. $prf\_edit$, an interactive comparison of the image and the ellipses). At the very least, a quick visual inspection of the ellipses seems required as a bad mismatch leads to strongly biased results (see cautionary tale in §4). A user can either step through a directory of data files (e.g. using the $probe$ tool) or a user can automatically produce a group of GIF images with a corresponding HTML page, then use a browser to skim through a large number of files. Calibration from image data numbers (DN) to fluxes (or magnitudes) is usually obtained through standard stars with corrections for airmass and instrumental absorption. If these values are in the FITS headers, then they are automatically added to the object’s XML file. Additional corrections for galactic absorption, k-corrections and surface brightness dimming are well documented in the literature and can be assigned automatically by grabbing XML data from NED. A chosen cosmology converts radius in pixel units into astrophysically meaningful values of kiloparsecs. A Python command line script ($cosmo$) based on Ned Wright’s cosmology calculator is included in the package All these values can be added to the XML file for automatic incorporation to the analysis programs. If they don’t exists, then instrumental mags will be used, which can easily be converted to real units later on.
Analysis of a 1D surface brightness profile (the job for the $bdd$ tool) depends on the scientific goals of the user. For example, early-type galaxies are typically fit to a de Vaucouleurs r$^{1/4}$ curve to extract a scale length (effective radius) and characteristic surface brightness (effective surface brightness). Irregular and dwarf galaxies are well fit by exponential profiles which provide a disk scale length and central surface brightness. Disk galaxies can be fit with a combination of bulge and disk fits, to extract B/D ratios and disk scale lengths.
Due to this combination of $r^{1/4}$ and exponential curves for large bulge spirals, it is computationally impossible to correctly determine which function, or combination of functions, best fits a particular galaxy’s profile. In the past, one would examine the 2D image of the galaxy and obvious disk-like galaxies would be fit to $r^{1/4}$ plus exponential. Objects with elliptical appearance were fit to a strict $r^{1/4}$ shape. This produces a problem for large bulge S0’s which are difficult to detect visually unless nearly edge-on.
The simplest solution to this problem, using only the 1D surface photometry, is to examine the profiles in a plot of mag arcsecs$^2$ versus linear radius. With this plot, exponential disks appear as straight lines, see Figure 6 as an example of a pure disk in NGC 2403. Bulge plus disk components are also straight forward in this mag/linear radius space, see Figure 7 a good example of a bulge plus disk fit in NGC 3983. If a profile displays too much curvature, with no clear linear disk portion, then it is a good candidate for a pure $r^{1/4}$ fit (see Figure 8, NGC 3193). This option is easily checked by plotting the profile in mag arcsecs$^2$ versus r$^{1/4}$ space as shown in Figure 8. Most r$^{1/4}$ profiles only have a linear region in the middle of the surface brightness profile, typically with a flattened core and fall-off at large radii (see Schombert 1987).
The Seric function is also popular for fitting surface brightness profiles (Graham & Driver 2005), although not currently supported by this package, any fitting function is easy to add to the reduction routines as the core search routine is a grid search $\chi^2$ minimization technique. However, there are issues with surface photometric data where the inner regions have the highest S/N but the outer regions better define a galaxy’s structure (Schombert & Bothun 1987). With user guidance, this grid search works well for any user defined function. Also, since there are a sufficient number of packages for fitting 1D data in the community, this package only provides a simple graphic plotting function. More sophisticated analysis needs guidance by the user, but this package provides the framework for just such additions.
Aperture Photometry
-------------------
Often the scientific goal of a galaxy project is to extract a total luminosity for the system (and colors for multiple filters). For small galaxies, a metric aperture or isophotal magnitude is suitable for comparison to other samples (certainly the dominate source of error will not be the aperture size). However, for galaxies with large angular size (i.e. many pixels), their very size makes total luminosity determination problematic.
Natively, one would think that a glut of pixels would make the problem of determining a galaxies luminosity easier, not more difficult. However, the problem here arises with the question of where does the galaxy stop? Or, even if you guess an outer radius, does your data contain all the galaxy’s light? The solution proposed by de Vaucouleurs’ decades ago is to use a curve of growth (de Vaucouleurs 1977). Almost all galaxies follow a particular luminosity distribution such that the total light of a galaxy can be estimated by using a standard growth curve to estimate the amount of light outside your largest aperture. For a vast majority of galaxies, selecting either an exponential or r$^{1/4}$ curve of growth is sufficient to adequately describe their total luminosities (Burstein 1987). However, for modern large scale CCD imaging, the entire galaxy can easily fit onto a single frame and there is no need for a curve of growth as all the data exists in the frame.
With adequate S/N, it would seem to be a simple task to place a large aperture around the galaxy and sum the total amount of light (minus the sky contribution). However, in practice, a galaxy’s luminosity distribution decreases as one goes to larger radii, when means the sky contribution (and, thus, error) increases. In most cases, larger and larger apertures simply introduce more sky noise (plus faint stars and other galaxies). And, to further complicate matters, the breakover point in the optical and near-IR, where the galaxy light is stronger than the sky contribution will not contain a majority of the galaxy’s light. So the choice of a safe, inner radius will underestimate the total light.
The procedure selected in this package, after some numerical experimentation, is to plot the aperture luminosity as a function of radius and attempt to determine a solution to an asymptotic limit of the galaxy’s light. This procedure begins by summing the pixel intensities inside the various ellipses determined by $efit$. For small radii, a partial pixel algorithm is used to determine aperture luminosity (using the surveyors technique to determine each pixel’s contribution to the aperture). At larger radii, a simply sum of the pixels, and the number used, is output. In addition, the intensity of the annulus based on the ellipse isophote and one based on the fit to the surface photometric profile are also outputted at these radii (see below).
Note that a correct aperture luminosity calculation requires that both a ellipse fit and a 1D fit to the resulting surface photometry has be made. The ellipse fit information is required as these ellipses will define the apertures, and masked pixels are filled with intensities given by the closest ellipse. A surface photometric fit allows the aperture routine to use a simple fit to the outer regions as a quick method to converge the curve of growth.
Once the aperture luminosities are calculated, there are two additional challenges to this procedure. The first is that an asymptotic fit is a difficult calculation to make as the smallest errors at large radii reflect into large errors for the fit. Two possible solutions are used to solve this dilemma. The first solution is to fit a 2nd or 3rd order polynomial to the outer radii in a luminosity versus radius plot. Most importantly for this fit, the error assigned the outer data points is the error on the knowledge of the sky, i.e. the RMS of the mean of the sky boxes. This is the dominant source of error in large apertures and the use of this error value results in a fast convergence for the asymptotic fit. The resulting values from the fit will be the total magnitude and total isophotal size, determined from the point where the fit has a slope of zero. A second solution is to use an obscure technique involving rational functions. A rational function is the ratio of two polynomial functions of the form
$$f(x) = {{a_nx^n+a_{n-1}x^{n-1}+ ... + a_2x^2+a_1x+a_0} \over
{b_mx^m+b_{m-1}x^{m-1}+ ... + b_2x^2+b_1x+1}}$$
where $n$ and $m$ are the degree of the function. Rational functions have a wide range in shape and have better interpolating properties than polynomial functions, particularly suited for fits to data where an asymptotic behavior is expected. A disadvantage is that rational functions are non-linear and, when unconstrained, produce vertical asymptotes due to roots in the denominator polynomial. A small amount of experimentation found that the best rational function for aperture luminosities is the quadratic/quadratic form, meaning a degree of 2 in the numerator and denominator. This is the simplest rational function and has the advantage that the asymptotic magnitude is simply $a_2/b_2$, although is best evaluated at some radii in the halo of the galaxy under study.
Usually the aperture luminosity values will not converge at the outer edges of a galaxy. This is the second challenge to aperture photometry, correct determination of the luminosity due to the faint galaxy halo. This is where the surface photometry profile comes in handy. Contained in that data is the relationship between isophotal luminosity and radius, using all the pixels around the galaxy. This is often a more accurate number than attempting to determine the integrated luminosity in an annulus at the same radius. This information can be used to constraint the curve of growth in two ways. One, we can use the actual surface brightness intensities and convert them to a luminosity for each annulus at large radii. Then, this value can be compared to the aperture value and a user (or script) can flag where the two begin to radically deviate. Often even the isophotal intensities will vary at large radii and, thus, a second, more stable method is to make a linear fit of an exponential, r$^{1/4}$ or combined function to the outer radii and interpolate/extrapolate that fit to correct the aperture numbers.
Figure 9 display the results for all three techniques for the galaxy NGC 1003. The black symbols are the raw intensities summed from the image file. The blue symbols are the intensities determined from the surface photometry. The orange symbols are the intensities determined from the fits to the surface photometric profile. This was one of the worst case scenarios due to the fact the original image is very LSB (in the near-IR $J$ band). Due to noise in the image and surface photometry, the outer intensities grow out of proportional to the light visible in the greyscale figure. A fit to the raw data does not converge (blue line). A 2nd order fit to the profile fit (orange line) also fails to capture the asymptotic nature. The rational function fit (pink line) does converge to an accurate value. If similar types of galaxies are being analyzed, it is a simple procedure to automate this process.
Data files and XML
==================
In the past, when disk space was at a premium and I/O rates were slow, astronomical data was stored in machine specific formats. However, today disk space is plentiful and file access times are similar to processing times on most desktop systems. Thus, a majority of simple astronomical databases are stored in flat file format, also called plain text or ASCII files (note: this is an interesting throwback to the original data methods from the end of the 19th century, where information is stored as a system of data and delimiters, such as spaces or commas). The endproduct data files for most packages rarely exceed a few kilobytes or a few hundred lines. The simplest access to these types of files is an editor such as vi or emacs. Sufficient documentation (i.e. header files) makes understanding the data, and writing applications for further analysis these data files, a relatively simple task.
However, there is a strong driver to migrate output files into XML format for even the simplest data files. Extensible Markup Language (XML) is a W3C-recommended general-purpose markup language that is designed store and process data. The core of XML is the use of tags, similar to HTML tags (i.e. `<tag>data</tag>`), to delineate data values and assign attributes to those values (for example, units of measure). XML is not terse and, therefore, somewhat human-legible (see Figure 1 for XML example of astronomical data).
XML has several key advantages over plain file formats. For one, XML format allows an endless amount of additional information to be stored in each file that would not have fit into the standard data plus delimiter style. For example, calibrating data, such as redshift or photometric zeropoint, can be stored in each file along with the raw data with very little increase in file size overhead as the tags handle the separation. There is no need to reserve space for these quantities nor is there any problem adding future parameters to the XML format. Using XML format puts all the reduction data into a single file for compactness and, in addition, since XML files are plain text files, there is no problem with machine to machine transfer. The reading of XML files is not a complication for either compiled or interpreted languages.
A disadvantage to XML format is that it’s clumsy to read. However, there exist a number of excellent XML editors on the market (for example, http://www.oxygenxml.com). These allow a GUI interface with an efficient query system to interact with the XML files. While many users would prefer to interact with the raw data files in plain text form, in fact, even a simple editor is GUI window into the bytes and bits of the actual data on machine hardware. A GUI XML editor is simply a more sophisticated version of vi or emacs.
An additional reason to migrate to XML is a new power that XML data files bring to data analysis. Many interpreted languages (i.e. Python and Perl) have an [*eval*]{} or [*exec*]{} function, a method to convert XML data into actual variables within the code at runtime (i.e. dynamically typed). This has a powerful aspect to analysis programs as one does not have to worry about formats or the type of data entries, this is handled in the code itself. Dynamical typing introduces a high level of flexibility to code. In Python, one can convert XML data (using Python’s own XML modules to read the data) into lists that contain the variable name and value, then transform these lists into actual code variables using an [*exec*]{} command. For example,
for var, value in xml_vars:
exec(var+'='+value)
produces a set of new variables in the running code. And Python’s unique try/except processing traps missing variables without aborting the routine. For example, if the variable ’redshift’ exists in the XML data file then
try:
distance=redshift/H_o
except:
print 'redshift undefined'
distance=std_distance
This same try/except processing also traps overflows and other security flaws that might be used by a malicious user attempting to penetrate your server using the XML files. Thus, XML brings a level of security as well as enhancing your code.
Lastly, another advantage to XML format is the fact that all of the reduction data (ellipse fitting, aperture photometry, calibration information, surface photometry) can be combined into a single file, e.g. galaxy\_name.xml, which can be interrogated by any analysis routine that understands XML. A simple switch at the end of the reduction process integrates the data into an XML file for transport, or access by plotting packages, etc.
A Cautionary Tale, the 2MASS Large Galaxy Atlas
===============================================
If you have read this far, and are still awake, this section walks through the reduction of part of the Revised Shapley-Ames sample (Schombert 2007) taken from the 2MASS database that overlaps the 2MASS Large Galaxy Atlas (Jarrett 2003). As a cautionary tale to the importance to doing large galaxy photometry with care, we also offer in this section a comparison of our technique with the results from an automated, but much cruder reduction pipeline from the 2MASS project.
Allowing the ellipses to vary from isophote to isophote, not only in eccentricity, but also in position angle and ellipse center, are critical to obtaining an accurate description of a galaxy’s luminosity profile. Shown in Figures 10, 11 and 12 are examples of $J$ images extracted from the 2MASS archives that were part of the Large Galaxy Atlas (Jarrett 2003). In each case, the 2MASS pipeline calculates a luminosity profile based on the isophotes around an ellipse from the mean moments of the whole galaxy. Thus, the fitted ellipses do not change in axial ratio or position angle and, for most spirals, this technique will result in an ellipse that is too flat in the core and, often, too round in the halo regions. If the galaxy has a bar, this technique will also underestimate the bar contribution, spreading its light into larger radii ellipses.
Given that the light is averaged around the ellipse, this effect may be minor if the galaxy is fairly smooth and uniform. However, galaxies that are smooth and regular are a minority in the local Universe. For the three examples, shown in Figure 10, 11 and 12, the 2MASS fits consistently underestimate the amount of disk light per isophote, as seen in comparison to the luminosity profile determined from the raw data using the ARCHANGEL routines. This, in turn, results in fitted central surface brightnesses that are too bright in central surface brightness, and fitted disk scale lengths that are too shallow. In fact, for 49 galaxies in common between the near-IR RSA sample (Schombert 2007) and the 2MASS Large Galaxy Atlas, Figure 13 displays the difference between the fitted disk scale lengths ($\alpha$) and the difference between the fitted disk central surface brightness ($\mu$). Given the typical $\alpha$’s, the error in 2MASS fits corresponds to a 50% error in a galaxy’s size. Likely, errors in the central surface brightness fits averages around 0.5 mags. Thus, not using the proper reduction technique not only increases the noise in the measured parameters, but produces a biased result.
{width="17cm"}
Network Tools
=============
One of the more powerful modules to the Python language is the $urllib$, the module that allows Python scripts to download any URL address. If address is a web page, there also exist several addition modules that parse HTML and convert HTML tables into arrays. This means a simple script can be written to pull down a web page, parse it HTML and extract a data into table format. And, on top of this procedure, the information could be then be used into a standard GET/POST web form used by many data archives.
As an example, the package contains [*dss\_read*]{}, a script that takes the standard name for a galaxy, queries NED for its coordinates and then goes to the DSS website and extracts the PSS-II image of the galaxy. While this sounds like a computationally intense task, in fact the script is composed of 49 lines. The downside to this network power is, of course, the possibility of abuse. Unrestricted application of such scripts will overload websites and given network speeds, the typical user doesn’t need their own personal digital sky at their home installation.
Lastly, various archives, in order to slow massive downloads, have an ID/password interface. To penetrate these sites requires the [*mechanize*]{} module which simulates the actions of a brower, following links, parsing ID’s and passwords and handling cookies. While these avatars are simple to build, the wise usage of them remains a key challenge for the future.
Package Summary
===============
The fastest way to learn a data reduction process is to jump in and try it. To this end, the tarball contains all the images discussed in this document, and several test images with known output. This allows the user to practice on images where the final results are known. Thus, we encourage the readers to download, compile and run! Tarballs are found at http://abyss.uoregon.edu/$\sim$js/archangel.
Another option, for the user who doesn’t wish to set-up the package on their own system (or perhaps only has a handful of galaxies to reduce), is the client/server version of this package available at http://http://abyss.uoregon.edu/$\sim$js/nexus (see Figure 14). Although more limited in its options, the web version has the advantage of speed (it’s run on a Solaris Sun Blade) and a fast learning curve.
As to the future, a number of tools need to be added to this package. For example, quantitative morphology uses the concentration and asymmetry indices to parameterize a galaxy’s global structure. While these values are easy to extract from small angular size objects, they are a challenge for large systems. Yet, a detailed comparison of these values to visual morphology is a key step in understanding quantitative morphology at higher redshifts. However, in order get the current tools out to the community, the package is frozen. Additional tools will be added to the package website as, in order of priority, 1) needed by the PI to meet various science goals, 2) requested by outside users to obtain their science goals, and 3) requested by outside users as possible new computational areas to explore. As with all evolving software, an interested user should contact the author to see where future directions lie (js@abyss.uoregon.edu).
This project was funded by Joe Bredekamp’s incredible NASA’s AIRS Program. I am grateful to all the suggestions I have gotten from AIRS PI’s at various workshops and panel reviews. The program is a mixed of technology plus science types and is one of NASA’s true gems for innovative research ideas.
Burstein, D., Davies, R. L., Dressler, A., Faber, S. M., Stone, R. P. S., Lynden-Bell, D., Terlevich, R. J., & Wegner, G. 1987, , 64, 601
Cawson, M. G. M., Kibblewhite, E. J., Disney, M. J., & Phillipps, S. 1987, , 224, 557
Graham, A. W., & Driver, S. P. 2005, Publications of the Astronomical Society of Australia, 22, 118
Jarrett, T. H., Chester, T., Cutri, R., Schneider, S. E., & Huchra, J. P. 2003, , 125, 525
Jedrzejewski, R. I. 1987, , 226, 747
Schombert, J. M., & Bothun, G. D. 1987, , 93, 60
de Vaucouleurs, G. 1977, , 33, 211
Package Management
==================
This package is a combination of FORTRAN and Python routines. The choice of these languages was not arbitrary. Python is well suited for high level command processing and decision making. It is a clear and expressive language for text processing. Therefore, its style is well suited to handling file names and data structures. Since it is a scripting language, it is extremely portable between OS’s. Currently, every flavor of Unix (Linux, Mac OS X and Solaris) comes packaged with Python. In addition, there is a hook between the traditional astronomy plotting package (PGPLOT) and Python (called ppgplot), which allows for easy GUI interfaces that do not need to be compiled.
The use of FORTRAN is driven by the fact that many of the original routines for this package were written in FORTRAN. For processing large arrays of numbers, C++ provides a faster routine, but current processor speeds are such that even a 2048x2048 image can be analyzed with a FORTRAN program on a dual processor architecture faster than the user can type the next command. STScI provides a hook to FITS formats and arrays (called pyfits and numarray), but Python is a factor of 100 slower than FORTRAN for array processing.
Currently there are three FORTRAN compilers in the wild, g77, gfortran and g95. The routines in this package can use any of these compilers plus a version of Python greater than 2.3. CFITSIO is required and avaliable for all OS’s from its GSFC website. The Python libaries pyfits and numarray are found at STScI’s PyRAF website. For any graphics routines, the user will need a verson of PGPLOT and install ppgplot as a Python library. The ppgplot source is avaliable at the same website as this package. The graphics routines are only needed for data inspection, the user should probably develop their own high-level graphics to match their specifics. In the directory /util one can find all the Python subroutines to fit 1D data surface or aperture photometry. The examples in this manual will guide you in constructing your own interface.
Lastly, the output data files for this package are all set in XML format. This format is extremely cumbersome and difficult to read (it is basically an extension of the HTML format that web browsers use). However, a simple command line routine is offered ($xml_archangel$) that will dump or add any parameter or array out of or into a XML file.
Core Analysis Routines
======================
To go from a raw data frame containing a galaxy image to a final stage containing ellipse fits, surface photometry, profile fits and aperture values requires three simple scripts, [*profile*]{}, [*bdd*]{} and [*el*]{}. The scripts [*profile*]{} and [*el*]{} are automatic and can be run as batch jobs. [*bdd*]{} is an interactive routine to fit the surface photometry and is a good mid-point to study the results of the ellipse fitting. In a majority of cases, the user simply needs to run those three scripts with default options to achieve their science results.
Note that command -h will provide a short summary of the commands usage.
[*sky\_box*]{}:
Usage: sky_box option file_name box_size prf_file
options: -h = this message
-f = first guess of border
-r = full search, needs box_size
and prf_file
-t = full search, needs box_size
-c = find sky for inner region (flats)
needs x1,x2,y1,y2 boundarys
Output: 1st mean, 1st sig, it mean, it sig npts, iterations
[$efit$]{}:
Usage: efit option file_name output_file other_ops
Ellipse fitting routine, needs a standard FITS file,
output in .prf file format (xml_archange converts this
format into XML)
options: -h = this mesage
-v= output each iteration
-q = quiet
-xy = use new xc and yc
-rx = max radius for fit
-sg = deletion sigma (0=no dets)
-ms = min slope (-0.5)
-rs = stopping radius
-st = starting radius
when deleting, output FITS file called file_name.jedsub
[$prf\_smooth$]{}:
Usage: prf_smooth option prf_file_name
parameter #6 set to -1 for cleaned ellipse, 0 for
unfixable ones
options: -x = delete unfixable ones
-s = spiral, low smooth
-d = neutral smooth
-q = quick smooth
[$prf\_edit$]{}:
Usage: prf_edit file_name
visual editor of isophote ellipses output from efit
note: needs a .xml file, works with cleaned images
cursor commands:
r = reset display z = zoom in
c = change cont x = flag ellipse/lum point
t = toggle wd cursor o = clean profile
q = exit h = this message
[$probe$]{}:
Usage: probe option master_file
quick grayscale display GUI
options: -f = do this image only
-m = do file of images
cursor commands:
/ = abort q = move to next frame
c = contrast r = reset zoom
z = zoom t = toggle ellipse plot
p = peek at values
a,1-9 = delete circle b = delete box
[$bdd$]{}:
Usage: bdd options file_name
quick surface photometry calibration and fitting GUI
options: -h = this message
-p = force sfb rebuild
window #1 cursor commands:
c = contrast control r = reset boundaries
z = zoom on points x = delete point
s = set sky (2 hits) i = show that ellipse
/ = write .sfb file q = abort
window #2 cursor commands:
x = erase point d = disk fit only
m = erase all min pts f = do bulge+disk fit
u = erase all max pts e = do r**1/4 fit only
b = redo boundaries p = toggle 3fit/4fit
q = abort r = reset graphics
/ = write .xml file and exit
[*xml\_archangel*]{}:
xml_archangel op file_name element data
add or delete data into xml format
-o = output element value or array
-d = delete element or array
-a = replace or add array, array header and data is cat'ed into routine
-e = replace or add element
-c = create xml file with root element
-k = list elements, attributes, children (no data)
[$el$]{}:
Usage: el options cleaned_file
script that takes cleaned FITS file and fills in NaN
pixels
from efit isophotes, then does elliptical apertures on
resulting .fake file
options: -v = verbose
[$asymptotic$]{}:
Usage: asymptotic xml_file
simply GUI that determines asymptotic fit on integrated
galaxy mag,
delivers mag/errors from apertures and curve of growth
fit into XML file
cursor commands:
r = reset a = adjust lum for better fit
f = linear fit x,1,2,3,4 = delete points
z = set profile extrapolation point
/ = exit b = change borders
|
---
abstract: 'We study the dynamical response of a two-dimensional Ising model subject to a square-wave oscillating external field. In contrast to earlier studies, the system evolves under a so-called soft Glauber dynamic \[P. A. Rikvold and M. Kolesik, J. Phys. A: Math.Gen. [**35**]{}, L117 (2002)\], for which both nucleation and interface propagation are slower and the interfaces smoother than for the standard Glauber dynamic. We choose the temperature and magnitude of the external field such that the metastable decay of the system following field reversal occurs through nucleation and growth of many droplets of the stable phase, i.e., the multidroplet regime. Using kinetic Monte Carlo simulations, we find that the system undergoes a nonequilibrium phase transition, in which the symmetry-broken dynamic phase corresponds to an asymmetric stationary limit cycle for the time-dependent magnetization. The critical point is located where the half-period of the external field is approximately equal to the metastable lifetime of the system. We employ finite-size scaling analysis to investigate the characteristics of this dynamical phase transition. The critical exponents and the fixed-point value of the fourth-order cumulant are found to be consistent with the universality class of the two-dimensional equilibrium Ising model. As this universality class has previously been established for the same nonequilibrium model evolving under the standard Glauber dynamic, our results indicate that this far-from-equilibrium phase transition is universal with respect to the choice of the stochastic dynamics.'
author:
- 'G. M. Buend[í]{}a$^{1,2}$'
- 'P. A. Rikvold$^{3,4}$'
title: ' Dynamic phase transition in the two-dimensional kinetic Ising model in an oscillating field: Universality with respect to the stochastic dynamic '
---
Introduction {#sec:INTRO}
============
Kinetic Ising or lattice-gas models with stochastic dynamics have been successfully applied to study a number of dynamical physical phenomena, including metastable decay [@RIKV94A; @RAMO99; @BERT04; @BERT04B; @FRAN05; @FRAN06], hysteretic responses [@SIDE98; @SIDE99; @KORN00], and magnetization switching in nanoscale ferromagnets [@RICH94; @NOVO02A]. Among the dynamic phenomena in such models that have attracted particular attention in recent years, is the dynamic phase transition (DPT) observed in systems with Ising-like symmetry that are driven far from equilibrium by an oscillatory applied force (typically a magnetic field or (electro)chemical potential). This phenomenon was first observed in kinetic simulations of a mean-field model [@TOME90; @MEND91] and later studied intensively by mean-field [@ZIMM93A; @BUEN98; @ACHA95; @CHAK99], Monte Carlo [@LO90; @ACHA95; @CHAK99; @SIDE98; @SIDE99; @KORN00; @KORN02B; @ROBB07], and analytical [@FUJI01; @TUTU04; @MEIL04; @DUTT04] methods. In this transition, the dynamic order parameter, which is the cycle-averaged magnetization, vanishes in a singular fashion at a critical value of the period of the applied field. Recently, strong experimental evidence has emerged that this nonequilibrium phase transition is observable in magnetic thin-film systems [@ROBB08], and an analogous phenomenon has been observed in simulations of a model of the heterogeneous catalytic oxidation of CO [@MACH05; @BUEN06C].
Perhaps the most fascinating aspect of this far-from-equilibrium phase transition is that it belongs to the same universality class as the corresponding equilibrium Ising model. This result is predicted from symmetry arguments [@GRIN85; @BASS94] and has been confirmed by exhaustive kinetic Monte Carlo simulations [@SIDE98; @SIDE99; @KORN00; @KORN02B] and analytical results [@FUJI01]. Very recently, the field conjugate to the dynamic order parameter was identified as the cycle-averaged applied field, and a fluctuation-dissipation relation valid near the nonequilibrium critical point was numerically established [@ROBB07].
The physics of equilibrium phase transitions is well understood, and it is well established that structures arising from different dynamics that obey detailed balance and respect the same conservation laws exhibit universal asymptotic large-scale features. However, the mechanisms behind nonequilibrium phase transitions are not that well known, and the dependence on the specific dynamic is still an open question. Except for the study of the model of CO oxidation [@MACH05; @BUEN06C], all previous kinetic Monte Carlo simulations in which this DPT was observed, were performed with the standard stochastic Glauber [@GLAU63] or Metropolis [@METR53] dynamics. All these studies, including the study of CO oxidation which used a very different dynamic, found critical exponent ratios consistent with the equilibrium Ising values, $\gamma/\nu = 7/4$ and $\beta/\nu = 1/8$. This gives a reasonable indication that this DPT is universal with respect to details of the model and the stochastic dynamics. However, a more direct test of just the universality with respect to the dynamics would be to use a significantly different stochastic dynamic for the two-dimensional kinetic Ising model. Such a test is the subject of the present paper.
All stochastic dynamics that respect detailed balance eventually lead to thermodynamic equilibrium [@LAND00], and all dynamics that obey the same conservation laws also give the same long-time dynamics (e.g., a $t^{1/2}$ dependence of the characteristic length for phase ordering with a non-conserved order parameter and a $t^{1/3}$ dependence for phase separation with a conserved order parameter [@GUNT83B]). However, it has recently been demonstrated that different stochastic dynamics give quantitatively dramatically different results for low-temperature nucleation [@PARK04; @BUEN04B], as well as for the nanostructure and mobility of field-driven interfaces [@RIKV02; @RIKV02B; @RIKV03; @BUEN06; @BUEN06B]. The differences are particularly striking between dynamics known as “hard," in which the effects of the configurational and field-related (“Zeeman") energy contributions in the transition rate do not factorize, and “soft," for which such factorization is possible [@RIKV02; @PARK04; @MARR99]. The class of hard dynamics includes the standard Glauber and Metropolis dynamics, while the soft dynamics here will be represented by the “soft Glauber dynamic" introduced in Ref. [@RIKV02], whose transition rate is given in Sec. \[sec:MODEL\]. Briefly, the nanostructure of field-driven “hard" interfaces is characterized by a local interface width and mobility that increase dramatically with the strength of the applied field, while “soft" interfaces remain relatively smooth and slow-moving, independent of the field [@RIKV02]. Similarly, low-temperature nucleation under hard dynamics becomes very fast for strong fields, while under soft dynamics it remains thermally activated and thus very slow, even for very strong fields [@PARK04]. While the soft Glauber dynamic is probably not particularly relevant to any specific physical system, it is ideally suited for comparison with the standard, hard Glauber dynamic in investigating universal properties of the DPT.
The rest of this paper is organized as follows. The kinetic Ising model and its dynamics are introduced in Sec. \[sec:MODEL\], and the numerical results and finite-size scaling analysis are presented in in Sec. \[sec:MC\]. A summary and conclusions are given in Sec. \[sec:DISC\].
Model and Dynamics {#sec:MODEL}
==================
For this study we choose a kinetic, nearest-neighbor, Ising ferromagnet on a square lattice with periodic boundary conditions. The Hamiltonian is given by $${\cal H}= -J\sum_{\langle ij \rangle} s_i s_j - H(t)\sum_{i} s_{i},
\label{eq:Ham}$$ where $s_{i}=\pm1$ is the state of the spin at the site $i$, $J>0$ is the ferromagnetic interaction, $\sum_{\langle ij \rangle}$ runs over all nearest-neighbor pairs, $\sum_{i}$ runs over all $L^2$ lattice sites, and $H(t)$ is an oscillating, spatially uniform applied field. The magnetization per site $$m(t)=\frac{1}{L^2}\sum_{i=1}^{L^2}s_{i}(t),
\label{eq:mag}$$ is the density conjugate to $H(t)$. The temperature $T$ (in this paper given in units such that Boltzmann’s constant equals unity) is fixed below its critical value ($k_{B}T_{c}= J/ \ln (1+\sqrt{2})$), so that, when there is no external field, the system has two degenerate equilibrium states with magnetizations of equal magnitude and opposite direction. When an external field is applied the degeneracy is lifted, and the equilibrium state is the one with magnetization in the same direction as the field. If the external field is not too strong, the state with opposite magnetization direction is metastable and eventually decays toward equilibrium [@RIKV94A]. This model is equivalent to a lattice-gas model with local occupation variables $c_i = (s_i+1)/2 \in \{0,1\}$ and (electro)chemical potential $\mu \propto H$ (for further details, see Ref. [@RIKV02B]).
The system evolves under the soft Glauber single-spin-flip (non-conservative) stochastic dynamic with updates at randomly chosen sites. In the lattice-gas representation, this corresponds to adsorption/desorption without lateral diffusion. The time unit is one Monte Carlo step per spin (MCSS). When the system is in contact with a heat bath at a temperature $T$, each proposed spin flip is accepted with probability $$W_{\rm SG}
= \frac{1}{1+ \exp (\beta\Delta E_J)}\frac{1}{1+ \exp (\beta\Delta E_H)}\;.
\label{eq:softg}$$ Here $\beta = 1/T$, $\Delta E_J$ is the energy change corresponding to the interaction term, and $\Delta E_H$ is the energy change corresponding to the field term in the Hamiltonian, Eq. (\[eq:Ham\]). This transition probability is to be contrasted with those of the standard, hard Glauber dynamic, $$W_{\rm HG}
= \frac{1}{1+ \exp (\beta\Delta E)}\;,
\label{eq:hardg}$$ and the Metropolis dynamic, $$W_{\rm M}
= {\rm Min} [1, \exp (-\beta\Delta E)]\;,
\label{eq:metro}$$ where $\Delta E = \Delta E_J + \Delta E_H$ is the total energy change that would result from a transition.
The dynamical order parameter is the time-averaged magnetization over the $k$th cycle of the oscillating field [@TOME90], $$Q_k =\frac{1}{2t_{1/2}} \int_{(k-1)2t_{1/2}}^{k2t_{1/2}} dt \, m(t) \;,
\label{eq:Q}$$ where $t_{1/2}$ is the half-period of the applied field. The cycle is chosen such that it starts when $H(t)$ changes sign. We also measured the normalized cycle-averaged internal energy, $$\frac{E}{J} = -\frac{1}{2t_{1/2}}\int_{(k-1)2t_{1/2}}^{k2t_{1/2}} dt
\frac{1}{L^2}\sum_{\langle ij \rangle}s_{i}(t)s_{j}(t)\;.
\label{eq:E}$$
As previous studies indicate [@SIDE98; @SIDE99; @KORN00], the DPT transition essentially depends on the competition between two time scales: the average lifetime of the metastable phase, $\langle \tau (T,H_{0}) \rangle$, and the half-period of the applied field, $t_{1/2}$. The metastable lifetime $\langle \tau \rangle$ is defined as the average time it takes the system to leave one of its two degenerate zero-field equilibrium states, when a field of magnitude $H_{0}$ opposite to the initial magnetization is applied. In practice the metastable lifetime is measured as the first-passage time to zero magnetization.
It is well known that metastable Ising models decay by different mechanisms depending on the magnitude of the applied field $H_{0}$, the system size $L$, and the temperature $T$. Detailed discussions of these different decays regimes are found in Ref. [@RIKV94A]. More recently it has also been shown that, contrary to some common beliefs, there is also a strong dependence on the specific stochastic dynamics [@PARK04; @BUEN04B]. For the purpose of this study the temperature, the system sizes and $H_0$ are chosen such that the metastable phase decays by random homogeneous nucleation of many critical droplets of the stable phase, which grow and coalesce, the so-called multidroplet (MD) regime. The metastable lifetime in the MD regime is independent of the system size [@RIKV94A].
Monte Carlo simulations {#sec:MC}
=======================
The numerical simulations reported in this work are performed on square lattices with $L$ between 64 and 256 at $T=0.8T_c$. The system is subjected to an square-wave field $H(t)$ of amplitude $H_{0}=0.3J$. The metastable lifetime was measured to be $\tau = 145\pm1$ MCSS, almost twice as long as for the kinetic Ising model evolving according to the standard (hard) Glauber dynamics under the same conditions [@SIDE98; @SIDE99; @ROBB07]. This is consistent with earlier observations of slow nucleation [@PARK04] and interface growth [@RIKV03] with this dynamic.
The system was initialized with all the spins up, and the square-wave external field started in the half-period in which $H=-H_0$. After the system relaxed, the magnetization and energy reached a limit cycle (except for thermal fluctuations), and all the period-averaged quantities became stationary stochastic processes. We discarded the first 2000 periods of the time series to exclude transients from the stationary-state averages.
The time evolution of the magnetization is shown in Fig. \[fig:mag\]. For slowly varying fields (Fig. \[fig:mag\](a)), the magnetization follows the field, switching every half-period. In this region, $Q\approx 0$. For rapidly varying fields (Fig. \[fig:mag\](b)), the magnetization does not have time to switch during a single half-period and remains nearly constant for many successive field cycles. As a result, the probability distribution of $Q$ becomes bimodal with two sharp peaks near the system’s spontaneous equilibrium magnetization, $\pm m_{\rm sp}(T)$, corresponding to the broken symmetry of the hysteresis loop. The transition between these two regimes is characterized by large fluctuations in $Q$. This behavior of the time series $Q_k$, shown in Fig. \[fig:q\], is a clear indication of the existence of a dynamical phase transition between a disordered dynamic phase (the region where $Q\approx 0$), and an ordered dynamic phase (where $Q\ne0$). Notice that the transition occurs at a critical value $\Theta_{\rm c} = t_{1/2}^{\rm c} / \tau$ that is very close to unity, the value at which the half-period of the external field is equal to the metastable lifetime of the system. To further explore the nature of the DPT, we perform a finite-size scaling analysis of the simulation data.
Finite-size scaling {#sec:FSS}
-------------------
Previous studies indicate that although scaling laws and finite-size scaling are tools designed for equilibrium systems with a well known Hamiltonian, they can be successfully applied to far-from-equilibrium systems like the one we are analyzing here [@SIDE98; @SIDE99; @KORN00; @MACH05; @ROBB07].
Since for finite systems in the dynamically ordered phase the probability distribution of the order parameter is bimodal, in order to capture symmetry breaking, the order parameter is better defined as the average norm of $Q$, i.e., $\langle |Q| \rangle$. To characterize and quantify this transition by using finite-size scaling we must define quantities analogous to the susceptibility with respect to the field conjugate to the order parameter in equilibrium systems. The scaled variance of the dynamic order parameter, $$\chi_{L}^Q=L^{2}(\langle Q^{2}\rangle_{L}-\langle |Q| \rangle_{L}^2) \;.
\label{eq:chiQ}$$ has long been used as a proxy for the non-equilibrium susceptibility. A fluctuation-dissipation relation was recently demonstrated, which justifies this practice by connecting $\chi_{L}^Q$ to the susceptibility with respect to an applied bias field for a two-dimensional kinetic Ising model evolving under the standard Glauber dynamic [@ROBB07].
In Fig. \[fig:qn\] we present the finite-size behavior of the order parameter and its fluctuations. Fig. \[fig:qn\](a) shows that this dynamic order parameter goes from unity to zero as $t_{1/2}$ increases, showing a sharp change around $t_{1/2}^c$, characterized by the peak in $\chi_{L}^Q$ shown in Fig. \[fig:qn\](b). The absence of finite-size effects below the critical point is the signature of the existence of a divergent length scale. The height and the location of the maximum in $\chi_{L}^Q$ change with $L$.
In Fig. \[fig:corr\] we show the normalized time-autocorrelation function of the order parameter, defined as $$C_{L}^{Q}(p)=\frac{\langle Q(i)Q(i+p) \rangle-
\langle Q(i)\rangle^{2}}{\langle Q(i)^{2}\rangle-\langle Q(i)\rangle^{2}}
.$$ The increasing correlation times with increasing system sizes are evidence of the critical slowing down of the system, and provides further support for the existence of a DPT.
We also measured the period-averaged internal energy, Eq. (\[eq:E\]), and its scaled variance $$\chi_{L}^E=L^{2}(\langle E^{2} \rangle_{L}- \langle E\rangle_{L}^2) \;.
\label{eq:chiE}$$ Both quantities are shown in Fig. \[fig:energ\]. Again, in the absence of a fluctuation-dissipation relation, we use the scaled variance as a proxy for the analog of the equilibrium heat capacity. The correlation time was used to estimate the proper sampling interval for estimating the fluctuation measures and their error bars as described in Ref. [@LAND00].
It is very difficult to locate with precision the maxima of $\chi_{L}^Q$ and $\chi_{L}^E$ for the individual finite system sizes. A more accurate estimation of the critical point at which the transition occurs in an $\it{infinite}$ system can be obtained from the fourth-order cumulant intersection method. In Fig. \[fig:cum\] we plot the fourth-order cumulant $U_L$ defined as [@LAND00] $$U_L=1-\frac{\langle Q^{4} \rangle_{L}}{3 \langle Q^{2}\rangle^{2}_{L}}
\label{eq:UL}$$ as a function of $t_{1/2}$ for several system sizes. Our estimate is $t_{1/2}^{\rm c}=(145 \pm 1)$ MCSS, with a fixed-point value $U^{*}=0.606\pm0.004$ for the cumulant (Fig. \[fig:cum\](b)). The latter is consistent with the universal value for the two-dimensional equilibrium Ising model, $U^* = 0.6106901(5)$ [@KAMI93].
Finite size-scaling theory for equilibrium systems [@PRIV84; @PRIV90] predicts the following scaling forms at the critical point, $$\langle |Q| \rangle_{L} \propto L^{-\beta/\nu},
\label{eq:QL}$$ $$\chi_{L}^Q \propto L^{\gamma/\nu},
\label{eq:chiQL}$$ which are also applicable to the far-from-equilibrium DPT [@SIDE98; @SIDE99; @KORN00; @ROBB07]. If the specific-heat critical exponent $\alpha=0$, as it is for the equilibrium Ising universality class, then we also expect the logarithmic divergence, $$\chi_{L}^{E} \propto A +B \ln (L)\;.
\label{eq:chiEL}$$ These relations enable us to estimate the critical exponent ratios $\beta/\nu$ and $\gamma/\nu$ and verify the logarithmic divergence in the period-averaged internal energy fluctuations. In Fig. \[fig:expo\] we present the results obtained by plotting the logarithm of $\langle |Q| \rangle_L$ (Fig. \[fig:expo\](a)), the logarithm of $\chi_{L}^Q$ (Fig. \[fig:expo\](b)), and $\chi_{L}^E$ (Fig. \[fig:expo\](c)), in term of the logarithm of $L$ at $t_{1/2}^{\rm c}$. We also plot the peak of the fluctuations, $\chi_{L}^Q({\rm Peak})$ (Fig. \[fig:expo\](b)), and $\chi_{L}^E({\rm Peak})$ (Fig. \[fig:expo\](c)), since they asymptotically should follow the same scaling laws. After fitting the data with a weighted, linear least-squares algorithm, our estimates for the critical exponents are: $\beta/\nu=1.44 \pm 0.06$, $\gamma/\nu=1.77 \pm 0.04$ (from the data at $t_{1/2}^{\rm c}$ ), the data from $\chi_{L}^Q(Peak)$ gives $\gamma/\nu=1.79\pm.02$ which agree within statistical error. Also, the straight line in Fig. \[fig:expo\](c) gives evidence of the logarithmic divergence of $\chi_{L}^E$ at the critical point. These results, together with our estimate for $U^*$, give strong support to the hypothesis that the DPT observed is in the same universality class of the [*[equilibrium]{}*]{} two-dimensional Ising model.
Discussion and Conclusions {#sec:DISC}
==========================
In this paper we have studied the dynamical response of a two-dimensional Ising model exposed to a square-wave oscillating external field. The system evolves under the so-called soft Glauber dynamic. In previous works it was established that, in the field and temperature regions when the metastable decay occurs via a multidroplet mechanism, this system evolving under a standard (hard) Glauber dynamics undergoes a continuous phase transition, with critical exponent ratios consistent with the equilibrium Ising values. The aim of the present study was to explore the universality of this far-from-equilibrium DPT with respect to the dynamics chosen to evolve the system.
Our numerical results clearly indicate the existence of a DPT in the multidroplet regime. The transition depends on the competition between two time scales: the half-period of the applied field and the metastable lifetime of the system. We found that the metastable lifetime of the system evolving under the soft Glauber dynamics is roughly twice that of the same system evolving under the standard Glauber dynamic. However, in both cases the transition occurs at a critical point where both times are of the same order of magnitude. If the half-period of the applied field increases much beyond the metastable lifetime, the system is in a dynamically disordered phase characterized by a vanishing dynamic order parameter. A study of the autocorrelation function of the order parameter at the critical point provides evidence of critical slowing down, showing increasing correlation times with increasing system sizes.
We applied the machinery of finite-size scaling, originally developed for equilibrium phase transitions, to estimate the critical point and the critical exponent ratios $\beta/\nu$ and $\gamma/\nu$ for system sizes between $64$ and $256$ at $T=0.8T_c$ and $H_{0}=0.3J$. Our estimates are $\beta/\nu \approx 1.44 \pm 0.06$ and $\gamma/\nu \approx 1.77 \pm 0.04$. These values are close to those of the two-dimensional equilibrium Ising model: $\beta/\nu=1/8=0.125$, $\gamma/\nu=7/4=1.75$. Furthermore, our data strongly indicate a slow logarithmic divergence with $L$ of the period-averaged energy fluctuations, consistent with the equilibrium Ising exponent $\alpha = 0$. The fixed-point value of the fourth-order cumulant, $U^*$, is also close to its expected universal Ising value, near 0.611.
This study provides further evidence of the universality class of the dynamic phase transition in kinetic Ising systems driven by an oscillating field, extending its domain to systems that evolve under different stochastic dynamics that lead to interfaces with significantly different structures on the nanoscale.
Acknowledgments {#sec:ACK .unnumbered}
===============
G. M. B. gratefully acknowledges many useful discussions with V. M. Kenkre and the hospitality of the Consortium of the Americas for Interdisciplinary Science at the University of New Mexico, and P. A. R. that of the Department of Physics of The University of Tokyo. Work at Florida State University was supported in part by NSF Grants No. DMR-0444051 and DMR-0802288, and work at the University of New Mexico was supported in part by NSF Grant No. INT-0336343.
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![ (Color online) Time series of the magnetization (solid curves) in the presence of a square-wave external field (dashed lines), for two values of the half-period $t_{1/2}$. (a) $t_{1/2}=400$ MCSS, corresponding to a dynamically disordered phase. (b) $t_{1/2}=40$ MCSS, corresponding to a dynamically ordered phase. The data were obtained for a system of size $L=128$ at $T=0.8T_c$ and field amplitude $H_{0}=0.3J$. []{data-label="fig:mag"}](mag400.eps "fig:"){width=".50\textwidth"}\
![ (Color online) Time series of the magnetization (solid curves) in the presence of a square-wave external field (dashed lines), for two values of the half-period $t_{1/2}$. (a) $t_{1/2}=400$ MCSS, corresponding to a dynamically disordered phase. (b) $t_{1/2}=40$ MCSS, corresponding to a dynamically ordered phase. The data were obtained for a system of size $L=128$ at $T=0.8T_c$ and field amplitude $H_{0}=0.3J$. []{data-label="fig:mag"}](mag40.eps "fig:"){width=".50\textwidth"}\
). The strongly fluctuating trace corresponds to $t_{1/2}=145$ MCSS, very close to the DPT. The horizontal trace near $Q=0$ corresponds to $t_{1/2}=400$ MCSS, well into the dynamically disordered phase. (See Fig. \[fig:mag\](a)). []{data-label="fig:q"}](qtime.eps "fig:"){width=".50\textwidth"}\
![(Color online) Dependence on the half-period $t_{1/2}$ of the order parameter $\langle |Q| \rangle$ (a), and of its scaled variance $\chi_{L}^Q$ (b), shown for various system sizes, $L$. All the results correspond to $T=0.8T_{c}$ and $H_{0}=0.3J$. []{data-label="fig:qn"}](q.eps "fig:"){width=".6\textwidth"}\
![(Color online) Dependence on the half-period $t_{1/2}$ of the order parameter $\langle |Q| \rangle$ (a), and of its scaled variance $\chi_{L}^Q$ (b), shown for various system sizes, $L$. All the results correspond to $T=0.8T_{c}$ and $H_{0}=0.3J$. []{data-label="fig:qn"}](chiq3.eps "fig:"){width=".6\textwidth"}
![ (Color online) Normalized autocorrelation function for the order parameter $Q$ for $t_{1/2}=145$ MCSS at $T=0.8T_c$ and $H_{0}=0.3J$, shown for different values of $L$. []{data-label="fig:corr"}](corr.eps "fig:"){width=".45\textwidth"}\
![(Color online) Dependence on the half-period $t_{1/2}$ of the period-averaged internal energy $\langle E \rangle$ (a), and its scaled variance $\chi_{L}^E$ (b) for various system sizes, $L$. All the results correspond to $T=0.8T_{c}$ and $H_{0}=0.3J$. []{data-label="fig:energ"}](energy.eps "fig:"){width=".45\textwidth"}\
![(Color online) Dependence on the half-period $t_{1/2}$ of the period-averaged internal energy $\langle E \rangle$ (a), and its scaled variance $\chi_{L}^E$ (b) for various system sizes, $L$. All the results correspond to $T=0.8T_{c}$ and $H_{0}=0.3J$. []{data-label="fig:energ"}](chiE2.eps "fig:"){width=".45\textwidth"}
![(Color online) (a) Dependence of the fourth-order cumulant $U_L$ on the half-period $t_{1/2}$, shown for various system sizes, $L$. (b) Enlargement of the region around the cumulant crossing. The horizontal and vertical dashed lines indicate the fixed point value $U^{*} \approx 0.606$ and the critical half-period, $t_{1/2}^{\rm c}=145$ MCSS, respectively. All the results correspond to $T=0.8T_{c}$ and $H_{0}=0.3J$. []{data-label="fig:cum"}](u.eps "fig:"){width=".50\textwidth"}\
![(Color online) (a) Dependence of the fourth-order cumulant $U_L$ on the half-period $t_{1/2}$, shown for various system sizes, $L$. (b) Enlargement of the region around the cumulant crossing. The horizontal and vertical dashed lines indicate the fixed point value $U^{*} \approx 0.606$ and the critical half-period, $t_{1/2}^{\rm c}=145$ MCSS, respectively. All the results correspond to $T=0.8T_{c}$ and $H_{0}=0.3J$. []{data-label="fig:cum"}](uzoom.eps "fig:"){width=".50\textwidth"}
![(Color online) Critical exponent estimates from the scaling relations. The symbols represent the MC data, the straight lines are weighted least-square fits. (a)Calculating $\beta/\nu$ from Eq. (\[eq:QL\]) calculating $\gamma/\nu$ from Eq. (\[eq:chiQL\]). (c) The logarithmic divergence of the period-averaged energy fluctuations, based on Eq. (\[eq:chiEL\]). From data at $t_{1/2}^{\rm c}=145$ MCSS, circles, at the peaks, squares. []{data-label="fig:expo"}](qexp2_145.eps "fig:"){width=".50\textwidth"}\
![(Color online) Critical exponent estimates from the scaling relations. The symbols represent the MC data, the straight lines are weighted least-square fits. (a)Calculating $\beta/\nu$ from Eq. (\[eq:QL\]) calculating $\gamma/\nu$ from Eq. (\[eq:chiQL\]). (c) The logarithmic divergence of the period-averaged energy fluctuations, based on Eq. (\[eq:chiEL\]). From data at $t_{1/2}^{\rm c}=145$ MCSS, circles, at the peaks, squares. []{data-label="fig:expo"}](expXqN.eps "fig:"){width=".50\textwidth"}\
![(Color online) Critical exponent estimates from the scaling relations. The symbols represent the MC data, the straight lines are weighted least-square fits. (a)Calculating $\beta/\nu$ from Eq. (\[eq:QL\]) calculating $\gamma/\nu$ from Eq. (\[eq:chiQL\]). (c) The logarithmic divergence of the period-averaged energy fluctuations, based on Eq. (\[eq:chiEL\]). From data at $t_{1/2}^{\rm c}=145$ MCSS, circles, at the peaks, squares. []{data-label="fig:expo"}](expXEN.eps "fig:"){width=".50\textwidth"}\
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abstract: 'For fixed $m \geq 1$, we consider the product of $m$ independent $n \times n$ random matrices with iid entries as $n \to \infty$. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the $m$-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Tao [@Tout] for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations.'
address:
- 'Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309 '
- 'Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309 '
- 'Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr., Madison, WI 53706 '
author:
- Natalie Coston
- 'Sean O’Rourke'
- Philip Matchett Wood
title: Outliers in the spectrum for products of independent random matrices
---
[^1]
Introduction {#Sec:Intro}
============
This paper is concerned with the asymptotic behavior of outliers in the spectrum of bounded-rank perturbations of large random matrices. We begin by fixing the following notation and introducing an ensemble of random matrices with independent entries.
The *eigenvalues* of an $n \times n$ matrix $M_n$ are the roots in $\mathbb{C}$ of the characteristic polynomial $\det (M_n- zI)$, where $I$ is the identity matrix. We let $\lambda_1(M_n), \ldots, \lambda_n(M_n)$ denote the eigenvalues of $M_n$ counted with (algebraic) multiplicity. The *empirical spectral measure* $\mu_{M_n}$ of $M_n$ is given by $$\mu_{M_n} := \frac{1}{n} \sum_{j=1}^n \delta_{\lambda_j(M_n)}.$$ If $M_n$ is a random $n \times n$ matrix, then $\mu_{M_n}$ is also random. In this case, we say $\mu_{M_n}$ converges weakly in probability (resp. weakly almost surely) to another Borel probability measure $\mu$ on the complex plane $\mathbb{C}$ if, for every bounded and continuous function $f:\mathbb{C} \to \mathbb{C}$, $$\int_{\mathbb{C}} f d \mu_{M_n} \longrightarrow \int_{\mathbb{C}} f d \mu$$ in probability (resp. almost surely) as $n \to \infty$.
Throughout the paper, we use asymptotic notation (such as $O, o$) under the assumption that $n \to \infty$; see Section \[sec:notation\] for a complete description of our asymptotic notation.
iid random matrices
-------------------
In this paper, we focus on random matrices whose entries are independent and identically distributed.
Let $\xi$ be a complex-valued random variable. We say $X_n$ is an $n \times n$ *iid random matrix* with atom variable $\xi$ if $X_n$ is an $n \times n$ matrix whose entries are independent and identically distributed (iid) copies of $\xi$.
The circular law describes the limiting distribution of the eigenvalues of an iid random matrix. For any matrix $M$, we denote the Hilbert-Schmidt norm $\|M\|_2$ by the formula $$\label{eq:def:hs}
\|M\|_2 := \sqrt{ \operatorname{tr}(M M^\ast) } = \sqrt{ \operatorname{tr}(M^\ast M)}.$$
\[thm:circ\] Let $\xi$ be a complex-valued random variable with mean zero and unit variance. For each $n \geq 1$, let $X_n$ be an $n \times n$ iid random matrix with atom variable $\xi$, and let $A_n$ be a deterministic $n \times n$ matrix. If $\operatorname{rank}(A_n) = o(n)$ and $\sup_{n \geq 1} \frac{1}{n} \|A_n\|^2_2 < \infty$, then the empirical measure $\mu_{\frac{1}{\sqrt{n}} X_n + A_n}$ of $\frac{1}{\sqrt{n}} X_n + A_n$ converges weakly almost surely to the uniform probability measure on the unit disk centered at the origin in the complex plane as $n \to \infty$.
This result appears as [@TVesd Corollary 1.12], but is the culmination of work by many authors. We refer the interested reader to the excellent survey [@BC] for further details.
From Theorem \[thm:circ\], we see that the low-rank perturbation $A_n$ does not affect the limiting spectral measure (i.e., the limiting measure is the same as the case when $A_n = 0$). However, the perturbation $A_n$ may create one or more outliers. An example of this phenomenon is illustrated in Figure \[Fig:GaussianM1\].
![On the left, we have plotted the eigenvalues of a $500\times 500$ random matrix with iid standard Gaussian entries scaled by $\frac{1}{\sqrt{500}}$. Additionally, the unit circle is plotted for reference. The image on the right contains the eigenvalues of $\frac{1}{\sqrt{500}} X + A$, where $X$ is a $500\times 500$ random matrix with iid symmetric $\pm 1$ Bernoulli entries and $A=\text{diag}(1+i,2i-1,2,-i-2,0,\ldots,0)$. For reference, we have also plotted each nonzero eigenvalue of $A$ with a cross.[]{data-label="Fig:GaussianM1"}](unpert_gauss-eps-converted-to.pdf "fig:") ![On the left, we have plotted the eigenvalues of a $500\times 500$ random matrix with iid standard Gaussian entries scaled by $\frac{1}{\sqrt{500}}$. Additionally, the unit circle is plotted for reference. The image on the right contains the eigenvalues of $\frac{1}{\sqrt{500}} X + A$, where $X$ is a $500\times 500$ random matrix with iid symmetric $\pm 1$ Bernoulli entries and $A=\text{diag}(1+i,2i-1,2,-i-2,0,\ldots,0)$. For reference, we have also plotted each nonzero eigenvalue of $A$ with a cross.[]{data-label="Fig:GaussianM1"}](pert_bern-eps-converted-to.pdf "fig:")
Recall that the spectral radius of a square matrix $M$ is the largest eigenvalue of $M$ in absolute value. Among other things, Theorem \[thm:circ\] implies that with probability tending to one, the spectral radius of $\frac{1}{\sqrt{n}}X_n$ is at least $1 - o(1)$. When the atom variable $\xi$ has finite fourth moment, it is possible to improve the lower bound on the spectral radius and give a matching upper bound.
\[thm:nooutlier:iid\] Let $\xi$ be a complex-valued random variable with mean zero, unit variance, and finite fourth moment. For each $n \geq 1$, let $X_n$ be an iid random matrix with atom variable $\xi$. Then the spectral radius of $\frac{1}{\sqrt{n}}X_n$ converges to $1$ almost surely as $n \to \infty$.
\[rem:twomomsr\] It is conjectured in [@BCCT] that the spectral radius of $\frac{1}{\sqrt{n}}X_n$ converges to $1$ in probability as $n \to \infty$ only assuming that $\xi$ has mean zero and unit variance.
Theorem \[thm:nooutlier:iid\] asserts that almost surely all eigenvalues of $\frac{1}{\sqrt{n}}X_n$ are contained in the disk of radius $1 + o(1)$ centered at the origin. However, as we saw in Figure \[Fig:GaussianM1\], this need not be the case for the eigenvalues of the additive perturbation $\frac{1}{\sqrt{n}}X_n + A_n$. In this case, it is possible for eigenvalues of the perturbation to be larger than $1 +o(1)$, and Tao precisely describes the location of these outlying eigenvalues in [@Tout].
\[thm:outlier:iid\] Let $\xi$ be a complex random variable with mean zero, unit variance, and finite fourth moment. For each $n \geq 1$, let $X_n$ be a $n \times n$ random matrix whose entries are iid copies of $\xi$, and let $A_n$ be a deterministic matrix with rank $O(1)$ and operator norm $O(1)$. Let ${\varepsilon}> 0$, and suppose that for all sufficiently large $n$, there are no eigenvalues of $A_n$ in the band $\{z \in {\mathbb{C}}: 1 + {\varepsilon}< |z| < 1 + 3{\varepsilon}\}$, and there are $j$ eigenvalues $\lambda_1(A_n), \ldots, \lambda_j(A_n)$ for some $j = O(1)$ in the region $\{z \in {\mathbb{C}}: |z| \geq 1+3 {\varepsilon}\}$. Then, almost surely, for sufficiently large $n$, there are precisely $j$ eigenvalues $\lambda_1(\frac{1}{\sqrt{n}} X_n + A_n), \ldots, \lambda_j(\frac{1}{\sqrt{n}} X_n + A_n)$ of $\frac{1}{\sqrt{n}}X_n + A_n$ in the region $\{z \in {\mathbb{C}}: |z| \geq 1 + 2{\varepsilon}\}$, and after labeling these eigenvalues properly, $$\lambda_i\left( \frac{1}{\sqrt{n}} X_n + A_n \right) = \lambda_i(A_n) + o(1)$$ as $n \rightarrow \infty$ for each $1 \leq i \leq j$.
Analogous results describing the location and fluctuation of the outlier eigenvalues have been obtained for many ensembles of random matrices; we refer the reader to [@BBC; @BBP; @BGCR; @BGM; @BGM2; @BR; @BR2; @BGR; @BCap; @BelC; @CDF1; @CDF; @CDFF; @FP; @KY; @KY2; @OR; @OW; @P; @PRS; @R; @RS; @Rochet; @Tout] and references therein. In particular, the results in [@R] extend Theorem \[thm:outlier:iid\] by also describing the joint fluctuations of the outlier eigenvalues about their asymptotic locations.
Products of independent iid matrices
------------------------------------
In this paper, we focus on the product of several independent iid matrices. In this case, the analogue of the circular law (Theorem \[thm:circ\]) is the following result from [@ORSV], due to Renfrew, Soshnikov, Vu, and the second author of the current paper.
\[thm:ORSV\] Fix an integer $m \geq 1$, and let $\tau > 0$. Let $\xi_1, \ldots, \xi_m$ be real-valued random variables with mean zero, and assume, for each $1 \leq k \leq m$, that $\xi_k$ has nonzero variance $\sigma^2_k$ and satisfies ${\mathbb{E}}|\xi_k|^{2 + \tau} < \infty$. For each $n \geq 1$ and $1 \leq k \leq m$, let $X_{n,k}$ be an $n \times n$ iid random matrix with atom variable $\xi_k$, and let $A_{n,k}$ be a deterministic $n \times n$ matrix. Assume $X_{n,1}, \ldots, X_{n,m}$ are independent. If $$\max_{1 \leq k \leq m} \operatorname{rank}(A_{n,k}) = O(n^{1 - {\varepsilon}}) \quad \text{and} \quad \sup_{n \geq 1} \max_{1 \leq k \leq m} \frac{1}{n} \|A_{n,k}\|_2^2 < \infty$$ for some fixed ${\varepsilon}> 0$, then the empirical spectral measure $\mu_{P_n}$ of the product[^2] $$P_n := \prod_{k=1}^m \left( \frac{1}{\sqrt{n}} X_{n,k} + A_{n,k} \right)$$ converges weakly almost surely to a (non-random) probability measure $\mu$ as $n \to \infty$. Here, the probability measure $\mu$ is absolutely continuous with respect to Lebesgue measure on $\mathbb{C}$ with density $$\label{eq:density}
f(z) := \left\{
\begin{array}{rr}
\frac{1}{m \pi} \sigma^{-2/m} |z|^{\frac{2}{m} - 2}, & \text{if } |z| \leq \sigma, \\
0, & \text{if } |z| > \sigma,
\end{array}
\right.$$ where $\sigma := \sigma_1 \cdots \sigma_m$.
When $\sigma = 1$, the density in is easily related to the circular law (Theorem \[thm:circ\]). Indeed, in this case, $f$ is the density of $\psi^m$, where $\psi$ is a complex-valued random variable uniformly distributed on the unit disk centered at the origin in the complex plane.
Theorem \[thm:ORSV\] is a special case of [@ORSV Theorem 2.4]. Indeed, [@ORSV Theorem 2.4] applies to so-called elliptic random matrices, which generalize iid matrices. Theorem \[thm:ORSV\] and the results in [@ORSV] are stated only for real random variables, but the proofs can be extended to the complex setting. Similar results have also been obtained in [@B; @GTprod; @OS]. The Gaussian case was originally considered by Burda, Janik, and Waclaw [@BJW]; see also [@Bsurv]. We refer the reader to [@AB; @ABK; @AIK; @AIK2; @AKW; @AS; @BJLNS; @F; @F2; @KZ; @S] and references therein for many other results concerning products of random matrices with Gaussian entries.
Main results {#Sec:Main}
============
From Theorem \[thm:ORSV\], we see that the low-rank deterministic perturbations $A_{n,k}$ do not affect the limiting empirical spectral measure. However, as was the case in Theorem \[thm:circ\], the perturbations may create one or more outlier eigenvalues. The goal of this paper is to study the asymptotic behavior of these outlier eigenvalues. In view of Theorem \[thm:outlier:iid\], we will assume the atom variables $\xi_1, \ldots, \xi_m$ have finite fourth moment.
\[assump:4th\] The complex-valued random variables $\xi_1, \ldots, \xi_m$ are said to satisfy Assumption \[assump:4th\] if, for each $1 \leq k \leq m$,
- the real and imaginary parts of $\xi_k$ are independent,
- $\xi_k$ has mean zero and finite fourth moment, and
- $\xi_k$ has nonzero variance $\sigma_k^2$.
We begin with the analogue of Theorem \[thm:nooutlier:iid\] for the product of $m$ independent iid matrices.
\[thm:nooutlier\] Let $m \geq 1$ be a fixed integer, and assume $\xi_1, \ldots, \xi_m$ are complex-valued random variables which satisfy Assumption \[assump:4th\]. For each $n \geq 1$, let $X_{n,1}, \ldots, X_{n,m}$ be independent $n \times n$ iid random matrices with atom variables $\xi_1, \ldots, \xi_m$, respectively. Define the products $$P_n := n^{-m/2} X_{n,1} \cdots X_{n,m}$$ and $\sigma := \sigma_1 \cdots \sigma_m$. Then, almost surely, the spectral radius of $P_n$ is bounded above by $\sigma + o(1)$ as $n \to \infty$. In particular, for any fixed ${\varepsilon}> 0$, almost surely, for $n$ sufficiently large, all eigenvalues of $P_n$ are contained in the disk $\{ z \in \mathbb{C} : |z| < \sigma + {\varepsilon}\}$.
A version of Theorem \[thm:nooutlier\] was proven by Nemish in [@N] under the additional assumption that the atom variables $\xi_1, \ldots, \xi_m$ satisfy a sub-exponential decay condition. In particular, this condition implies that all moments of $\xi_1, \ldots, \xi_m$ are finite. Theorem \[thm:nooutlier\] only requires the fourth moments of the atom variables to be finite.
In view of Remark \[rem:twomomsr\], it is natural to also conjecture that the spectral radius of $P_n$ is bounded above by $\sigma + o(1)$ in probability as $n \to \infty$ only assuming the atom variables $\xi_1, \ldots, \xi_m$ have mean zero and unit variance. Here, we need the result to hold almost surely, and hence require the atom variables have finite fourth moment.
In view of Theorem \[thm:ORSV\], it is natural to consider perturbations of the form $$P_n := \prod_{k=1}^m \left( \frac{1}{\sqrt{n}} X_{n,k} + A_{n,k} \right).$$ However, there are many other types of perturbations one might consider, such as multiplicative perturbations $$\label{eq:prodmult}
P_n := \frac{1}{\sqrt{n}} X_{n,1} (I + A_{n,1}) \frac{1}{\sqrt{n}} X_{n,2} (I + A_{n,2}) \cdots \frac{1}{\sqrt{n}} X_{n,m} (I + A_{n,m})$$ or perturbations of the form $$P_n := n^{-m/2} \prod_{k=1}^m X_{n,k} + A_n.$$ In any of these cases, the product $P_n$ can be written as $$P_n = n^{-m/2} X_{n,1} \cdots X_{n,m} + M_n + A_n,$$ where $A_n$ is deterministic and $M_n$ represents the “mixed” terms, each containing at least one random factor and one deterministic factor. Our main results below show that only the deterministic term $A_n$ determines the location of the outliers. The “mixed” terms $M_n$ do not effect the asymptotic location of the outliers.
This phenomenon is most easily observed in the case of multiplicative perturbations , for which there is no deterministic term (i.e., $A_n=0$ and the perturbation consists entirely of “mixed” terms). In this case, the heuristic above suggests that there should be no outliers, and this is the content of the following theorem.
Let $m \geq 1$ be a fixed integer, and assume $\xi_1, \ldots, \xi_m$ are complex-valued random variables which satisfy Assumption \[assump:4th\]. For each $n \geq 1$, let $X_{n,1}, \ldots, X_{n,m}$ be independent $n \times n$ iid random matrices with atom variables $\xi_1, \ldots, \xi_m$, respectively. In addition for any fixed integer $s\geq 1$, let ${{A}_{n,1}},{{A}_{n,2}},\ldots,{{A}_{n,s}}$ be $n\times n$ deterministic matrices, each of which has rank $O(1)$ and operator norm $O(1)$. Define the product $P_{n}$ to be the product of the terms $$\frac{1}{\sqrt{n}}{{X}_{n,1}},\ldots,\frac{1}{\sqrt{n}}{{X}_{n,m}},\left(I+{{A}_{n,1}}\right),\ldots,\left(I+{{A}_{n,s}}\right)$$ in some fixed order. Then for any $\delta > 0$, almost surely, for sufficiently large $n$, $P_n$ has no eigenvalues in the region $\{z\in{\mathbb{C}}\;:\;|z|>\sigma+\delta\}$ where $\sigma := \sigma_1 \cdots \sigma_m$. \[Thm:NoOutlierInProductPert\]
We now consider the case when there is a deterministic term and no “mixed” terms.
\[thm:nomixed\] Let $m \geq 1$ be a fixed integer, and assume $\xi_1, \ldots, \xi_m$ are complex-valued random variables which satisfy Assumption \[assump:4th\]. For each $n \geq 1$, let $X_{n,1}, \ldots, X_{n,m}$ be independent $n \times n$ iid random matrices with atom variables $\xi_1, \ldots, \xi_m$, respectively. In addition, let $A_n$ be an $n \times n$ deterministic matrix with rank $O(1)$ and operator norm $O(1)$. Define $$\label{eq:nomixedproduct}
P_n := n^{-m/2} \prod_{k=1}^m X_{n,k} + A_n$$ and $\sigma := \sigma_1 \cdots \sigma_m$. Let ${\varepsilon}> 0$, and suppose that for all sufficiently large $n$, there are no eigenvalues of $A_n$ in the band $\{z \in {\mathbb{C}}: \sigma + {\varepsilon}< |z| < \sigma + 3{\varepsilon}\}$, and there are $j$ eigenvalues $\lambda_1(A_n), \ldots, \lambda_j(A_n)$ for some $j = O(1)$ in the region $\{z \in {\mathbb{C}}: |z| \geq \sigma+3 {\varepsilon}\}$. Then, almost surely, for sufficiently large $n$, there are precisely $j$ eigenvalues $\lambda_1(P_n), \ldots, \lambda_j(P_n)$ of $P_n$ in the region $\{z \in {\mathbb{C}}: |z| \geq \sigma + 2{\varepsilon}\}$, and after labeling these eigenvalues properly, $$\lambda_i\left(P_n \right) = \lambda_i(A_n) + o(1)$$ as $n \rightarrow \infty$ for each $1 \leq i \leq j$.
![In this figure, we have plotted the eigenvalues of $(500)^{-2} X_1 X_2 X_3 X_4 + A$, where $X_1, \ldots, X_4$ are independent $500\times 500$ iid random matrices with symmetric $\pm 1$ Bernoulli entries and $A=\text{diag}(-1+i,-2,2,0,\ldots,0)$. The majority of the eigenvalues cluster inside the unit disc with the exception of three outliers. These outliers are close to the eigenvalues of $A$, each of which is marked with a cross.[]{data-label="Fig:PertProdWithOne"}](pert_prod_once_bern_with_circle-eps-converted-to.pdf)
Figure \[Fig:PertProdWithOne\] presents a numerical simulation of Theorem \[thm:nomixed\]. In the case that all the entries of $A_n$ take the same value, the product $P_n$ in can be viewed as a product matrix whose entries have the same nonzero mean. Technically, Theorem \[thm:nomixed\] cannot be applied in this case, since such a matrix $A_n$ does not have operator norm $O(1)$. However, using a similar proof, we establish the following result.
\[thm:nonzeromean\] Let $m \geq 1$ be an integer, and let $\mu \in \mathbb{C}$ be nonzero. Assume $\xi_1, \ldots, \xi_m$ are complex-valued random variables which satisfy Assumption \[assump:4th\]. For each $n \geq 1$, let $X_{n,1}, \ldots, X_{n,m}$ be independent $n \times n$ iid random matrices with atom variables $\xi_1, \ldots, \xi_m$, respectively. Let $\phi_n := \frac{1}{\sqrt{n}} (1, \ldots, 1)^\ast$ and fix $\gamma > 0$. Define $$P_n := n^{-m/2} \prod_{k=1}^m X_{n,k} + \mu n^{\gamma} \phi_n \phi_n^\ast$$ and $\sigma := \sigma_1 \cdots \sigma_m$, and fix ${\varepsilon}> 0$. Then, almost surely, for $n$ sufficiently large, all eigenvalues of $P_n$ lie in the disk $\{z \in \mathbb{C} : |z| \leq \sigma + {\varepsilon}\}$ with a single exception taking the value $\mu n^{\gamma} + o(1)$.
Lastly, we consider the case of Theorem \[thm:ORSV\], where there are both “mixed” terms and a deterministic term.
\[thm:outliers\] Let $m \geq 1$ be an integer, and assume $\xi_1, \ldots, \xi_m$ are complex-valued random variables which satisfy Assumption \[assump:4th\]. For each $n \geq 1$, let $X_{n,1}, \ldots, X_{n,m}$ be independent $n \times n$ iid random matrices with atom variables $\xi_1, \ldots, \xi_m$, respectively. In addition, for each $1 \leq k \leq m$, let $A_{n,k}$ be a deterministic $n \times n$ matrix with rank $O(1)$ and operator norm $O(1)$. Define the products $$\label{eq:products}
P_n := \prod_{k=1}^m \left( \frac{1}{\sqrt{n}} X_{n,k} + A_{n,k} \right), \quad A_n := \prod_{k=1}^m A_{n,k},$$ and $\sigma := \sigma_1 \cdots \sigma_m$. Let ${\varepsilon}> 0$, and suppose that for all sufficiently large $n$, there are no eigenvalues of $A_n$ in the band $\{z \in {\mathbb{C}}: \sigma + {\varepsilon}< |z| < \sigma + 3{\varepsilon}\}$, and there are $j$ eigenvalues $\lambda_1(A_n), \ldots, \lambda_j(A_n)$ for some $j = O(1)$ in the region $\{z \in {\mathbb{C}}: |z| \geq \sigma+3 {\varepsilon}\}$. Then, almost surely, for sufficiently large $n$, there are precisely $j$ eigenvalues $\lambda_1(P_n), \ldots, \lambda_j(P_n)$ of the product $P_n$ in the region $\{z \in {\mathbb{C}}: |z| \geq \sigma + 2{\varepsilon}\}$, and after labeling these eigenvalues properly, $$\lambda_i\left(P_n \right) = \lambda_i(A_n) + o(1)$$ as $n \rightarrow \infty$ for each $1 \leq i \leq j$.
Theorem \[thm:outliers\] can be viewed as a generalization of Theorem \[thm:outlier:iid\]. In fact, when $m = 1$, Theorem \[thm:outliers\] is just a restatement of Theorem \[thm:outlier:iid\]. However, the most interesting cases occur when $m \geq 2$. Indeed, in these cases, Theorem \[thm:outliers\] implies that the outliers of $P_n$ are asymptotically close to the outliers of the product $A_n$. Specifically, if even one of the deterministic matrices $A_{n,k}$ is zero, asymptotically, there cannot be any outliers for the product $P_n$. Figure \[Fig:PertProdEach\] presents a numerical simulation of Theorem \[thm:outliers\].
![In the above figure, we display the eigenvalues of products of random matrices of the form $\prod_{k=1}^{5}\left(\frac{1}{\sqrt{1000}}X_{k}+A_{k}\right)$, where $X_1, \ldots, X_5$ are $1000\times 1000$ independent iid matrices with symmetric $\pm 1$ Bernoulli entries, and the product of the deterministic matrices $A_1, \ldots, A_5$ is $\text{diag}(-2,-1+2i,2,0,\ldots,0)$. Each nonzero eigenvalue of the product $A_1 \cdots A_5$ is marked with a cross. []{data-label="Fig:PertProdEach"}](pert_prod_each_bern_with_circle-eps-converted-to.pdf)
Outline
-------
The main results of this paper focus on iid matrices. However, one may ask if these results may be generalized to other matrix ensembles. These questions and others are discussed in Section \[Sec:RelatedResults\]. In Sections \[Sec:PreToolsAndNotation\] and \[sec:overview\], we present some preliminary results and tools. In particular, Section \[Sec:PreToolsAndNotation\] presents our notation and some standard linearization results that will be used throughout the paper. Section \[sec:overview\] contains a key eigenvalue criterion lemma and a brief overview of the proofs of the main results. In Section \[sec:isotropic\], we state an isotropic limit law and use it to prove our main results. The majority of the paper (Sections \[Sec:TruncAndTools\]–\[sec:combin\]) is devoted to the proof of this isotropic limit law. Section \[Sec:RelatedResultProofs\] contains the proofs of the related results from Section \[Sec:RelatedResults\]. A few auxiliary results and proofs are presented in the appendices.
Related results, applications, and open questions {#Sec:RelatedResults}
=================================================
There are a number of related results which are similar to the main results of this paper, but do not directly follow from the theorems in Section 2. We discuss these results as well as some applications of our main results in this section.
Related Results
---------------
While our main results have focused on independent iid matrices, it is also possible to consider the case when the random matrices $X_{n,1}, \ldots, X_{n,m}$ are no longer independent. In particular, we consider the extreme case where $X_{n,1} = \cdots = X_{n,m}$ almost surely. In this case, we obtain the following results, which are analogs of the results from Section \[Sec:Main\].
Assume that $\xi$ is a complex-valued random variable which satisfies Assumption \[assump:4th\] with $\sigma^2:=\operatorname{Var}(\xi)$. For each $n \geq 1$, let $X_{n}$ be an $n \times n$ iid random matrix with atom variable $\xi$. In addition for any finite integer $s\geq 1$, let ${{A}_{n,1}},{{A}_{n,2}},\ldots,{{A}_{n,s}}$ be $n\times n$ deterministic matrices, each of which has rank $O(1)$ and operator norm $O(1)$. Define the product $P_{n}$ to be the product of $m$ copies of $\frac{1}{\sqrt{n}}X_{n}$ with the terms $$\left(I+{{A}_{n,1}}\right),\left(I+{{A}_{n,2}}\right),\ldots,\left(I+{{A}_{n,s}}\right)$$ in some fixed order. Then for any $\delta > 0$, almost surely, for sufficiently large $n$, $P_n$ has no eigenvalues in the region $\{z\in{\mathbb{C}}\;:\;|z|>\sigma^{m}+\delta\}$. \[Thm:NoOutlierInPowerPert\]
For a single additive perturbation, we have the following analog of Theorem \[thm:nomixed\].
\[thm:nomixedinpower\] Assume $\xi$ is a complex-valued random variable which satisfies Assumption \[assump:4th\] with $\sigma^2 := \operatorname{Var}(\xi)$. For each $n \geq 1$, let $X_{n}$ be an $n \times n$ iid random matrix with atom variable $\xi$. In addition, let $A_n$ be an $n \times n$ deterministic matrix with rank $O(1)$ and operator norm $O(1)$. Define $$\label{eq:nomixedpower}
P_n := n^{-m/2} X_{n}^{m} + A_n .$$ Let ${\varepsilon}> 0$, and suppose that for all sufficiently large $n$, there are no eigenvalues of $A_n$ in the band $\{z \in {\mathbb{C}}: \sigma^{m} + {\varepsilon}< |z| < \sigma^{m} + 3{\varepsilon}\}$, and there are $j$ eigenvalues $\lambda_1(A_n), \ldots, \lambda_j(A_n)$ for some $j = O(1)$ in the region $\{z \in {\mathbb{C}}: |z| \geq \sigma^{m}+3 {\varepsilon}\}$. Then, almost surely, for sufficiently large $n$, there are precisely $j$ eigenvalues $\lambda_1(P_n), \ldots, \lambda_j(P_n)$ of $P_n$ in the region $\{z \in {\mathbb{C}}: |z| \geq \sigma^{m} + 2{\varepsilon}\}$, and after labeling these eigenvalues properly, $$\lambda_i\left(P_n \right) = \lambda_i(A_n) + o(1)$$ as $n \rightarrow \infty$ for each $1 \leq i \leq j$.
Note that Theorem \[thm:nonzeromean\] can also be generalized in an analogous way to Theorem \[thm:nomixedinpower\] above.
Lastly, we have the following analog of Theorem \[thm:outliers\].
\[Thm:RepeatedProdOutliers\] Assume $\xi$ is a complex-valued random variable which satisfies Assumption \[assump:4th\] with $\sigma^2 := \operatorname{Var}(\xi)$. For each $n \geq 1$, let $X_{n}$ be an $n \times n$ iid random matrix with atom variable $\xi$. In addition, let $m \geq 1$ be an integer and for each $1 \leq k \leq m$, let $A_{n,k}$ be a deterministic $n \times n$ matrix with rank $O(1)$ and operator norm $O(1)$. Define the products $$\label{Equ:RepeatedProds}
P_n := \prod_{k=1}^m \left( \frac{1}{\sqrt{n}} X_{n} + A_{n,k} \right), \quad A_n := \prod_{k=1}^m A_{n,k}.$$ Let ${\varepsilon}> 0$, and suppose that for all sufficiently large $n$, there are no eigenvalues of $A_n$ in the band $\{z \in {\mathbb{C}}: \sigma^{m} + {\varepsilon}< |z| < \sigma^{m} + 3{\varepsilon}\}$, and there are $j$ eigenvalues $\lambda_1(A_n), \ldots, \lambda_j(A_n)$ for some $j = O(1)$ in the region $\{z \in {\mathbb{C}}: |z| \geq \sigma^{m}+3 {\varepsilon}\}$. Then, almost surely, for sufficiently large $n$, there are precisely $j$ eigenvalues $\lambda_1(P_n), \ldots, \lambda_j(P_n)$ of the product $P_n$ in the region $\{z \in {\mathbb{C}}: |z| \geq \sigma^{m} + 2{\varepsilon}\}$, and after labeling these eigenvalues properly, $$\lambda_i\left(P_n \right) = \lambda_i(A_n) + o(1)$$ as $n \rightarrow \infty$ for each $1 \leq i \leq j$.
The proofs of these results are presented in Section \[Sec:RelatedResultProofs\] and use similar techniques to the proofs of the main results from Section \[Sec:Main\].
Applications
------------
Random matrices are useful tools in the study of many physically motivated systems, and we note here two potential applications for products of perturbed random matrices. First, iid Gaussian matrices can be used to model neural networks as in, for example, [@AR; @AbSu; @ACHLM]. In the case of a linear version of the feed-forward networks in [@AbSu], the model becomes a perturbation of a product of iid random matrices, which, if the interactions in the model were fixed, could potentially be analyzed using approaches in the current paper. Second, one can conceive of a dynamical system (see, for example, [@I]) evolving according to a matrix equation, which, when iterated, would lead a matrix product of the form discussed in Theorem \[thm:outliers\].
Open Questions
--------------
While our main results have focused on iid matrices, it is natural to ask if the same results can be extended to other matrix models. For example, Theorem \[thm:ORSV\] and the results in [@ORSV] also hold for products of so-called elliptic random matrices. However, the techniques used in this paper (in particular, the combinatorial techniques in Section \[sec:combin\]) rely heavily on the independence of the entries of each matrix. It is an interesting question whether an alternative proof can be found for the case when the entries of each matrix are allowed to be dependent.
In this paper, we have focused on the asymptotic location of the outlier eigenvalues. One can also ask about the fluctuations of the outliers. For instance, in [@R], the joint fluctuations of the outlier eigenvalues from Theorem \[thm:outlier:iid\] were studied. We plan to pursue this question elsewhere.
Preliminary tools and notation {#Sec:PreToolsAndNotation}
==============================
This section is devoted to introducing some additional concepts and notation required for the proofs of our main results. In Section \[sec:overview\], we present a brief overview of our proofs and explain how these concepts will be used.
Notation {#sec:notation}
--------
We use asymptotic notation (such as $O,o, \Omega$) under the assumption that $n \to \infty$. In particular, $X= O(Y)$, $Y = \Omega(X)$, $X \ll Y$, and $Y \gg X$ denote the estimate $|X| \leq C Y$, for some constant $C > 0$ independent of $n$ and for all $n \geq C$. If we need the constant $C$ to depend on another constant, e.g. $C = C_k$, we indicate this with subscripts, e.g. $X = O_{k}(Y)$, $Y = \Omega_k(X)$, $X \ll_k Y$, and $Y\gg_k X$. We write $X = o(Y)$ if $|X| \leq c(n) Y$ for some sequence $c(n)$ that goes to zero as $n \to \infty$. Specifically, $o(1)$ denotes a term which tends to zero as $n \to \infty$. If we need the sequence $c(n)$ to depend on another constant, e.g. $c(n) = c_k(n)$, we indicate this with subscripts, e.g. $X = o_k(Y)$.
Throughout the paper, we view $m$ as a fixed integer. Thus, when using asymptotic notation, we will allow the implicit constants (and implicit rates of convergence) to depend on $m$ without including $m$ as a subscript (i.e. we will not write $O_m$ or $o_m$).
An event $E$, which depends on $n$, is said to hold with *overwhelming probability* if ${\mathbb{P}}(E) \geq 1 - O_C(n^{-C})$ for every constant $C > 0$. We let ${\ensuremath{\mathbf{1}_{{E}}}}$ denote the indicator function of the event $E$, and we let $E^{c}$ denote the complement of the event $E$. We write a.s. for almost surely.
For a matrix $M$, we let $\|M\|$ denote the spectral norm of $M$, and we let $\|M\|_2$ denote the Hilbert-Schmidt norm of $M$ (defined in ). We denote the eigenvalues of an $n \times n$ matrix $M$ by $\lambda_{1}(M),\ldots,\lambda_{n}(M)$, and we let $\rho(M):=\max\{|\lambda_{1}(M)|,\ldots,|\lambda_{n}(M)|\}$ denote its spectral radius. We let $I_n$ denote the $n \times n$ identity matrix and $0_n$ denote the $n \times n$ zero matrix. Often we will just write $I$ (or $0$) for the identity matrix (alternatively, zero matrix) when the size can be deduced from context.
The singular values of an $n\times n$ matrix $M_{n}$ are the non-negative square roots of the eigenvalues of the matrix $M_{n}^{*}M_{n}$ and we will denote their ordered values $s_{1}(M_{n})\geq s_{2}(M_{n})\geq \cdots \geq s_{n}(M_{n})$.
We let $C$ and $K$ denote constants that are non-random and may take on different values from one appearance to the next. The notation $K_p$ means that the constant $K$ depends on another parameter $p$. We allow these constants to depend on the fixed integer $m$ without explicitly denoting or mentioning this dependence. For a positive integer $N$, we let $[N]$ denote the discrete interval $\{1, \ldots, N\}$. For a finite set $S$, we let $|S|$ denote its cardinality. We let $\sqrt{-1}$ denote the imaginary unit and reserve $i$ as an index.
Linearization
-------------
Let $M_1, \ldots, M_m$ be $n \times n$ matrices, and suppose we wish to study the product $M_1 \cdots M_m$. A useful trick is to linearize this product and instead consider the $mn \times mn$ block matrix $$\label{def:M}
\mathcal{M} := \begin{bmatrix}
0 & {M}_{1} & & & 0 \\
0 & 0 & {M}_{2} & & 0 \\
& & \ddots & \ddots & \\
0 & & & 0 & {M}_{m-1} \\
{M}_{m} & & & & 0
\end{bmatrix}.$$ The following proposition relates the eigenvalues of $\mathcal{M}$ to the eigenvalues of the product $M_1 \cdots M_m$. We note that similar linearization tricks have been used previously; see, for example, [@A; @BJW; @HT; @ORSV; @OS] and references therein.
\[prop:linear\] Let $M_1, \ldots, M_m$ be $n \times n$ matrices. Let $P := M_1 \cdots M_m$, and assume $\mathcal{M}$ is the $mn \times mn$ block matrix defined in . Then $$\det(\mathcal{M}^m - z I) = [\det( P - z I)]^m$$ for every $z \in \mathbb{C}$. In other words, the eigenvalues of $\mathcal{M}^m$ are the eigenvalues of $P$, each with multiplicity $m$.
A simple computation reveals that $\mathcal{M}^m$ is a block diagonal matrix of the form $$\mathcal{M}^m = \begin{bmatrix}
Z_1 & & 0 \\
& \ddots & \\
0 & & Z_m
\end{bmatrix},$$ where $Z_1 := P$ and $$Z_k := M_k \cdots M_m M_1 \cdots M_{k-1}$$ for $1 < k \leq m$. Since each product $Z_2, \ldots, Z_m$ has the same characteristic polynomial[^3] as $P$, it follows that $$\det( \mathcal{M}^m - zI) = \prod_{k=1}^m \det(Z_k - z I) = [\det (P - zI)]^m$$ for all $z \in \mathbb{C}$.
We will exploit Proposition \[prop:linear\] many times in the coming proofs. For instance, in order to study the product $X_{n,1} \cdots X_{n,m}$, we will consider the $mn \times mn$ block matrix $$\label{def:Y}
\mathcal{Y}_n := \begin{bmatrix}
0 & {X}_{n,1} & & & 0 \\
0 & 0 & {X}_{n,2} & & 0 \\
& & \ddots & \ddots & \\
0 & & & 0 & {X}_{n,m-1} \\
{X}_{n,m} & & & & 0
\end{bmatrix}$$ and its resolvent $$\label{def:G}
\mathcal{G}_n(z) := \left( \frac{1}{\sqrt{n}} \mathcal{Y}_n - z I \right)^{-1},$$ defined for $z \in \mathbb{C}$ provided $z$ is not an eigenvalue of $\frac{1}{\sqrt{n}} \mathcal{Y}_n$. We study the location of the eigenvalues of $\frac{1}{\sqrt{n}} \mathcal{Y}_n$ in Theorem \[thm:isotropic\] below.
Similarly, when we deal with the deterministic $n \times n$ matrices $A_{n,1}, \ldots, A_{n,m}$, it will be useful to consider the analogous $mn \times mn$ block matrix $$\label{def:A}
\mathcal{A}_n := \begin{bmatrix}
0 & {A}_{n,1} & & & 0 \\
0 & 0 & {A}_{n,2} & & 0 \\
& & \ddots & \ddots & \\
0 & & & 0 & {A}_{n,m-1} \\
{A}_{n,m} & & & & 0
\end{bmatrix}.$$
Matrix notation
---------------
Here and in the sequel, we will deal with matrices of various sizes. The most common dimensions are $n \times n$ and $N \times N$, where we take $N := mn$. Unless otherwise noted, we denote $n \times n$ matrices by capital letters (such as $M, X, A$) and larger $N \times N$ matrices using calligraphic symbols (such as $\mathcal{M}$, $\mathcal{Y}$, $\mathcal{A}$).
If $M$ is an $n \times n$ matrix and $1 \leq i,j \leq n$, we let $M_{ij}$ and $M_{(i,j)}$ denote the $(i,j)$-entry of $M$. Similarly, if $\mathcal{M}$ is an $N \times N$ matrix, we let $\mathcal{M}_{ij}$ and $\mathcal{M}_{(i,j)}$ denote the $(i,j)$-entry of $\mathcal{M}$ for $1 \leq i,j \leq N$. However, in many instances, it is best to view $N \times N$ matrices as block matrices with $n \times n$ entries. To this end, we introduce the following notation. Let $\mathcal{M}$ be an $N \times N$ matrix. For $1 \leq a, b \leq m$, we let $\mathcal{M}^{[a,b]}$ denote the $n \times n$ matrix which is the $(a,b)$-block of $\mathcal{M}$. For convenience, we extend this notation to include the cases where $a = m+1$ or $b= m+1$ by taking the value $m+1$ to mean $1$ (i.e., modulo $m$). For instance, $\mathcal{M}^{[m+1, m]} = \mathcal{M}^{[1,m]}$. For $1 \leq i,j \leq n$, the notation $\mathcal{M}^{[a,b]}_{ij}$ or $\mathcal{M}^{[a,b]}_{(i,j)}$ denotes the $(i,j)$-entry of $\mathcal{M}^{[a,b]}$.
Sometimes we will deal with $n \times n$ matrices notated with a subscript such as $M_n$. In this case, for $1 \leq i,j \leq n$, we write $(M_n)_{ij}$ or $M_{n,(i,j)}$ to denote the $(i,j)$-entry of $M_n$. Similarly, if $\mathcal{M}_n$ is an $N \times N$ matrix, we write $\mathcal{M}^{[a,b]}_{n, (i,j)}$ to denote the $(i,j)$-entry of the block $\mathcal{M}_n^{[a,b]}$.
In the special case where we deal with a vector, the notation is the same, but only one index or block will be specified. In particular, if $v$ is a vector in ${\mathbb{C}}^{N}$, then $v_{i}$ denotes the $i$-th entry of $v$. If we consider $v$ to be a block vector with $m$ blocks of size $n$, then $v^{[1]}, \ldots, v^{[m]}$ denote these blocks, i.e., each $v^{[a]}$ is an $n$-vector, and $$v = \begin{pmatrix} v^{[1]} \\ \vdots \\ v^{[m]} \end{pmatrix}.$$ In addition, $v^{[a]}_{i}$ denotes the $i$-th entry in block $a$.
Singular Value Inequalities
---------------------------
For an $n \times n$ matrix $M$, recall that $s_{1}(M)\geq\dots\geq s_{n}(M)$ denote its ordered singular values. We will need the following elementary bound concerning the largest and smallest singular values.
Let $M$ be an $n\times n$ matrix and assume $E\subseteq{\mathbb{C}}$ such that $$\inf_{z\in E} s_{n}(M-zI)\geq c$$ for some constant $c>0$. Then $$\sup_{z\in E} \|G(z) \|\leq \frac{1}{c}$$ where $G(z)=(M-zI)^{-1}$. \[Prop:LargeAndSmallSingVals\]
First, observe that for any $z\in E$, we have that $z$ is not an eigenvalue of $M$ and so $M-zI$ is invertible and $G(z)$ exists. Recall that if $s$ is a singular value of $M-zI$ and $M - zI$ is invertible, then $1/s$ is a singular value of $(M-zI)^{-1}$. Thus, we conclude that $$\begin{aligned}
\sup_{z\in E} s_{1}(G(z))&=\sup_{z\in E}\frac{1}{s_{n}(M-zI)}\\
&=\frac{1}{\inf_{z\in E}s_{n}(M-zI)}\\
&\leq \frac{1}{c},
\end{aligned}$$ as desired.
Eigenvalue criterion lemma and an overview of the proof {#sec:overview}
=======================================================
Let us now briefly overview the proofs of our main results. One of the key ingredients is the eigenvalue criterion lemma presented below (Lemma \[lemma:eigenvalue\]), which is based on Sylvester’s determinant theorem: $$\label{eq:sylvester}
\det (I + AB) = \det (I + BA)$$ whenever $A$ is an $n \times k$ matrix and $B$ is a $k \times n$ matrix. In particular, the left-hand side of is an $n \times n$ determinant and the right-hand side is a $k \times k$ determinant.
For concreteness, let us focus on the proof of Theorem \[thm:outliers\]. That is, we wish to study the eigenvalues of $$P_n := \prod_{k=1}^m \left( \frac{1}{\sqrt{n}} X_{n,k} + A_{n,k} \right)$$ outside the disk $\{z \in \mathbb{C} : |z| \leq \sigma + 2{\varepsilon}\}$. We first linearize the problem by invoking Proposition \[prop:linear\] with the matrix $\frac{1}{\sqrt{n}} \mathcal{Y}_n + \mathcal{A}_n$, where $\mathcal{Y}_n$ and $\mathcal{A}_n$ are defined in and . Indeed, by Proposition \[prop:linear\], it suffices to study the eigenvalues of $\frac{1}{\sqrt{n}} \mathcal{Y} _n + \mathcal{A}_n$ outside of the disk $\{z \in \mathbb{C} : |z| \leq \sigma^{1/m} + \delta\}$ for some $\delta > 0$ (depending on $\sigma$, ${\varepsilon}$, and $m$). Let us suppose that $\mathcal{A}_n$ is rank one. In other words, assume $\mathcal{A}_n = v u^\ast$ for some $u, v \in \mathbb{C}^{mn}$. In order to study the outlier eigenvalues, we will need to solve the equation $$\label{eq:firstdet}
\det \left( \frac{1}{\sqrt{n}} \mathcal{Y}_n + \mathcal{A}_n - zI \right) = 0$$ for $z \in \mathbb{C}$ with $|z| > \sigma^{1/m} + \delta$. Assuming $z$ is not eigenvalue of $\mathcal{Y}_n$, we can rewrite as $$\det \left( I + \mathcal{G}_n(z) \mathcal{A}_n \right) = 0,$$ where the resolvent $\mathcal{G}_n(z)$ is defined in . From and the fact that $\mathcal{A}_n = v u^\ast$, we find that this reduces to solving $$1 + u^\ast \mathcal{G}_n(z) v = 0.$$ Thus, the problem of locating the outlier eigenvalues reduces to studying the resolvent $\mathcal{G}_n(z)$. In particular, we develop an isotropic limit law in Section \[sec:isotropic\] to compute the limit of $u^\ast \mathcal{G}_n(z) v$. This limit law is inspired by the isotropic semicircle law developed by Knowles and Yin in [@KY; @KY2] for Wigner random matrices as well as the isotropic law verified in [@OR] for elliptic matrices.
The general case, when $\mathcal{A}_n$ is not necessarily rank one, is similar. In this case, we will exploit the following criterion to characterize the outlier eigenvalues.
\[lemma:eigenvalue\] Let $Y$ and $A$ be $n \times n$ matrices, and assume $A = BC$, where $B$ is an $n \times k$ matrix and $C$ is a $k \times n$ matrix. Let $z$ be a complex number which is not an eigenvalue of $Y$. Then $z$ is an eigenvalue of $Y + A$ if and only if $$\det \left(I_k + C \left( Y - z I_n \right)^{-1} B \right) = 0.$$
\[rem:arg\] The proof of Lemma \[lemma:eigenvalue\] actually reveals that $$\label{eq:iddet}
\det \left(I_k + C \left( Y - z I_n \right)^{-1} B \right) = \frac{\det \left( Y + A - zI \right) }{\det \left( Y - z I \right) }$$ provided the denominator does not vanish. Versions of this identity have appeared in previous publications including [@AGG; @BGM; @BGM2; @BR; @CDF1; @CDF; @KY; @KY2; @OR; @OW; @PRS; @RS; @Tout].
Assume $z$ is not an eigenvalue of $Y$. Then $\det (Y - z I) \neq 0$ and $$\begin{aligned}
\det (Y + A - z I) &= \det (Y - z I) \det ( I + (Y - z I)^{-1} A) \\
&= \det (Y- z I) \det (I + (Y - zI)^{-1} BC) .\end{aligned}$$ Thus, by , $z$ is an eigenvalue of $Y +A$ if and only if $$\det (I + C (Y-zI)^{-1} B) = 0,$$ as desired.
Another identity we will make use of is the Resolvent Identity, which states that $$A^{-1}-B^{-1}=A^{-1}(B-A)B^{-1}
\label{Equ:ResolventIndentity}$$ whenever $A$ and $B$ are invertible.
Isotropic limit law and the proofs of the main theorems {#sec:isotropic}
=======================================================
This section is devoted to the proofs of Theorems \[thm:nooutlier\], \[Thm:NoOutlierInProductPert\], \[thm:nomixed\], \[thm:nonzeromean\], and \[thm:outliers\]. The key ingredient is the following result concerning the properties of the resolvent $\mathcal{G}_n(z)$.
\[thm:isotropic\] Let $m \geq 1$ be a fixed integer, and assume $\xi_1, \ldots, \xi_m$ are complex-valued random variables with mean zero, unit variance, finite fourth moments, and independent real and imaginary parts. For each $n \geq 1$, let $X_{n,1}, \ldots, X_{n,m}$ be independent $n \times n$ iid random matrices with atom variables $\xi_1, \ldots, \xi_m$, respectively. Recall that $\mathcal{Y}_n$ is defined in and its resolvent $\mathcal{G}_n(z)$ is defined in . Then, for any fixed $\delta > 0$, the following statements hold.
1. \[item:invertible\] Almost surely, for $n$ sufficiently large, the eigenvalues of $\frac{1}{\sqrt{n}} \mathcal{Y}_n$ are contained in the disk $\{z \in \mathbb{C} : |z| \leq 1 + \delta \}$. In particular, this implies that almost surely, for $n$ sufficiently large, the matrix $\frac{1}{\sqrt{n}} \mathcal{Y}_n - z I$ is invertible for every $z \in \mathbb{C}$ with $|z| > 1 + \delta$.
2. \[item:invtbnd\] There exists a constant $c > 0$ (depending only on $\delta$) such that almost surely, for $n$ sufficiently large, $$\sup_{z \in \mathbb{C} : |z| > 1 + \delta} \| \mathcal{G}_n(z) \| \leq c.$$
3. \[item:isotropic\] For each $n \geq 1$, let $u_n, v_n \in \mathbb{C}^{mn}$ be deterministic unit vectors. Then $$\sup_{z \in \mathbb{C} : |z| > 1 + \delta} \left| u_n^\ast \mathcal{G}_n(z) v_n + \frac{1}{z} u_n^\ast v_n \right| \longrightarrow 0$$ almost surely as $n \to \infty$.
We conclude this section with the proofs of Theorems \[thm:nooutlier\], \[Thm:NoOutlierInProductPert\], \[thm:nomixed\], \[thm:nonzeromean\], and \[thm:outliers\] assuming Theorem \[thm:isotropic\]. Sections \[Sec:TruncAndTools\]–\[sec:combin\] are devoted to the proof of Theorem \[thm:isotropic\].
Proof of Theorem \[thm:nooutlier\]
----------------------------------
Consider $$P_{n}:=n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}$$ and note that by rescaling by $\frac{1}{\sigma}$, it is sufficient to assume that $\sigma_{i}=1$ for all $1\leq i\leq m$. By Proposition \[prop:linear\], the eigenvalues of $P_{n}$ are precisely the eigenvalues of $n^{-m/2}{\mathcal{Y}_{n}}^{m}$, each with multiplicity $m$. Additionally, the eigenvalues of $n^{-m/2}{\mathcal{Y}_{n}}^{m}$ are exactly the $m$-th powers of the eigenvalues of $n^{-1/2}{\mathcal{Y}_{n}}$. Thus, it is sufficient to study the spectral radius of $n^{-1/2}{\mathcal{Y}_{n}}$. By part \[item:invertible\] of Theorem \[thm:isotropic\], we conclude that almost surely, $$\limsup_{n\rightarrow\infty}\rho\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}\right)\leq 1$$ where $\rho(M)$ denotes the spectral radius of the matrix $M$. This completes the proof of Theorem \[thm:nooutlier\].
Proof of Theorem \[Thm:NoOutlierInProductPert\]
-----------------------------------------------
By rescaling by $\frac{1}{\sigma}$, it is sufficient to assume that $\sigma_{i}=1$ for $1\leq i\leq m$. Observe that if two deterministic terms $$I+{{A}_{n,i}}\text{ and }I+ {{A}_{n,j}}$$ appeared consecutively in the product $P_{n}$, then they could be rewritten $$\left(I+{{A}_{n,i}}\right)\cdot\left(I+ {{A}_{n,j}}\right)=I+{{A}_{n,j}}'$$ where all non-identity terms are lumped into the new deterministic matrix ${{A}_{n,j}}'$ which still satisfies the assumptions on rank and norm. Additionally, if two random matrices ${{X}_{n,i}}$ and ${{X}_{n,j}}$ appeared in the product $P_{n}$ consecutively, then we could write $$\left(\frac{1}{\sqrt{n}}{{X}_{n,i}}\right)\left(\frac{1}{\sqrt{n}}{{X}_{n,j}}\right)=\left(\frac{1}{\sqrt{n}}{{X}_{n,i}}\right)(I_{n}+0_{n})\left(\frac{1}{\sqrt{n}}{{X}_{n,j}}\right)$$ where $0_{n}$ denotes the $n\times n$ zero matrix. Therefore, it is sufficient to consider products in which terms alternate between a random term $\frac{1}{\sqrt{n}}{{X}_{n,i}}$ and a deterministic term $I+{{A}_{n,j}}$. Next, observe that the eigenvalues of the product $P_{n}$ remain the same when the matrices in the product are cyclically permuted. Thus, without loss of generality and up to reordering the indices, we may assume that the product $P_{n}$ appears as $$P_{n}= \left(I+{{A}_{n,1}}\right)\left(\frac{1}{\sqrt{n}}{{X}_{n,1}}\right)\left(I+{{A}_{n,2}}\right)\left(\frac{1}{\sqrt{n}}{{X}_{n,2}}\right)\cdots\left(I+{{A}_{n,m}}\right)\left(\frac{1}{\sqrt{n}}{{X}_{n,m}}\right).
\label{equ:ProdAnyOrder}$$ Next, define the $2mn\times 2mn$ matrix $$\mathcal{L}_{n} :=\left[\begin{array}{cc}
0_{mn} & I_{mn}+{\operatorname{diag}({{A}_{n,1}}, \dots, {{A}_{n,m}})}\\
\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}} & 0_{mn}\\
\end{array} \right],
\label{Def:Ln}$$ where $0_{mn}$ denotes the $mn\times mn$ zero matrix, ${\mathcal{Y}_{n}}$ is as defined in , and ${\operatorname{diag}({{A}_{n,1}}, \dots, {{A}_{n,m}})}$ is defined so that $$I_{mn}+{\operatorname{diag}({{A}_{n,1}}, \dots, {{A}_{n,m}})}:=\left[\begin{array}{cccc}
I+{{A}_{n,1}} & 0 & \dots & 0\\
0 & I+{{A}_{n,2}} & \dots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \dots & I+{{A}_{n,m}}\\
\end{array}\right].$$ Here, we have to slightly adjust our notation to deal with the fact that ${\mathcal{L}_{n}}$ is a $2mn \times 2mn$ matrix instead of $mn \times mn$. For the remainder of the proof, we view ${\mathcal{L}_{n}}$ as a $2\times 2$ matrix with entries which are $mn\times mn$ matrices, and we denote the $mn \times mn$ blocks as ${\mathcal{L}_{n}}^{[a,b]}$ for $1 \leq a,b \leq 2$. We will use analogous notation for other $2mn \times 2mn$ matrices and $2mn$-vectors.
For $1 \le k \le 2m$, let $W_{k}$ denote the product $P_{n}$, but with terms cyclically permuted so that the product starts on the $k$th term. Note that there are $2m$ such products and each results in an $n\times n$ matrix. For instance, $$W_{2}=\left(\frac{1}{\sqrt{n}}{{X}_{n,1}}\right)(I+{{A}_{n,2}})\left(\frac{1}{\sqrt{n}}{{X}_{n,2}}\right)(I+{{A}_{n,3}})\cdots\left(\frac{1}{\sqrt{n}}{{X}_{n,m}}\right)(I+{{A}_{n,1}})$$ and $$W_{3}=(I+{{A}_{n,2}})\left(\frac{1}{\sqrt{n}}{{X}_{n,2}}\right)(I+{{A}_{n,3}})\cdots\left(\frac{1}{\sqrt{n}}{{X}_{n,m}}\right)(I+{{A}_{n,1}})\left(\frac{1}{\sqrt{n}}{{X}_{n,1}}\right).$$ A simple computation reveals that $${\mathcal{L}_{n}}^{2m}=\left[\begin{array}{cc}
\mathcal{W} & 0_{mn}\\
0_{mn} & \tilde{\mathcal{W}}\\
\end{array}\right]$$ where $$\mathcal{W}=\left[\begin{array}{cccc}
W_{1} & 0_{n} & \dots & 0_{n}\\
0_{n} & W_{3} & \dots & 0_{n}\\
\vdots & \vdots & \ddots & \vdots\\
0_{n} & 0_{n} & \dots & W_{2m-1}\\
\end{array}\right]\;\;\;\;\text{and}\;\;\;\;\tilde{\mathcal{W}}=\left[\begin{array}{cccc}
W_{2} & 0_{n} & \dots & 0_{n}\\
0_{n} & W_{4} & \dots & 0_{n}\\
\vdots & \vdots & \ddots & \vdots\\
0_{n} & 0_{n} & \dots & W_{2m}\\
\end{array}\right].$$ Thus, the eigenvalues of ${\mathcal{L}_{n}}^{2m}$ are precisely the eigenvalues of the product $P_{n}$, each with multiplicity $2m$.
Define $${\mathcal{X}_{n}}:=\left[\begin{array}{cc}
0_{mn} & I_{mn}\\
\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}} & 0_{mn}\\
\end{array}\right]\;\;\;\;\text{ and }\;\;\;\;
{{\mathcal{A}_{n}}^{\square}}:=\left[\begin{array}{cc}
0_{mn} & {\operatorname{diag}({{A}_{n,1}}, \dots, {{A}_{n,m}})}\\
0_{mn} & 0_{mn}\\
\end{array}\right].$$ Then we can rewrite ${\mathcal{L}_{n}}={\mathcal{X}_{n}}+{{\mathcal{A}_{n}}^{\square}}$.
For $1 \leq k \leq m$, let $Z_{k}:={{X}_{n,k}}{{X}_{n,k+1}}\cdots{{X}_{n,m}}{{X}_{n,1}}\cdots{{X}_{n,k-1}}$. Then $$\label{eq:XnZn}
{\mathcal{X}_{n}}^{2m}=n^{-m/2}
\left[\begin{array}{cccccccc}
\mathcal{Z}_{n} & 0_{mn}\\
0_{mn} & \mathcal{Z}_{n}\\
\end{array}\right]$$ where $$\mathcal{Z}_{n}=\left[\begin{array}{ccc}
Z_{1} & \dots & 0_{n}\\
\vdots & \ddots & \vdots\\
0_{n} & \dots & Z_{m}\\
\end{array}\right].$$ Thus, the eigenvalues of ${\mathcal{X}_{n}}^{2m}$ are precisely the eigenvalues of $n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}$, each with multiplicity $2m$.
Fix $\delta > 0$. By part \[item:invertible\] of Theorem \[thm:isotropic\] and Proposition \[prop:linear\], we find that almost surely, for $n$ sufficiently large, the eigenvalues of $n^{-m/2}X_{n,1}\cdots X_{n,m}$ are contained in the disk $\{ z \in {\mathbb{C}}: |z| \leq 1 + \delta\}$. By , we conclude that almost surely, for $n$ sufficiently large, the eigenvalues of ${\mathcal{X}_{n}}$ are contained in the disk $\{z \in {\mathbb{C}}: |z| \leq 1 + \delta\}$. Therefore, almost surely, for $n$ sufficiently large ${\mathcal{X}_{n}} - zI$ is invertible for all $|z| > 1 + \delta$. For all such values of $z$, we define $$\mathcal{R}_{n}(z):= \left({\mathcal{X}_{n}}-zI_{2mn}\right)^{-1}.$$ Since ${{\mathcal{A}_{n}}^{\square}}$ has rank $O(1)$ and operator norm $O(1)$, we can decompose (by the singular value decomposition) ${{\mathcal{A}_{n}}^{\square}}= {\mathcal{B}_{n}} {\mathcal{C}_{n}}$, where ${\mathcal{B}_{n}}$ is a $mn \times k$ matrix, ${\mathcal{C}_{n}}$ is a $k \times mn$ matrix, $k = O(1)$, and both ${\mathcal{B}_{n}}$ and ${\mathcal{C}_{n}}$ have rank $O(1)$ and operator norm $O(1)$.
Thus, for $|z| > 1 + \delta$, almost surely, for $n$ sufficiently large, $$\det({\mathcal{L}_{n}}-zI_{2mn})=\det({\mathcal{X}_{n}}+{{\mathcal{A}_{n}}^{\square}}-zI_{2mn})=0$$ if and only if $$\label{eq:proddecompcb}
\det(I_{k}+{\mathcal{C}_{n}}\mathcal{R}_{n}(z){\mathcal{B}_{n}})=0$$ by Lemma \[lemma:eigenvalue\].
Using Schur’s Compliment to calculate the $mn\times mn$ blocks of $\mathcal{R}_n(z)$, we can see that $\mathcal{R}_n(z)=\begin{pmatrix}
z\mathcal{G}_{n}(z^{2}) & \mathcal{G}_{n}(z^{2})\\
I+z^{2}\mathcal{G}_{n}(z^{2}) & z\mathcal{G}_{n}(z^{2})
\end{pmatrix}$ , where $\mathcal{G}_{n}(z):=\left(\frac{1}{\sqrt{n}}\mathcal{Y}_{n}-zI\right)^{-1}$ (which is defined for $|z| > 1 +\delta$ by Theorem \[thm:isotropic\]); hence $$\mathcal{R}_{n}(z)^{[a,b]}=\begin{cases}
z\mathcal{G}_{n}(z^{2}) & \text{ if } a=b\\
\mathcal{G}_{n}(z^{2}) & \text{ if } a=1,\;b=2\\
I+z^{2}\mathcal{G}_{n}(z^{2}) & \text{ if } a=2,\;b=1.\\
\end{cases}$$ Note that for $u=u_{2mn}$ and $v=v_{2mn}$ in ${\mathbb{C}}^{2mn}$, we have $$u^{*}\mathcal{R}_{n}(z)v = \sum_{1\leq a,b\leq 2}\left(u^{*}\right)^{[a]}\mathcal{R}_{n}(z)^{[a,b]}v^{[b]}$$ where $v^{[1]}, v^{[2]}$ denote the $mn\times 1$ sub-blocks of the vector $v$ and $\left(u^{*}\right)^{[1]}, \left(u^{*}\right)^{[2]}$ denote the $1 \times mn$ sub-blocks of the vector $u^*$. Additionally, if $u$ and $v$ have uniformly bounded norm for all $n$, then by Theorem \[thm:isotropic\], almost surely $$\begin{aligned}
&\sup_{|z|>1+\delta}\left|\left(u^{*}\right)^{[1]}z\mathcal{G}_{n}(z^{2})v^{[1]}-z\left(-\frac{1}{z^{2}}\right)\left(u^{*}\right)^{[1]}v^{[1]}\right|=o(1),\\
&\sup_{|z|>1+\delta}\left|\left(u^{*}\right)^{[2]}z\mathcal{G}_{n}(z^{2})v^{[2]}-z\left(-\frac{1}{z^{2}}\right)\left(u^{*}\right)^{[2]}v^{[2]}\right|=o(1),\\
&\sup_{|z|>1+\delta}\left|(u^{*})^{[1]}\mathcal{G}_{n}(z^{2})v^{[2]}-\left(-\frac{1}{z^{2}}(u^{*})^{[1]}v^{[2]}\right)\right|=o(1), \text{ and }\\
&\sup_{|z|>1+\delta}\left|(u^{*})^{[2]}(I+z^{2}\mathcal{G}_{n}(z^{2}))v^{[1]}-\left((u^{*})^{[2]}v^{[1]}-z^{2}\frac{1}{z^{2}}(u^{*})^{[2]}v^{[1]}\right)\right|=o(1).\\
\end{aligned}$$ We note that the off-diagonal blocks of $\mathcal{R}_n(z)$ have a much different behavior than $\mathcal{G}_n(z)$. In order to keep track of this behavior, define $$\mathcal{H}_{n}=\left[\begin{array}{cc}
0_{mn} & I_{mn}\\
0_{mn} & 0_{mn}\\
\end{array}\right].$$ Then, for deterministic $u$ and $v$ in ${\mathbb{C}}^{2mn}$ with uniformly bounded norm for all $n$, almost surely $$\begin{aligned}
\sup_{|z|>1+\delta}\left|u^{*}\mathcal{R}_{n}(z)v-\left(-\frac{1}{z}u^{*}v -\frac{1}{z^{2}}u^{*}\mathcal{H}_{n}v\right)\right|=o(1).
\end{aligned}$$ Applying this to , we obtain that $$\sup_{|z|>1+\delta}{\left\lVert}{\mathcal{C}_{n}}\mathcal{R}_{n}(z){\mathcal{B}_{n}}-\left(-\frac{1}{z}{\mathcal{C}_{n}}{\mathcal{B}_{n}}-\frac{1}{z^{2}}{\mathcal{C}_{n}} \mathcal{H}_{n} {\mathcal{B}_{n}}\right){\right\rVert}=o(1)$$ almost surely. Hence, Lemma \[Lem:normtodet\] reveals that almost surely $$\sup_{|z|>1+\delta}\left|\det\left(I-\mathcal{C}_{n}\mathcal{R}_{n}\mathcal{B}_{n}\right)-\det\left(I-\frac{1}{z}{\mathcal{C}_{n}}{\mathcal{B}_{n}}-\frac{1}{z^{2}}{\mathcal{C}_{n}}\mathcal{H}_{n}{\mathcal{B}_{n}}\right)\right|=o(1).$$ By another application of Sylvester’s determinant formula and by noticing that $\mathcal{H}_{n}{{\mathcal{A}_{n}}^{\square}}$ is the zero matrix, this can be rewritten as $$\sup_{|z|>1+\delta}\left|\det\left(I+\mathcal{R}_{n}(z){{\mathcal{A}_{n}}^{\square}}\right) - \det\left(I-\frac{1}{z}{{\mathcal{A}_{n}}^{\square}}\right)\right|=o(1).$$ Finally, since the eigenvalues of ${{\mathcal{A}_{n}}^{\square}}$ are all zero, $\det\left(I-\frac{1}{z}{{\mathcal{A}_{n}}^{\square}}\right)=1$ so we can write the above statement as $$\sup_{|z|>1+\delta}\left|\det \left( \mathcal R_n\right)\det\left(\mathcal{L}_{n}-zI\right) -1\right|=o(1)$$ almost surely. Almost surely, for $n$ sufficiently large, and for $|z|> 1+\delta$, we know that $\det(R_n(z))$ is finite, and thus $\det\left(\mathcal{L}_{n}-zI\right)$ is nonzero for all $|z| > 1 + \delta$, implying that $\mathcal{L}_n$ has no eigenvalues outside the disk $\{z \in {\mathbb{C}}: |z| \leq 1 + \delta\}$. By the previous observations, this implies the same conclusion for $P_n$ (since $\delta$ is arbitrary), and the proof is complete.
Proof of Theorem \[thm:outliers\]
---------------------------------
Recall that $$P_{n} :=\prod_{k=1}^{m}\left(\frac{1}{\sqrt{n}}{{X}_{n,k}}+A_{n,k}\right), \qquad A_n := \prod_{k=1}^m A_{n,k}.$$ By rescaling $P_{n}$ by $\frac{1}{\sigma}$, we may assume that $\sigma_{i}=1$ for $1\leq i\leq n$. Let $\varepsilon>0$, and assume that for sufficiently large $n$, no eigenvalues of $A_{n}$ fall in the band $\{z\in{\mathbb{C}}\;:\;1+\varepsilon <|z|<1+3\varepsilon\}$. Assume that for some $j=O(1)$, there are $j$ eigenvalues $\lambda_{1}(A_{n}),\ldots,\lambda_{j}(A_{n})$ that lie in the region $\{z\in {\mathbb{C}}\;:\;|z|\geq 1+3\varepsilon\}$.
Let $\mathcal{Y}_n$ and $\mathcal{A}_n$ be defined as in and . Using Proposition \[prop:linear\], it will suffice to study the eigenvalues of $\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}+{\mathcal{A}_{n}}$. In particular, let ${\varepsilon}' > 0$ such that $(1 + 2 {\varepsilon})^{1/m} = 1 + {\varepsilon}'$. Then we want to find solutions to $$\det\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}+{\mathcal{A}_{n}}-zI\right)=0
\label{Equ:detZero}$$ for $|z|\geq 1+{\varepsilon}'$. By part \[item:invertible\] of Theorem \[thm:isotropic\], almost surely, for $n$ sufficiently large $\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}} - zI$ is invertible for all $|z| \geq 1 + {\varepsilon}'$. By supposition, we can decompose (using the singular value decomposition) $\mathcal{A}_n = \mathcal{B}_n \mathcal{C}_n$, where $\mathcal{B}_n$ is $mn \times k$, $\mathcal{C}_n$ is $k \times mn$, $k = O(1)$, and both $\mathcal{B}_n$ and $\mathcal{C}_n$ have rank $O(1)$ and operator norm $O(1)$. Thus, by Lemma \[lemma:eigenvalue\], we need to investigate the values of $z \in {\mathbb{C}}$ with $|z| \geq 1 + {\varepsilon}'$ such that $$\det(I_{k}+\mathcal{C}_{n}\mathcal{G}_{n}(z)\mathcal{B}_{n})=0,$$ where $\mathcal{G}_n(z)$ is defined in .
Since $k = O(1)$, Theorem \[thm:isotropic\] implies that $$\sup_{|z| \geq 1 + {\varepsilon}'} {\left\lVert}\mathcal{C}_{n}\mathcal{G}_{n}(z)\mathcal{B}_{n}-\left(-\frac{1}{z}\right)\mathcal{C}_{n}\mathcal{B}_{n}{\right\rVert}\longrightarrow 0$$ almost surely as $n\rightarrow\infty$. By Lemma \[Lem:normtodet\], this implies that $$\sup_{|z| \geq 1 + {\varepsilon}'} \left|\det\left(I_{k}+ \mathcal{C}_{n}\mathcal{G}_{n}(z)\mathcal{B}_{n}\right)-\det\left(I_{k}-\frac{1}{z}\mathcal{C}_{n}\mathcal{B}_{n}\right)\right|\longrightarrow 0$$ almost surely. By an application of Sylvester’s determinant theorem , this is equivalent to $$\label{eq:rouchedet}
\sup_{|z| \geq 1 + {\varepsilon}'} \left| \det\left(I_{k}+ \mathcal{C}_{n}\mathcal{G}_{n}\mathcal{B}_{n}\right) - \det\left(I_{n}-\frac{1}{z}{\mathcal{A}_{n}}\right)\right|\longrightarrow 0$$ almost surely as $n\rightarrow\infty$. Define $$g(z) := \det\left(I_{n}-\frac{1}{z}{\mathcal{A}_{n}}\right)=\prod_{i=1}^{k}\left(1-\frac{\lambda_{i}({\mathcal{A}_{n}})}{z}\right).$$ Since the eigenvalues of ${\mathcal{A}_{n}}^m$ are precisely the eigenvalues of $A_n$, each with multiplicity $m$, it follows that $g$ has precisely $l := jm$ roots $\lambda_1({\mathcal{A}_{n}}), \ldots, \lambda_l({\mathcal{A}_{n}})$ outside the disk $\{ z \in {\mathbb{C}}: |z| < 1 + {\varepsilon}'\}$. Thus, by and Rouché’s theorem, almost surely, for $n$ sufficiently large, $$f(z):=\det\left(I_{k}+ \mathcal{C}_{n}\mathcal{G}_{n}\mathcal{B}_{n}\right)$$ has exactly $l$ roots outside the disk $\{ z \in {\mathbb{C}}: |z| < 1 + {\varepsilon}' \}$ and these roots take the values $\lambda_i({\mathcal{A}_{n}}) + o(1)$ for $1 \leq i \leq l$.
Returning to , we conclude that almost surely for $n$ sufficiently large, $\frac{1}{\sqrt{n}} {\mathcal{Y}_{n}} + {\mathcal{A}_{n}}$ has exactly $l$ roots outside the disk $\{z \in \mathbb{C} : |z| < 1 + {\varepsilon}'\}$, and after possibly reordering the eigenvalues, these roots take the values $$\lambda_i \left( \frac{1}{\sqrt{n}} {\mathcal{Y}_{n}} + {\mathcal{A}_{n}} \right) = \lambda_i ( {\mathcal{A}_{n}} ) + o(1)$$ for $1 \leq i \leq l$.
We now relate these eigenvalues back to the eigenvalues of $P_n$. Recall that $\left( \frac{1}{\sqrt{n}} {\mathcal{Y}_{n}} + {\mathcal{A}_{n}} \right)^m$ has the same eigenvalues as $P_n$, each with multiplicity $m$; and ${\mathcal{A}_{n}}^m$ has the same eigenvalues of $A_n$, each with multiplicity $m$. Taking this additional multiplicity into account and using the fact that $$\left( \lambda_i( {\mathcal{A}_{n}} ) + o(1) \right)^m = \lambda_i( {\mathcal{A}_{n}}^m ) + o(1)$$ since ${\mathcal{A}_{n}}$ has spectral norm $O(1)$, we conclude that almost surely, for $n$ sufficiently large, $P_n$ has exactly $j$ eigenvalues in the region $\{ z \in \mathbb{C} : |z| \geq 1 + 2 {\varepsilon}\}$, and after reordering the indices correctly $$\lambda_i(P_n) = \lambda_i(A_n) + o(1)$$ for $1 \leq i \leq j$. This completes the proof of Theorem \[thm:outliers\].
Proof of Theorems \[thm:nomixed\] and \[thm:nonzeromean\]
---------------------------------------------------------
In the proofs of Theorems \[thm:nomixed\] and \[thm:nonzeromean\] we will make use of an alternative isotropic law which is a corollary of Theorem \[thm:isotropic\]. We state and prove the result now.
\[cor:ProductIsotropic\] Let $m \geq 1$ be a fixed integer, and assume $\xi_1, \ldots, \xi_m$ are complex-valued random variables with mean zero, unit variance, finite fourth moments, and independent real and imaginary parts. For each $n \geq 1$, let $X_{n,1}, \ldots, X_{n,m}$ be independent $n \times n$ iid random matrices with atom variables $\xi_1, \ldots, \xi_m$, respectively. Then, for any fixed $\delta > 0$, the following statements hold.
1. \[item:alt:invertible\] Almost surely, for $n$ sufficiently large, the eigenvalues of $n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}$ are contained in the disk $\{z \in \mathbb{C} : |z| \leq 1 + \delta \}$. In particular, this implies that almost surely, for $n$ sufficiently large, the matrix\
$n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}- z I$ is invertible for every $z \in \mathbb{C}$ with $|z| > 1 + \delta$.
2. There exists a constant $c > 0$ (depending only on $\delta$) such that almost surely, for $n$ sufficiently large, $$\sup_{z \in \mathbb{C} : |z| > 1 + \delta} {\left\lVert}\left(n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}-zI\right)^{-1} {\right\rVert}\leq c.$$
3. For each $n \geq 1$, let $u_n, v_n \in \mathbb{C}^{n}$ be deterministic unit vectors. Then $$\sup_{z \in \mathbb{C} : |z| > 1 + \delta} \left| u_n^\ast \left(n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}-zI\right)^{-1} v_n + \frac{1}{z} u_n^\ast v_n \right| \longrightarrow 0$$ almost surely as $n \to \infty$.
Part \[item:alt:invertible\] follows from Theorem \[thm:nooutlier\]. Let ${\mathcal{Y}_{n}}$ be defined by , and let ${\mathcal{G}_{n}}(z)$ be defined by . Then the last two parts of Corollary \[cor:ProductIsotropic\] follow from the last two parts of Theorem \[thm:isotropic\] due to the fact that $$\mathcal{G}_n^{[1,1]}(z) = z^{m-1} ( n^{-m/2} X_{n,1} \cdots X_{n,m} - z^m I)^{-1}.$$
With this result in hand, we proceed to the remainder of the proofs.
By rescaling by $\frac{1}{\sigma}$ it suffices to assume that $\sigma_i = 1$ for $1 \leq i \leq m$. Let $\varepsilon >0$. Assume that for sufficiently large $n$ there are no eigenvalues of $A_{n}$ in the band $\{z\in{\mathbb{C}}\;:\;1+\varepsilon < |z| < 1+3\varepsilon\}$ and there are $j$ eigenvalues, $\lambda_{1}(A_{n})$,…,$\lambda_{j}(A_{n})$, in the region $\{z\in{\mathbb{C}}\;:\;|z|\geq 1+3\varepsilon\}$.
By Corollary \[cor:ProductIsotropic\], almost surely, for $n$ sufficiently large, $n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}} - zI$ is invertible for all $|z| > 1 + {\varepsilon}$. We decompose (using the singular value decomposition) $A_n = B_n C_n$, where $B_n$ is $n \times k$, $C_n$ is $k \times n$, $k = O(1)$, and both $B_n$ and $C_n$ have rank $O(1)$ and spectral norm $O(1)$. By Lemma \[lemma:eigenvalue\], the eigenvalues of $P_n$ outside $\{ z \in {\mathbb{C}}: |z| < 1 + 2 {\varepsilon}\}$ are precisely the values of $z \in \mathbb{C}$ with $|z| > 1 + 2 {\varepsilon}$ such that $$\det\left(I_{k}+C_{n}\left(n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}-zI_{n}\right)^{-1}B_{n}\right) = 0.$$
By Corollary \[cor:ProductIsotropic\], $$\sup_{|z|>1+{\varepsilon}}{\left\lVert}C_{n}\left(n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}-zI_{n}\right)^{-1}B_{n}+\frac{1}{z}C_{n}B_{n}{\right\rVert}\longrightarrow 0$$ almost surely as $n\rightarrow\infty$. By applying and Lemma \[Lem:normtodet\], this gives $$\sup_{|z|>1+{\varepsilon}}\left|\det\left(I_{k}+C_{n}\left(n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}-zI_{n}\right)^{-1}B_{n}\right) - \det\left(I_{n}-\frac{1}{z}A_{n}\right)\right|\longrightarrow 0$$ almost surely as $n\rightarrow\infty$. By recognizing the roots of $\det\left(I_{n}-\frac{1}{z}A_{n}\right)$ as the eigenvalues of $A_{n}$, Rouché’s Theorem implies that almost surely for $n$ sufficiently large $P_n$ has exactly $j$ eigenvalues in the region $\{z \in {\mathbb{C}}: |z| > 1 + 2 {\varepsilon}\}$, and after labeling the eigenvalues properly, $$\lambda_{i}(P_{n})=\lambda_{i}(A_{n})+o(1)$$ for $1 \leq i \leq j$.
The proof of Theorem \[thm:nonzeromean\] will require the following corollary of [@Tout Lemma 2.3].
\[Lem:ProductToZero\] Let $\varphi_{n}$ and $X_{n,1}$,…,$X_{n,m}$ be as in Theorem \[thm:nonzeromean\]. Then almost surely, $$n^{-m/2} \left| \varphi_{n}^{*}X_{n,1}X_{n,2}\cdots X_{n,m}\varphi_{n} \right| = o(1).$$
Let $u := n^{-(m-1)/2}\varphi_{n}^{*}X_{n,1}X_{n,2}\cdots X_{n,m-1}$. In view of [@Tout Theorem 1.4], it follows that $\|u \| = O(1)$ almost surely. We now condition on $X_{n,1}, \ldots, X_{n,m-1}$ so that $\| u \| = O(1)$. As $X_{n,m}$ is independent of $u$, we apply [@Tout Lemma 2.3] to conclude that $$u \left(\frac{1}{\sqrt{n}}X_{n,m}\right)\varphi_{n}=o(1)$$ almost surely, concluding the proof.
With this result, we may proceed to the proof of Theorem \[thm:nonzeromean\].
By rescaling by $\frac{1}{\sigma}$ it suffices to assume that $\sigma_i = 1$ for $1 \leq i \leq m$. Fix $\gamma > 0$ and let $\varepsilon>0$. By Corollary \[cor:ProductIsotropic\] and Lemma \[lemma:eigenvalue\], almost surely, for $n$ sufficiently large, the only eigenvalues of $$P_{n}:= n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}+\mu n^{\gamma}\phi_{n}\phi_{n}^{*}$$ in the region $\{z \in {\mathbb{C}}: |z| > 1 + {\varepsilon}\}$ are the values of $z \in {\mathbb{C}}$ with $|z| > 1 + {\varepsilon}$ such that $$1+\mu n^{\gamma} \phi^{*}_{n}\left(n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}-zI_{n}\right)^{-1}\phi_{n}=0.
\label{equ:detInNonZeroMean}$$
Define the functions $$\begin{aligned}
&f(z):=1+\mu n^{\gamma}\phi^{*}_{n}\left(n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}-zI_{n}\right)^{-1}\phi_{n}\\
&g(z):=1-\frac{\mu n^{\gamma}}{z}.
\end{aligned}$$ Observe that $g(z)$ has one zero located at $\mu n^{\gamma}$ which will be outside the disk $\{z\in {\mathbb{C}}\;:\;|z| \leq 1+\varepsilon\}$ for large enough $n$. By Corollary \[cor:ProductIsotropic\], it follows that almost surely $$\sup_{|z|>1+{\varepsilon}}|f(z)-g(z)| = o(n^{\gamma}).$$ Thus, almost surely $$\begin{aligned}
f(z)=g(z)+o(n^{\gamma})
\end{aligned}$$ for all $z \in {\mathbb{C}}$ with $|z| > 1+{\varepsilon}$.
Observe that if $z$ is a root of $f$ with $|z| > 1 + {\varepsilon}$, then $|g(z)|=\left|1-\frac{\mu n^{\gamma}}{z}\right|=o(n^{\gamma})$. We conclude that if $z$ is a root of $f$ outside the disk $\{z \in {\mathbb{C}}: |z| > 1 + {\varepsilon}\}$, then the root must tend to infinity with $n$ almost surely. We will return to this fact shortly.
For the next step of the proof, we will need to bound the spectral norm of $n^{-m/2} X_{n,1} \cdots X_{n,m}$. To do so, we apply [@Tout Theorem 1.4] and obtain that, almost surely, for $n$ sufficiently large, $$\label{eq:tout:normbound}
n^{-m/2} \| X_{n,1} \cdots X_{n,m} \| \leq (2.1)^m.$$ Thus, by a Neumann series expansion, for all $|z| > (2.5)^m$, we have $$\begin{aligned}
f(z)&=1-\frac{\mu n^{\gamma}}{z}+\frac{\mu n^{\gamma}}{z^{2}}\phi^{*}_{n}\left(n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}\right)\phi_{n}+O\left(\frac{ n^{\gamma}}{|z|^{3}}\right)\\
&= g(z)+\frac{\mu n^{\gamma}}{z^{2}}\phi^{*}_{n}\left(n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}\right)\phi_{n}+O\left(\frac{n^{\gamma}}{|z|^{3}}\right).
\end{aligned}$$ By Lemma \[Lem:ProductToZero\], $\varphi^{*}_{n}\left(n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}\right)\varphi_{n}=o(1)$ almost surely, so one can see $$f(z)=g(z)+o\left(\frac{n^{\gamma}}{|z|^{2}}\right)+O\left(\frac{n^{\gamma}}{|z|^{3}}\right)$$ uniformly for all $|z|>(2.5)^{m}$. In particular, when $|z| \to \infty$, we obtain $$\label{eq:fztendinfty}
|z| |f(z) - g(z)| = o \left( \frac{n^\gamma}{|z|} \right)$$ almost surely. Since any root of $zf(z)$ outside $\{z \in {\mathbb{C}}: |z| \leq 1 + {\varepsilon}\}$ must tend to infinity with $n$, it follows from Rouché’s theorem that almost surely, for $n$ sufficiently large, $zf(z)$ has precisely one root outside the disk $\{ z \in {\mathbb{C}}: |z| \leq 1 + {\varepsilon}\}$ and that root takes the value $\mu n^\gamma + o(n^\gamma)$.
It remains to reduce the error from $o(n^\gamma)$ to $o(1)$. Fix $\delta > 0$, and let $\Gamma$ be the circle around $\mu n^\gamma$ with radius $\delta$. Then from we see that almost surely $$\sup_{z \in \Gamma} |z||f(z) - g(z)| = o(1).$$ Hence, almost surely, for $n$ sufficiently large, $$|z||f(z) - g(z)| < \delta = |z||g(z)|$$ for all $z \in \Gamma$. Therefore, by another application of Rouché’s theorem, we conclude that almost surely, for $n$ sufficiently large, $zf(z)$ contains precisely one root outside of $\{z \in {\mathbb{C}}: |z| \leq 1 + {\varepsilon}\}$ and that root is located in the interior of $\Gamma$. Since $\delta$ was arbitrary, this completes the proof.
Truncation and useful tools {#Sec:TruncAndTools}
===========================
We now turn to the proof of Theorem \[thm:isotropic\]. We will require the following standard truncation results for iid random matrices.
Let $\xi$ be a complex-valued random variable with mean zero, unit variance, finite fourth moment, and independent real and imaginary parts. Let ${\operatorname{Re}}(\xi)$ and ${\operatorname{Im}}(\xi)$ denote the real and imaginary parts of $\xi$ respectively, and let $\sqrt{-1}$ denote the imaginary unit. For $L > 0$, define $$\begin{aligned}
\tilde{\xi}&:={\operatorname{Re}}(\xi){\ensuremath{\mathbf{1}_{\{|{\operatorname{Re}}(\xi)|\leq L/\sqrt{2}\}}}}-{\mathbb{E}}\left[{\operatorname{Re}}(\xi){\ensuremath{\mathbf{1}_{\{|{\operatorname{Re}}(\xi)|\leq L/\sqrt{2}\}}}}\right]\\
&\quad \quad \quad +\sqrt{-1}\left({\operatorname{Im}}(\xi){\ensuremath{\mathbf{1}_{\{|{\operatorname{Im}}(\xi)|\leq L/\sqrt{2}\}}}}-{\mathbb{E}}\left[{\operatorname{Im}}(\xi){\ensuremath{\mathbf{1}_{\{|{\operatorname{Im}}(\xi)|\leq L/\sqrt{2}\}}}}\right]\right)
\end{aligned}$$ and $$\hat{\xi}:=\frac{\tilde{\xi}}{\sqrt{\emph{Var}(\tilde{\xi})}}.$$ Then there exists a constant $L_0 > 0$ (depending only on ${\mathbb{E}}|\xi|^{4}$) such that the following statements hold for all $L > L_0$.
1. \[item:truncation:i\] $\emph{Var}(\tilde{\xi})\geq \frac{1}{2}$
2. \[item:truncation:ii\] $|1-\emph{Var}(\tilde{\xi})|\leq\frac{4}{L^2}{\mathbb{E}}|\xi|^{4}$
3. \[item:truncation:iii\] Almost surely, $|\hat{\xi}|\leq 4L$
4. \[item:truncation:iv\] $\hat{\xi}$ has mean zero, unit variance, ${\mathbb{E}}|\hat{\xi}|^{4}\leq C{\mathbb{E}}|\xi|^{4}$ for some absolute constant $C>0$, and the real and imaginary parts of $\hat{\xi}$ are independent.
\[lem:Truncate\]
The proof of this theorem is a standard truncation argument. The full details of the proof can be found in Appendix \[Sec:ProofOfTruncation\].
Let $X$ be an $n\times n$ random matrix filled with iid copies of a random variable $\xi$ which has mean zero, unit variance, finite fourth moment, and independent real and imaginary parts. We define matrices $\tilde{X}$ and $\hat{X}$ to be the $n\times n$ matrices with entries defined by $$\begin{aligned}
\tilde{X}_{(i,j)}&:={\operatorname{Re}}(X_{(i,j)}){\ensuremath{\mathbf{1}_{\{|{\operatorname{Re}}(X_{(i,j)})|\leq L/\sqrt{2}\}}}}-{\mathbb{E}}\left[{\operatorname{Re}}(X_{(i,j)}){\ensuremath{\mathbf{1}_{\{|{\operatorname{Re}}(X_{(i,j)})|\leq L/\sqrt{2}\}}}}\right]\notag\\
& \quad +\sqrt{-1}\left({\operatorname{Im}}(X_{(i,j)}){\ensuremath{\mathbf{1}_{\{|{\operatorname{Im}}(X_{(i,j)})|\leq L/\sqrt{2}\}}}}-{\mathbb{E}}\left[{\operatorname{Im}}(X_{(i,j)}){\ensuremath{\mathbf{1}_{\{|{\operatorname{Im}}(X_{(i,j)})|\leq L/\sqrt{2}\}}}}\right]\right)\label{def:Xtilde}\end{aligned}$$ and $$\hat{X}_{(i,j)}:= \frac{\tilde{X}_{(i,j)}}{\sqrt{{\text{Var}}(\tilde{X}_{(i,j)})}}
\label{def:Xhat}$$ for $1 \leq i,j \leq n$.
Let $X$ be an iid random matrix with atom variable $\xi$ which has mean zero, unit variance, $m_{4}:={\mathbb{E}}|\xi|^4< \infty$, and independent real and imaginary parts. Let $\hat{X}$ be as defined in . Then, there exist constants $C, L_0 > 0$ (depending only on $m_4$) such that for all $L > L_0$ $$\limsup_{n\rightarrow \infty} \frac{1}{\sqrt n} {\left\lVert}X-\hat{X}{\right\rVert}\leq \frac{C}{L}$$ almost surely. \[lem:XcloseXhat\]
Let $\tilde{X}$ be defined as in , and let $L_0$ be the value from Lemma \[lem:Truncate\]. Begin by noting that $${\left\lVert}X-\hat{X}{\right\rVert}\leq {\left\lVert}X-\tilde{X}{\right\rVert}+{\left\lVert}\tilde{X}-\hat{X}{\right\rVert}$$ and thus it suffices to show that $$\underset{n\rightarrow\infty}{\limsup}\frac{1}{\sqrt{n}}{\left\lVert}X-\tilde{X}{\right\rVert}\leq \frac{C}{L}$$ and $$\underset{n\rightarrow\infty}{\limsup}\frac{1}{\sqrt{n}}{\left\lVert}\tilde{X}-\hat{X}{\right\rVert}\leq \frac{C}{L}$$ almost surely. First, by Lemma \[lem:Truncate\], $$\begin{aligned}
\frac{1}{\sqrt{n}}{\left\lVert}\tilde{X}-\hat{X}{\right\rVert}&=
\frac{1}{\sqrt{n}}{\left\lVert}\tilde{X}{\right\rVert}\left|1-\frac{1}{\sqrt{{\text{Var}}(\tilde{\xi})}}\right|\notag\\
&\leq \frac{1}{\sqrt{n}}{\left\lVert}\tilde{X}{\right\rVert}\sqrt{2}\left|\sqrt{{\text{Var}}(\tilde{\xi})}-1\right|\notag\\
&\leq \frac{1}{\sqrt{n}}{\left\lVert}\tilde{X}{\right\rVert}\sqrt{2}\left|{\text{Var}}(\tilde{\xi})-1\right|\notag\\
&\leq \frac{1}{\sqrt{n}}{\left\lVert}\tilde{X}{\right\rVert}\sqrt{2}\left(\frac{4}{L^2}{\mathbb{E}}|\xi|^{4}\right).
\label{Equ:XsCloseStep}
\end{aligned}$$ By [@Tout Theorem 1.4], we find that almost surely $$\limsup_{n\rightarrow\infty}\frac{1}{\sqrt{n}}{\left\lVert}\tilde{X}{\right\rVert}\leq 2,$$ and thus by $$\underset{n\rightarrow\infty}{\limsup}\frac{1}{\sqrt{n}}{\left\lVert}\tilde{X}-\hat{X}{\right\rVert}\leq\frac{C}{L}$$ almost surely for all $L \geq \max\{1, L_0\}$.
Next consider $\underset{n\rightarrow\infty}{\limsup}\frac{1}{\sqrt{n}}{\left\lVert}X-\tilde{X}{\right\rVert}$. Note that $X - \tilde{X}$ is an iid matrix with atom variable $$\begin{aligned}
{\operatorname{Re}}(X_{(i,j)})&{\ensuremath{\mathbf{1}_{\{|{\operatorname{Re}}(X_{(i,j)})|> L/\sqrt{2}\}}}}-{\mathbb{E}}\left[{\operatorname{Re}}(X_{(i,j)}){\ensuremath{\mathbf{1}_{\{|{\operatorname{Re}}(X_{(i,j)})|> L/\sqrt{2}\}}}}\right]\\
&+\sqrt{-1}\left({\operatorname{Im}}(X_{(i,j)}){\ensuremath{\mathbf{1}_{\{|{\operatorname{Im}}(X_{(i,j)})|> L/\sqrt{2}\}}}}-{\mathbb{E}}\left[{\operatorname{Im}}(X_{(i,j)}){\ensuremath{\mathbf{1}_{\{|{\operatorname{Im}}(X_{(i,j)})|> L/\sqrt{2}\}}}}\right]\right).
\end{aligned}$$ Thus, each entry has mean zero, variance $${\text{Var}}((X-\tilde{X})_{i,j}) \leq \frac{8}{L^{2}}{\mathbb{E}}|\xi|^{4},$$ and finite fourth moment. Thus, again by [@Tout Theorem 1.4], $$\underset{n\rightarrow\infty}{\limsup}\frac{1}{\sqrt{n}}{\left\lVert}X-\tilde{X}{\right\rVert}\leq \frac{C}{L}$$ almost surely, and the proof is complete.
We now consider the iid random matrices $X_{n,1}, \ldots, X_{n,m}$ from Theorem \[thm:isotropic\]. For each $1\leq k\leq m$, define the truncation ${\hat{{X}}_{n,k}}$ as was done above for $\hat{X}$ in . Define ${\hat{\mathcal{Y}}_{n}}$ by $${\hat{\mathcal{Y}}_{n}}:=\left[\begin{matrix}
0 & {\hat{{X}}_{n,1}} & \dots & 0\\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \dots & {\hat{{X}}_{n,m-1}}\\
{\hat{{X}}_{n,m}} & 0 & \dots & 0\\
\end{matrix}\right].
\label{Equ:trBlockMat}$$ Using Theorem \[lem:Truncate\], we have the following corollary.
Let $X_{n,1},\ldots,X_{n,m}$ be independent iid random matrices with atom variables $\xi_{1},\ldots,\xi_{m}$, each of which has mean zero, unit variance, finite fourth moment, and independent real and imaginary parts. Let $\hat{X}_{n,1},\ldots,\hat{X}_{n,m}$ be the truncations of $X_{n,1}, \ldots, X_{n,m}$ as was done in . In addition, let ${\mathcal{Y}_{n}}$ be as defined in and ${\hat{\mathcal{Y}}_{n}}$ be as defined in . Then there exist constants $C, L_0 > 0$ (depending only on the atom variables $\xi_1, \ldots, \xi_m$) such that $$\limsup_{n\rightarrow\infty}\frac{1}{\sqrt{{n}}}{\left\lVert}{\mathcal{Y}_{n}}-{\hat{\mathcal{Y}}_{n}}{\right\rVert}\leq \frac{C}{L}$$ almost surely for all $L > L_0$. \[Cor:YsClose\]
Due to the block structure of ${\mathcal{Y}_{n}}$ and ${\hat{\mathcal{Y}}_{n}}$, it follows that $${\left\lVert}{\mathcal{Y}_{n}}-{\hat{\mathcal{Y}}_{n}}{\right\rVert}\leq \max_{k}{\left\lVert}{{X}_{n,k}}-{\hat{{X}}_{n,k}}{\right\rVert}.$$ Therefore, the claim follows from Lemma \[lem:XcloseXhat\].
Least singular value bounds {#Sec:LeastSingVal}
===========================
In this section, we study the least singular value of $\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}-zI$. We begin by recalling Weyl’s inequality for the singular values (see, for example, [@Bhatia Problem III.6.5]), which states that for $n\times n$ matrices $A$ and $B$, $$\label{eq:weyl}
\max_{1 \leq i \leq n}\left|s_{i}(A)-s_{i}(B)\right|\leq {\left\lVert}A-B{\right\rVert}.$$
We require the following theorem, which is based on [@N Theorem 2].
\[Thm:LeastTruncSingValNonZero\] Fix $L > 0$, and let $\xi_1, \ldots, \xi_m$ be complex-valued random variables, each having mean zero, unit variance, independent real and imaginary parts, and which satisfy $$\sup_{1 \leq k \leq m} |\xi_k| \leq L$$ almost surely. Let $X_{n,1}, \ldots, X_{n,m}$ be independent iid random matrices with atom variables $\xi_1, \ldots, \xi_m$, respectively. Define $\mathcal{Y}_n$ as in , and fix $\delta > 0$. Then there exists a constant $c > 0$ (depending only on $\delta$) such that $$\inf_{|z|\geq 1+\delta}s_{mn}\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}-zI\right)\geq c$$ with overwhelming probability.
A similar statement was proven in [@N Theorem 2], where the same conclusion was shown to hold almost surely rather than with overwhelming probability. However, many of the intermediate steps in [@N] are proven to hold with overwhelming probability. We use these intermediate steps to prove Theorem \[Thm:LeastTruncSingValNonZero\] in Appendix \[sec:singoutlier\].
\[Lem:LeastSingValAwayFromZero\] Let $X_{n,1},\ldots,X_{n,m}$ satisfy the assumptions of Theorem \[thm:isotropic\], and let ${\mathcal{Y}_{n}}$ be as defined in . Fix $\delta > 0$. Then there exists a constant $c>0$ such that almost surely, for $n$ sufficiently large, $$\inf_{|z|\geq 1+\delta}s_{mn}\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}-zI\right)\geq c.$$
Let $L > 0$ be a large constant to be chosen later. Let $\hat{X}_{n,1}, \ldots, \hat{X}_{n,m}$ be defined as in , and let $\hat{\mathcal{Y}}_n$ be defined as in . By Theorem \[Thm:LeastTruncSingValNonZero\] and the Borel–Cantelli lemma, there exists a constant $c'>0$ such that almost surely, for $n$ sufficiently large, $$\inf_{|z|\geq 1+\delta}s_{mn}\left(\frac{1}{\sqrt{n}}{\hat{\mathcal{Y}}_{n}}-zI\right)\geq c'.$$ By Corollary \[Cor:YsClose\] and we may choose $L$ sufficiently large to ensure that almost surely, for $n$ sufficiently large, $$\left|s_{mn}\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}-zI\right)-s_{mn}\left(\frac{1}{\sqrt{n}}{\hat{\mathcal{Y}}_{n}}-zI\right)\right|\leq \frac{c'}{2},$$ uniformly in $z$. We conclude that almost surely, for $n$ sufficiently large, $$\inf_{|z|\geq 1+\delta}s_{mn}\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}-zI\right)\geq \frac{c'}{2},$$ and the proof is complete.
With this result we can prove the following lemma.
\[Cor:LeastSingValOfProductAwayFromZero\] Let $X_{n,1},\ldots,X_{n,m}$ satisfy the assumptions of Theorem \[thm:isotropic\], and fix $\delta > 0$. Then there exists a constant $c>0$ such that almost surely, for $n$ sufficiently large, $$\inf_{|z|\geq 1+\delta}s_{mn}\left(n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}-zI\right)\geq c.$$
Let $\mathcal{Y}_n$ be defined as in . Then Lemma \[Lem:LeastSingValAwayFromZero\] implies that almost surely, for $n$ sufficiently large, $\frac{1}{\sqrt{n}} \mathcal{Y}_n - z I$ is invertible for all $|z| \geq 1 + \delta$. By computing the block inverse of this matrix, we find $$\left(\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}-zI\right)^{-1}\right)^{[1,1]}=z^{m-1}\left(n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}-z^{m}I\right)^{-1}.$$ Thus, for $|z| \geq 1 + \delta$, $$\begin{aligned}
\left\| \left(n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}-z^{m}I\right)^{-1} \right\| &\leq |z|^{m-1} \left\| \left(n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}-z^{m}I\right)^{-1} \right\| \\
&\leq \left\| \left( \frac{1}{\sqrt{n}} \mathcal{Y}_n - z I \right)^{-1} \right \|, \end{aligned}$$ where the last step used the fact that the operator norm of a matrix bounds above the operator norm of any sub-matrix.
Recall that if $M$ is an invertible $N \times N$ matrix, then $s_N(M) = \| M^{-1} \|^{-1}$. Applying this fact to the matrices above, we conclude that $$s_{mn}\left(n^{-m/2}{{X}_{n,1}}\cdots{{X}_{n,m}}-z^{m}I\right) \geq s_{mn}\left( \frac{1}{\sqrt{n}} \mathcal{Y}_n - z I \right),$$ and the claim follows from Lemma \[Lem:LeastSingValAwayFromZero\].
Observe that $z$ is an eigenvalue of $\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}$ if and only if $\det\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}-zI\right)=0$. Also, recall that $\left|\det\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}-zI\right)\right|=\prod_i s_{i}\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}-zI\right)$; this product is zero if and only if the smallest singular value is zero. Since Lemma \[Lem:LeastSingValAwayFromZero\] and Lemma \[Cor:LeastSingValOfProductAwayFromZero\] bound the least singular values of $\frac{1}{\sqrt{n}} \mathcal{Y}_n - z I$ and $n^{-m/2}X_{n,1} \cdots X_{n,m} - zI$ away from zero for $|z| \geq 1 + \delta$, we can conclude that such values of $z$ are almost surely, for $n$ sufficiently large, not eigenvalues of these matrices. \[Rem:ZnotEval\]
Reductions to the proof of Theorem \[thm:isotropic\] {#Sec:Reductions}
====================================================
In this section, we will prove that it is sufficient to reduce the proof of Theorem \[thm:isotropic\] to the case in which the entries of each matrix are truncated and where we restrict $z$ to the band $5\leq |z|\leq 6$.
Let $X_{n,1}, \ldots, X_{n,m}$ be as in Theorem \[thm:isotropic\]. Then there exists a constant $L_0$ such that the following holds for all $L > L_0$. Let $\hat{X}_{n,1}, \ldots, \hat{X}_{n,m}$ be defined as in , let $\hat{\mathcal{Y}}_n$ be given by , and let $\hat{\mathcal{G}}_n(z) := \left( \frac{1}{\sqrt{n}} \hat{\mathcal{Y}}_n - z I \right)^{-1}$.
1. \[item:tr:invertible\] For any fixed $\delta > 0$, almost surely, for $n$ sufficiently large, the eigenvalues of $\frac{1}{\sqrt{n}} {\hat{\mathcal{Y}}_{n}}$ are contained in the disk $\{z \in \mathbb{C} : |z| \leq 1 + \delta \}$. In particular, $\frac{1}{\sqrt{n}} {\hat{\mathcal{Y}}_{n}}-zI$ is almost surely invertible for every $z\in{\mathbb{C}}$ with $|z|>1+\delta$.
2. \[item:tr:invtbnd\]For any fixed $\delta > 0$, there exists a constant $c > 0$ (depending only on $\delta$) such that almost surely, for $n$ sufficiently large $$\sup_{z \in \mathbb{C} : |z| > 1 + \delta} \| \hat{\mathcal{G}}_n(z) \| \leq c.$$
3. \[item:tr:isotropic\] For each $n \geq 1$, let $u_n, v_n \in \mathbb{C}^{mn}$ be deterministic unit vectors. Then $$\sup_{z \in \mathbb{C} :5\leq |z|\leq 6} \left| u_n^\ast \hat{\mathcal{G}}_n(z) v_n + \frac{1}{z} u_n^\ast v_n \right| \longrightarrow 0$$ almost surely as $n \to \infty$.
\[Thm:reductions\]
We now prove Theorem \[thm:isotropic\] assuming Theorem \[Thm:reductions\].
Part \[item:invertible\] of Theorem \[thm:isotropic\] follows from Lemma \[Lem:LeastSingValAwayFromZero\] (see Remark \[Rem:ZnotEval\]). In addition, part \[item:invtbnd\] of Theorem \[thm:isotropic\] follows from an application of Lemma \[Lem:LeastSingValAwayFromZero\] and Proposition \[Prop:LargeAndSmallSingVals\].
We now turn to the proof of part \[item:isotropic\]. Fix $0 < \delta < 1$. Let $\mathcal{Y}_n$ be given by , and let $\mathcal{G}_n$ be given by . Let ${\varepsilon}, {\varepsilon}' > 0$, and observe that there exists a positive constant $M_{1}$ such that for $|z|\geq M_{1}$, $${\left\lVert}\left(-\frac{1}{z}\right)u^{*}v{\right\rVert}\leq \left|\frac{1}{z}\right|{\left\lVert}u^{*}{\right\rVert}{\left\lVert}v{\right\rVert}\leq \frac{\varepsilon}{2}.$$ Also, by [@Tout Theorem 1.4] and Lemma \[Lem:specnorm\], there exists a constant $M_{2} > 0$ such that almost surely, for $n$ sufficiently large, $$\sup_{|z| \geq M_2} {\left\lVert}u^{*}\mathcal{G}_{n}(z)v{\right\rVert}\leq \sup_{|z| \geq M_2} \| \mathcal{G}_n(z) \| \leq \frac{{\varepsilon}}{2}.$$ Let $M:=\max\{M_{1},M_{2}, 6\}$. Then, almost surely, for $n$ sufficiently large, $$\sup_{|z| \geq M} \left|u^{*}\mathcal{G}_{n}(z)v+\frac{1}{z}u^{*}v\right| \leq \varepsilon.
\label{equ:MtoInf}$$
We now work on the region where $1 + \delta < |z| \leq M$. By the resolvent identity , we note that $${\left\lVert}\mathcal{G}_{n}(z)-\hat{\mathcal{G}}_{n}(z){\right\rVert}\leq {\left\lVert}\mathcal{G}_n(z) {\right\rVert}{\left\lVert}\hat{\mathcal{G}}_n(z) {\right\rVert}\frac{1}{\sqrt{n}} {\left\lVert}{\hat{\mathcal{Y}}_{n}} - {\mathcal{Y}_{n}} {\right\rVert}.$$ Thus, by part \[item:invtbnd\] of Theorem \[thm:isotropic\] (proven above), Theorem \[Thm:reductions\], and Corollary \[Cor:YsClose\], there exist constants $C,c > 0$ such that $$\limsup_{n\rightarrow\infty} \sup_{|z| > 1 + \delta} {\left\lVert}\mathcal{G}_{n}(z)-\hat{\mathcal{G}}_{n}(z){\right\rVert}\leq \limsup_{n\rightarrow\infty} c^2\frac{1}{\sqrt{n}}{\left\lVert}{\hat{\mathcal{Y}}_{n}}-{\mathcal{Y}_{n}}{\right\rVert}\leq c^2 \frac{C}{L} \leq \frac{{\varepsilon}'}{2}
\label{equ:ResolventsColse}$$ almost surely for $L$ sufficiently large.
From and Theorem \[Thm:reductions\], almost surely, for $n$ sufficiently large, $$\begin{aligned}
\sup_{5 \leq |z| \leq 6} \left|u^{*}\mathcal{G}_{n}(z)v+\frac{1}{z}u^{*}v\right| &\leq \sup_{5 \leq |z| \leq 6} \left|u^{*}\mathcal{G}_{n}(z)v-u^{*}\hat{\mathcal{G}}_{n}(z)v\right| \\
&\qquad \qquad + \sup_{5 \leq |z| \leq 6} \left|u^{*}\hat{\mathcal{G}}_{n}(z)v+\frac{1}{z}u^{*}v\right| \\
&\leq \varepsilon'.
\end{aligned}$$ Since ${\varepsilon}' > 0$ was arbitrary, this implies that $$\limsup_{n \to \infty} \sup_{5 \leq |z| \leq 6} \left|u^{*}\mathcal{G}_{n}(z)v+\frac{1}{z}u^{*}v\right| = 0$$ almost surely. Since the region $\{z\in{\mathbb{C}}\;:\;1+\delta\leq |z|\leq M\}$ is compact and contains the region $\{z\in{\mathbb{C}}\;:\;5\leq |z|\leq 6\}$, Vitali’s Convergence Theorem[^4] (see, for instance, [@BSbook Lemma 2.14]) implies that we can extend this convergence to the larger region, and we obtain $$\limsup_{n \to \infty} \sup_{1 + \delta \leq |z| \leq M} \left|u^{*}\mathcal{G}_{n}(z)v+\frac{1}{z}u^{*}v\right| = 0$$ almost surely. In particular, this implies that, almost surely, for $n$ sufficiently large, $$\sup_{1 + \delta \leq |z| \leq M} \left|u^{*}\mathcal{G}_{n}(z)v+\frac{1}{z}u^{*}v\right| \leq {\varepsilon}.$$ Combined with , this completes the proof of Theorem \[thm:isotropic\] (since ${\varepsilon}> 0$ was arbitrary).
It remains to prove Theorem \[Thm:reductions\]. We prove parts \[item:tr:invertible\] and \[item:tr:invtbnd\] of Theorem \[Thm:reductions\] now. The proof of part \[item:tr:isotropic\] is lengthy and will be addressed in the forthcoming sections.
Let $\delta>0$, and observe that by Theorem \[Thm:LeastTruncSingValNonZero\] and the Borel–Cantelli lemma, there exists a constant $c>0$ (depending only on $\delta$) such that almost surely, for $n$ sufficiently large, $$\label{eq:lsvtrbnd}
\inf_{|z| > 1+\delta}s_{mn}\left(\frac{1}{\sqrt{n}}{\hat{\mathcal{Y}}_{n}}-zI\right)\geq c.$$ This implies (see Remark \[Rem:ZnotEval\]) that almost surely, for $n$ sufficiently large, the eigenvalues $\frac{1}{\sqrt{n}}{\hat{\mathcal{Y}}_{n}}$ are contained in the disk $\{z \in \mathbb{C} : |z| \leq 1 + \delta \}$, proving \[item:tr:invertible\]. From and Proposition \[Prop:LargeAndSmallSingVals\], we conclude that almost surely, for $n$ sufficiently large, $$\sup_{|z|>1+\delta}{\left\lVert}\hat{\mathcal{G}}_{n}(z){\right\rVert}\leq \frac{1}{c},$$ proving \[item:tr:invtbnd\].
Concentration of bilinear forms involving the resolvent $\mathcal{G}_n$ {#Sec:Concentration}
=======================================================================
Sections \[Sec:Concentration\] and \[sec:combin\] are devoted to the proof of part \[item:tr:isotropic\] of Theorem \[Thm:reductions\]. Let $\hat{X}_{n,1}, \ldots, \hat X_{n,m}$ be the truncated matrices from Theorem \[Thm:reductions\], and let $u_n, v_n \in {\mathbb{C}}^{mn}$ be deterministic unit vectors. For ease of notation, in Sections \[Sec:Concentration\] and \[sec:combin\], we drop the decorations and write $X_{n,1}, \ldots, X_{n,m}$ for $\hat{X}_{n,1}, \ldots, \hat X_{n,m}$. Similarly, we write ${\mathcal{Y}_{n}}$ for ${\hat{\mathcal{Y}}_{n}}$ and ${\mathcal{G}_{n}}$ for ${\hat{\mathcal{G}}_{n}}$. Recall that all constants and asymptotic notation may depend on $m$ without explicitly showing the dependence.
Define the following event: $$\Omega_{n}:=\left\{ \frac{1}{\sqrt{n}} {\left\lVert}{\mathcal{Y}_{n}} {\right\rVert}\leq 4.5\right\}.
\label{Equ:condition}$$
Under the assumptions above, the event $\Omega_n$ holds with overwhelming probability. \[Lem:specnormbound\]
Based on the block structure of ${\mathcal{Y}_{n}}$, it follows that $${\left\lVert}{\mathcal{Y}_{n}} {\right\rVert}\leq \max_{i} {\left\lVert}X_{n,i} {\right\rVert}.$$ Therefore, the claim follows from [@BSbook Theorem 5.9] (alternatively, [@Tout Theorem 1.4]). In fact, the constant $4.5$ can be replaced with any constant strictly larger than $2$; $4.5$ will suffice for what follows.
By Lemma \[Lem:specnormbound\], $\Omega_{n}$ holds with overwhelming probability, i.e., for every $A>0$, $$\label{eq:owpOmega_n}
{\mathbb{P}}\left(\Omega_{n}\right)= 1-O_{A}(n^{-A}).$$ We will return to this fact several times throughout the proof. The remainder of this section is devoted to proving the following lemma.
Let $u_{n}, v_{n} \in \mathbb{C}^{mn}$ be deterministic unit vectors. Then, under the assumptions above, for any $\varepsilon >0$, almost surely $$\sup_{5 \leq |z| \leq 6} \left|u_{n}^{*}\mathcal{G}_{n}(z)v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}-{\mathbb{E}}\left[u_{n}^{*}\mathcal{G}_{n}(z)v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right]\right|<\varepsilon$$ for $n$ sufficiently large. \[Lem:concentration\]
The proof of Lemma \[Lem:concentration\] follows the arguments of [@BP; @OR]. Before we begin the proof, we present some notation. Define ${\mathcal{Y}_{n}}^{(k)}$ to be the matrix ${\mathcal{Y}_{n}}$ with all entries in the $k$th row and the $k$th column filled with zeros. Note that ${\mathcal{Y}_{n}}^{(k)}$ is still an $mn\times mn$ matrix. Define $$\mathcal{G}_{n}^{(k)}:=\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}^{(k)}-zI\right)^{-1},
\label{Equ:resolventmodified}$$ let $r_{k}$ denote the $k$th row of ${\mathcal{Y}_{n}}$, and let $c_{k}$ denote $k$th column of ${\mathcal{Y}_{n}}$. Also define the $\sigma$-algebra $$\mathcal{F}_{k}:=\sigma(r_{1},\ldots,r_{k},c_{1},\ldots,c_{k})$$ generated by the first $k$ rows and the first $k$ columns of ${\mathcal{Y}_{n}}$. Note that $\mathcal{F}_{0}$ is the trivial $\sigma$-algebra and $\mathcal{F}_{mn}$ is the $\sigma$-algebra generated by all rows and columns. Next define $$\label{def:conditionalExpectation}
{\mathbb{E}}_{k}[\;\cdot\;]:={\mathbb{E}}[\;\cdot\;|\;\mathcal{F}_{k}]$$ to be the conditional expectation given the first $k$ rows and columns, and $$\Omega_{n}^{(k)}:=\left\{\frac{1}{\sqrt{n}}\lVert {\mathcal{Y}_{n}}^{(k)}\rVert\leq 4.5\right\}.$$ Observe that $\Omega_{n}\subseteq\Omega_{n}^{(k)}$ and therefore, by Lemma \[Lem:specnormbound\], $\Omega_{n}^{(k)}$ holds with overwhelming probability as well.
By Lemma \[Lem:specnorm\], we have that $$\sup_{5 \leq |z| \leq 6} {\left\lVert}\mathcal{G}_{n}(z) {\right\rVert}\leq 2, \qquad \sup_{5 \leq |z| \leq 6} {\left\lVert}\mathcal{G}_{n}^{(k)}(z) {\right\rVert}\leq 2$$ on the event $\Omega_{n}$, and $\sup_{5 \leq |z| \leq 6} {\left\lVert}\mathcal{G}_{n}^{(k)}(z) {\right\rVert}\leq 2$ on $\Omega_{n}^{(k)}$. \[Equ:ResolventNormBdd\]
We will now collect some preliminary calculations and lemmata that will be required for the proof of Lemma \[Lem:concentration\].
Let $\{x_{k}\}$ be a complex martingale difference sequence with respect to the filtration $\mathcal{F}_{k}$. Then for $p\geq 2$, $${\mathbb{E}}\left|\sum x_{k}\right|^{p}\leq K_{p}\left({\mathbb{E}}\left(\sum {\mathbb{E}}\left[|x_{k}|^{2}\;|\;\mathcal{F}_{k-1}\right]\right)^{p/2}+{\mathbb{E}}\sum |x_{k}|^{p}\right)$$ for a constant $K_{p}>0$ which depends only on $p$. \[Lem:rosenthals\]
Let $A$ be an $n\times n$ Hermitian positive semidefinite matrix, and let $S \subset [n]$. Then $\sum_{i \in S} A_{ii} \leq \operatorname{tr}A$. \[Lem:patrialtrace\]
The claim follows from the fact that, by definition of $A$ being Hermitian positive semidefinite, the diagonal entries of $A$ are real and non-negative.
\[Lem:conjLessThanConstant\] Let $A$ be an $N\times N$ Hermitian positive semidefinite matrix with rank at most one. Suppose that $\xi$ is a complex-valued random variable with mean zero, unit variance, and which satisfies $|\xi| \leq L$ almost surely for some constant $L > 0$. Let $S\subseteq [N]$, and let $w = (w_i)_{i=1}^N$ be a vector with the following properties:
1. $\{w_i : i \in S \}$ is a collection of iid copies of $\xi$,
2. $w_{i}=0$ for $i\not\in S$.
Then for any $p\geq 1$, $${\mathbb{E}}\left|w^{*}Aw\right|^{p} \ll_{L,p}{\left\lVert}A{\right\rVert}^{p}.$$
Let $w_{S}$ denote the $|S|$-vector which contains entries $w_{i}$ for $i\in S$, and let $A_{S\times S}$ denote the $|S|\times|S|$ matrix which has entries $A_{(i,j)}$ for $i,j\in S$. Then we observe $$w^{*}Aw = \sum_{i,j}\bar{w}_{i}A_{(i,j)}w_{j}= w_{S}^{*}A_{S\times S}w_{S}.$$ By Lemma \[Lem:BilinearForms\], we get $${\mathbb{E}}\left|w^{*}Aw\right|^{p} \ll_{p}\left( \operatorname{tr}A_{S\times S}\right)^{p}+{\mathbb{E}}|\xi|^{2p}\operatorname{tr}A^{p}_{S\times S} \leq \left(\operatorname{tr}A_{S \times S} \right)^{p}+L^{2p}\text{tr}A_{S \times S}^{p}.$$ Since the rank of $A_{S \times S}$ is at most one, we find $$\operatorname{tr}A_{S \times S} \leq \| A \|$$ and $$\operatorname{tr}A_{S \times S}^p \leq \| A \|^p,$$ where we used the fact that the operator norm of a matrix bounds the operator norm of any sub-matrix. We conclude that $${\mathbb{E}}\left|w^{*}Aw\right|^{p} \ll_{p}{\left\lVert}A{\right\rVert}^{p}+L^{2p}{\left\lVert}A{\right\rVert}^p \ll_{L,p}{\left\lVert}A{\right\rVert}^{p},$$ as desired.
\[lem:BilinearFormWithDifferentVectorsBig\] Let $A$ be a deterministic complex $N\times N$ matrix. Suppose that $\xi$ is a complex-valued random variable with mean zero, unit variance, and which satisfies $|\xi| \leq L$ almost surely for some constant $L > 0$. Let $S,R\subseteq [N]$, and let $w = (w_i)_{i=1}^N$ and $t = (t_i)_{i=1}^N$ be independent vectors with the following properties:
1. $\{w_i : i \in S \}$ and $\{t_j : j \in R \}$ are collections of iid copies of $\xi$,
2. $w_{i}=0$ for $i\not\in S$, and $t_{j}=0$ for $j\not\in R$.
Then for any $p\geq 1$, $${\mathbb{E}}\left|w^{*}At\right|^{p}\ll_{L,p}(\operatorname{tr}(A^{*}A))^{p/2}.$$
Let $w_{S}$ denote the $|S|$-vector which contains entries $w_{i}$ for $i\in S$, and let $t_{R}$ denote the $|R|$-vector which contains entries $t_{j}$ for $j\in R$. For an $N \times N$ matrix $B$, we let $B_{S \times S}$ denote the $|S| \times |S|$ matrix with entries $B_{(i,j)}$ for $i,j \in S$. Similarly, we let $B_{R \times R}$ denote the $|R| \times |R|$ matrix with entries $B_{(i,j)}$ for $i,j \in R$.
We first note that, by the Cauchy–Schwarz inequality, it suffices to assume $p$ is even. In this case, since $w$ is independent of $t$, Lemma \[Lem:BilinearForms\] implies that $$\begin{aligned}
{\mathbb{E}}|w^{*}At|^{p} &= {\mathbb{E}}|w^{*}Att^{*}A^{*}w|^{p/2}\\
&= {\mathbb{E}}\left|w^{*}_{S}(Att^{*}A^{*})_{S\times S}w_{S}\right|^{p/2}\\
&\ll_{p}{\mathbb{E}}\left[\left(\operatorname{tr}(Att^{*}A^{*})_{S\times S}\right)^{p/2}+L^{p}\operatorname{tr}(Att^{*}A^{*})_{S\times S}^{p/2}\right]. \end{aligned}$$ Recall that for any matrix $B$, $\operatorname{tr}(B^{*}B)^{p/2}\leq (\operatorname{tr}(B^{*}B))^{p/2}$. By this fact and by Lemma \[Lem:patrialtrace\], we observe that $${\mathbb{E}}\left[\left(\operatorname{tr}(Att^{*}A^{*})_{S\times S}\right)^{p/2}+L^{p}\operatorname{tr}(Att^{*}A^{*})_{S\times S}^{p/2}\right]\ll_{L,p}{\mathbb{E}}\left[(\operatorname{tr}(Att^{*}A^{*}))^{p/2}\right].$$ By a cyclic permutation of the trace, we have $${\mathbb{E}}\left[(\operatorname{tr}(Att^{*}A^{*}))^{p/2}\right] = {\mathbb{E}}\left[(t^{*}A^{*}At)^{p/2}\right]\leq{\mathbb{E}}\left|t^{*}A^{*}At\right|^{p/2}.$$ By Lemma \[Lem:BilinearForms\], Lemma \[Lem:patrialtrace\], and a similar argument as above, we obtain $$\begin{aligned}
{\mathbb{E}}\left|t^{*}A^{*}At\right|^{p/2}&={\mathbb{E}}\left|t_{R}^{*}(A^{*}A)_{R\times R}t_{R}\right|^{p/2}\\
&\ll_{p}(\operatorname{tr}(A^{*}A)_{R\times R})^{p/2}+L^{p}\operatorname{tr}(A^{*}A)_{R\times R}^{p/2}\\
&\ll_{L,p}(\operatorname{tr}(A^{*}A))^{p/2}, \end{aligned}$$ completing the proof.
Let $r_{k}$ be the $k$th row of ${\mathcal{Y}_{n}}$, $c_k$ be the $k$th column of ${\mathcal{Y}_{n}}$, $\mathcal{G}_{n}^{(k)}(z)$ be the resolvent of ${\mathcal{Y}_{n}}^{(k)}$, and $u_{n}\in{\mathbb{C}}^{mn}$ be a deterministic unit vector. Then, under the assumptions above, $$\label{eq:concsimp1}
\sup_{5 \leq |z| \leq 6} {\mathbb{E}}_{k-1}\left|\frac{1}{n}r_{k}\mathcal{G}_{n}^{(k)}(z)u_{n}u_{n}^{*}\mathcal{G}_{n}^{(k)*}(z){\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}r_{k}^{*}\right|^{p}\ll_{L,p}n^{-p}$$ and $$\label{eq:concsimp2}
\sup_{5 \leq |z| \leq 6} {\mathbb{E}}_{k-1}\left|\frac{1}{n}c_{k}^{*}\mathcal{G}_{n}^{(k)*}(z)u_{n}u_{n}^{*}\mathcal{G}_{n}^{(k)*}(z){\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}c_{k}\right|^{p}\ll_{L,p}n^{-p}.$$ almost surely. \[Lem:concentrationSimplification\]
We will only prove the bound in as the proof of is identical. For each fixed $z$ in the band $5 \leq |z| \leq 6$, we will show that $${\mathbb{E}}_{k-1} \left|\frac{1}{n}r_{k}\mathcal{G}_{n}^{(k)}(z)u_{n}u_{n}^{*}\mathcal{G}_{n}^{(k)*}(z){\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}r_{k}^{*}\right|^{p}\ll_{L,p}n^{-p}$$ surely, where the implicit constant does not depend on $z$. The claim then follows by taking the supremum over all $z$ in the band $5 \leq |z| \leq 6$.
To this end, fix $z$ with $5\leq |z|\leq 6$. Throughout the proof, we drop the dependence on $z$ in the resolvent as it is clear from context. Note that $${\mathbb{E}}_{k-1}\left|\frac{1}{n}r_{k}\mathcal{G}_{n}^{(k)}u_{n}u_{n}^{*}\mathcal{G}_{n}^{(k)*}{\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}r_{k}^{*}\right|^{p}=\frac{1}{n^{p}}{\mathbb{E}}_{k-1}\left|r_{k}\left(\mathcal{G}_{n}^{(k)}u_{n}u_{n}^{*}\mathcal{G}_{n}^{(k)*}{\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}\right)r_{k}^{*}\right|^{p},$$ and $\mathcal{G}_{n}^{(k)}u_{n}u_{n}^{*}\mathcal{G}_{n}^{(k)*}{\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}$ is independent of $r_{k}$. In addition, observe that $\mathcal{G}_{n}^{(k)}u_{n}u_{n}^{*}\mathcal{G}_{n}^{(k)*}{\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}$ is Hermitian positive semidefinite matrix with rank at most one. Applying Lemma \[Lem:conjLessThanConstant\] and Remark \[Equ:ResolventNormBdd\], we obtain $${\mathbb{E}}_{k-1}\left|\frac{1}{n}r_{k}\mathcal{G}_{n}^{(k)}u_{n}u_{n}^{*}\mathcal{G}_{n}^{(k)*}{\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}r_{k}^{*}\right|^{p}\ll_{L,p} \frac{1}{n^{p}}{\mathbb{E}}_{k-1}{\left\lVert}\mathcal{G}_{n}^{(k)}u_{n}u_{n}^{*}\mathcal{G}_{n}^{(k)*}{\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}{\right\rVert}^{p}\ll_{L,p}n^{-p}$$ surely, and the proof is complete.
\[lemma:indicator\] Let $\zeta_1, \ldots, \zeta_{mn}$ be complex-valued random variables (not necessarily independent) which depend on $\mathcal{Y}_{n}$. Assume $$\sup_{k} |\zeta_{k}|{\ensuremath{\mathbf{1}_{{\Omega_n^{(k)}}}}} = O(1)$$ almost surely. Then, under the assumptions above, for any $p \geq 1$, $${\mathbb{E}}\left| \sum_{k=1}^{mn} \left( \zeta_k {\ensuremath{\mathbf{1}_{{\Omega_n^{(k)}}}}} - \zeta_k {\ensuremath{\mathbf{1}_{{\Omega_n \cap \Omega^{(k)}_n}}}}\right) \right|^p = O_p(n^{-p}).$$
We will exploit the fact that $\Omega_n \subseteq \Omega_n^{(k)}$ for any $1 \leq k \leq mn$. Indeed, we have $$\begin{aligned}
{\mathbb{E}}\left| \sum_{k=1}^{mn} \left( \zeta_k {\ensuremath{\mathbf{1}_{{\Omega_n^{(k)}}}}} - \zeta_k {\ensuremath{\mathbf{1}_{{\Omega_n \cap \Omega^{(k)}_n}}}}\right) \right|^p & \leq {\mathbb{E}}\left| \sum_{k=1}^{mn} \zeta_k {\ensuremath{\mathbf{1}_{{\Omega_n^{(k)} \cap \Omega_n^c}}}} \right|^p \\
& \ll_p {\mathbb{E}}\left( \sum_{k=1}^{mn} {\ensuremath{\mathbf{1}_{{\Omega_n^{(k)} \cap \Omega_n^c}}}} \right)^p \\
&\ll_p n^p {\mathbb{P}}( \Omega_n^c),\end{aligned}$$ and the claim follows from .
We will made use of the Sherman–Morrison rank one perturbation formula (see [@HJ Section 0.7.4]). Suppose $A$ is an invertible square matrix, and let $u$, $v$ be vectors. If $1+v^{*}A^{-1}u\neq 0$, then $$(A+uv^{*})^{-1}=A^{-1}-\frac{A^{-1}uv^{*}A^{-1}}{1+v^{*}A^{-1}u}
\label{equ:ShermanMorrison1}$$ and $$(A+uv^{*})^{-1}u=\frac{A^{-1}u}{1+v^{*}A^{-1}u}.
\label{equ:ShermanMorrison2}$$
Now, we proceed to prove the main result of this section, Lemma \[Lem:concentration\].
Define $${\mathcal{Y}_{n}}^{(k1)}:= {\mathcal{Y}_{n}}^{(k)}+e_{k}r_{k},\quad{\mathcal{Y}_{n}}^{(k2)}:= {\mathcal{Y}_{n}}^{(k)}+c_{k}e_{k}^{*}
\label{def:Y(kj)}$$ where $e_{1},\ldots,e_{mn}$ are the standard basis elements of ${\mathbb{C}}^{mn}$. Also define $$\mathcal{G}_{n}^{(kj)}:=\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}^{(kj)}-zI\right)^{-1},\quad j=1,2,$$ and set $$\begin{aligned}
& \alpha_{n}^{(k)} := \frac{1}{1+z^{-1}n^{-1}r_{k}\mathcal{G}_{n}^{(k)}c_{k}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}},\\
& \zeta_{n}^{(k)} :=n^{-1}r_{k}\mathcal{G}_{n}^{(k)}c_{k},\\
&\eta_{n}^{(k)} := n^{-1}r_{k}\mathcal{G}_{n}^{(k)}v_{n}u_{n}^{*}\mathcal{G}_{n}^{(k)}c_{k}. \end{aligned}$$ Using these definitions, we make the following observations.
1. \[item:F1\] Since the only nonzero element in the $k$-th row and $k$-th column of ${\mathcal{Y}_{n}}^{(k)}-zI$ is on the diagonal, $$e_{k}^{*}\mathcal{G}_{n}^{(k)}e_{k}=-z^{-1},\quad e_{k}^{*}\mathcal{G}_{n}^{(k)}v_{n}= -z^{-1}v_{n,k},\quad u_{n}^{*}\mathcal{G}_{n}^{(k)}e_{k}=-z^{-1}\bar{u}_{n,k}$$ where $u_{n}=(u_{n,k})_{k=1}^{mn}$ and $v_{n}=(v_{n,k})_{k=1}^{mn}$.
2. \[item:F2\] Since the $k$-th elements of $c_{k}$ and $r_{k}$ are zero, $$e_{k}^{*}\mathcal{G}_{n}^{(k)}c_{k}=0,\quad r_{k}\mathcal{G}_{n}^{(k)}e_{k}=0.$$
3. \[item:F3\] By , \[item:F1\], and \[item:F2\], $$\begin{aligned}
e_{k}^{*}\mathcal{G}_{n}^{(k1)}n^{-1/2}c_{k} & = e_{k}^{*}\mathcal{G}_{n}^{(k)}n^{-1/2}c_{k}-\frac{e_{k}^{*}\mathcal{G}_{n}^{(k)}e_{k}n^{-1/2}r_{k}\mathcal{G}_{n}^{(k)}n^{-1/2}c_{k}}{1+n^{-1/2}r_{k}\mathcal{G}_{n}^{(k)}e_{k}}\\
&=z^{-1}n^{-1}r_{k}\mathcal{G}_{n}^{(k)}c_{k},
\end{aligned}$$ so that $$\frac{1}{1+e_{k}^{*}\mathcal{G}_{n}^{(k1)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}n^{-1/2}c_{k}}=\alpha_{n}^{(k)}.$$
4. \[item:F4\] By the same argument as \[item:F3\], $$n^{-1/2}r_{k}\mathcal{G}_{n}^{(k2)}e_{k}=z^{-1}n^{-1}r_{k}\mathcal{G}_{n}^{(k)}c_{k},$$ so that $$\frac{1}{1+n^{-1/2}r_{k}\mathcal{G}_{n}^{(k2)}e_{k}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}}=\alpha_{n}^{(k)}.$$
5. \[item:F5\] By Schur’s compliment, $$(\mathcal{G}_{n})_{(k,k)} = -\frac{1}{z+n^{-1}r_{k}\mathcal{G}_{n}^{(k)}c_{k}}$$ provided the necessary inverses exist (which is the case on the event $\Omega_n$). Thus, on $\Omega_{n}=\Omega_{n}\cap\Omega_{n}^{(k)}$ and uniformly for $5\leq |z|\leq 6$, by Remark \[Equ:ResolventNormBdd\], $$\begin{aligned}
\left|\alpha_{n}^{(k)}\right| &=\left|\frac{z}{z+n^{-1}r_{k}\mathcal{G}_{n}^{(k)}c_{k}}\right|\\
&=|z|\left|(\mathcal{G}_{n})_{(k,k)}\right|\\
&\leq 12.
\end{aligned}$$ On $\Omega_{n}^{c}$, $\alpha_{n}^{(k)}=1$, so we have that, almost surely, $$\left|\alpha_{n}^{(k)}\right|\leq 12.$$
6. \[item:F6\] By and \[item:F3\], $$\begin{aligned}
u_{n}^{*}\mathcal{G}_{n}n^{-1/2}c_{k}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}} & =\frac{u_{n}^{*}\mathcal{G}_{n}^{(k1)}n^{-1/2}c_{k}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}}{1+e_{k}^{*}\mathcal{G}_{n}^{(k1)}n^{-1/2}c_{k}}\\
& =\frac{u_{n}^{*}\mathcal{G}_{n}^{(k1)}n^{-1/2}c_{k}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}}{1+e_{k}^{*}\mathcal{G}_{n}^{(k1)}n^{-1/2}c_{k}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}}\\
& =u_{n}^{*}\mathcal{G}_{n}^{(k1)}n^{-1/2}c_{k}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\alpha_{n}^{(k)}.
\end{aligned}$$
7. \[item:F7\] By and \[item:F4\], a similar argument as above gives $$u_{n}^{*}\mathcal{G}_{n}e_{k}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}} = u_{n}^{*}\mathcal{G}_{n}^{(k2)}e_{k}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\alpha_{n}^{(k)}.$$
8. \[item:F8\] By and \[item:F2\], $$\begin{aligned}
\mathcal{G}_{n}^{(k1)} &=\mathcal{G}_{n}^{(k)}-\frac{\mathcal{G}_{n}^{(k)}e_{k}n^{-1/2}r_{k}\mathcal{G}_{n}^{(k)}}{1+n^{-1/2}r_{k}\mathcal{G}_{n}^{(k)}e_{k}}\\
&=\mathcal{G}_{n}^{(k)}-\mathcal{G}_{n}^{(k)}e_{k}n^{-1/2}r_{k}\mathcal{G}_{n}^{(k)}.
\end{aligned}$$
9. \[item:F9\] By and \[item:F2\], and by the same calculation as in \[item:F8\], $$\mathcal{G}_{n}^{(k2)}=\mathcal{G}_{n}^{(k)}-\mathcal{G}_{n}^{(k)}n^{-1/2}c_{k}e_{k}^{*}\mathcal{G}_{n}^{(k)}.$$
10. \[item:F10\] By definition of $\alpha_{n}^{(k)}$, $$z^{-1}({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1})[n^{-1}r_{k}\mathcal{G}_{n}^{(k)}c_{k}\alpha_{n}^{(k)}]=-({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1})[\alpha_{n}^{(k)}].$$
11. \[item:F11\] By definition of $\alpha_{n}^{(k)}$ and $\zeta_{n}^{(k)}$, $$\begin{aligned}
\alpha_{n}^{(k)}-1&=\frac{-z^{-1}n^{-1}r_{k}\mathcal{G}_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}c_{k}}{1+z^{-1}n^{-1}r_{k}\mathcal{G}_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}c_{k}}\\
&=-z^{-1}\zeta_{n}^{(k)}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}.
\end{aligned}$$
12. \[item:F12\] The entries of $c_{k}$ and $r_{k}$ have mean zero, unit variance, and are bounded by $4L$ almost surely. In addition, $(r_{k}^{T},c_{k})$ and $\mathcal{G}_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}$ are independent. By Remark \[Equ:ResolventNormBdd\], ${\left\lVert}\mathcal{G}_{n}^{(k)*}u_{n}v_{n}^{*}\mathcal{G}_{n}^{(k)*}\mathcal{G}_{n}^{(k)}v_{n}u_{n}^{*}\mathcal{G}_{n}^{(k)}{\right\rVert}\leq 16$ on $\Omega_{n}^{(k)}$. Thus, by Lemma \[lem:BilinearFormWithDifferentVectorsBig\], for any $p\geq2$, $$\begin{aligned}
\sup_{1\leq k\leq n}{\mathbb{E}}_{k-1}[|\eta_{n}^{(k)}|^{p}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}] &\leq \sup_{1\leq k\leq n}{\mathbb{E}}_{k-1}[|\eta_{n}^{(k)}|^{p}{\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}]\\
&\ll_{L,p}\sup_{1\leq k\leq n} n^{-p}{\mathbb{E}}_{k-1}\left[\left(\operatorname{tr}(\mathcal{G}_{n}^{(k)*}u_{n}v_{n}^{*}\mathcal{G}_{n}^{(k)*}\mathcal{G}_{n}^{(k)}v_{n}u_{n}^{*}\mathcal{G}_{n}^{(k)})\right)^{p/2}{\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}\right]\\
&\ll_{L,p} n^{-p}
\end{aligned}$$ since $$\mathcal{G}_{n}^{(k)*}u_{n}v_{n}^{*}\mathcal{G}_{n}^{(k)*}\mathcal{G}_{n}^{(k)}v_{n}u_{n}^{*}\mathcal{G}_{n}^{(k)}$$ is at most rank one. Similarly, Remark \[Equ:ResolventNormBdd\] give the almost sure bound ${\left\lVert}\mathcal{G}_{n}^{(k)*}\mathcal{G}_{n}^{(k)}{\right\rVert}\leq 4$ on $\Omega_{n}^{(k)}$, which gives $$\begin{aligned}
\sup_{1\leq k\leq n}{\mathbb{E}}_{k-1}[|\zeta_{n}^{(k)}|^{p}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}] &\leq \sup_{1\leq k\leq n}{\mathbb{E}}_{k-1}[|\zeta_{n}^{(k)}|^{p}{\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}]\\
&\ll_{L,p}\sup_{1\leq k\leq n} n^{-p}{\mathbb{E}}_{k-1}\left[\left(\operatorname{tr}(\mathcal{G}_{n}^{(k)*}\mathcal{G}_{n}^{(k)})\right)^{p/2}{\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}\right]\\
&\ll_{L,p} n^{-p/2}.
\end{aligned}$$
With the above observations in hand, we now complete the proof. By the resolvent identity and Remark \[Equ:ResolventNormBdd\], it follows that the function $$z \mapsto \left|u_{n}^{*}\mathcal{G}_{n}(z)v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}-{\mathbb{E}}\left[u_{n}^{*}\mathcal{G}_{n}(z)v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right]\right|$$ is Lipschitz continuous in the region $\{z \in {\mathbb{C}}: 5 \leq |z| \leq 6 \}$. Thus, by a standard net argument, it suffices to prove that almost surely, for $n$ sufficiently large, $$\left|u_{n}^{*}\mathcal{G}_{n}(z)v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}-{\mathbb{E}}\left[u_{n}^{*}\mathcal{G}_{n}(z)v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right]\right|<\varepsilon$$ for each fixed $z \in \mathbb{C}$ with $5 \leq |z| \leq 6$. To this end, fix such a value of $z$. Throughout the proof, we drop the dependence on $z$ in the resolvent as it is clear from context. Note that by Markov’s inequality and the Borel–Cantelli lemma, it is sufficient to prove that $${\mathbb{E}}\left|u_{n}^{*}\mathcal{G}_{n}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}-{\mathbb{E}}\left[u_{n}^{*}\mathcal{G}_{n}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right]\right|^{p}=O_{L,p}(n^{-p/2})$$ for all $p> 2$. We now rewrite the above expression as a martingale difference sequence: $$u_{n}^{*}\mathcal{G}_{n}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}-{\mathbb{E}}[u_{n}^{*}\mathcal{G}_{n}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}] =\sum_{k=1}^{mn}\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)[u_{n}^{*}\mathcal{G}_{n}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}].$$ Since $u_{n}^{*}\mathcal{G}_{n}^{(k)}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}$ is independent of $r_k$ and $c_{k}$, one can see that $$({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1})[u_{n}^{*}\mathcal{G}_{n}^{(k)}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}]=0,$$ and so $$u_{n}^{*}\mathcal{G}_{n}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}-{\mathbb{E}}[u_{n}^{*}\mathcal{G}_{n}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}] =\sum_{k=1}^{mn}\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[u_{n}^{*}\mathcal{G}_{n}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}-u_{n}^{*}\mathcal{G}_{n}^{(k)}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}\right].$$ In view of Lemma \[lemma:indicator\] and since $\Omega_{n}\cap\Omega_{n}^{(k)}=\Omega_{n}$, it suffices to prove that $${\mathbb{E}}\left| \sum_{k=1}^{mn}\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[u_{n}^{*}\mathcal{G}_{n}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n} }}}}-u_{n}^{*}\mathcal{G}_{n}^{(k)}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_n}}}}\right] \right|^p = O_{L,p}(n^{-p/2})$$ for all $p > 2$. Define $$W_{k}:=\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[u_{n}^{*}\mathcal{G}_{n}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}-u_{n}^{*}\mathcal{G}_{n}^{(k)}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right]
\label{def:W_k}$$ and observe that $\{W_{k}\}_{k=1}^{mn}$ is a martingale difference sequence with respect to $\{ \mathcal{F}_{k} \}$.
From the resolvent identity , we observe that $$\begin{aligned}
\sum_{k=1}^{mn}W_{k}&=\sum_{k=1}^{mn}\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[u_{n}^{*}\mathcal{G}_{n}\frac{1}{\sqrt{n}} ({\mathcal{Y}_{n}}^{(k)}-{\mathcal{Y}_{n}})\mathcal{G}_{n}^{(k)}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right]\\
&=-\sum_{k=1}^{mn}\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[u_{n}^{*}\mathcal{G}_{n}\left(\frac{1}{\sqrt{n}}e_{k}r_{k}+\frac{1}{\sqrt{n}}c_{k}e_{k}^{*}\right)\mathcal{G}_{n}^{(k)}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right]\\
&:=-\sum_{k=1}^{mn}(W_{k1}+W_{k2})
\end{aligned}$$ where we define $$W_{k1}:=\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[u_{n}^{*}\mathcal{G}_{n}n^{-1/2}e_{k}r_{k}\mathcal{G}_{n}^{(k)}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_n }}}}\right],
\label{def:W_k1}$$ and $$W_{k2}:=\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[u_{n}^{*}\mathcal{G}_{n}n^{-1/2}c_{k}e_{k}^{*}\mathcal{G}_{n}^{(k)}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_n}}}}\right].
\label{def:W_k2}$$
By \[item:F1\], \[item:F7\], and \[item:F9\], we can further decompose $$\begin{aligned}
\sum_{k=1}^{mn} W_{k1} &=\sum_{k=1}^{mn}\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[u_{n}^{*}\mathcal{G}_{n}n^{-1/2}e_{k}r_{k}\mathcal{G}_{n}^{(k)}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_n }}}}\right]\\
&=\sum_{k=1}^{mn}\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[u_{n}^{*}\mathcal{G}_{n}^{(k2)}n^{-1/2}e_{k}r_{k}\mathcal{G}_{n}^{(k)}v_{n}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_n }}}}\right]\\
&=-\sum_{k=1}^{mn}\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[z^{-1}(\bar{u}_{n,k}-u_{n}^{*}\mathcal{G}_{n}^{(k)}n^{-1/2}c_{k})n^{-1/2}r_{k}\mathcal{G}_{n}^{(k)}v_{n}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_n }}}}\right]\\
&=-\sum_{k=1}^{mn}(W_{k11}+W_{k12})
\end{aligned}$$ where $$W_{k11}:= \left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[z^{-1}\bar{u}_{n,k}n^{-1/2}r_{k}\mathcal{G}_{n}^{(k)}v_{n}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_n }}}}\right]
\label{def:W_k11}$$ and $$W_{k12}:= -\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[z^{-1}u_{n}^{*}\mathcal{G}_{n}^{(k)}n^{-1}c_{k}r_{k}\mathcal{G}_{n}^{(k)}v_{n}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_n }}}}\right].
\label{def:W_k112}$$ Similarly, by \[item:F1\],\[item:F6\], \[item:F8\], and \[item:F10\], $$\begin{aligned}
\sum_{k=1}^{mn}W_{k2}&=\sum_{k=1}^{mn}\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[u_{n}^{*}\mathcal{G}_{n}n^{-1/2}c_{k}e_{k}^{*}\mathcal{G}_{n}^{(k)}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_n}}}}\right]\\
&=\sum_{k=1}^{mn}\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[u_{n}^{*}\frac{\mathcal{G}_{n}^{(k1)}n^{-1/2}c_{k}}{1+r_{k}^{*}\mathcal{G}_{n}^{(k1)}c_{k}}e_{k}^{*}\mathcal{G}_{n}^{(k)}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_n}}}}\right]\\
&=\sum_{k=1}^{mn}\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[u_{n}^{*}\frac{\mathcal{G}_{n}^{(k1)}n^{-1/2}c_{k}}{1+r_{k}^{*}\mathcal{G}_{n}^{(k1)}c_{k}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}}e_{k}^{*}\mathcal{G}_{n}^{(k)}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_n}}}}\right]\\
&=\sum_{k=1}^{mn}\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[u_{n}^{*}\mathcal{G}_{n}^{(k1)}n^{-1/2}c_{k}e_{k}^{*}\mathcal{G}_{n}^{(k)}v_{n}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_n}}}}\right]\\
&=-\sum_{k=1}^{mn}\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[z^{-1}v_{n,k}u_{n}^{*}\mathcal{G}_{n}^{(k1)}n^{-1/2}c_{k}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_n}}}}\right]\\
&=-\sum_{k=1}^{mn}\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[z^{-1}v_{n,k}u_{n}^{*}(\mathcal{G}_{n}^{(k)}-\mathcal{G}_{n}^{(k)}e_{k}n^{-1/2}r_{k}\mathcal{G}_{n}^{(k)})n^{-1/2}c_{k}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_n}}}}\right]\\
&=-\sum_{k=1}^{mn}\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[z^{-1}v_{n,k}(u_{n}^{*}\mathcal{G}_{n}^{(k)}n^{-1/2}c_{k}\alpha_{n}^{(k)}-u_{n}^{*}\mathcal{G}_{n}^{(k)}e_{k}n^{-1}r_{k}\mathcal{G}_{n}^{(k)}c_{k}\alpha_{n}^{(k)}){\ensuremath{\mathbf{1}_{{\Omega_n}}}}\right]\\
&=-\sum_{k=1}^{mn}\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[z^{-1}v_{n,k}(u_{n}^{*}\mathcal{G}_{n}^{(k)}n^{-1/2}c_{k}\alpha_{n}^{(k)}+\bar{u}_{n,k}z^{-1}n^{-1}r_{k}\mathcal{G}_{n}^{(k)}c_{k}\alpha_{n}^{(k)}){\ensuremath{\mathbf{1}_{{\Omega_n}}}}\right]\\
&=-\sum_{k=1}^{mn}\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[z^{-1}v_{n,k}u_{n}^{*}\mathcal{G}_{n}^{(k)}n^{-1/2}c_{k}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_n}}}}-z^{-1}\bar{u}_{n,k}v_{n,k}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_n}}}}\right]\\
&=-\sum_{k=1}^{mn}(W_{k21}+W_{k22})
\end{aligned}$$ where $$W_{k21}:=\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[z^{-1}v_{n,k}u_{n}^{*}\mathcal{G}_{n}^{(k)}n^{-1/2}c_{k}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_n}}}}\right]
\label{def:W_k21}$$ and $$W_{k22}:=-\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[z^{-1}\bar{u}_{n,k}v_{n,k}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_n}}}}\right].
\label{def:W_k22}$$ Thus, in order to complete the proof, it suffices to show that for all $p> 2$, $$\label{eq:concshow}
{\mathbb{E}}\left|\sum_{k=1}^{mn}W_{k11}\right|^{p}+{\mathbb{E}}\left|\sum_{k=1}^{mn}W_{k12}\right|^{p}+{\mathbb{E}}\left|\sum_{k=1}^{mn}W_{k21}\right|^{p}+{\mathbb{E}}\left|\sum_{k=1}^{mn}W_{k22}\right|^{p}=O_{L,p}\left(n^{-p/2}\right).$$ We bound each term individually. To begin, observe that by Lemma \[Lem:rosenthals\], Lemma \[Lem:concentrationSimplification\], and \[item:F5\], for any $p>2$, $$\begin{aligned}
{\mathbb{E}}\left|\sum_{k=1}^{mn}W_{k11}\right|^{p} &={\mathbb{E}}\left|\sum_{k=1}^{mn}\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[z^{-1}\bar{u}_{n,k}n^{-1/2}r_{k}\mathcal{G}_{n}^{(k)}v_{n}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_n }}}}\right]\right|^{p}\\
&\ll_{p} {\mathbb{E}}\left[\sum_{k=1}^{mn}{\mathbb{E}}_{k-1}\left|({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1})[z^{-1}\bar{u}_{n,k}n^{-1/2}r_{k}\mathcal{G}_{n}^{(k)}v_{n}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}]\right|^{2}\right]^{p/2}\\
&\quad \quad \quad \quad +{\mathbb{E}}\left[\sum_{k=1}^{mn}\left|({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1})[z^{-1}\bar{u}_{n,k}n^{-1/2}r_{k}\mathcal{G}_{n}^{(k)}v_{n}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}]\right|^{p}\right]\\
&\ll_{p} {\mathbb{E}}\left[\sum_{k=1}^{mn}|\bar{u}_{n,k}|^{2}{\mathbb{E}}_{k-1}\left|n^{-1/2}r_{k}\mathcal{G}_{n}^{(k)}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right|^{2}\right]^{p/2}\\
&\quad \quad \quad \quad +\sum_{k=1}^{mn}|\bar{u}_{n,k}|^{p}{\mathbb{E}}\left|n^{-1/2}r_{k}\mathcal{G}_{n}^{(k)}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right|^{p}\\
&\ll_{p} {\mathbb{E}}\left[\sum_{k=1}^{mn}|\bar{u}_{n,k}|^{2}{\mathbb{E}}_{k-1}\left|n^{-1/2}r_{k}\mathcal{G}_{n}^{(k)}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}\right|^{2}\right]^{p/2}\\
&\quad \quad \quad \quad +\sum_{k=1}^{mn}|\bar{u}_{n,k}|^{p}{\mathbb{E}}\left|n^{-1/2}r_{k}\mathcal{G}_{n}^{(k)}v_{n}{\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}\right|^{p}\\
&\ll_{L,p}\left({\mathbb{E}}\left[\sum_{k=1}^{mn}|\bar{u}_{n,k}|^{2}\cdot n^{-1}\right]^{p/2}+\sum_{k=1}^{mn}|\bar{u}_{n,k}|^{p}\cdot n^{-p/2}\right)\\
&\ll_{L,p} n^{-p/2},
\end{aligned}$$ where we also used Jensen’s inequality and the fact that $|z| \geq 5$. Similarly, by Lemma \[Lem:rosenthals\], Lemma \[Lem:concentrationSimplification\], and \[item:F5\], for any $p>2$, $$\begin{aligned}
{\mathbb{E}}\left|\sum_{k=1}^{mn}W_{k21}\right|^{p} &={\mathbb{E}}\left|\sum_{k=1}^{mn}\left({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1}\right)\left[z^{-1}v_{n,k}u_{n}^{*}\mathcal{G}_{n}^{(k)}n^{-1/2}c_{k}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_n }}}}\right]\right|^{p}\\
&\ll_{p} {\mathbb{E}}\left[\sum_{k=1}^{mn}{\mathbb{E}}_{k-1}\left|({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1})[z^{-1}v_{n,k}u_{n}^{*}\mathcal{G}_{n}^{(k)}n^{-1/2}c_{k}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_n }}}}]\right|^{2}\right]^{p/2}\\
&\quad \quad \quad \quad +{\mathbb{E}}\left[\sum_{k=1}^{mn}\left|({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1})[z^{-1}v_{n,k}u_{n}^{*}\mathcal{G}_{n}^{(k)}n^{-1/2}c_{k}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_n }}}}]\right|^{p}\right]\\
&\ll_{p} {\mathbb{E}}\left[\sum_{k=1}^{mn}|v_{n,k}|^{2}{\mathbb{E}}_{k-1}\left|u_{n}^{*}\mathcal{G}_{n}^{(k)}n^{-1/2}c_{k}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right|^{2}\right]^{p/2}\\
&\quad \quad \quad \quad +\sum_{k=1}^{mn}|v_{n,k}|^{p}{\mathbb{E}}\left|u_{n}^{*}\mathcal{G}_{n}^{(k)}n^{-1/2}c_{k}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right|^{p}\\
&\ll_{p} {\mathbb{E}}\left[\sum_{k=1}^{mn}|v_{n,k}|^{2}{\mathbb{E}}_{k-1}\left|u_{n}^{*}\mathcal{G}_{n}^{(k)}n^{-1/2}c_{k}{\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}\right|^{2}\right]^{p/2}\\
&\quad \quad \quad \quad +\sum_{k=1}^{mn}|v_{n,k}|^{p}{\mathbb{E}}\left|u_{n}^{*}\mathcal{G}_{n}^{(k)}n^{-1/2}c_{k}{\ensuremath{\mathbf{1}_{{\Omega_{n}^{(k)}}}}}\right|^{p}\\
&\ll_{L,p}\left({\mathbb{E}}\left[\sum_{k=1}^{mn}|v_{n,k}|^{2}\cdot n^{-1}\right]^{p/2}+\sum_{k=1}^{mn}|v_{n,k}|^{p}\cdot n^{-p/2}\right)\\
&\ll_{L,p} n^{-p/2}.
\end{aligned}$$ Next, by Lemma \[Lem:rosenthals\], \[item:F5\], and \[item:F12\], for any $p>2$, $$\begin{aligned}
{\mathbb{E}}\left|\sum_{k=1}^{mn}W_{k12}\right|^{p} &={\mathbb{E}}\left|\sum_{k=1}^{mn}(E_{k}-{\mathbb{E}}_{k-1})[z^{-1}u_{n}^{*}\mathcal{G}_{n}^{(k)}n^{-1}c_{k}r_{k}\mathcal{G}_{n}^{(k)}v_{n}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}]\right|^{p}\\
&={\mathbb{E}}\left|\sum_{k=1}^{mn}({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1})[z^{-1}\alpha_{n}^{(k)}\eta_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}]\right|^p\\
&\ll_{p} {\mathbb{E}}\left[\sum_{k=1}^{mn}{\mathbb{E}}_{k-1}\left|({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1})[z^{-1}\alpha_{n}^{(k)}\eta_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}]\right|^{2}\right]^{p/2}\\
&\quad \quad \quad \quad +{\mathbb{E}}\left[\left|({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1})[z^{-1}\alpha_{n}^{(k)}\eta_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}]\right|^{p}\right]\\
&\ll_{p} \left({\mathbb{E}}\left[\sum_{k=1}^{mn}{\mathbb{E}}_{k-1}\left|\eta_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right|^2\right]^{p/2}+\sum_{k=1}^{mn}{\mathbb{E}}\left|\eta_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right|^{p}\right)\\
&\ll_{L,p} \left({\mathbb{E}}\left[\sum_{k=1}^{mn}n^{-2}\right]^{p/2}+\sum_{k=1}^{mn}n^{-p}\right)\\
&\ll_{L,p} n^{-p/2}+n^{-p+1}\\
&\ll_{L,p} n^{-p/2},
\end{aligned}$$ where we also used Jensen’s inequality and the fact that $|z| \geq 5$; the last inequality follows from the fact that $p > 2$.
Finally, moving on to $W_{k22}$, by \[item:F11\] we can decompose this further as $$\begin{aligned}
\sum_{k=1}^{mn}W_{k22}&=\sum_{k=1}^{mn}({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1})[z^{-1}\bar{u}_{n,k}v_{n,k}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}]\\
&=\sum_{i=1}^{mn}({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1})[z^{-1}\bar{u}_{n,k}v_{n,k}(1-z^{-1}\zeta_{n}^{(k)}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}})]\\
&=\sum_{k=1}^{mn}(W_{k221}+W_{k222})
\end{aligned}$$ where $$W_{k221}:=({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1})[z^{-1}\bar{u}_{n,k}v_{n,k}]
\label{def:W_k221}$$ and $$W_{k222}:=-({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1})[z^{-2}\bar{u}_{n,k}v_{n,k}\zeta_{n}^{(k)}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}].
\label{def:W_k222}$$ Since $z^{-1}\bar{u}_{n,k}v_{n,k}$ is deterministic, it follows that $$\left|\sum_{k=1}^{mn}W_{k221}\right|^{p}=0.$$ Thus, it suffices to show that ${\mathbb{E}}\left|\sum_{k=1}^{mn}W_{k222}\right|^p=O_L,p(n^{-p/2})$ for $p>2$. By Lemma \[Lem:rosenthals\], \[item:F5\], and \[item:F12\], we have that for any $p>2$, $$\begin{aligned}
{\mathbb{E}}\left|\sum_{k=1}^{mn}W_{k222}\right|^{p}&={\mathbb{E}}\left|\sum_{k=1}^{mn}({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1})[z^{-2}\bar{u}_{n,k}v_{n,k}\zeta_{n}^{(k)}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}]\right|^{p}\\
&\ll_{p}{\mathbb{E}}\left[\sum_{k=1}^{mn}{\mathbb{E}}_{k-1}\left|({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1})[z^{-2}\bar{u}_{n,k}v_{n,k}\zeta_{n}^{(k)}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}]\right|^{2}\right]^{p/2}\\
&\quad \quad \quad \quad +{\mathbb{E}}\left[\sum_{k=1}^{mn}\left|({\mathbb{E}}_{k}-{\mathbb{E}}_{k-1})[z^{-2}\bar{u}_{n,k}v_{n,k}\zeta_{n}^{(k)}\alpha_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}]\right|^{p}\right]\\
&\ll_{p} {\mathbb{E}}\left[\sum_{k=1}^{mn}|\bar{u}_{n,k}|^{2}|v_{n,k}|^{2}{\mathbb{E}}_{k-1}\left|\zeta_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right|^{2}\right]^{p/2}+\sum_{k=1}^{mn}|\bar{u}_{n,k}|^{p}|v_{n,k}|^{p}{\mathbb{E}}\left|\zeta_{n}^{(k)}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right|^{p}\\
&\ll_{L,p} \left[\sum_{k=1}^{mn}|\bar{u}_{n,k}|^{2}|v_{n,k}|^{2}n^{-1}\right]^{p/2}+\sum_{k=1}^{mn}|\bar{u}_{n,k}|^{p}|v_{n,k}|^{p}n^{-p/2}\\
&\ll_{L,p} n^{-p/2},
\end{aligned}$$ where we again used Jensen’s inequality, the bound $|z| \geq 5$, and the fact that $u_n$ and $v_n$ are unit vectors. This completes the proof of , and hence the proof of Lemma \[Lem:concentration\] is complete.
Proof of Theorem \[Thm:reductions\] {#sec:combin}
===================================
In this section we complete the proof of Theorem \[Thm:reductions\]. We continue to work under the assumptions and notation introduced in Section \[Sec:Concentration\].
It remains to prove part \[item:tr:isotropic\] of Theorem \[Thm:reductions\]. In view of Lemma \[Lem:concentration\] and , it suffices to show that $$\label{eq:tr:isotropic:show}
\sup_{5 \leq |z| \leq 6} \left| {\mathbb{E}}[u_n^\ast \mathcal{G}_n(z) v_n {\ensuremath{\mathbf{1}_{{\Omega_n}}}}] + \frac{1}{z} u_n^\ast v_n \right| = o(1).$$
Neumann series
--------------
We will rewrite the resolvent, $\mathcal{G}_{n}$, as a Neumann series. Indeed, for $|z|\geq 5$, $$\label{eq:neumannnormbnd}
\frac{1}{\sqrt{n}}{\left\lVert}\frac{{\mathcal{Y}_{n}}}{z}{\right\rVert}\leq \frac{9}{10} < 1$$ on the event $\Omega_n$, so we may write $$\mathcal{G}_n(z) {\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}
=-\frac{1}{z}\left(I{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}} + \sum_{k=1}^{\infty}\left(\frac{1}{\sqrt{n}}\frac{{\mathcal{Y}_{n}}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}}{z}\right)^{k}\right)
=-\frac{1}{z}I{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}-\sum_{k=1}^{\infty}\frac{\left(\frac{{\mathcal{Y}_{n}}}{\sqrt{n}}{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right)^{k}}{z^{k+1}}$$ almost surely. Therefore, using , we obtain $$\begin{aligned}
{\mathbb{E}}\left[u_n^{*}\mathcal{G}_{n}v_n{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right]&= {\mathbb{E}}\left[-\frac{1}{z}u_n^{*}v_n{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}-u_n^{*}\sum_{k=1}^{\infty}\frac{\left(\frac{{\mathcal{Y}_{n}}}{\sqrt{n}}\right)^{k}}{z^{k+1}}v_n{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right]\notag\\
&=-\frac{1}{z}u_n^{*}v_n{\mathbb{P}}(\Omega_{n})-\sum_{k=1}^{\infty}\frac{1}{z^{k+1}}{\mathbb{E}}\left[u_n^{*}\left(\frac{{\mathcal{Y}_{n}}}{\sqrt{n}}\right)^{k}v_n{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right]\notag\\
&= -\frac{1}{z}u_n^{*}v_n + o(1) -\sum_{k=1}^{\infty}\frac{1}{z^{k+1}}{\mathbb{E}}\left[u_n^{*}\left(\frac{{\mathcal{Y}_{n}}}{\sqrt{n}}\right)^{k}v_n{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right].\label{Equ:Neumann}\end{aligned}$$ Showing the sum in converges to zero uniformly in the region $\{z \in {\mathbb{C}}: 5 \leq |z| \leq 6 \}$ will complete the proof of .
Removing the indicator function
-------------------------------
In this subsection, we prove the following.
\[lem:removeindicator\] Under the assumptions above, for any integer $k \geq 1$, $$\left|{\mathbb{E}}\left[u_n^{*}\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}\right)^{k}v_n\right]-{\mathbb{E}}\left[u_n^{*}\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}\right)^{k}v_n{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right]\right| = o_{k,L}(1).$$
Since the entries of $\mathcal{Y}_n$ are truncated, it follows that $${\left\lVert}\mathcal{Y}_n {\right\rVert}\leq {\left\lVert}\mathcal{Y}_n {\right\rVert}_2 \ll_{L} n$$ almost surely. Therefore, as ${\mathbb{P}}(\Omega_n^c) = O_A(n^{-A})$ for any $A > 0$, we obtain $$\begin{aligned}
\left|{\mathbb{E}}\left[u_n^{*}\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}\right)^{k}v_n\right]-{\mathbb{E}}\left[u_n^{*}\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}\right)^{k}v_n{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right]\right| &\leq {\mathbb{E}}\left[ {\left\lVert}\mathcal{Y}_n {\right\rVert}^k {\ensuremath{\mathbf{1}_{{\Omega_n^c}}}} \right] \\
&\ll_{L} n^{k} {\mathbb{P}}(\Omega_n^c) \\
&\ll_{L,A} n^{k-A}.\end{aligned}$$ Choosing $A$ sufficiently large (in terms of $k$), completes the proof.
Combinatorial arguments
-----------------------
In this section, we will show that $$\sup_{5 \leq |z| \leq 6} \left| \sum_{k=1}^{\infty}\frac{{\mathbb{E}}\left[u_n^{*}\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}\right)^{k}v_n{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right]}{z^{k+1}} \right| =o_{L}(1).
\label{Equ:sumtozero}$$ In view of , the tail of the series can easily be controlled. Thus, it suffices to show that, for each integer $k \geq 1$, $${\mathbb{E}}\left[u_n^{*}\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}\right)^{k}v_n{\ensuremath{\mathbf{1}_{{\Omega_{n}}}}}\right] = o_{L,k}(1). $$ By Lemma \[lem:removeindicator\], it suffices to prove the statement above without the indicator function. In particular, the following lemma completes the proof of (and hence completes the proof of Theorem \[Thm:reductions\]).
\[Lem:momentstozero\] Under the assumptions above, for any integer $k \geq 1$, $${\mathbb{E}}\left[u_{n}^{*}\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}\right)^{k}v_{n}\right] = o_{L,k}(1).$$
To prove Lemma \[Lem:momentstozero\], we will expand the above expression in terms of the entries of the random matrices ${{X}_{n,1}}$,…,${{X}_{n,m}}$. For brevity, in this section we will drop the subscript $n$ from our notation and just write $X_{1}, \ldots, X_m$ for $X_{n,1}, \ldots, X_{n,m}$. Similarly, we write the vectors $u_{n}$ and $v_{n}$ as $u$ and $v$, respectively.
To begin, we exploit the block structure of ${\mathcal{Y}_{n}}$ and write $$\begin{aligned}
{\mathbb{E}}\left[u^{*}\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}\right)^{k}v\right]&=n^{-k/2}{\mathbb{E}}\left[\sum_{1\leq a,b\leq\notag m}(u^{*})^{[a]}\left({\mathcal{Y}_{n}}^{k}\right)^{[a,b]}v^{[b]}\right]\\
&=n^{-k/2}\sum_{1\leq a,b\leq m}(u^{*})^{[a]}{\mathbb{E}}\left[\left({\mathcal{Y}_{n}}^{k}\right)^{[a,b]}\right]v^{[b]}.\label{Equ:BlockSum}\end{aligned}$$ Due to the block structure of ${\mathcal{Y}_{n}}$, for each $1\leq a\leq m$, there exists some $1\leq b\leq m$ which depends on $a$ and $k$ such that the $[a,b]$ block of ${\mathcal{Y}_{n}}^{k}$ is nonzero and all other blocks, $[a,c]$ for $c\neq b$, are zero. Additionally, the nonzero block entry $({\mathcal{Y}_{n}}^{k})^{[a,b]}$ is $X_{a}X_{a+1}\cdots X_{a+k}$ where the subscripts are reduced modulo $m$ and we use modular class representatives $\{1,\ldots,m\}$ (as opposed to the usual $\{0,..,m-1\}$). To show that the expectation of this sum is $o_{L,k}(1)$, we need to systematically count all terms which have a nonzero expectation. To do so, we use graphs to characterize the terms which appear in the sum. In particular we develop path graphs, each of which corresponds uniquely to a term in the expansion of . For each term, the corresponding path graph will record the matrix entries which appear, the order in which they appear, and the matrix from which they come. We begin with the following definitions.
We consider graphs where each vertex is specified by the ordered pair $(t,i_{t})$ for integers $1\leq t\leq k+1$ and $1\leq i_{t}\leq n$. We call $t$ the *time* coordinate of the vertex and $i_{t}$ the *height* coordinate of the vertex. Let $(i_{1},i_{2},\ldots,i_{k+1})$ be a $k+1$ tuple of integers from the set $\{1,2,\ldots,n\}$ and let $1\leq a\leq m$. We define an *$m$-colored $k$-path graph* $G^{a}(i_{1},i_{2},\ldots,i_{k+1})$ to be the edge-colored directed graph with vertex set $V=\{(1,i_{1}),\;(2,i_{2}),\ldots,(k+1,i_{k+1})\}$ and directed edges from $(t,i_{t})$ to $(t+1,i_{t+1})$ for $1\leq t\leq k$, where the edge from $(t,i_{t})$ to $(t+1,i_{t+1})$ is color $a+t-1$, with the convention that colors are reduced modulo $m$, and the modulo class representatives are $\{1,\ldots,m\}$. The graph is said to *visit* a vertex $(t,i_{t})$ if there exists an edge which begins or terminates at that vertex, so that $(t,i_{t})\in V$. We say an edge of $G^{a}(i_{1},\ldots,i_{k+1})$ is of *type I* if it terminates on a vertex $(t,i_{t})$ such that $i_{t}\neq i_{s}$ for all $s<t$. We say an edge is of *type II* if it terminates on a vertex $(t,i_{t})$ such that there exists some $s<t$ with $i_{t} = i_{s}$.
We make some observations about this definition.
We view $k$ and $n$ as specified parameters. Once these parameters are specified, the notation $G^{a}(i_{1},\ldots,i_{k+1})$ completely determines the graph. The vertex set $V$ is a subset of the vertices of the $(k+1)\times n$ integer lattice and each graph has exactly $k$ directed edges. In each graph, there is an edge which begins on vertex $(1,i_{1})$ for some $1\leq i_{1}\leq n$ and there is an edge which terminates on vertex $(k+1,i_{k+1})$ for some $1\leq i_{k+1}\leq n$. Each edge begins at $(t,i_{t})$ and terminates on $(t+1,i_{t+1})$ for some integers $1\leq t\leq k$ and $1\leq i_{t},i_{t+1}\leq n$. Additionally, each edge is one of $m$ possible colors. Note that if we think about the edges as ordered by the time coordinate, then the order of the colors is a cyclic permutation of the coloring $1,2,\ldots,m$, beginning with $a$. This cycle is repeated as many times as necessary in order to cover all edges.
Notice that we call $G^{a}(i_{1},i_{2},\ldots,i_{k+1})$ a path graph because it can be thought of as a path through the integer lattice from vertex $(1,i_{1})$ to $(k+1,i_{k+1})$ for some $1\leq i_{1},i_{k+1}\leq n$. Indeed, by the requirements in the definition, this graph must be one continuous path and no vertex can be visited more than once. We may call an $m$-colored $k$-path graph a *path graph* when $m$ and $k$ are clear from context.
Finally, we can think of an edge as type I if it terminates at a height not previously visited. It is of type II if it terminates at a height that has been previously visited.
For a given $m$-colored $k$-path graph $G^{a}(i_{1},i_{2},\ldots,i_{k+1})$, we say two edges $e_{1}$ and $e_{2}$ of $G^{a}(i_{1},i_{2},\ldots,i_{k+1})$ are *time-translate parallel* if $e_{1}$ begins at vertex $(t,i_{t})$ and terminates at vertex $(t+1,i_{t+1})$ and edge $e_{2}$ begins at vertex $(t',i_{t'})$ and terminates at vertex $(t'+1,i_{t'+1})$ where $i_{t}=i_{t'}$ and $i_{t+1}=i_{t'+1}$ for some $1\leq t,t'\leq k$ with $t\neq t'$.
Intuitively, two edges are time-translate parallel if they span the same two hight coordinates at different times. Throughout this section, we shorten the term “time-translate parallel" and refer to edges with this property as “parallel" for brevity. We warn the reader that by this, we mean that the edges must span the same heights at two different times. For instance, the edge from $(1,2)$ to $(2,4)$ is not parallel to the edge from $(3,1)$ to $(4,3)$ since they don’t span the same heights, although these edges might appear parallel in the geometric interpretation of the word. Also note that two parallel edges need not have the same color. See Figure \[Fig:graph8\_2M1\] for examples of parallel and non parallel edges.
For a fixed $k$, we say two $m$-colored $k$-path graphs $G^{a}(i_{1},\ldots,i_{k+1})$ and $G^{a}(i'_{1},\ldots,i'_{k+1})$ are *equivalent*, denoted $G^{a}(i_{1},\ldots,i_{k+1})\sim G^{a}(i'_{1},\ldots,i'_{k+1})$, if there exists some permutation $\sigma$ of $\{1,\ldots,n\}$ such that $(i_{1},\ldots,i_{k+1})=(\sigma(i'_{1}),\ldots,\sigma(i'_{k+1}))$. Note here that for two path graphs to be equivalent, the color of the first edge, and hence the color of all edges sequentially, must be the same in both.
One can check that the above definition of equivalent $m$-colored $k$-path graphs is an equivalence relation. Thus, the set of all $m$-colored $k$-path graphs can be split into equivalence classes.
For each equivalence class of graphs, the *canonical $m$-colored $k$-path graph* is the unique graph from an equivalence class which satisfies the following condition: If $G^{a}(i_{1},\ldots,i_{k+1})$ visits vertex $(t,i_{t})$, then for every $0 < i<i_{t}$ there exists $s<t$ such that $G^{a}(i_{1},\ldots,i_{k+1})$ visits $(s,i)$.
Observe that, intuitively, the canonical representation for $G^{a}(i_{1},\ldots,i_{k+1})$ is the graph which does not “skip over" any height coordinates. Namely, the canonical graph necessarily begins at vertex $(1,1)$ and at each time step the height of the next vertex can be a most one larger than the maximum height of all previous vertices.
![The graphs featured here are two possible 2-colored 3-path graphs, $G^{1}(4,1,4,3)$ and $G^{1}(1,2,1,3)$, which correspond to the terms $X_{1,(4,1)}X_{2,(1,4)}X_{1,(4,3)}$ and $X_{1,(1,2)}X_{2,(2,1)}X_{1,(1,3)}$, respectively. These two graphs are equivalent. The first graph is not a canonical graph while the second graph is a canonical graph. For ease of notation and clarity, we have drawn all vertices on the integer lattice as dots, with the time axis appearing horizontally and the height axis vertically. While each dot represents a vertex in the integer latter, not all of the dots represent vertices in the path graphs. Only dots from which an edge begins or terminates are vertices of the path graph. We have also colored the edges with blue and green to represent the two colors. The colors of each edge are also represented by the number in parenthesis, e.g., $(1)$ or $(2)$. In either graph, the first and third edges are type I edges, while the second edge in each graph is type II. See Example \[Example:TwoEquiv\] for further discussion.[]{data-label="Fig:3path1and2"}](m2k3h3noncanon.png "fig:") ![The graphs featured here are two possible 2-colored 3-path graphs, $G^{1}(4,1,4,3)$ and $G^{1}(1,2,1,3)$, which correspond to the terms $X_{1,(4,1)}X_{2,(1,4)}X_{1,(4,3)}$ and $X_{1,(1,2)}X_{2,(2,1)}X_{1,(1,3)}$, respectively. These two graphs are equivalent. The first graph is not a canonical graph while the second graph is a canonical graph. For ease of notation and clarity, we have drawn all vertices on the integer lattice as dots, with the time axis appearing horizontally and the height axis vertically. While each dot represents a vertex in the integer latter, not all of the dots represent vertices in the path graphs. Only dots from which an edge begins or terminates are vertices of the path graph. We have also colored the edges with blue and green to represent the two colors. The colors of each edge are also represented by the number in parenthesis, e.g., $(1)$ or $(2)$. In either graph, the first and third edges are type I edges, while the second edge in each graph is type II. See Example \[Example:TwoEquiv\] for further discussion.[]{data-label="Fig:3path1and2"}](m2k3h3canon.png "fig:")
Now, we return to the task at hand: the proof of Lemma \[Lem:momentstozero\]. We fix a positive integer $k$, and we expand as in . For a fixed value $1 \leq a \leq m$, consider the nonzero block $\left({\mathcal{Y}_{n}}^{k} \right)^{[a, b]}$. We further expand to see $$\label{Equ:ExpandedTerm}
(u^{*})^{[a]}{\mathbb{E}}\left[\left({\mathcal{Y}_{n}}^{k}\right)^{[a,b]}\right]v^{[b]} = \sum_{i_{1},\ldots,i_{k+1}}\overline{u}^{[a]}_{i_{1}}{\mathbb{E}}[X_{a,(i_{1},i_{2})}X_{a+1,(i_{2},i_{3})}\cdots X_{a+k,(i_{k},i_{k+1})}]v^{[b]}_{i_{k+1}},$$ where the subscripts $a,\ldots,a+k$ are reduced mod $m$ with representatives $\{1,\ldots,m\}$. Observe that by the structure of ${\mathcal{Y}_{n}}$, these subscripts must appear cyclically in the order $a,a+1,\ldots,m,1,\ldots,a-1$, with the order repeating as many times as necessary before ending at $b$. In particular, the subscripts are uniquely determined by the starting subscript $a$ and the value of $k$.
We now consider the expectation on the right-hand side of . Since all entries of each matrix are independent, if an index appears only once in a product, that product will have expectation zero. Therefore, only terms in which every index appears more than once will have a nonzero expectation, and only such terms will contribute to the expected value of the sum. Note that for an entry to appear more than once, we not only need the index of the entries to match but also which of the $m$ matrices the entries came from. The following definition will assist in encoding each entry on the right-hand side of as a unique $m$-colored $k$-path graph.
We say the term ${X_{a,(i_{1},i_{2})}}{X_{a+1,(i_{2},i_{3})}}\cdots{X_{a+k,(i_{k},i_{k+1})}}$ from the expansion of *corresponds* to the $m$-colored $k$-path graph $G^{a}(i_{1},\ldots,i_{k+1})$. We use the notation $$x_{G} := X_{a,(i_{1},i_{2})}X_{a+1,(i_{2},i_{3})}\cdots X_{a+k,(i_{k},i_{k+1})}$$ whenever $X_{a,(i_{1},i_{2})}X_{a+1,(i_{2},i_{3})}\cdots X_{a+k,(i_{k},i_{k+1})}$ corresponds to the path graph $G$. In this case, we also write $u^{*}_{G}$ and $v_{G}$ for $\bar{u}_{i_{1}}^{[a]}$ and $v_{i_{k+1}}^{[a+k]}$, respectively.
Each term in the expansion of corresponds uniquely to an $m$-colored $k$-path graph. In terms of the corresponding $m$-colored $k$-path graph, if an edge spans two vertices $(t,i_{t})$ and $(t+1,i_{t+1})$, and has color $a$ then the corresponding matrix product must contain the entry $X_{a,(i_{t},i_{t+1})}$. Thus, the color corresponds to the matrix from which the entry came and the height coordinates correspond to the matrix indices. Repeating indices is analogous to parallel edges, and entries coming from the same matrix corresponds to edges sharing a color. For example, if $X_{4,(3,5)}$ appears at some point in a term, the corresponding $m$-colored $k$-path graph will have an edge from $(t,3)$ to $(t+1,5)$ for some $t$, and the edge will be colored with color $4$. Thus, a graph corresponds to a term with nonzero expectation if for every edge $e_{1}$, there exists at least one other edge in the graph which is parallel to $e_{1}$ and which has the same color as $e_{1}$. We must systematically count the terms which have nonzero expectation.
Since two equivalent graphs correspond to two terms which differ only by a permutation of indices, and since entries in a given matrix are independent and identically distributed, the expectation of the corresponding terms will be equal. This leads to the following lemma.
\[Lem:EquivGraphsEqualInExpectation\] If two path graphs $G^{a}(i_{1},\ldots,i_{k+1})$ and $G^{a}(i'_{1},\ldots,i'_{k+1})$ are equivalent, then $${\mathbb{E}}[x_{G^{a}(i_{1},\ldots,i_{k+1})}] = {\mathbb{E}}[x_{G^{a}(i'_{1},\ldots,i'_{k+1})}].$$
This lemma allows us to characterize graphs with non-zero expectation based on their canonical representation. Before we begin counting the graphs which correspond to terms with nonzero expectation, we present some examples.
\[Example:TwoEquiv\] Consider two $2$-colored $3$-path graphs: $G^{1}(4,1,4,3)$ and $G^{1}(1,2,1,3)$. $G^{1}(4,1,4,3)$ is the leftmost graph in Figure \[Fig:3path1and2\] and $G^{1}(1,2,1,3)$ is the rightmost graph in Figure \[Fig:3path1and2\]. They correspond to the terms $X_{1,(4,1)}X_{2,(1,4)}X_{1,(4,3)}$ and $X_{1,(1,2)}X_{2,(2,1)}X_{1,(1,3)}$ respectively from the expansion of $u^{*}\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}\right)^{3}v$ where $m=2$. Note that these graphs are equivalent by the permutation which maps $4\mapsto1\mapsto2\mapsto4$. $G^{1}(4,1,4,3)$ is not a canonical graph, while $G^{1}(1,2,1,3)$ is a canonical graph. Observe that since $X_{1,(4,1)}$ appeared first in the product $X_{1,(4,1)}X_{2,(1,4)}X_{1,(4,3)}$, the first edge in the corresponding path graph is an edge of color $1$ spanning from height coordinate $4$ to height coordinate $1$.
Consider a product of the form $${X_{1,(i_{1},i_{2})}} {X_{1,(i_{2},i_{1})}}{X_{1,(i_{1},i_{2})}}{X_{1,(i_{2},i_{1})}}{X_{1,(i_{1},i_{2})}}{X_{1,(i_{2},i_{1})}},{X_{1,(i_{1},i_{2})}}$$ where $i_1, i_2$ are distinct. This corresponds to a $1$-colored $7$-path graph whose canonical representative can be seen in Figure \[Fig:graph8M1\]. Since all entries come from matrix $X_{1}$, we know $m=1$. Since there are 7 terms in the product, $k=7$. This term has expected value ${\mathbb{E}}\left[ ({X_{1,(i_{1},i_{2})}})^{4}\right]{\mathbb{E}}\left[ ({X_{1,(i_{2},i_{1})}})^{3}\right]$. Since $X_1$ is an iid matrix, the particular choice of $i_1 \neq i_2$ is irrelevant to the expected value. \[Ex:8M1\]
Consider a product of the form $${X_{1,(i_{1},i_{2})}}{X_{1,(i_{2},i_{3})}}{X_{1,(i_{3},i_{1})}}{X_{1,(i_{1},i_{3})}}{X_{1,(i_{3},i_{4})}}{X_{1,(i_{4},i_{3})}}{X_{1,(i_{3},i_{4})}}$$ where $i_{1},i_{2},i_{3}$ and $i_{4}$ are distinct. This corresponds to a $1$-colored $7$-path graph whose canonical representative is featured in Figure \[Fig:graph8\_2M1\]. Since the only index pair which is appears more than once is $(i_{3},i_{4})$, the corresponding term will have zero expectation. \[Ex:8\_2M1\]
\[Ex:8M4\] Let $i_{1},i_{2},i_{3},i_{4}$ be distinct, and consider the product $${X_{1,(i_{1},i_{2})}} {X_{2,(i_{2},i_{1})}}{X_{3,(i_{1},i_{2})}}{X_{4,(i_{2},i_{1})}}{X_{1,(i_{1},i_{2})}}{X_{2,(i_{2},i_{1})}}{X_{3,(i_{1},i_{2})}}{X_{4,(i_{2},i_{1})}}.$$ Since there are 8 entries in this product, $k=8$, and as there are entries from 4 matrices, $m=4$. The corresponding canonical $4$-colored $8$-path graph representative is shown in Figure \[Fig:8M4\]. This term has expectation $${\mathbb{E}}\left[({X_{1,(i_{1},i_{2})}})^{2}( {X_{2,(i_{2},i_{1})}})^{2}({X_{3,(i_{1},i_{2})}})^{2}({X_{4,(i_{2},i_{1})}})^{2}\right].$$
Consider the product $${X_{3,(i_{1},i_{2})}} {X_{1,(i_{2},i_{3})}}{X_{2,(i_{3},i_{4})}}{X_{3,(i_{4},i_{1})}}{X_{1,(i_{1},i_{2})}}{X_{2,(i_{2},i_{4})}}{X_{3,(i_{4},i_{2})}}{X_{1,(i_{2},i_{1})}}{X_{2,(i_{1},i_{5})}}.$$ Since there are 9 entries in this product, $k=9$ and we can see that there are 3 different matrices so that $m=3$. For any distinct indices $i_{1},i_{2},i_{3},i_{4},i_{5}$, the corresponding canonical $3$-colored $9$-path graph representative is shown in Figure \[Fig:9M3\]. In this product, every entry appears only once. Thus the expectation of this product factors, and the term will have expectation zero. \[Ex:9M3\]
.
We now complete the proof of Lemma \[Lem:momentstozero\], which will occupy the remainder of the section. Let $$\Delta_{n,k}^{a} := \{G^{a}(i_{1},\ldots,i_{k+1})\;:\;1\leq i_{1},\ldots,i_{k+1}\leq n\},$$ and let $\tilde{\Delta}_{n,k}^{a}$ be the set of all canonical graphs in $\Delta_{n,k}^{a}$. We now divide the proof into cases based on the value of $k$.
Case where $k$ is a multiple of $m$
: If $k$ is a multiple of $m$, then by the block structure of ${\mathcal{Y}_{n}}$, it follows that ${\mathcal{Y}_{n}}^{k}$ is a block diagonal matrix. Since the diagonal blocks are the only nonzero blocks in this case, simplifies to $$\begin{aligned}
{\mathbb{E}}\left[u^{*}\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}\right)^{k}v\right]&=n^{-k/2}\sum_{1\leq a\leq m}(u^{*})^{[a]}{\mathbb{E}}\left[\left({\mathcal{Y}_{n}}^{k}\right)^{[a,a]}\right]v^{[a]}\notag\\
&=n^{-k/2}\sum_{1\leq a\leq m}\sum_{G\in\Delta_{n,k}^{a}}u^{*}_{G}{\mathbb{E}}[x_{G}]v_{G}\label{Equ:FirstSumExpansion}
\end{aligned}$$ Recall that if $G\in\Delta_{n,k}^{a}$, then $G$ is an $m$-colored $k$-path graph which starts with color, $a$, i.e., $G=G^{a}(i_{1},\ldots,i_{k+1})$ for some $i_{1},\ldots,i_{k+1}\in [n]$. By Lemma \[Lem:EquivGraphsEqualInExpectation\], we can reduce the task of counting all terms with nonzero expectation to counting canonical graphs and the cardinality of each equivalence class.
Observe that if $k=m$, then any term in will be of the form $$\overline{u}_{i_{1}}{\mathbb{E}}[X_{a,(i_{1},i_{2})}X_{a+1,(i_{2},i_{3})}\cdots X_{a-1,(i_{k},i_{k+1})}]v_{i_{k+1}}$$ where each matrix contributes only one entry to the above expression. In this case, all terms are independent and the expectation in is zero.
Now consider the case where $k=cm$ for some integer $c\geq 2$. Define $$h(G)=:\max\{i_{t}\;:\;(t,i_{t})\in V_{\tilde{G}}\},
\label{Def:h(G)}$$ where $\tilde{G}$ is the canonical representative for the graph $G$ and $V_{\tilde{G}}$ is the vertex set for the graph $\tilde{G}$. We call $h(G)$ the *maximal height* (or sometimes just *height*) of a graph $G$. Intuitively, $h(G)$ is the number of distinct height coordinates $G$ visits. In terms of the canonical graph, $\tilde{G}$, this is the largest height coordinate visited by an edge in $\tilde{G}$. For each $a$, define $$\begin{aligned}
& (\Delta_{n,k}^{a})_{1}:=\{G\in \Delta_{n,k}^{a}\;:\; h(G)>k/2\}\label{Def:Delta1}\\
& (\Delta_{n,k}^{a})_{2}:=\{G\in \Delta_{n,k}^{a}\;:\;h(G)=k/2\}\label{Def:Delta2}\\
& (\Delta_{n,k}^{a})_{3}:=\{G\in \Delta_{n,k}^{a}\;:\;h(G)<k/2\}\label{Def:Delta3}.
\end{aligned}$$ This partitions $\Delta_{n,k}^{a}$ into disjoint subsets. Without loss of generality, we assume that $a=1$ since the argument will be the same for any permutation of the coloring. In this case, all path graphs start with color $1$ and since $k=cm$, the colors $1,2,\ldots,m$ will each repeat $c$ times. We analyze each set of graphs separately.\
First, consider the set $(\Delta_{n,k}^{1})_{1}$ and recall that each $G\in(\Delta_{n,k}^{1})_{1}$ must have exactly $k$ edges. Since the expectation of all equivalent graphs is the same, it is sufficient to assume that $G$ is canonical. If $h(G)>k/2$, there must be more than $k/2$ type I edges. If each edge were parallel to at least one other edge in $G$, then there would be more than $k$ edges, a contradiction. Hence there will be at least one edge that is not parallel to any other edge. This implies ${\mathbb{E}}[x_{G}]=0$ whenever $G\in (\Delta_{n,k}^{1})_{1}$ and thus $$\sum_{G\in(\Delta_{n,k}^{a})_{1}}u^{*}_{G}{\mathbb{E}}[x_{G}]v_{G}=0.
\label{Equ:MMultKG1}$$
Note that if $k$ is odd, then this set will be empty; so assume $k$ is even. Now consider a graph $G\in(\Delta_{n,k}^{1})_{2}$. By Lemma \[Lem:EquivGraphsEqualInExpectation\], we can assume that $G$ is canonical. If $G$ has any edges which are not parallel to any other edges, then ${\mathbb{E}}[x_{G}]=0$ and it does not contribute to the expectation. Thus we can consider only graphs in which every edge is parallel to at least one other edge. Since any $G\in (\Delta_{n,k}^{1})_{2}$ must visit exactly $k/2$ distinct height coordinates and since there must be precisely $k$ edges in $G$, a counting argument reveals that every edge in $G$ must be parallel to exactly one other edge in $G$. This gives way to the following lemma.
Let $k \geq 2$ be any even integer (not necessarily a multiple of $m$). Then there is only one canonical $k$-path graph in $\Delta^{1}_{n,k}$ for which $h(G)=\frac{k}{2}$ and in which each edge is parallel to exactly one other edge. \[Lem:OnlyOne\]
The proof of this lemma, which relies on a counting argument and induction, is detailed in Appendix \[Sec:OnlyOne\]. In fact, the proof reveals that this one canonical $m$-colored $k$-path graph starting with color $1$ is $$G^{1}(1,2,\ldots,k/2,1,2,\ldots,k/2,1).$$ If two edges are parallel but are not the same color then the expectation of terms with corresponding canonical graph will be zero.
If $c$ is odd and $m$ is even, then the edge from $(k/2,k/2)$ to $(k/2+1,1)$ will have color $\frac{m}{2}$ and thus edge from $(k/2+1,1)$ to $(k/2+2,2)$ will have color $\frac{m}{2}+1$. This edge is necessarily parallel to the edge from $(1,1)$ to $(2,2)$, and it is the only edge parallel to the edge from $(1,1)$ to $(2,2)$. But note that the edge from $(1,1)$ to $(2,2)$ had color 1 and $\frac{m}{2}+1$ is not congruent to 1 $\mod$ $m$. Therefore in the case where $c$ is odd and $m$ is even, the canonical $m$-colored $k$-path graph corresponds to a term in the product which has expectation zero.
Finally, if $c$ is even, then the edge from $(k/2,k/2)$ to $(k/2+1,1)$ must have color $m$. Hence the edge from $(k/2+1,1)$ to $(k/2+2,2)$ will have color 1, which is the same color as the edge from $(1,1)$ to $(2,2)$. This means that when $k=cm$ and $c$ is even, every edge in $G^{1}(1,2,\ldots,k/2,1,2,\ldots, k/2,1)$ will be parallel to exactly one other edge of the same color. In particular, note that for this graph $$\begin{aligned}
\left| u^{*}_{G}{\mathbb{E}}[x_{G}]v_{G} \right| &\leq \left|\overline{u}_{1}^{[1]} \right| {\mathbb{E}}\left| X_{1,(1,2)}\cdots X_{m,(k/2,1)} \right| \left| v_{1}^{[m]} \right| \notag\\
&\leq \left|\overline{u}_{1}^{[1]} \right|{\mathbb{E}}\left|X_{1,(1,2)} \right|^{2}\cdots {\mathbb{E}}\left|X_{m,(k/2,1)}\right|^{2} \left|v_{1}^{[m]} \right| \notag\\
&\leq \left|\overline{u}_{1}^{[1]}\right| \left|v_{1}^{[m]}\right|. \label{Equ:SimpleInnerProd}
\end{aligned}$$
For ease of notation, let $\tilde{G}:=G^{1}(1,2,\ldots,k/2,1,2,\ldots,k/2,1)$, and consider $G^{1}(i_{1},\ldots,i_{k+1})\in (\Delta_{n,k}^{1})_{2}$ such that $G^{1}(i_{1},\ldots,i_{k+1})\sim \tilde{G}$. Observe that there are $n$ options for the first coordinate $i_{1}$ of $G^{1}(i_{1},\ldots,i_{k+1})$. If we fix the first coordinate, then there are at most $(n-1)(n-2)\cdots(n-k/2-1)\leq n^{k/2-1}$ graphs with first coordinate $i_{1}$ which are equivalent to $\tilde{G}$. If we repeated the computation of the expectation of any of these equivalent graphs, we would get a term similar to but with different starting and ending coordinates, yielding an upper bound of $\left| \overline{u}_{i_{1}} \right| \left| v_{i_{1}} \right|$. Therefore, by the above argument and the Cauchy–Schwarz inequality, we obtain $$\begin{aligned}
\left|\sum_{G\in(\Delta_{n,k}^{1})_{2}}u^{*}_{G}{\mathbb{E}}[x_{G}]v_{G}\right| & \leq \sum_{1\leq i_{1}\leq n}n^{k/2-1}\left|\overline{u}_{i_{1}}\right|\left|v_{i_{1}}\right|\notag\\
&\leq n^{k/2-1}{\left\lVert}u{\right\rVert}{\left\lVert}v{\right\rVert}\notag\\
& \leq n^{k/2-1}.\label{Equ:MMultKG2}
\end{aligned}$$
Consider an $m$-colored $k$-path graphs $G\in (\Delta_{n,k}^{1})_{3}$, and assume that $G$ is canonical. If $G$ contains any edges which were not parallel to another edge, then the graph will correspond to a term with expectation zero. So consider a canonical graph $G\in (\Delta_{n,k}^{1})_{3}$ such that all edges are parallel to at least one other edge. If $h(G)=1$, then $G=G^{1}(1,1,\ldots,1)$ and so $$\left| {\mathbb{E}}[x_{G}] \right| \leq {\mathbb{E}}\left| {X_{1,(1,1)}}\cdots{X_{1,(1,1)}}\right| ={\mathbb{E}}\left| {X_{1,(1,1)}} \right|^{k} \leq (4L)^{k}.$$ Note that this is the highest possible moment in a term. Let $M:=(4L)^{k}$. For any canonical $m$-colored $k$-path graph $G\in (\tilde{\Delta}_{n,k}^{1})_{3}$, $${\mathbb{E}}|x_{G} | \leq M.$$ Also note that this bound holds for graphs of all starting colors, not just starting color 1. In addition, for any $G$ with maximal height $h(G)$, there are $n(n-1)\cdots (n-h(G)-1)<n^{h(G)}$ graphs in the equivalence class of $G$. By over counting, we can bound the number of distinct equivalence classes by $k^{k}$ since there are $k$ edges and at each time coordinate, the edge which starts at that time coordinate can terminate at most one height coordinate larger than it started, so any edge has a most $k$ options for an ending coordinate.
Based on the above observations, we have $$\begin{aligned}
\left|\sum_{G\in (\Delta_{n,k}^{1})_{3}}u^{*}_{G}{\mathbb{E}}[x_{G}]v_{G}\right|&\leq \sum_{G\in (\tilde{\Delta}_{n,k}^{1})_{3}}\left|n^{h(G)}u^{*}_{G}{\mathbb{E}}[x_{G}]v_{G}\right|\notag\\
&\leq n^{\frac{k}{2}-\frac{1}{2}}\sum_{G\in (\tilde{\Delta}_{n,k}^{1})_{3}}\left|u^{*}_{G}\right| {\mathbb{E}}\left|x_{G} \right| \left| v_{G}\right|\notag\\
&\leq n^{\frac{k}{2}-\frac{1}{2}}\sum_{G\in (\tilde{\Delta}_{n,k}^{1})_{3}}M\notag\\
& \ll_{L,k}n^{\frac{k}{2}-\frac{1}{2}}. \label{Equ:MMultKG3}
\end{aligned}$$
By , , and we conclude that $$\begin{aligned}
\left|\sum_{G\in\Delta_{n,k}^{1}}u^{*}_{G}{\mathbb{E}}[x_{G}]v_{G}\right| & \leq \sum_{i=1}^{3}\left|\sum_{G\in(\Delta_{n,k}^{1})_{i}}u^{*}_{G}{\mathbb{E}}[x_{G}]v_{G}\right|\\
&\ll_{L,k} n^{k/2-1/2}.
\end{aligned}$$ While the bounds above were calculated for $a=1$, the same argument applies for any $a$ by simply permuting the colors. Therefore, in the case where $k$ is a multiple of $m$, from we have $$\begin{aligned}
\left|{\mathbb{E}}\left[u^{*}\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}\right)^{k}v\right]\right|&\leq n^{-k/2}\sum_{1\leq a\leq m}\left|\sum_{G\in\Delta_{n,k}^{a}}u^{*}_{G}{\mathbb{E}}[x_{G}]v_{G}\right|\\
&\ll_{L,k}n^{-k/2}\sum_{1\leq a\leq m} n^{k/2-1/2}\\
&\ll_{L,k}\frac{1}{\sqrt{n}}.
\end{aligned}$$
Case where $k$ is not a multiple of $m$
: Now assume that $k$ is not a multiple of $m$. If $k<m$, then each matrix has at most one entry in the product on the right-hand side of and all terms will be independent. Hence the expectation will be zero. Therefore, consider the case when $k>m$. Then there must exist some positive integer $c$ such that $$cm<k<(c+1)m.$$ We can write $k=cm+r$ for some $0<r<c$ and in this case, a computation reveals that the only nonzero blocks in ${\mathcal{Y}_{n}}^{k}$ are blocks of the form $[a,a+r]$ where $a$ and $a+r$ are reduced modulo $m$, and the modulo class representatives are $\{1,2,\ldots,m\}$. In this case we can write $$\begin{aligned}
{\mathbb{E}}\left[u^{*}\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}\right)^{k}v\right]&=n^{-k/2}\sum_{1\leq a\leq m}(u^{*})^{[a]}{\mathbb{E}}\left[\left({\mathcal{Y}_{n}}^{k}\right)^{[a,a+r]}\right]v^{[a+r]}\notag\\
&=n^{-k/2}\sum_{1\leq a\leq m}\sum_{G\in\Delta_{n,k}^{a}}u^{*}_{G}{\mathbb{E}}[x_{G}]v_{G}\label{Equ:SecondSumExpansion}
\end{aligned}$$
Again, define $h(G)$ as in and define $(\Delta_{n,k}^{a})_{1}$, $(\Delta_{n,k}^{a})_{2}$, and $(\Delta_{n,k}^{a})_{3}$ as in , , and , respectively. Without loss of generality, assume that $a=1$.
If a graph $G$ has height greater than $k/2$, by the same argument in the previous case we can see that there must be an edge which is not parallel to any other edge. Therefore when $k$ is not a multiple of $m$ we still have $$\sum_{G\in(\Delta_{n,k}^{1})_{1}}u^{*}_{G}{\mathbb{E}}[x_{G}]v_{G}=0.
\label{Equ:MNotMultKG1}$$ If $G\in(\Delta_{n,k}^{1})_{3}$ has height less than $k/2$, then we may still bound ${\mathbb{E}}\left|x_{G}\right|\leq M$. Therefore, we may use the same argument as in the previous case to conclude that $$\left|\sum_{G\in(\Delta_{n,k}^{1})_{3}}u^{*}_{G}{\mathbb{E}}[x_{G}]v_{G}\right|\ll_{L,k}n^{k/2-1}.
\label{Equ:MNotMultKG3}$$ Thus, we need only to consider graphs in $(\Delta_{n,k}^{a})_{2}$.\
If $k$ is odd, then $(\Delta_{n,k}^{1})_{2}$ is empty, so assume that $k$ is even. Consider a graph $G\in(\Delta_{n,k}^{1})_{2}$ and by Lemma \[Lem:EquivGraphsEqualInExpectation\], we may assume that $G$ is canonical. If $G$ has any edges which are not parallel to any other edge, then ${\mathbb{E}}[x_{G}]=0$, so assume each edge is parallel to at least one other edge. A counting argument reveals that in fact each edge must be parallel to exactly one other edge and by Lemma \[Lem:OnlyOne\], we can conclude that in fact $G=G^{1}(1,2,\ldots,k/2,1,2,\ldots,k/2,1)$.
In order for this graph to correspond to a term with nonzero expectation, the colors on the pairs of parallel edges must match. In order for this to happen, we would need the edge from $(k/2+1,1)$ to $(k/2+2,2)$ to have color 1. This would force the edge from $(k/2,k/2)$ to $(k/2+1,1)$ to have color $m$. Note that if we think about drawing edges sequentially with the time coordinate, then this implies that the $k/2$th edge drawn from $(k/2,k/2)$ to $(k/2+1,1)$ is of color $m$, forcing $k/2$ to be a multiple of $m$. However, this would imply that $k$ is also a multiple of $m$, a contradiction. Hence in this case, if $G\in(\tilde{\Delta}_{n,k}^{a})_{2}$, then ${\mathbb{E}}[x_{G}]=0$. By Lemma \[Lem:EquivGraphsEqualInExpectation\], this gives $$\sum_{G\in(\Delta_{n,k}^{a})_{2}}u^{*}_{G}{\mathbb{E}}[x_{G}]v_{G}=0.
\label{Equ:MNotMultKG2}$$
By , , and we can see that $$\begin{aligned}
\left|\sum_{G\in\Delta_{n,k}^{1}}u^{*}_{G}{\mathbb{E}}[x_{G}]v_{G}\right| & \leq \sum_{i=1}^{3}\left| \sum_{G\in(\Delta_{n,k}^{1})_{i}}u^{*}_{G}{\mathbb{E}}[x_{G}]v_{G}\right|\\
&\ll_{L,k} n^{k/2-1}.
\end{aligned}$$ While the bounds above were calculated for $a=1$, the same arguments apply for any $a$ by simply permuting the colors. Thus, from we have $$\begin{aligned}
\left|{\mathbb{E}}\left[u^{*}\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}\right)^{k}v\right]\right|&\leq n^{-k/2}\sum_{1\leq a\leq m}\left|\sum_{G\in\Delta_{n,k}^{a}}u^{*}_{G}{\mathbb{E}}[x_{G}]v_{G}\right|\\
&\ll_{L,k}n^{-k/2}\sum_{1\leq a\leq m} n^{k/2-1}\\
&\ll_{L,k}\frac{1}{n}
\end{aligned}$$ in the case where $k$ is not a multiple of $m$.
Combining the cases above completes the proof of Lemma \[Lem:momentstozero\].
Note that if $m=1$, then $k$ is trivially a multiple of $m$. Hence, the case where ${\mathcal{Y}_{n}}$ is an $n\times n$ matrix follows as a special case of the above argument.
Proofs of results from Section \[Sec:RelatedResults\] {#Sec:RelatedResultProofs}
=====================================================
Before we prove the results from Section \[Sec:RelatedResults\], we must prove an isotropic limit law for products of repeated matrices.
\[Thm:RepeatedProdIsotropic\] Assume $\xi$ is a complex-valued random variable with mean zero, unit variance, finite fourth moment, and independent real and imaginary parts. For each $n \geq 1$, let $X_{n}$ be an $n \times n$ iid random matrix with atom variable $\xi$. Define ${\mathcal{Y}_{n}}$ as in and define $\mathcal{G}_{n}(z)$ as in but with ${{X}_{n,1}}={{X}_{n,2}}=\cdots={{X}_{n,m}}=X_{n}$. Then, for any fixed $\delta > 0$, the following statements hold.
1. \[item:pw:invertible\] Almost surely, for $n$ sufficiently large, the eigenvalues of $\frac{1}{\sqrt{n}} \mathcal{Y}_n$ are contained in the disk $\{z \in \mathbb{C} : |z| \leq 1 + \delta \}$. In particular, this implies that almost surely, for $n$ sufficiently large, the matrix $\frac{1}{\sqrt{n}} \mathcal{Y}_n - z I$ is invertible for every $z \in \mathbb{C}$ with $|z| > 1 + \delta$.
2. \[item:pw:invtbnd\] There exists a constant $c>0$ (depending only on $\delta$ and $m$) such that almost surely, for $n$ sufficiently large, $$\sup_{z\in{\mathbb{C}}:|z|>1+\delta}{\left\lVert}\mathcal{G}_{n}(z){\right\rVert}\leq c.$$
3. \[item:pw:isortopic\] For each $n \geq 1$, let $u_n, v_n \in \mathbb{C}^{mn}$ be deterministic unit vectors. Then $$\sup_{z \in \mathbb{C} : |z| > 1 + \delta} \left| u_n^\ast \mathcal{G}_n(z) v_n + \frac{1}{z} u_n^\ast v_n \right| \longrightarrow 0$$ almost surely as $n \to \infty$.
\[Thm:isotropic power\]
Fix $\delta>0$. From [@Tout Theorem 1.4], the spectral radius of $\frac{1}{\sqrt{n}}X_{n}$ converges to $1$ almost surely as $n \to \infty$. Thus, $n^{-m/2}(X_{n})^{m}$ has spectral radius converging to $1$ almost surely as well. It follows that the spectral radius of $n^{-m/2}({\mathcal{Y}_{n}})^{m}$ converges to $1$ almost surely, which in turn implies that the spectral radius of $\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}$ converges to $1$ almost surely as $n \to \infty$, proving claim \[item:pw:invertible\].
To prove part \[item:pw:invtbnd\], we consider two events, both of which hold almost surely. By [@Tout Theorem 1.4], there exists a constant $K> 0$ such that almost surely, for $n$ sufficiently large, $n^{-1/2} \|X_n \| \leq K$, and hence, on the same event, $n^{-1/2} {\left\lVert}{\mathcal{Y}_{n}}{\right\rVert}\leq K$. By Lemma \[Lem:specnorm\] this implies that almost surely, for $n$ sufficiently large, $$\sup_{z\in{\mathbb{C}}:|z|\geq K+1}{\left\lVert}\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}-zI\right)^{-1}{\right\rVert}\leq 1.$$ To deal with $1+\delta\leq |z|\leq K+1$, we observe that $$\label{eq:inverseblockbnd}
\left(\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}-zI\right)^{-1}\right)^{[a,b]} = z^{(m-1)-\alpha}n^{-\alpha/2}X_{n}^{\alpha}\left(n^{-m/2}X_{n}^{m}-z^{m}I\right)^{-1}$$ by a block inverse computation, where $\alpha = (b-a) \;(\mathrm{mod}\ m)$. Thus, we have $$\begin{aligned}
&{\left\lVert}\left(\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}-zI\right)^{-1}\right)^{[a,b]}{\right\rVert}\\
& \quad \leq | z |^{(m-1)-\alpha} {\left\lVert}n^{-\alpha/2}X_{n}^{\alpha}{\right\rVert}{\left\lVert}\left(n^{-m/2}X_{n}^{m}-z^{m}I\right)^{-1}{\right\rVert}\\
& \quad \leq | z |^{(m-1)-\alpha} {\left\lVert}n^{-\alpha/2}X_{n}^{\alpha}{\right\rVert}\prod_{k=1}^{m}{\left\lVert}\left(n^{-1/2}X_{n}-ze^{2\pi k\sqrt{-1}/m}I\right)^{-1}{\right\rVert}.
\end{aligned}$$ We now bound $$\sup_{z\in{\mathbb{C}}:1+\delta<|z|<K+1}| z|^{(m-1)-\alpha}{\left\lVert}n^{-\alpha/2}X_{n}^{\alpha}{\right\rVert}\prod_{k=1}^{m}{\left\lVert}\left(n^{-1/2}X_{n}-ze^{2\pi k\sqrt{-1}/m}I\right)^{-1}{\right\rVert}.$$ Note that almost surely, for $n$ sufficiently large ${\left\lVert}n^{-\alpha/2}X_{n}^{\alpha}{\right\rVert}\leq K^\alpha \leq K^{m-1}$. Hence, we obtain $$\begin{aligned}
&\sup_{z\in{\mathbb{C}}:1+\delta<|z|<K+1}| z|^{(m-1)-\alpha}{\left\lVert}n^{-\alpha/2}X_{n}^{\alpha}{\right\rVert}\prod_{k=1}^{m}{\left\lVert}\left(n^{-1/2}X_{n}-ze^{2\pi k\sqrt{-1}/m}I\right)^{-1}{\right\rVert}\\
&\qquad\qquad \leq (K+1)^{m-1} K^{m-1} \sup_{z\in{\mathbb{C}}:1+\delta<|z|<K+1}\prod_{k=1}^{m}{\left\lVert}\left(n^{-1/2}X_{n}-ze^{2\pi k\sqrt{-1}/m}I\right)^{-1}{\right\rVert}\end{aligned}$$ almost surely, for $n$ sufficiently large. The bound for $$\sup_{z\in{\mathbb{C}}:1+\delta<|z|<K+1} \prod_{k=1}^{m}{\left\lVert}\left(n^{-1/2}X_{n}-ze^{2\pi k\sqrt{-1}/m}I\right)^{-1}{\right\rVert}$$ follows from Lemma \[Lem:LeastSingValAwayFromZero\] (taking $m = 1$). Returning to , we conclude that almost surely, for $n$ sufficiently large $$\sup_{z \in \mathbb{C} : 1 + \delta \leq |z| \leq K + 1} \left \| \left(\left(\frac{1}{\sqrt{n}}{\mathcal{Y}_{n}}-zI\right)^{-1}\right)^{[a,b]} \right\| \leq c$$ for some constant $c > 0$ (depending only on $\delta$ and $m$). Since $1 \leq a,b \leq m$ are arbitrary, the proof of property \[item:pw:invtbnd\] is complete.
For \[item:pw:isortopic\], [@Tout Theorem 1.4] yields that almost surely, for $n$ sufficiently large, $$\label{eq:910bnd}
\sup_{|z| \geq 5} \frac{1}{\sqrt{n}}{\left\lVert}\frac{{\mathcal{Y}_{n}}}{z}{\right\rVert}\leq \frac{9}{10}<1.$$ Thus, we expand the resolvent as a Neumann series to obtain $$\mathcal{G}_{n}(z)=-\frac{1}{z}\left(I+\sum_{k=1}^{\infty}\left(\frac{1}{\sqrt{n}}\frac{{\mathcal{Y}_{n}}}{z}\right)^{k}\right)=-\frac{1}{z}I-\sum_{k=1}^{\infty}\frac{\left(\frac{{\mathcal{Y}_{n}}}{\sqrt{n}}\right)^{k}}{z^{k+1}}.$$ Thus, we have almost surely, for $n$ sufficiently large, $$u^{*}\mathcal{G}_{n}(z)v=-\frac{1}{z}u^{*}v-\sum_{k=1}^{\infty}\frac{u^{*}\left(\frac{{\mathcal{Y}_{n}}}{\sqrt{n}}\right)^{k}v}{z^{k+1}}.$$ We will show that the series on the right-hand side converges to zero almost surely uniformly in the region $\{z \in {\mathbb{C}}: 5 \leq |z| \leq 6\}$. Indeed, from , the tail of the series is easily controlled. Thus, it suffices to show that, for each fixed integer $k \geq 1$, $$\left| u_n^\ast \left( \frac{1}{\sqrt{n}} \mathcal{Y}_n \right)^k v_n \right| = o_k(1).$$ But this follows from the block structure of $\mathcal{Y}_n$ and [@Tout Lemma 2.3].
We now extend this convergence to the region $\{z \in {\mathbb{C}}: |z| \geq 1 + \delta\}$. Let ${\varepsilon}> 0$. Let $M \geq 6$ be a constant to be chosen later. By Vitali’s convergence theorem (see, for instance [@BSbook Lemma 2.14]), it follows that $$\sup_{1 + \delta \leq |z| \leq M} \left| u_n^\ast \mathcal{G}_n(z) v_n + \frac{1}{z} u_n^\ast v_n \right| \longrightarrow 0$$ almost surely. In particular, almost surely, for $n$ sufficiently large, $$\label{eq:altiscond1}
\sup_{1 + \delta \leq |z| \leq M} \left| u_n^\ast \mathcal{G}_n(z) v_n + \frac{1}{z} u_n^\ast v_n \right| \leq {\varepsilon}.$$
Choose $M_1 > 0$ such that, for all $|z| \geq M_1$, $${\left\lVert}\left(-\frac{1}{z}\right)u^{*}v{\right\rVert}\leq \left|\frac{1}{z}\right|{\left\lVert}u^{*}{\right\rVert}{\left\lVert}v{\right\rVert}\leq \frac{\varepsilon}{2}.$$ Also there exists a constant $M_{2} > 0$ such that $$\sup_{|z| \geq M_2} {\left\lVert}u^{*}\mathcal{G}_{n}(z)v{\right\rVert}\leq \frac{\varepsilon}{2}$$ almost surely, for $n$ sufficiently large, by Lemma \[Lem:specnorm\] and [@Tout Theorem 1.4]. Take $M:=\max\{M_{1},M_{2}, 6\}$. Then almost surely, for $n$ sufficiently large, $$\label{equ:MtoInfProds}
\sup_{|z| \geq M} \left|u^{*}\mathcal{G}_{n}(z)v+\frac{1}{z}u^{*}v\right|\leq\varepsilon.$$
Combining and , we obtain almost surely, for $n$ sufficiently large, $$\sup_{|z| \geq 1 + \delta} \left|u^{*}\mathcal{G}_{n}(z)v+\frac{1}{z}u^{*}v\right|\leq\varepsilon.$$ Since ${\varepsilon}> 0$ was arbitrary, the proof is complete.
We also need the following lemma in order to prove Theorem \[thm:nomixedinpower\].
Let $\xi$ be a complex-valued random variable with mean zero, unit variance, finite fourth moment, and independent real and imaginary parts. For each $n \geq 1$, let $X_{n}$ be an $n \times n$ iid random matrix with atom variable $\xi$. Let $m$ be a positive integer. Then, for any fixed $\delta > 0$, the following statements hold.
1. Almost surely, for $n$ sufficiently large, the eigenvalues of $n^{-m/2}{X}_{n}^{m}$ are contained in the disk $\{z \in \mathbb{C} : |z| \leq 1 + \delta \}$. In particular, this implies that almost surely, for $n$ sufficiently large, the matrix $n^{-m/2}{X}_{n}^{m}- z I$ is invertible for every $z \in \mathbb{C}$ with $|z| > 1 + \delta$.
2. There exists a constant $c > 0$ such that almost surely, for $n$ sufficiently large, $$\sup_{z \in \mathbb{C} : |z| > 1 + \delta} {\left\lVert}\left(n^{-m/2}{X}_{n}^{m}-zI\right)^{-1} {\right\rVert}\leq c.$$
3. For each $n \geq 1$, let $u_n, v_n \in \mathbb{C}^{n}$ be deterministic unit vectors. Then $$\sup_{z \in \mathbb{C} : |z| > 1 + \delta} \left| u_n^\ast \left(n^{-m/2}{X}_{n}^{m}-zI\right)^{-1} v_n + \frac{1}{z} u_n^\ast v_n \right| \longrightarrow 0$$ almost surely as $n \to \infty$.
\[lem:PowerIsotropic\]
The proof of this lemma is similar to the proof of Corollary \[cor:ProductIsotropic\]; we omit the details.
With these results, we may proceed to the proofs of the results in Section \[Sec:RelatedResults\]. The proofs of Theorems \[Thm:NoOutlierInPowerPert\], \[thm:nomixedinpower\], and \[Thm:RepeatedProdOutliers\] follow the proofs of Theorems \[Thm:NoOutlierInProductPert\], \[thm:nomixed\], and \[thm:outliers\], respectively, verbatim, except for the following changes:
- Take $X_{n,1}=\cdots = X_{n,m} = X_{n}$,
- Replace all occurrences of Theorem \[thm:isotropic\] by Theorem \[Thm:RepeatedProdIsotropic\]
- Replace all occurrences of Corollary \[cor:ProductIsotropic\] by Lemma \[lem:PowerIsotropic\].
- The scaling factor of $\sigma$ needs to be replaced by $\sigma^{m}$.
Proof of Lemma \[lem:Truncate\] {#Sec:ProofOfTruncation}
===============================
In this section, we present the proof of Lemma \[lem:Truncate\].
Take $L_{0} := \sqrt{8 {\mathbb{E}}|\xi|^{4}}$. We begin by proving \[item:truncation:ii\]. Observe that $$\begin{aligned}
1 &= {\mathbb{E}}|\xi|^{2}\\
&={\mathbb{E}}|{\operatorname{Re}}(\xi)|^{2}+{\mathbb{E}}|{\operatorname{Im}}(\xi)|^{2}\\
&={\mathbb{E}}\left[|{\operatorname{Re}}(\xi)|^{2}{\ensuremath{\mathbf{1}_{\{|{\operatorname{Re}}(\xi)|\leq L/\sqrt{2}\}}}}\right]+{\mathbb{E}}\left[|{\operatorname{Re}}(\xi)|^{2}{\ensuremath{\mathbf{1}_{\{|{\operatorname{Re}}(\xi)|> L/\sqrt{2}\}}}}\right]\\
&\quad \quad +{\mathbb{E}}\left[|{\operatorname{Im}}(\xi)|^{2}{\ensuremath{\mathbf{1}_{\{|{\operatorname{Im}}(\xi)|\leq L/\sqrt{2}\}}}}\right]+{\mathbb{E}}\left[|{\operatorname{Im}}(\xi)|^{2}{\ensuremath{\mathbf{1}_{\{|{\operatorname{Im}}(\xi)|> L/\sqrt{2}\}}}}\right]\\
&={\text{Var}}({\operatorname{Re}}(\tilde{\xi}))+{\text{Var}}({\operatorname{Im}}(\tilde{\xi}))\\
&\quad \quad+\left|{\mathbb{E}}\left[{\operatorname{Re}}(\xi){\ensuremath{\mathbf{1}_{\{|{\operatorname{Re}}(\xi)|\leq L/\sqrt{2}\}}}}\right]\right|^{2}+{\mathbb{E}}\left[|{\operatorname{Re}}(\xi)|^{2}{\ensuremath{\mathbf{1}_{\{|{\operatorname{Re}}(\xi)|> L/\sqrt{2}\}}}}\right]\\
&\quad \quad +\left|{\mathbb{E}}\left[{\operatorname{Im}}(\xi){\ensuremath{\mathbf{1}_{\{|{\operatorname{Im}}(\xi)|\leq L/\sqrt{2}\}}}}\right]\right|^{2}+{\mathbb{E}}\left[|{\operatorname{Im}}(\xi)|^{2}{\ensuremath{\mathbf{1}_{\{|{\operatorname{Im}}(\xi)|> L/\sqrt{2}\}}}}\right],
\end{aligned}$$ which implies $$\begin{aligned}
1-{\text{Var}}(\tilde{\xi})=&\left|{\mathbb{E}}\left[{\operatorname{Re}}(\xi){\ensuremath{\mathbf{1}_{\{|{\operatorname{Re}}(\xi)|\leq L/\sqrt{2}\}}}}\right]\right|^{2}+{\mathbb{E}}\left[|{\operatorname{Re}}(\xi)|^{2}{\ensuremath{\mathbf{1}_{\{|{\operatorname{Re}}(\xi)|> L/\sqrt{2}\}}}}\right]\\
& +\left|{\mathbb{E}}\left[{\operatorname{Im}}(\xi){\ensuremath{\mathbf{1}_{\{|{\operatorname{Im}}(\xi)|\leq L/\sqrt{2}\}}}}\right]\right|^{2}+{\mathbb{E}}\left[|{\operatorname{Im}}(\xi)|^{2}{\ensuremath{\mathbf{1}_{\{|{\operatorname{Im}}(\xi)|> L/\sqrt{2}\}}}}\right].
\end{aligned}$$ Thus, using the fact that ${\operatorname{Re}}(\xi)$ and ${\operatorname{Im}}(\xi)$ both have mean zero (so, for example, ${\mathbb{E}}\left[{\operatorname{Re}}(\xi){\ensuremath{\mathbf{1}_{\{|{\operatorname{Re}}(\xi)|\leq L/\sqrt{2}\}}}}\right] = -{\mathbb{E}}\left[{\operatorname{Re}}(\xi){\ensuremath{\mathbf{1}_{\{|{\operatorname{Re}}(\xi)|> L/\sqrt{2}\}}}}\right]$) and then applying Jensen’s inequality, we obtain $$\begin{aligned}
|1-{\text{Var}}(\tilde{\xi})| &\leq 2{\mathbb{E}}[|{\operatorname{Re}}(\xi)|^{2}{\ensuremath{\mathbf{1}_{\{|{\operatorname{Re}}(\xi)|> L/\sqrt{2}\}}}}] +2{\mathbb{E}}[|{\operatorname{Im}}(\xi)|^{2}{\ensuremath{\mathbf{1}_{\{|{\operatorname{Im}}(\xi)|> L/\sqrt{2}\}}}}]\\
&\leq 2 {\mathbb{E}}[| \xi |^2 {\ensuremath{\mathbf{1}_{\{|\xi| > L /\sqrt{2}\}}}}] \\
&\leq \frac{4}{L^2} {\mathbb{E}}|\xi|^4.
\end{aligned}$$ This concludes the proof of \[item:truncation:ii\].
Property \[item:truncation:i\] follows easily from \[item:truncation:ii\] by the choice of $L_{0}$. Next we move onto the proof of \[item:truncation:iii\]. One can see that since ${\text{Var}}(\tilde{\xi})\geq \frac{1}{2}$, $$\begin{aligned}
\left|\hat{\xi}\right|
&\leq \frac{\left|{\operatorname{Re}}(\xi)\right|{\ensuremath{\mathbf{1}_{\{|{\operatorname{Re}}(\xi)|\leq L/\sqrt{2}\}}}}+{\mathbb{E}}\left[\left|{\operatorname{Re}}(\xi)\right|{\ensuremath{\mathbf{1}_{\{|{\operatorname{Re}}(\xi)|\leq L/\sqrt{2}\}}}}\right]}{\sqrt{{\text{Var}}(\tilde{\xi})}}\\
&\quad \quad \quad +\frac{\left|{\operatorname{Im}}(\xi)\right|{\ensuremath{\mathbf{1}_{\{|{\operatorname{Im}}(\xi)|\leq L/\sqrt{2}\}}}}+{\mathbb{E}}\left[\left|{\operatorname{Im}}(\xi)\right|{\ensuremath{\mathbf{1}_{\{|{\operatorname{Im}}(\xi)|\leq L/\sqrt{2}\}}}}\right]}{\sqrt{{\text{Var}}(\tilde{\xi})}}\\
&\leq 4L
\end{aligned}$$ almost surely.
For \[item:truncation:iv\], we observe that $\hat{\xi}$ has mean zero and unit variance by construction. Additionally, since the real and imaginary parts of $\hat{\xi}$ depend only on the real and imaginary parts of $\xi$ respectively, they are independent by construction. For the fourth moment, we use that ${\text{Var}}(\tilde{\xi})\geq \frac{1}{2}$ and Jensen’s inequality inequality to obtain $$\begin{aligned}
{\mathbb{E}}|\hat{\xi}|^{4} &\ll \frac{1}{{\text{Var}}(\tilde{\xi})^{2}}\left({\mathbb{E}}\left[\left|{\operatorname{Re}}(\xi)\right|^{4}{\ensuremath{\mathbf{1}_{\{|{\operatorname{Re}}(\xi)|\leq L/\sqrt{2}\}}}}\right]+{\mathbb{E}}\left[\left|{\operatorname{Im}}(\xi)\right|^{4}{\ensuremath{\mathbf{1}_{\{|{\operatorname{Im}}(\xi)|\leq L/\sqrt{2}\}}}}\right]\right)\\
& \ll {\mathbb{E}}|\xi|^4,
\end{aligned}$$ as desired.
Proof of Theorem \[Thm:LeastTruncSingValNonZero\] {#sec:singoutlier}
=================================================
This section is devoted to the proof of Theorem \[Thm:LeastTruncSingValNonZero\]. We begin with Lemma \[lemma:singoutlier\] below, which is based on [@N Theorem 4]. Throughout this section, we use $\sqrt{-1}$ for the imaginary unit and reserve $i$ as an index.
\[lemma:singoutlier\] Let $\mu$ be a probability measure on $[0, \infty)$, and for each $n \geq 1$, let $$\mu_n := \frac{1}{n} \sum_{i=1}^n \delta_{\lambda_{n,i}}$$ for some triangular array $\{ \lambda_{n,i} \}_{i \leq n}$ of nonnegative real numbers. Let $m_n$ by the Stieltjes transform of $\mu_n$ and $m$ be the Stieltjes transform of $\mu$, i.e., $$m_n(z) := \int \frac{ d \mu_n(x) }{ x - z }, \quad m(z) := \int \frac{ d \mu(x) }{x - z }$$ for all $z \in \mathbb{C}$ with ${\operatorname{Im}}(z) > 0$. Assume
1. $\mu_n \to \mu$ as $n \to \infty$,
2. there exists a constant $c > 0$ such that $\mu([0,c]) = 0$,
3. \[item:strate\] $\sup_{E \in [0,c]} \left| m_n(E + \sqrt{-1} n^{-1/2}) - m(E + \sqrt{-1} n^{-1/2}) \right| = o(n^{-1/2})$.
Then there exists a constant $n_0 \geq 1$ such that $\mu_n([0,c/2]) = 0$ for all $n > n_0$.
Observe that $$\begin{aligned}
{\operatorname{Im}}m_n(E + \sqrt{-1} n^{-1/2}) - {\operatorname{Im}}m(E + \sqrt{-1} n^{-1/2}) = \int \frac{ n^{-1/2} d (\mu_n - \mu)(x) }{ (E - x)^2 + n^{-1} }.\end{aligned}$$ From assumption \[item:strate\], we conclude that $$\sup_{E \in [0,c]} \left| \int \frac{ d (\mu_n - \mu)(x) }{ (E - x)^2 + n^{-1} } \right| = o(1).$$ We decompose this integral into two parts $$\int \frac{ d (\mu_n - \mu)(x) }{ (E - x)^2 + n^{-1} } = \int_0^c \frac{ d \mu_n(x) }{(E - x)^2 + n^{-1}} + \int_c^\infty \frac{ d (\mu_n - \mu)(x) }{ (E - x)^2 + n^{-1} },$$ where we used the assumption that $\mu([0,c]) = 0$.
Observe that $$\int_c^\infty \frac{ d (\mu_n - \mu)(x) }{ (E - x)^2 + n^{-1} } \longrightarrow 0$$ uniformly for any $E \in [0,c/2]$ by the assumption that $\mu_n \to \mu$. Therefore, it must be the case that $$\sup_{E \in [0,c/2]} \int_0^c \frac{ d \mu_n(x) }{(E - x)^2 + n^{-1}} \longrightarrow 0.$$ Take $n_0 \geq 1$ such that $$\label{eq:convcont}
\sup_{E \in [0,c/2]} \int_0^c \frac{ d \mu_n(x) }{(E - x)^2 + n^{-1}} \leq 1/2$$ for all $n \geq n_0$.
In order to reach a contradiction, assume there exists $n > n_0$ and $i \in [n]$ such that $\lambda_{n,i} \in [0,c/2]$. Then $$\begin{aligned}
\sup_{E \in [0,c/2]} \int_0^c \frac{ d \mu_n(x) }{(E - x)^2 + n^{-1}} &= \sup_{E \in [0,c/2]} \frac{1}{n} \sum_{j=1}^n \frac{1}{(E - \lambda_{n,j})^2 + n^{-1}} \\
&\geq \sup_{E \in [0,c/2]} \frac{1}{n} \frac{1}{(E - \lambda_{n,i})^2 + n^{-1}} \\
&\geq 1, \end{aligned}$$ a contradiction of . We conclude that $\mu_n([0,c/2]) = 0$ for all $n > n_0$.
With Lemma \[lemma:singoutlier\] in hand, we are now prepared to prove Theorem \[Thm:LeastTruncSingValNonZero\]. The proof below is based on a slight modification to the arguments from [@N; @N2]. As such, in some places we will omit technical computations and only provide appropriate references and necessary changes to results from [@N; @N2].
Fix $\delta > 0$. It suffices to prove that $$\label{eq:singvalshow1}
\inf_{1 + \delta \leq |z| \leq 6} s_{mn} \left( \frac{1}{\sqrt{n}} \mathcal{Y}_n - z I \right) \geq c$$ and $$\label{eq:singvalshow2}
\inf_{|z| > 6} s_{mn} \left( \frac{1}{\sqrt{n}} \mathcal{Y}_n - zI \right) \geq c'$$ with overwhelming probability for some constants $c, c' > 0$ depending only on $\delta$.
The second bound follows by Lemma \[Lem:specnorm\]. Indeed, a bound on the spectral norm of $\mathcal{Y}_n$ (which follows from standard bounds on the spectral norms of $X_{n,k}$; see, for example, [@Tout Theorem 1.4]) gives $$\| \mathcal{Y}_n \| \leq 3 \sqrt{n}$$ with overwhelming probability. The bound in then follows by applying Lemma \[Lem:specnorm\].
We now turn to the bound in . To prove this bound, we will use Lemma \[lemma:singoutlier\]. Let $\mu_{n,z}$ be the empirical spectral measure constructed from the eigenvalues of $$\left( \frac{1}{\sqrt{n}} \mathcal{Y}_n - z I \right) \left( \frac{1}{\sqrt{n}} \mathcal{Y}_n - z I \right)^\ast.$$ From [@N2 Theorem 2.6], for all $|z| \geq 1 + \delta$, there exists a probability measure $\mu_z$ supported on $[0, \infty)$ such that $\mu_{n,z} \to \mu_z$ with overwhelming probability. Moreover, from [@Bcirc Lemma 4.2] there exists a constant $c > 0$ (depending only on $\delta$) such that $\mu_z([0,c]) = 0$ for all $|z| \geq 1 + \delta$. Lastly, condition \[item:strate\] in Lemma \[lemma:singoutlier\] follows for all $1 + \delta \leq |z| \leq 6$ with overwhelming probability from [@N Theorem 5]. Applying Lemma \[lemma:singoutlier\], we conclude that $$s_{mn} \left( \frac{1}{\sqrt{n}} \mathcal{Y}_n - z I \right) \geq c/2$$ with overwhelming probability uniformly for all $1+ \delta \leq |z| \leq 6$. The bound for the infimum can now be obtained by a simple net argument and Weyl’s inequality . The proof of Theorem \[Thm:LeastTruncSingValNonZero\] is complete.
Proof of Lemma \[Lem:OnlyOne\] {#Sec:OnlyOne}
==============================
This section is devoted to the proof of Lemma \[Lem:OnlyOne\].
The proof proceeds inductively. We begin with a graph only containing the vertex $(1,1)$ and then add vertices and edges sequentially with time. First, edge 1 is added; it will span from $(1,1)$ to $(2,i_{2})$. Next, edge 2 is added and will span from $(2,i_{2})$ to $(3,i_{3})$, and so on. We use induction to prove that at each time step $t$, there is only one possible choice for $i_{t+1}$, resulting in a unique canonical graph with maximal height $k/2$ and in which each edge is parallel to exactly one other edge.
The edge starting at vertex $(1,1)$ can either be of type II (terminating on $(2,1)$) or of type I (terminating on $(2,2)$). By way of contradiction, assume the edge is type II. Since $G$ still has $k/2-1$ more height coordinates left to reach, it would require at least $k/2-1$ type I edges to reach hight coordinate $k/2$. Since each edge must be parallel to exactly one other edge, at some point there must be a type II edge, returning to a height coordinate previously visited. This edge will also need to be parallel to another edge. Counting all pairs of parallel edges shows that $G$ must have at least $k+1$ more edges, a contradiction. Hence the edge starting at vertex $(1,1)$ must be of type I.
Assume that all edges up to time coordinate $t$, where $1\leq t < k/2-1$, are type I edges. Then $G$ must have an edge starting at vertex $(t+1,t+1)$. This edge can either of type I or type II. In order to reach a contradiction, assume that the edge is type II. Then $G$ must have at least $k/2-t-1$ more type I edges in order to reach the height $k/2$, and $G$ has exactly $k-t-1$ more edges to be added. Visiting each unvisited height coordinate would require at least $k/2-t-1$ more type I edges, and at some point after visiting new height coordinates, $G$ must return to a smaller hight coordinate, resulting in a type II edge. None of these edges could be parallel to any previous edges. Thus, overall $G$ would need to have at least $k-t+1$ more edges, a contradiction to the fact that $G$ must have exactly $k-t-1$ more edges. We conclude that each edge of $G$ must be type I until the hight coordinate $k/2$ is reached. Namely, we have vertices $(1,1),\ldots, (k/2,k/2)$.
At this point $G$ must have an edge starting at vertex $(k/2,k/2)$. Note that $G$ has $k/2-1$ edges up to this point, none of which are parallel to any other edge. $G$ must have edges parallel to the edges previously introduced and $G$ has exactly $k/2+1$ edges remaining to do so. Since there are no remaining unvisited height coordinates, the edge which starts at vertex $(k/2,k/2)$ must terminate at $(k/2+1,i_{k/2+1})$ for some $1\leq i_{k/2+1}\leq k/2$, resulting in the first type II edge. See Figure \[fig:proofExample\] for a visual representation of the graph up to this point.
We now claim that this first type II edge must in fact terminate at $(k/2+1, 1)$. By way of contradiction, suppose this edge terminates at vertex $(k/2+1,i)$ for any $1<i\leq \frac{k}{2}$. Since this is the first type II edge, it cannot be parallel to any other previously drawn edge. Up to this point $G$ has $k/2$ edges drawn and $k/2$ edges remaining to be drawn. Since all edges are by assumption to be parallel to exactly one other, a simple counting argument reveals that each edge drawn from this point on must be parallel to an existing edge. Since there is only one edge which starts at height coordinate $i$, we must now draw the edge starting at $(k/2+1,i)$ and terminating at $(k/2+2,i+1)$. By continuing this argument inductively, we must draw edges which start at $(k/2+j+1, i+j)$ and terminate at vertex $(k/2+j+2, i+j+1)$ for $0\leq j\leq k/2-i$, until the height coordinate $k/2$ is reached again. Now, since $k-i+1<k$, we must draw at least one more edge, and this edge must start at vertex $(k-i,k/2)$. In order to draw an edge parallel to an existing edge, this edge must terminate at vertex $(k-i+1, i)$. However, if we do this, we must now draw an edge parallel to an existing edge which starts a height coordinate $i$, a contradiction because the only previous edges which began at height coordinate $i$ are parallel to one another and there cannot be three edges parallel. This concludes the proof of the claim.
By the previous claim, the first type II edge must terminate at $(k/2+1, 1)$. Again, since this is the first type II edge it cannot be parallel to any other edge. Up to this point $G$ has $k/2$ edges drawn and $k/2$ edges remaining to be drawn. Since all edges are by assumption parallel to exactly one other, a simple counting argument reveals that each edge drawn from this point on must be parallel to an existing edge. Since there is only one edge which starts at height coordinate $1$, we must now draw the edge starting at $(k/2+1,1)$ and terminating at $(k/2+2,2)$. By continuing this argument inductively, we must draw edges which start at $(k/2+j+1, j+1)$ and terminate at vertex $(k/2+j+2, j+2)$ for $0\leq j\leq k/2-1$, until the height coordinate $k/2$ is reached again. Up to this point, $k-1$ edges of $G$ have been drawn and we must draw one more edge which starts at vertex $(k,k/2)$. Since there is only one previous edge in the graph which starts at height coordinate $k/2$, the final edge must terminate at height coordinate $1$ and all edges are parallel to exactly one other edge. This results in the vertex set $$V=\{(1,1),\;(2,2),\;\dots,(k/2,k/2),\;(k/2+1,1),\;(k/2+2,2),\dots (k,k/2),\;(k+1,1)\}.$$ The corresponding canonical $m$-colored $k$-path graph would be $$G^{1}(1,2,\ldots,k/2,1,2,\ldots,k/2, 1),$$ and the proof is complete.
Useful inequalities
===================
For $X = (x_{1},x_{2},\ldots,x_{N})^{T}$ iid standardized complex entries, $B$ an $N\times N$ Hermitian nonnegative definite matrix, we have, for any $p\geq 1$, $${\mathbb{E}}\left|X^{*}BX\right|^{p}\leq K_{p}\left(\left(\emph{tr} B\right)^{p}+{\mathbb{E}}|x_{1}|^{2p}\emph{tr}B^{p}\right).$$ where $K_{p}>0$ depends only on $p$. \[Lem:BilinearForms\]
Let $A$ and $B$ be $k\times k$ matrices with ${\left\lVert}A{\right\rVert}, {\left\lVert}B{\right\rVert}= O(1)$. Then $$\left| \det(A)-\det(B)\right| \ll_{k} {\left\lVert}A-B{\right\rVert}.$$ \[Lem:normtodet\]
Let $A$ be a square matrix that satisfies ${\left\lVert}A{\right\rVert}\leq \mathcal{K}$. Then $${\left\lVert}\left(A-zI\right)^{-1}{\right\rVert}\leq \frac{1}{\varepsilon}$$ for all $z\in{\mathbb{C}}$ with $|z|\geq \mathcal{K}+\varepsilon$. \[Lem:specnorm\]
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[^1]: S. O’€™Rourke has been supported in part by NSF grant ECCS-1610003. P. Wood was partially supported by National Security Agency (NSA) Young Investigator Grant number H98230-14-1-0149.
[^2]: Here and in the sequel, we use Pi (product) notation for products of matrices. To avoid any ambiguity, if $M_1, \ldots, M_m$ are $n \times n$ matrices, we define the order of the product $$\prod_{k=1}^m M_k := M_1 \cdots M_m.$$ In many cases, such as in Theorem \[thm:ORSV\], the order of matrices in the product is irrelevant by simply relabeling indices.
[^3]: This fact can easily be deduced from Sylvester’s determinant theorem; see .
[^4]: The hypothesis of Vitali’s Convergence Theorem are satisfied almost surely, for $n$ sufficiently large, by parts \[item:invertible\] and \[item:invtbnd\] of Theorem \[thm:isotropic\]. In addition, one can check that $(u_n)^\ast \mathcal{G}_n(z) v_n$ is holomorphic in the region $\{z \in {\mathbb{C}}: |z| > 1 + \delta\}$ almost surely, for $n$ sufficiently large, using the resolvent identity .
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---
abstract: 'Matching pursuit, especially its orthogonal version and other variations, is a greedy algorithm widely used in signal processing, compressed sensing, and sparse modeling and approximation. Inspired by constrained sparse signal recovery, this paper proposes a constrained matching pursuit algorithm and develops conditions for exact support and vector recovery on constraint sets via this algorithm. We show that exact recovery via constrained matching pursuit not only depends on a measurement matrix but also critically relies on a constraint set. We thus identify an important class of constraint sets, called coordinate projection admissible set, or simply CP admissible sets. This class of sets includes the Euclidean space, the nonnegative orthant, and many others arising from various applications; analytic and geometric properties of these sets are established. We then study exact vector recovery on convex, CP admissible cones for a fixed support. We provide sufficient exact recovery conditions for a general fixed support as well as necessary and sufficient recovery conditions for a fixed support of small size. As a byproduct of our results, we construct a nontrivial counterexample to the necessary conditions of exact vector recovery via the orthogonal matching pursuit given by Foucart, Rauhut, and Tropp, when the a given support is of size three. Moreover, by making use of cone properties and conic hull structure of CP admissible sets and constrained optimization techniques, we also establish sufficient conditions for uniform exact recovery on CP admissible sets in terms of the restricted isometry-like constant and the restricted orthogonality-like constant.'
author:
- 'Jinglai Shen[^1] and Seyedahmad Mousavi'
title: Exact Support and Vector Recovery of Constrained Sparse Vectors via Constrained Matching Pursuit
---
Introduction
============
Sparse models and representations find broad applications in numerous fields of contemporary interest [@ElderK_book2012], e.g., signal and image processing, high dimensional statistics, compressed sensing, and machine learning. Effective recovery of sparse signals from a few measurements poses challenging theoretical and numerical questions. A variety of sparse recovery schemes have been proposed and studied, including the basis pursuit and its extensions, greedy algorithms, and thresholding based algorithms [@FoucartRauhut_book2013; @ShenMousavi_SIOPT18].
Originally introduced in signal processing and statistics, matching pursuit [@MallatZ_TSP93], and particularly the orthogonal matching pursuit (OMP) [@PRKrishnaprasad_Asilomar93], is a greedy algorithm widely used in sparse signal recovery. At each step, the OMP uses the current target vector to select an additional “best” index via coordinate-wise optimization and adds it to the target support, and then updates the target vector over the new support via optimal fitting of a measurement vector. The deterministic and statistical performance of the OMP has been extensively studied in the literature [@CaiWang_TIT2011; @Tropp_ITI04; @TroppGilbert_TIT2007; @Zhang_JMLR09; @Zhang_TIT2011]. In particular, the exact support and vector recovery via the OMP has been characterized in term of the restricted isometry constant with extensions to noisy measurements [@MoS_TIT12; @WenZWTM_SIT16]. Besides, many variations and extensions of the OMP have been developed in order to improve the recovery accuracy, effectiveness, and robustness under noise and errors; representative examples of these variations and extensions include compressive sampling matching pursuit [@NeedellTropp_ACHA09; @NeedellV_JSTSP2010], simultaneous OMP [@TroppGS_SP06], stagewise OMP [@DonohoTDS_TIT12], subspace pursuit [@DaiMilenk_TIT2009], generalized OMP [@WangKS_TSP2012], grouped OMP [@SwirszczAL_NIPS09], and multipath matching pursuit [@KwonWS_TIT14], just to name a few; see [@FoucartRauhut_book2013] and the references therein for more details. Sparse signals arising from diverse applications are subject to constraints, for example, the nonnegative constraint in nonnegative factorization in signal and image processing [@RrucksteinEZ_TIT2008], the polyhedral constraint in index tracking problems in finance [@XuLX_OMS16], and the monotone or shape constraint in order statistics and shape constrained estimation [@ShenLebair_Auto15; @ShenWang_SICON11]. Hence, constrained sparse recovery has attracted increasing interest from different areas, such as machine learning and sparse optimization [@BahmaniRaj_JMLR2013; @BeckE_SIOPT2013; @BeckH_MOR15; @FoucartKoslicki_ISPL14; @IDP_ISP17; @Locatello_NIPS2017; @MouShen_COCV18; @WangXTang_TSP11]. While matching pursuit, particularly the OMP and its variations or extensions, has been extensively studied on $\mathbb R^N$, its constrained version has received much less attention, especially the exact recovery on a general constraint set; exceptions include [@RrucksteinEZ_TIT2008] where the uniqueness of the OMP recovery on the nonnegative orthant is considered. Inspired by the constrained sparse recovery, this paper proposes a constrained matching pursuit algorithm for a general constraint set, and develops conditions for exact support and vector recovery on constraint sets via this algorithm. Similar to the OMP, the constrained matching pursuit algorithm selects a new optimal index by solving a constrained coordinate-wise optimization problem at each step, and then updates its target vector over the updated support by solving another constrained optimization problem for the best fitting of a measurement vector. We show that exact recovery via the constrained matching pursuit not only depends on a measurement matrix but also critically relies on a constraint set. This motivates us to introduce an important class of constraint sets, called coordinate projection admissible sets, or simply CP admissible sets. This class of sets includes the Cartesian product of arbitrary copies of $\mathbb R$, $\mathbb R_+$, and $\mathbb R_-$, and many others arising from applications. We establish analytic and geometric properties of these sets to be used for exact recovery analysis. We then study exact vector recovery on convex, CP admissible cones for a fixed support. When a fixed support has the size of two and three, we develop necessary and sufficient recovery conditions; when the support size is large, we provide sufficient exact recovery conditions. As a byproduct of our results, we construct a nontrivial counterexample to the necessary conditions of exact vector recovery via the OMP given by Foucart, Rauhut, and Tropp, when the size of a given support is three (cf. Section \[subsect:counterexample\_S=3\]). Moreover, we establish sufficient conditions for uniform exact recovery on general CP admissible sets in terms of the restricted isometry-like constant and the restricted orthogonality-like constant, by leveraging cone properties and conic hull structure of CP admissible sets, the positive homogeneity of the aforementioned constants, as well as constrained optimization techniques. Its extensions are also discussed.
The rest of the paper is organized as follows. Section \[sect:Constrained\_MP\] presents the constrained matching pursuit algorithm and discusses underlying optimization problems in this algorithm. Section \[sect:exact\_supp\_recover\] studies basic properties of exact support recovery via constrained matching pursuit. In Section \[sect:CP\_admissible\_set\], the CP admissible sets are introduced, and their properties are established. Section \[sect:exact\_vector\_recovery\] is concerned with the exact vector recovery of convex, CP admissible cones for a fixed support. In Section \[sect:suff\_cond\_exact\_recovery\], sufficient conditions for uniform exact recovery on general convex, CP admissible sets are derived with conclusions made in Section \[sect:conclusion\].
[*Notation*]{}. Let $A$ be an $m\times N$ real matrix. For any index set ${\mathcal S}\subseteq \{1, \ldots, N\}$, let $|{\mathcal S}|$ denote the cardinality of ${\mathcal S}$, $\mathcal S^c$ denote the complement of ${\mathcal S}$, and $A_{\bullet{\mathcal S}}$ be the matrix formed by the columns of $A$ indexed by elements of ${\mathcal S}$. We write the $i$th column of $A$ as $A_{\bullet i}$ instead of $A_{\bullet \{i\} } $. Further, $\mathbb R^N_+$ and $\mathbb R^N_{++}$ denote the nonnegative and positive orthants of $\mathbb R^N$ respectively, and $\mathbf e_j$ denotes the $j$th column of the $N\times N$ identity matrix. For $a \in \mathbb R$, let $a_+:=\max(a, 0) \ge 0$ and $a_-:=\max(-a, 0) \ge 0$. For a given $x \in \mathbb R^N$, ${ \mbox{supp} }(x)$ denotes the support of $x$, i.e., ${ \mbox{supp} }(x)=\{ i \, | \, x_i \ne 0 \}$. The standard inner product on $\mathbb R^n$ is denoted by $\langle \cdot, \cdot \rangle$. When a minimization problem has multiple solutions, $x \in {\mbox{Argmin}}$ denotes an arbitrary optimal solution; if there is a unique optimal solution, then we use $x ={\operatornamewithlimits{\arg\min}}$. Let $\mbox{cone}(S)$ denote the conic hull of a set $S$ in $\mathbb R^N$, i.e., the collection of nonnegative combinations of finitely many vectors in $S$. We always assume that a cone in $\mathbb R^n$ contains the zero vector. For two sets $A$ and $B$, $A \subseteq B$ means that $A$ is a subset of $B$ and $A$ possibly equals to $B$, while $A \subset B$ means that $A$ is a proper subset of $B$. For $K \in \mathbb N$, let $\Sigma_K$ be the set of all vectors $x \in \mathbb R^N$ satisfying $|{ \mbox{supp} }(x)|\le K$. For $u, v \in \mathbb R^n$, $u \perp v$ stands for the orthogonality of $u$ and $v$, i.e., $u^T v =0$.
Constrained Matching Pursuit: Algorithm and Preliminary Results {#sect:Constrained_MP}
===============================================================
Consider the following constrained sparse recovery problem: $$\label{eqn:constrained_L0}
\min_{x \in \mathbb R^N} \, \| x \|_0 \qquad \mbox{subject to } \quad A x =y, \quad x \in {\mathcal P},$$ where $\| x\|_0:=|{ \mbox{supp} }(x)|$, $A \in \mathbb R^{m\times N}$ with $N> m$, $y \in \mathbb R^m$, and ${\mathcal P}$ is a closed constraint set in $\mathbb R^N$. Throughout this paper, we assume that ${\mathcal P}$ contains the zero vector, there is no measurement error so that $y$ is in the range of $A$, and each column of $A$ is nonzero, i.e., $\| A_{\bullet i}\|_2>0$ for each $i=1, \ldots, N$. To solve the problem (\[eqn:constrained\_L0\]), we introduce the constrained matching pursuit scheme given below.
\[algo:constrained\_MP\] Input: $A \in \mathbb R^{m\times N}$, $y \in \mathbb R^m$, ${\mathcal P}\subseteq \mathbb R^N$, and a stopping criteria
Initialize: $k=0$, $x^0=0$, and ${\mathcal J}_0=\emptyset$
$g^*_j = \min_{t \in \mathbb R} \| y - A (x^k + t \, \mathbf e_j ) \|^2_2 \ \ \mbox{ subject to } \ x^k + t \, \mathbf e_j \in \mathcal P$, $\forall \, j =1, \ldots, N$
$j^*_{k+1} \in {\mbox{Argmin}}_{j \in \{1, \ldots, N \} } \, g^*_j$ ${\mathcal J}_{k+1} = {\mathcal J}_k \cup \{ j^*_{k+1}\}$
$x^{k+1} \in {\mbox{Argmin}}_{w \in {\mathcal P}, \ { \mbox{supp} }(w) \subseteq {\mathcal J}_{k+1} } \, \| A w - y\|^2_2$
$k\leftarrow k+1$
Output: $x^* = x^k$
At each step in the constrained matching pursuit algorithm, two constrained optimization problems are solved. The first problem, given in Line 4 of Algorithm \[algo:constrained\_MP\], is a constrained coordinate-wise minimization problem; the second problem, given in Line 7 of Algorithm \[algo:constrained\_MP\], is a minimization problem on the constraint set ${\mathcal P}$ subject to an additional support constraint ${ \mbox{supp} }(w) \subseteq {\mathcal J}_{k+1}$. In what follows, we discuss these two underlying problems and their solution properties.
For a given $x \in \mathcal P$ and an index $j=1, \ldots, N$, the first minimization problem can be written as $$(\mbox{P}_{x, j}): \quad \min_{t \in \mathbb R} \| y - A (x + t \, \mathbf e_j ) \|^2_2 \qquad \mbox{ subject to } \quad x + t \, \mathbf e_j \in \mathcal P.$$ Since $\mathcal P$ is closed, it is easy to verify that the constraint set of $(\mbox{P}_{x, j})$ given by $$\label{eqn:interval_j}
\mathbb I_j(x) \, := \, \big \{ \, t \in \mathbb R \, | \, x + \mathbf e_j t \in \mathcal P \, \big\}$$ is a closed set in $\mathbb R$. Besides, for any $x \in \mathcal P$ and $j=1, \ldots, N$, we have $0 \in \mathbb I_{j}(x)$, and $(\mbox{P}_{x, j})$ attains an optimal solution because $\| A_{\bullet j} \|_2 >0$. Motivated by the fact that $y$ is given by $y=A u$ for some $u\in {\mathcal P}$, we define, for any $u, v \in \mathcal P$ and $j=1, \ldots, N$, $$f^*_j(u, v) \, := \, \min_{t \in \mathbb I_j(v)} \| A u - A (v + t \, \mathbf e_j ) \|^2_2 = \min_{t \in \mathbb I_j(v)} \| A( u -v) - t A_{\bullet j} \|^2_2.$$
A particularly interesting and important case is when $\mathcal P$ is closed and convex. In this case, for any $v\in {\mathcal P}$ and any index $j$, $\mathbb I_j(v)$ is also closed and convex and thus is a closed interval in $\mathbb R$. Letting $
a_j(v):=\inf \, \mathbb I_j(v)$ and $b_j(v) := \sup \, \mathbb I_j(v)$, where $a_j(v)\in \mathbb R_-\cup\{-\infty\}$ and $b_j(v) \in \mathbb R_+\cup \{+\infty\}$, $\mathbb I_j(v)$ can be written as $\mathbb I_j(v)=[a_j(v), b_j(v)]$. For any given $u, v \in {\mathcal P}$, since $A_{\bullet j} \ne 0$, the minimization problem $\min_{t \in [a_j(v), b_j(v)]} \|A(u-v) - t \, A_{\bullet j}\|^2_2$ attains a unique optimal solution $$t^*_j(u, v) \, = \, \left\{\begin{array}{llc} a_j(v), & \mbox{ if } \ {\widetilde}t_j(u, v) \le a_j(v) \\ {\widetilde}t_j(u, v), & \mbox{ if } \ {\widetilde}t_j(u, v) \in [a_j(v), b_j(v)] \\ b_j(v), & \mbox{ if } \ {\widetilde}t_j(v) \ge b_j(v) \end{array} \right.,$$ where $$\label{eqn:def_tilda_t}
{\widetilde}t_j(u, v) \, := \, \langle A(u-v), A_{\bullet j} \rangle/\|A_{\bullet j}\|^2_2.$$ Consequently, $$\label{eqn:f*_j_convex}
f^*_j(u, v) \, = \, \left\{\begin{array}{llc} \|A(u-v)\|^2_2 - \| A_{\bullet j}\|^2_2 \cdot [ 2 a_j(v) {\widetilde}t_j(u, v) - a^2_j(v)], & \mbox{ if }\ {\widetilde}t_j(u, v) \le a_j(v) \\ \|A(u-v)\|^2_2 -\| A_{\bullet j}\|^2_2 \cdot {\widetilde}t^2_j(u, v), & \mbox{ if } \ {\widetilde}t_j(u, v) \in [a_j(v), b_j(v)] \\ \|A(u-v)\|^2_2 - \| A_{\bullet j}\|^2_2 \cdot [ 2 b_j(v) {\widetilde}t_j(u, v) - b^2_j(v)], & \mbox{ if } \ {\widetilde}t_j(v) \ge b_j(v) \end{array} \right.$$ For illustration, we show the expressions of $f^*_j(u, v)$ for two special cases below. (i) $\mathbb I_j(v)=\mathbb R$, i.e., $a_j(v)=-\infty$ and $b_j(v)=+\infty$. In this case, $$\label{eqn:f*_RN}
f^*_j(u, v) \, = \, \|A(u-v)\|^2_2 -\| A_{\bullet j}\|^2_2 \cdot {\widetilde}t^2_j(u, v).$$
\(ii) $\mathbb I_j(v)=\mathbb R_+$, i.e., $a_j(v)=0$ and $b_j(v)=+\infty$. In this case, $$\label{eqn:f*_RN+}
f^*_j(u, v) \, = \, \|A(u-v)\|^2_2 -\| A_{\bullet j}\|^2_2 \cdot \big([{\widetilde}t_j(u, v)]_+\big)^2.$$
We next study the constrained minimization problem pertaining to that in Line 7 of Algorithm \[algo:constrained\_MP\] for a given $y \in \mathbb R^m$ and a given index set ${\mathcal J}\subseteq\{1, \ldots, N\}$: $$\label{eqn:P_y_J}
(\mbox{P}_{y, {\mathcal J}}): \quad \min_{w \in \mathbb R^N} \| A w - y \|^2_2 \qquad \mbox{ subject to } \quad w \in \mathcal P \quad \mbox{ and } \quad { \mbox{supp} }(w) \subseteq {\mathcal J}.$$ Since ${\mathcal P}$ contains the zero vector, $(\mbox{P}_{y, {\mathcal J}})$ is always feasible for any index set ${\mathcal J}$, even if ${\mathcal J}$ is empty. Note that we always assume that the minimization problem in Line 7 of Algorithm \[algo:constrained\_MP\] has a solution in each step. Moreover, certain solution existence and uniqueness results for $(\mbox{P}_{y, {\mathcal J}})$ can be established under mild assumptions on $A$ and ${\mathcal P}$ as shown below.
\[lemma:sol\_existence\] Let the set $\mathcal P \subseteq \mathbb R^N$ and the matrix $A \in \mathbb R^{m\times N}$. The following hold:
- If $A \mathcal P$ is closed, then for any index set $\mathcal J$ and any $y \in \mathbb R^m$, $(\mbox{P}_{y, {\mathcal J}})$ attains an optimal solution.
- If ${\mathcal P}$ is closed and an index set ${\mathcal I}$ is such that $A_{\bullet {\mathcal I}}$ has linearly independent columns, then $(\mbox{P}_{y, {\mathcal I}})$ has an optimal solution. If, in addition, $\mathcal P$ is convex, then such an optimal solution is unique.
\(i) Given any $y \in \mathbb R^m$ and any index set $\mathcal J$, $(\mbox{P}_{y, {\mathcal J}})$ is equivalent to $\min_{w \in {\mathcal P}\cap \mathcal V} \|A w - y\|^2_2$, where $\mathcal V:=\{ z=(z_{\mathcal J}, z_{{\mathcal J}^c}) \, | \, z_{{\mathcal J}^c} = 0\}$ is a subspace of $\mathbb R^N$. Note that $A \mathcal V$ is a subspace and thus closed. Since $A ({\mathcal P}\cap \mathcal V)=(A{\mathcal P}) \cap (A \mathcal V)$ and $A{\mathcal P}$ is closed, $A ({\mathcal P}\cap \mathcal V)$ is also closed. Moreover, the function $\| \cdot \|^2_2$ is continuous, coercive, and bounded below on $\mathbb R^m$. By [@MouShen_COCV18 Lemma 4.1], $(\mbox{P}_{y, {\mathcal J}})$ has an optimal solution.
\(ii) Suppose ${\mathcal P}$ is closed. Then the set ${\mathcal P}_{\mathcal J}:={\mathcal P}\cap \mathcal V$ is closed for any index set ${\mathcal J}$, where $\mathcal V$ is the subspace associated with ${\mathcal J}$ defined in the proof for (i). Since $A_{\bullet {\mathcal I}}$ has linearly independent columns, it is easy to see that $\{A_{\bullet {\mathcal I}} \, w_{\mathcal I}\, | \, (w_{\mathcal I}, 0) \in {\mathcal P}_{\mathcal I}\}$ is closed. By the similar argument for (i), $(\mbox{P}_{y, {\mathcal I}})$ attains an optimal solution. If, in addition, ${\mathcal P}$ is convex, then $(\mbox{P}_{y, {\mathcal I}})$ is a convex optimization problem with a strongly convex objective function in $w_{\mathcal I}$. This yields a unique optimal solution for any $y\in \mathbb R^m$.
Typical constraint sets $\mathcal P$ satisfying the closedness assumption given in statement (i) of Lemma \[lemma:sol\_existence\] for an arbitrary matrix $A \in \mathbb R^{m\times N}$ include compact sets and polyhedral sets, e.g., $\mathbb R^N$ and $\mathbb R^N_+$. Also see Corollary \[coro:sol\_existence\_CP\_adm\] in Section \[sect:CP\_admissible\_set\] for a general class of sets on which $(\mbox{P}_{y, {\mathcal J}})$ attains a solution.
When $(\mbox{P}_{y, {\mathcal J}})$ is a convex optimization problem (whose ${\mathcal P}$ is closed and convex), well developed numerical solvers can be exploited to solve $(\mbox{P}_{y, {\mathcal J}})$, e.g., the gradient projection method and primal-dual schemes, provided that it has a solution. In particular, the necessary and sufficient optimality condition for an optimal solution $w^*=(w^*_{\mathcal J}, 0)\in {\mathcal P}$ of $(\mbox{P}_{y, {\mathcal J}})$ is given by the variational inequality (VI): $\langle A^T_{\bullet {\mathcal J}}( A_{\bullet {\mathcal J}} w^*_{\mathcal J}-y), w_{\mathcal J}- w^*_{\mathcal J}\rangle \ge 0$ for all $(w_{\mathcal J}, 0) \in {\mathcal P}$. When ${\mathcal P}$ is a closed convex cone, the above VI is equivalent to the cone complementarity problem: $ \mathcal C \ni w^*_{\mathcal J}\perp A^T_{\bullet {\mathcal J}}( A_{\bullet {\mathcal J}} w^*_{\mathcal J}-y) \in \mathcal C^*$, where the closed convex cone $\mathcal C:=\{ w_{\mathcal J}\, | \, (w_{{\mathcal J}}, 0) \in {\mathcal P}\}$ and $\mathcal C^*$ denotes the dual cone of $\mathcal C$. Especially, when ${\mathcal P}=\mathbb R^N_+$, it is further equivalent to the linear complementarity problem (LCP): $0 \le w^*_{\mathcal J}\perp A^T_{\bullet {\mathcal J}}( A_{\bullet {\mathcal J}} w^*_{\mathcal J}-y) \ge 0$. These optimality conditions will be invoked in the subsequent sections.
Exact Support Recovery via Constrained Matching Pursuit {#sect:exact_supp_recover}
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Fix $K \in \mathbb N$ with $K < N$ throughout the rest of the paper. For a given $z \in \Sigma_K \cap \mathcal P$, let $\big( (x^k, j^*_k, {\mathcal J}_k) \big)_{k \in \mathbb N}$ be a sequence of triples generated by Algorithm \[algo:constrained\_MP\] with $y=A z$ starting from $x^0=0$ and ${\mathcal J}_0=\emptyset$, where ${\mathcal J}_{k+1} ={\mathcal J}_k \cup \{ j^*_{k+1} \}$ such that ${\mathcal J}_0 \subseteq {\mathcal J}_1 \subseteq \cdots \subseteq {\mathcal J}_k \subseteq \cdots$. Note that there are multiple sequences in general for a given $z$, since the optimization problems in Lines 5 and 7 of Algorithm \[algo:constrained\_MP\] may attain multiple solutions at each step. For example, if the underlying optimization problem (\[eqn:P\_y\_J\]) is a convex minimization problem with non-unique solutions for some ${\mathcal J}={\mathcal J}_k$ and $y=Az$, then it attains infinitely many $x^k$’s. In this case, there are infinitely many sequences $\big( (x^k, j^*_k, {\mathcal J}_k) \big)_{k \in \mathbb N}$.
\[def:exact\_suppt\_recovery\] Given a matrix $A \in \mathbb R^{m\times N}$ and a constraint set ${\mathcal P}$, we say that [*the exact support recovery*]{} of a vector $z \in \Sigma_K \cap \mathcal P$ is achieved from $y=A z$ via constrained matching pursuit given by Algorithm \[algo:constrained\_MP\], if along an [*arbitrary*]{} sequence $\big( (x^k, j^*_k, {\mathcal J}_k) \big)_{k \in \mathbb N}$ for the given $z$, there exists an index $s \in \mathbb N$ such that $\mathcal J_s = { \mbox{supp} }(z)$. If the exact support recovery of any $z \in \Sigma_K \cap \mathcal P$ is achieved, then we call [*the exact support recovery on $\Sigma_K \cap \mathcal P$*]{} (or simply the exact support recovery) is achieved via constrained matching pursuit.
Necessary and sufficient conditions for the exact support recovery are given as follows.
\[lem:exact\_suppt\_recovery\_index\] Given $0 \ne u \in \sum_K \cap \, \mathcal P$ and an index set $\mathcal J \subseteq { \mbox{supp} }(u)$, let $v$ be an optimal solution to $\min_{w \in \mathcal P, \ { \mbox{supp} }(w)\subseteq \mathcal J} \| A (u - w) \|^2_2$, where we assume that such a solution exists. Then $f^*_j(u, v) = \| A (u -v) \|^2_2$ for each $j\in \mathcal J$, and $f^*_j(u, v) \le \| A (u -v) \|^2_2$ for each $j \notin \mathcal J$.
Consider an arbitrary $j \notin \mathcal J$. Noting that $0\in \mathbb I_j(v)$, we have $f^*_j(u, v) \le \| A(u -v) \|^2_2$. We then consider an arbitrary $j \in \mathcal J$. For any $t \in \mathbb I_j(v)$, we have $v + \mathbf e_j t \in \mathcal P$ and ${ \mbox{supp} }(v + \mathbf e_j t) \subseteq \mathcal J$. Since $v$ is an optimal solution to $\min_{w \in \mathcal P, \ { \mbox{supp} }(w)\subseteq\mathcal J} \| A (u - w) \|^2_2$, we have $\|A (u - v)\|^2_2 \le \|A u - A(v + \mathbf e_j t) \|^2_2$ for all $t\in \mathbb I_{j}(v)$. This shows that $\| A (u -v) \|^2_2 \le f^*_j(u, v)$. Furthermore, $f^*_j(u, v) \le \| A(u -v) \|^2_2 $ since $0\in \mathbb I_j(v)$. Therefore, $f^*_j(u, v) = \| A (u -v) \|^2_2$ for each $j \in {\mathcal J}$.
\[thm:nec\_suf\_condition\_for\_exact\_supp\_recovery\] Given a matrix $A \in \mathbb R^{m\times N}$ and a constraint set ${\mathcal P}$, let $0 \ne z \in \Sigma_K \cap {\mathcal P}$ with $|{ \mbox{supp} }(z)|=r$. Then the exact support recovery of $z$ is achieved via constrained matching pursuit if and only if for any sequence $\big( (x^k, j^*_k, {\mathcal J}_k) \big)_{k \in \mathbb N}$ generated by Algorithm \[algo:constrained\_MP\] with $y=A z$, the following holds $$\label{eqn:exact_sppt_recovery_inequality}
\min_{j \in { \mbox{supp} }(z)\setminus {\mathcal J}_k} f^*_j(z, x^k) \, < \, \min_{j\in [{ \mbox{supp} }(z)]^c} f^*_j(z, x^k), \qquad \forall \ k=0, 1, \ldots, r-1.$$ Moreover, when the exact support recovery of $z$ is achieved, the support of $z$ is firstly attained at the $r$th step along any sequence $\big( (x^k, j^*_k, {\mathcal J}_k) \big)_{k \in \mathbb N}$, i.e., ${\mathcal J}_r={ \mbox{supp} }(z)$ and ${\mathcal J}_k \subset { \mbox{supp} }(z)$ for each $k < r$.
“If”. For the given $z$, suppose an arbitrary sequence $\big( (x^k, j^*_k, {\mathcal J}_k) \big)_{k \in \mathbb N}$ generated by Algorithm \[algo:constrained\_MP\] satisfies (\[eqn:exact\_sppt\_recovery\_inequality\]). We prove below by induction on iterative steps of Algorithm \[algo:constrained\_MP\] that ${\mathcal J}_k \subseteq { \mbox{supp} }(z)$ with $|{\mathcal J}_k|=k$ and $j^*_{k+1} \in { \mbox{supp} }(z)\setminus{\mathcal J}_k$ for each $k=1, \ldots, r-1$. At Step 1, since $x^0=0$ and ${\mathcal J}_0$ is the empty set, we deduce from (\[eqn:exact\_sppt\_recovery\_inequality\]) that $\min_{j \in { \mbox{supp} }(z)} f^*_{j}(z, 0) \, < \, \min_{j \in [{ \mbox{supp} }(z) ]^c} f^*_{j}(z, 0)$. It follows from Algorithm \[algo:constrained\_MP\] that the optimal index $j^*_1 \in {\mbox{Argmin}}_{j=1, \ldots, N} f^*_j(z, 0)$ satisfies $j^*_1 \in { \mbox{supp} }(z)$ such that ${\mathcal J}_1=\{ j^*_1 \} \subseteq { \mbox{supp} }(z)$ and $|{\mathcal J}_1|=1$. Now suppose ${\mathcal J}_k \subseteq { \mbox{supp} }(z)$ with $|{\mathcal J}_k|=k$ and $j^*_k \in { \mbox{supp} }(z) \setminus {\mathcal J}_{k-1}$ for $1\le k \le r-2$. Consider Step $(k+1)$. In view of Lemma \[lem:exact\_suppt\_recovery\_index\], the optimal index $j^*_{k+1} \in {\mbox{Argmin}}_{j=1, \ldots, N} f^*_j(z, x^k)$ satisfies $j^*_{k+1} \notin {\mathcal J}_k$. Since ${\mathcal J}_k \subseteq { \mbox{supp} }(z)$, $j^*_{k+1} \in [{ \mbox{supp} }(z) \setminus {\mathcal J}_k]\cup [{ \mbox{supp} }(z)]^c$. Further, it follows from (\[eqn:exact\_sppt\_recovery\_inequality\]) that $j^*_{k+1} \in { \mbox{supp} }(z)\setminus {\mathcal J}_k$. Therefore, ${\mathcal J}_{k+1}:={\mathcal J}_k \cup \{ j^*_{k+1} \}$ satisfies ${\mathcal J}_{k+1} \subseteq { \mbox{supp} }(z)$ and $|{\mathcal J}_{k+1}|=k+1$. By the induction principle, we see that ${\mathcal J}_r\subseteq { \mbox{supp} }(z)$ and $|{\mathcal J}_r|=r=|{ \mbox{supp} }(z)|$. This implies that ${\mathcal J}_r={ \mbox{supp} }(z)$ and ${\mathcal J}_k \subset { \mbox{supp} }(z)$ for each $k < r$.
“Only if”. Suppose the exact support recovery of $z$ is achieved via Algorithm \[algo:constrained\_MP\]. By Definition \[def:exact\_suppt\_recovery\], we claim that for any given sequence $\big( (x^k, j^*_k, {\mathcal J}_k) \big)_{k \in \mathbb N}$ generated by Algorithm \[algo:constrained\_MP\] with $y=A z$ starting from $x^0=0$ and ${\mathcal J}_0=\emptyset$, the following must hold: $$\min_{j \in { \mbox{supp} }(z)} f^*_j(z, x^k) \, < \, \min_{j\in [{ \mbox{supp} }(z)]^c} f^*_j(z, x^k), \qquad \forall \ k=0, 1, \ldots, r-1,$$ This is because otherwise, $\min_{j \in { \mbox{supp} }(z)} f^*_j(z, x^\ell) \ge \min_{j\in [{ \mbox{supp} }(z)]^c} f^*_j(z, x^\ell)$ for some $\ell=0,1, \ldots, r-1$. Hence, there exists an optimal index $j^*_{\ell+1}\notin { \mbox{supp} }(z)$ such that ${\mathcal J}_{\ell+1} \neq { \mbox{supp} }(z)$ (along a possibly different sequence), leading to ${\mathcal J}_{s} \neq { \mbox{supp} }(z)$ for all $s \ge \ell$. Note that ${\mathcal J}_k\ne { \mbox{supp} }(z)$ for each $k=1, \ldots, \ell$ since each $|{\mathcal J}_k|<r$. Therefore, there exists a sequence so that ${\mathcal J}_k \ne { \mbox{supp} }(z)$ for all $k \in \mathbb N$, yielding a contradiction. Finally, since each $x^k$ is a minimizer of $\min_{w\in {\mathcal P}, { \mbox{supp} }(w) \subseteq {\mathcal J}_k} \|A (z - w) \|^2_2$, we deduce via Lemma \[lem:exact\_suppt\_recovery\_index\] that $\min_{j \in { \mbox{supp} }(z)} f^*_j(z, x^k)=\min_{j \in { \mbox{supp} }(z)\setminus {\mathcal J}_k} f^*_j(z, x^k)$. This leads to (\[eqn:exact\_sppt\_recovery\_inequality\]).
In what follows, we show the implications of the exact support recovery.
\[prop:index\_set\] Given a matrix $A$ and a constraint set ${\mathcal P}$, let $0\ne z \in \Sigma_K \cap \mathcal P$ with $|{ \mbox{supp} }(z)|=r$ be such that the exact support recovery of $z$ is achieved. Then for any sequence $\big( (x^k, j^*_k, {\mathcal J}_k) \big)_{k \in \mathbb N}$ generated by Algorithm \[algo:constrained\_MP\] with $y=A z$, the following hold:
- $\| A (z - x^{k+1}) \|^2_2 \le f^*_{j^*_{k+1}}(z, x^k) < \| A (z - x^k) \|^2_2$ for each $k=0, 1, \ldots, r-1$;
- For each $k=1, \ldots, r$, $(x^k)_{j^*_k} \ne 0$, and $x^k_{{\mathcal J}_{k-1}} \ne 0$ when $k>1$. Hence, ${ \mbox{supp} }(x^k)={\mathcal J}_k$ for $k=1, 2$.
\(i) Fix $k \in \{0, 1, \ldots, r-1\}$. Since $x^k$ is an optimal solution to $\min_{w \in {\mathcal P}, \, { \mbox{supp} }(w)\subseteq {\mathcal J}_k} \| A(z - w)\|^2_2$, it follows from Lemma \[lem:exact\_suppt\_recovery\_index\] that $f^*_j(z, x^k) \le \| A (z - x^k)\|^2_2$ for all $j=1, \ldots, N$. In light of the inequality given by (\[eqn:exact\_sppt\_recovery\_inequality\]), we have $\min_{j \in { \mbox{supp} }(z)\setminus \mathcal J_k} f^*_{j}(z, x^k) \, < \, \min_{j \in [{ \mbox{supp} }(z) ]^c} f^*_{j}(z, x^k) \le \| A (z - x^k)\|^2_2$. Since $j^*_{k+1} \in {\mbox{Argmin}}_{j \in { \mbox{supp} }(z)\setminus \mathcal J_k} f^*_{j}(z, x^k)$, we have $ f^*_{j^*_{k+1}}(z, x^k) < \| A (z - x^k) \|^2_2$. Besides, by virtue of the definition of $f^*_{j}(\cdot, \cdot)$, we deduce that there exists $0 \ne t_* \in \mathbb I_{j^*_{k+1}}$ such that $$f^*_{j^*_{k+1}}(z, x^k) \, = \, \big \|A z- A \big(x^k+ t_* \mathbf e_{j^*_{k+1}} \big) \big\|^2_2.$$ Note that $x^k + t_* \mathbf e_{j^*_{k+1}} \in \mathcal P$ and ${ \mbox{supp} }( x^k + t_* \mathbf e_{j^*_{k+1}} ) = {\mathcal J}_k \cup \{ j^*_{k+1}\}= {\mathcal J}_{k+1}$. Since $x^{k+1}$ is an optimal solution to $\min_{w \in {\mathcal P}, \, { \mbox{supp} }(w)\subseteq {\mathcal J}_{k+1}} \| A(z - w)\|^2_2$, we have $\|A(z - x^{k+1} ) \|^2_2 \le \|A z- A (x^k+ t_* {\mathbf e}_{j^*_{k+1}} ) \|^2_2 = f^*_{j^*_{k+1}}(z, x^k)$.
\(ii) Fix $k \in \{1, \ldots, r\}$. We first show the following claim: $x^k_{{\mathcal J}_k \setminus {\mathcal J}_s} \ne 0$ for each $s \in \{0, 1, \ldots, k-1 \}$. Suppose, in contrast, that $(x^k)_{{\mathcal J}_k \setminus {\mathcal J}_s} = 0$ for some $s\in \{0, 1, \ldots, k-1\}$. In light of ${\mathcal J}_s \subset {\mathcal J}_k$, we have ${ \mbox{supp} }(x^k)\subseteq {\mathcal J}_{s}$. Since $x^k \in {\mathcal P}$ and $x^{s}$ is an optimal solution to $\min_{w \in {\mathcal P}, \, { \mbox{supp} }(w)\subseteq {\mathcal J}_{s}} \| A(z - w)\|^2_2$, we deduce that $\| A(z - x^{s})\|^2_2 \le \| A(z - x^k)\|^2_2$. Since $s<k$, this yields a contradiction to statement (i). Hence, the claim holds. In view of ${\mathcal J}_k\setminus {\mathcal J}_{k-1} = \{ j^*_k \}$, we obtain $(x^k)_{j^*_k} \ne 0$.
We then show that $x^k_{{\mathcal J}_{k-1}} \ne 0$ when $k>1$. Suppose, in contrast, that $x^k_{{\mathcal J}_{k-1}} = 0$. Then ${ \mbox{supp} }(x^k)=\{ j^*_{k} \}$ since $(x^k)_{j^*_k} \ne 0$. By the definition of $f^*_{j}(\cdot, \cdot)$, we have that $f^*_{j^*_k}(z, 0) \le \| A( z - x^k) \|^2_2$. Furthermore, we deduce via $x^0=0$ that $f^*_{j^*_1}(z, x^0)\le f^*_{j^*_k}(z, 0)$. Therefore, $f^*_{j^*_1}(z, x^0) \le \| A( z - x^k) \|^2_2$. On the other hand, it follows from statement (i) that $\| A(z - x^1)\|^2_2 \le f^*_{j^*_1}(z, x^0)$. This leads to $\| A(z - x^1)\|^2_2 \le \| A( z - x^k) \|^2_2$. Since $k>1$, we attain a contradiction to statement (i). Consequently, $x^k_{{\mathcal J}_{k-1}} \ne 0$ when $k>1$.
We specify particular conditions for the exact support recovery on $\mathbb R^N$ and $\mathbb R^N_+$, respectively.
\[coro:Nec\_Suf\_suppt\_recovery\_RN\_RN+\] Given a matrix $A \in \mathbb R^{m\times N}$ with unit columns (i.e., $\|A_{\bullet i}\|_2=1$ for all $i$) and a constraint set ${\mathcal P}$, let $0 \ne z \in \Sigma_K \cap {\mathcal P}$ with $|{ \mbox{supp} }(z)|=r$. The following hold:
- When ${\mathcal P}=\mathbb R^N$, the exact support recovery of $z$ is achieved if and only if for any sequence $\big( (x^k, j^*_k, {\mathcal J}_k) \big)_{k \in \mathbb N}$ generated by Algorithm \[algo:constrained\_MP\] with $y=A z$, $$\max_{j \in { \mbox{supp} }(z)\setminus {\mathcal J}_k} |A^T_{\bullet j} A(z - x^k)| \, > \, \max_{j\in [{ \mbox{supp} }(z)]^c} |A^T_{\bullet j} A(z - x^k)|, \qquad \forall \ k=0, 1, \ldots, r-1;$$
- When ${\mathcal P}=\mathbb R^N_+$, the exact support recovery of $z$ is achieved if and only if for any sequence $\big( (x^k, j^*_k, {\mathcal J}_k) \big)_{k \in \mathbb N}$ generated by Algorithm \[algo:constrained\_MP\] with $y=A z$, $$\max_{j \in { \mbox{supp} }(z)\setminus {\mathcal J}_k} [A^T_{\bullet j} A(z - x^k) ]_+ \, > \, \max_{j\in [{ \mbox{supp} }(z)]^c} [ A^T_{\bullet j} A(z - x^k) ]_+, \qquad \forall \ k=0, 1, \ldots, r-1.$$
\(i) Let ${\mathcal P}=\mathbb R^N$. Then for any $v \in \mathbb R^N$ and any index $j$, $\mathbb I_j(v)=\mathbb R$. It follows from the definition of ${\widetilde}t_j(u, v)$ given by (\[eqn:def\_tilda\_t\]) and $f^*_j(u, v)$ given by (\[eqn:f\*\_RN\]) that (\[eqn:exact\_sppt\_recovery\_inequality\]) holds if and only if for each $k=0,1, \ldots, r-1$, $$\max_{j \in { \mbox{supp} }(z)\setminus {\mathcal J}_k} \langle A(z- x^k), A_{\bullet j} \rangle^2 > \max_{j \in [{ \mbox{supp} }(z)]^c} \langle A(z- x^k), A_{\bullet j} \rangle^2.$$ The latter is equivalent to $\max_{j \in { \mbox{supp} }(z)\setminus {\mathcal J}_k} |A^T_{\bullet j} A(z - x^k)| \, > \, \max_{j\in [{ \mbox{supp} }(z)]^c} |A^T_{\bullet j} A(z - x^k)|$. (ii) Let ${\mathcal P}=\mathbb R^N_+$. Consider the pair $(x^k, {\mathcal J}_k)$ for any fixed $k\in \{0, 1, \ldots, r-1\}$. For each $j \in { \mbox{supp} }(z)\setminus {\mathcal J}_k$, we have $(x^k)_j=0$ such that $\mathbb I_j(x^k)=\mathbb R_+$. Further, since ${ \mbox{supp} }(x^k) \subset { \mbox{supp} }(z)$ as shown in Theorem \[thm:nec\_suf\_condition\_for\_exact\_supp\_recovery\], we see that for any $j\in [{ \mbox{supp} }(z)]^c$, $j \notin { \mbox{supp} }(x^k)$ such that $(x^k)_j=0$ and $\mathbb I_j(x^k)=\mathbb R_+$. Hence, in view of $f^*_j(\cdot, \cdot)$ given by (\[eqn:f\*\_RN+\]), we see that $\min_{j \in { \mbox{supp} }(z)\setminus {\mathcal J}_k} f^*_j(z, x^k) < \min_{j\in [{ \mbox{supp} }(z)]^c} f^*_j(z, x^k)$ if and only if $\max_{j \in { \mbox{supp} }(z)\setminus {\mathcal J}_k} ([A^T_{\bullet j} A(z - x^k) ]_+)^2 > \max_{j\in [{ \mbox{supp} }(z)]^c} ([ A^T_{\bullet j} A(z - x^k) ]_+)^2$, which is equivalent to $\max_{j \in { \mbox{supp} }(z)\setminus {\mathcal J}_k} [A^T_{\bullet j} A(z - x^k) ]_+ > \max_{j\in [{ \mbox{supp} }(z)]^c} [ A^T_{\bullet j} A(z - x^k) ]_+$. This yields the desired result.
Inspired by Theorem \[thm:nec\_suf\_condition\_for\_exact\_supp\_recovery\], we introduce the following condition for a matrix $A$ and a constraint set ${\mathcal P}$: $$\begin{aligned}
(\mathbf H): \quad & \mbox{For any $0 \ne u \in \Sigma_K \cap \, \mathcal P$, any index set ${\mathcal J}\subset { \mbox{supp} }(u)$ (where ${\mathcal J}$ is possibly the empty set),} \notag \\
& \mbox{and an arbitrary optimal solution $v$ of $\min_{w \in \mathcal P, \ { \mbox{supp} }(w)\subseteq \mathcal J} \| A (u - w) \|^2_2$, the following holds:} \notag \\
& \qquad \qquad \qquad \qquad \quad \min_{j \in { \mbox{supp} }(u)\setminus \mathcal J} f^*_{j}(u, v) \, < \, \min_{j \in [{ \mbox{supp} }(u) ]^c} f^*_{j}(u, v). \label{eqn:condition_H'}\end{aligned}$$ The next proposition states that $(\mathbf H)$ is a sufficient condition for the exact support recovery. We omit its proof since it follows directly from the fact that the inequality in (\[eqn:condition\_H’\]) implies (\[eqn:exact\_sppt\_recovery\_inequality\]) given in Theorem \[thm:nec\_suf\_condition\_for\_exact\_supp\_recovery\].
\[prop:condition\_H\_suppt\_recovery\] Given a matrix $A\in \mathbb R^{m\times N}$ and a constraint set ${\mathcal P}$, suppose condition $(\mathbf H)$ holds. Then the exact support recovery is achieved on $\Sigma_K \cap {\mathcal P}$.
In general, condition $(\mathbf H)$ is [*not*]{} necessary for the exact support recovery. This is because the exact support recovery of a vector $z \in \Sigma_K \cap {\mathcal P}$ requires that the inequality (\[eqn:exact\_sppt\_recovery\_inequality\]) hold for ${\mathcal J}_k$’s only along a sequence $\big( (x^k, j^*_k, {\mathcal J}_k) \big)_{k \in \mathbb N}$ for $z$, while condition $(\mathbf H)$ says that the inequality (\[eqn:condition\_H’\]) hold for [*all*]{} proper subsets ${\mathcal J}\subset { \mbox{supp} }(z)$. Nevertheless, condition $(\mathbf H)$ is necessary for the exact support recovery when $K$ is small; see Corollary \[coro:condition\_H’\_necessary\_RN\_S2\] for $\Sigma_2\cap \mathbb R^N$ and Corollary \[coro:condition\_H’\_necessary\_RN+\_S2\] for $\Sigma_2\cap \mathbb R^N_+$, respectively.
Before ending this section, we give an example of a closed convex set ${\mathcal P}$, on which no matrix $A$ can achieve the exact support recovery. It demonstrates that the exact support recovery and condition $(\mathbf H)$ not only depend on the measurement matrix $A$ but also critically rely on the constraint set $\mathcal P$.
\[example:counter01\] Let $d=(d_1, \ldots, d_N)^T \in \mathbb R^N$ be such that $d_i \ne 0$ for each $i$. Consider the hyperplane $\mathcal P:=\{ x \in \mathbb R^N \, | \, d^T x = 0 \}$. Clearly, $\mathcal P$ is closed and convex, and it contains the zero vector and other sparse vectors. Since each $d_i\ne 0$, it is easy to verify that for any $v\in \mathcal P$ and any index $j$, the set $\mathbb I_j(v) =\{ 0 \}$. This shows that for any $u, v \in \mathcal P$ and any index $j$, $f^*_j(u,v)=\| A(u-v)\|^2_2$ for any matrix $A$. Hence, for any $z \in \Sigma_K \cap {\mathcal P}$, we deduce that at Step 1 of Algorithm \[algo:constrained\_MP\], ${\mbox{Argmin}}_{j \in \{1, \ldots, N\}} f^*(z, 0)=\{1, \ldots, N \}$. Thus $j^*_1$ can be chosen as $j^*_1 \notin{ \mbox{supp} }(z)$. This means that no matrix $A$ achieves the exact support recovery of any $z \in\Sigma_K \cap {\mathcal P}$. It also implies that no matrix $A$ satisfies condition $(\mathbf H)$ on ${\mathcal P}$.
Coordinate Projection Admissible Sets {#sect:CP_admissible_set}
=====================================
Since the exact recovery via constrained matching pursuit critically relies on a constraint set, it is essential to find a class of constraint sets to which the constrained matching pursuit can be successfully applied for exact recovery. An ideal class of constraint sets is expected to satisfy some crucial conditions, including but not limited to: (i) each set in this class contains sufficiently many sparse vectors; (ii) this class of sets is broad enough to include important sets arising from applications, such as $\mathbb R^N$ and $\mathbb R^N_+$; and (iii) (relatively) easily verifiable sufficient recovery conditions can be established using general properties of this class of sets. Motivated by these requirements, we identify an important class of constraint sets in this section and study their analytic properties to be used for the exact recovery.
We introduce some notation first. Let $\mathcal U$ be a nonempty set in $\mathbb R$, and ${\mathcal I}$ be an index subset of $\{1, \ldots, N\}$. We let $\mathcal U^{\mathcal I}:= \{ x =(x_1, \ldots, x_N)^T \in \mathbb R^N \, | \, x_i \in \mathcal U, \forall \, i \in {\mathcal I}, \mbox{ and } \ x_{{\mathcal I}^c}=0 \}$, and $\mathcal U_{\mathcal I}:=\{ u \in \mathbb R^{|{\mathcal I}|} \, | \, u_i \in \mathcal U, \forall \, i \in {\mathcal I}\}$. For each $x\in \mathbb R^N$ and an index set ${\mathcal I}$, define the coordinate projection operator $\pi_{{\mathcal I}}:\mathbb R^N \rightarrow \mathbb R^N$ as $\pi_{\mathcal I}(x):=z$, where $z_i=x_i, \forall \, i\in {\mathcal I}$ and $z_{{\mathcal I}^c}=0$. If ${\mathcal I}$ is the empty set, then $\pi_{\mathcal I}(x)=0, \forall \, x$. We often write $\pi_{\mathcal I}(x)=(x_{\mathcal I}, 0)$ with $x_{{\mathcal I}^c}=0$ for notational simplicity. We also write $\pi_{\{i\}}$ as $\pi_i$ for $i=1, \ldots, N$ when the context is clear. For each index set ${\mathcal I}$, $\pi_{\mathcal I}$ is obviously a linear operator on $\mathbb R^N$ given by $\pi_{\mathcal I}(x) = W x$ for $x =(x_{\mathcal I}, x_{{\mathcal I}^c})\in \mathbb R^N$, where the matrix $
W \, = \, \begin{bmatrix} W_{{\mathcal I}{\mathcal I}} & W_{{\mathcal I}{\mathcal I}^c} \\ W_{{\mathcal I}^c {\mathcal I}} & W_{{\mathcal I}^c {\mathcal I}^c} \end{bmatrix} \, = \, \begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix} \in \mathbb R^{N\times N}.
$ For any index sets ${\mathcal I}, {\mathcal J}\subseteq\{1, \ldots, N\}$, the following results can be easily established: $$\label{eqn:pi_two_index sets}
\pi_{\mathcal I}\circ \pi_{\mathcal J}\, = \, \pi_{{\mathcal I}\, \cap {\mathcal J}} \, = \, \pi_{\mathcal J}\circ \pi_{\mathcal I},$$ where $\circ$ denotes the composition of two functions.
We call a nonempty set $\mathcal P \subseteq \mathbb R^N$ [*coordinate projection admissible*]{} or simply [*CP admissible*]{} if for any $x \in \mathcal P$ and any index set ${\mathcal J}\subseteq { \mbox{supp} }(x)$, $\pi_{\mathcal J}(x)=(x_{\mathcal J}, 0) \in \mathcal P$, where ${\mathcal J}$ may be the empty set.
Clearly, $\mathcal P$ must contain the zero vector (by setting ${\mathcal J}= \emptyset$). An equivalent geometric condition for a CP admissible set is shown in the following lemma.
\[lem:CP\_adm\_geometry\] $\mathcal P$ is CP admissible if and only if $\pi_{\mathcal I}(\mathcal P) \subseteq \mathcal P$ for any index set ${\mathcal I}\subseteq \{1, \ldots, N\}$.
“If”. Since $\pi_{\mathcal I}(\mathcal P) \subseteq \mathcal P$ for any index set ${\mathcal I}$, we have $\pi_{\mathcal I}(x) \in \mathcal P$ for any $x \in \mathcal P$ and any ${\mathcal I}$. Hence, for any $x \in \mathcal P$ and any index set ${\mathcal J}\subseteq { \mbox{supp} }(x)$, we have $\pi_{\mathcal J}(x) \in \mathcal P$. This shows that $\mathcal P$ is CP admissible.
“Only If”. Suppose $\mathcal P$ is CP admissible, and let ${\mathcal I}$ be an arbitrary index set. It suffices to show that $\pi_{\mathcal I}(x) \in \mathcal P$ for any given $x \in \mathcal P$. Toward this end, in view of $
{\mathcal I}= ({\mathcal I}\cap { \mbox{supp} }(x)) \cup ({\mathcal I}\setminus { \mbox{supp} }(x))$ and $x_{{\mathcal I}\setminus { \mbox{supp} }(x)}=0$, we have $\pi_{\mathcal I}(x) = (x_{{\mathcal I}\cap { \mbox{supp} }(x)}, x_{{\mathcal I}\setminus { \mbox{supp} }(x)}, x_{{\mathcal I}^c}) = (x_{{\mathcal I}\cap { \mbox{supp} }(x)}, 0, 0) = \pi_{ {\mathcal I}\cap { \mbox{supp} }(x)}(x) \in \mathcal P$, where the last membership is due to the facts that ${\mathcal I}\cap { \mbox{supp} }(x) \subseteq { \mbox{supp} }(x)$ and that $\mathcal P$ is CP admissible.
Examples of bounded CP admissible sets include $\{ x \in \mathbb R^N \, | \, a^T x \le 1, \mbox{ and } x \ge 0 \}$ for a vector $a\in \mathbb R^N_{++}$, and any $\ell_p$-ball $\{ x \in \mathbb R^N \, | \, \| \| x \|_p \le \varepsilon \}$ with $p > 0$ and $\varepsilon > 0$, and $\mathcal P=[a_1, b_1]\times [a_2, b_2] \times \cdots \times [a_N, b_N]$ where $a_i \le 0 \le b_i$ for each $i$. Examples of unbounded CP admissible sets include $\mathbb R^N$, $\mathbb R^N_+$, and $\Sigma_K=\{ x \in \mathbb R^N \, | \, \| x \|_0 \le K \}$ for some $K \in \mathbb N$, Note that $\Sigma_K$ and the $\ell_p$-ball with $0< p <1$ are non-convex. Another example of non-convex CP admissible set is $\mathcal P=\mathbb R^N_+\cup \mathbb R^N_-$. Further, a CP admissible set may be neither open nor closed, e.g., $\mathcal P=[0, 1)\times (-1, 2]$ in $\mathbb R^2$.
The following proposition provides a list of important properties of CP admissible sets.
\[prop:properties\_CP\_admissible\] The following hold:
- The set $\mathcal P$ is CP admissible if and only if $\lambda \mathcal P$ is CP admissible for any real number $\lambda \ne 0$, and the intersection and union of CP admissible sets are CP admissible;
- The algebraic sum of two CP admissible sets is CP admissible;
- If $\mathcal P$ is CP admissible, then for any index set ${\mathcal I}$, $\pi_{\mathcal I}(\mathcal P)$ is also CP admissible;
- If $\mathcal P$ is a convex and CP admissible set, then $\dim(\mathcal P)=\max\{ |{ \mbox{supp} }(x)| \, : \, x \in \mathcal P \}$.
\(i) This is a direct consequence of the definition of a CP admissible set.
\(ii) Let $\mathcal P_1$ and $\mathcal P_2$ be two CP admissible sets, and $z$ be an arbitrary vector in ${\mathcal P}_1+{\mathcal P}_2$. Hence, $z=x+y$, where $x\in {\mathcal P}_1$ and $y\in {\mathcal P}_2$. For any index set ${\mathcal I}$, it follows from Lemma \[lem:CP\_adm\_geometry\] that $\pi_{\mathcal I}(x) \in {\mathcal P}_1$ and $\pi_{\mathcal I}(y) \in {\mathcal P}_2$. Therefore, $\pi_{\mathcal I}(z)= \pi_{\mathcal I}(x) + \pi_{\mathcal I}(y) \in \mathcal P_1+\mathcal P_2$. By Lemma \[lem:CP\_adm\_geometry\] again, we deduce that $\mathcal P_1+{\mathcal P}_2$ is CP admissible. (iii) Let $\mathcal P$ be CP admissible, and ${\mathcal I}$ be an arbitrary but fixed index set. Then for any index set ${\mathcal J}$, we deduce via equation (\[eqn:pi\_two\_index sets\]) that $\pi_{\mathcal J}(\pi_{\mathcal I}({\mathcal P})) = \pi_{{\mathcal I}} (\pi_{\mathcal J}({\mathcal P}))$. Since ${\mathcal P}$ is CP admissible, $\pi_{\mathcal J}({\mathcal P}) \subseteq {\mathcal P}$. Hence, by Lemma \[lem:CP\_adm\_geometry\], we have $\pi_{{\mathcal I}} (\pi_{\mathcal J}({\mathcal P}))
\subseteq \pi_{\mathcal I}({\mathcal P})$. This shows that $\pi_{\mathcal I}({\mathcal P})$ is CP admissible.
\(iv) Suppose $\mathcal P$ is a convex and CP admissible set. Let ${\widehat}x\in \mathcal P$ be such that $|{ \mbox{supp} }({\widehat}x)| \ge |{ \mbox{supp} }(x)$ for all $x \in \mathcal P$. We claim that for any $x \in \mathcal P$, ${ \mbox{supp} }(x) \subseteq { \mbox{supp} }({\widehat}x)$. Suppose not. Then there exist a point $x' \in \mathcal P$ and an index $i\in { \mbox{supp} }(x')$ such that $i\notin { \mbox{supp} }({\widehat}x)$. Since $\mathcal P$ is convex, $z(\lambda):=\lambda x' + (1-\lambda) {\widehat}x \in \mathcal P$ for all $\lambda \in [ 0, 1]$. However, for all $\lambda>0$ sufficiently small, $({ \mbox{supp} }({\widehat}x) \cup \{ i \}) \subseteq { \mbox{supp} }(z(\lambda))$. This shows that $|{ \mbox{supp} }(z(\lambda))|> |{ \mbox{supp} }({\widehat}x)|$, leading to a contradiction. Therefore, ${ \mbox{supp} }(x) \subseteq { \mbox{supp} }({\widehat}x)$ for all $x \in \mathcal P$. Furthermore, it is known that $\dim(\mathcal P) = \dim(\mbox{aff}(\mathcal P))$, where $\mbox{aff}(\cdot)$ denotes the affine hull of a set. Since ${\mathcal P}$ contains the zero vector, $\mbox{aff}(\mathcal P)=\mbox{span}({\mathcal P})$. In view of the claim that ${ \mbox{supp} }(x) \subseteq { \mbox{supp} }({\widehat}x)$ for any $x \in \mathcal P$, we deduce that $\dim(\mathcal P) =\dim(\mbox{span}({\mathcal P}))\le |{ \mbox{supp} }({\widehat}x)|$. Letting $p:=|{ \mbox{supp} }({\widehat}x)|$, we assume without loss of generality that ${ \mbox{supp} }({\widehat}x)=\{1, \ldots, p\}$. For each $s \in \{1, \ldots, p\}$, let ${\widehat}{\mathcal J}_s:=\{1, 2, \ldots, s\}$ and $z^s:=({\widehat}x_{{\widehat}{\mathcal J}_s}, 0)$. Therefore, $z^p = {\widehat}x$. Since ${\mathcal P}$ is CP admissible, each $z^s \in \mathcal P$. Besides, $\{z^1, z^2, \ldots, z^p \}$ is linearly independent. Since $\mathcal P$ is convex and $\{ 0, z^1, z^2, \ldots, z^p\}$ is affinely independent, the convex hull of $\{ 0, z^1, z^2, \ldots, z^p\}$ is a simplex of dimension $p$ and is contained in ${\mathcal P}$. Therefore, it follows from [@Rockafellar_book70 Theorem 2.4] that $\dim(\mathcal P) \ge p=|{ \mbox{supp} }({\widehat}x)|$. Consequently, $\dim(\mathcal P)= |{ \mbox{supp} }({\widehat}x)|$.
Using (iv) of Proposition \[prop:properties\_CP\_admissible\], we see that the hyperplane $\mathcal P =\{ x \in \mathbb R^N \, | \, d^T x = 0 \}$ with each $d_i \ne 0$ given in Example \[example:counter01\] is [*not*]{} CP admissible, since $\dim(\mathcal P)=N-1$ but $\max\{ |{ \mbox{supp} }(x)| : x \in \mathcal P \}=N$.
\[lem:CP\_adm\_closedness\] Let $\mathcal P$ be a closed and CP admissible set. Then for any index set ${\mathcal J}$, $\pi_{\mathcal J}(\mathcal P)$ is closed.
Fix an index set ${\mathcal J}$. Let $(z^k)$ be a convergent sequence in $\pi_{\mathcal J}(\mathcal P)$ such that $(z^k) \rightarrow z^*$. Hence, for each $k$, $z^k=(z^k_{\mathcal J}, z^k_{{\mathcal J}^c}) \in \pi_{\mathcal J}(\mathcal P)$ with $z^k_{{\mathcal J}^c}=0$. Since $(z^k)$ converges to $z^*$, we have $z^*=(z^*_{\mathcal J}, 0)$ and $(z^k_{\mathcal J}) \rightarrow z^*_{\mathcal J}$. Since $\mathcal P$ is CP admissible, $\pi_{\mathcal J}(\mathcal P) \subseteq \mathcal P$ such that $z^k \in \mathcal P$ for each $k$. Further, since $\mathcal P$ is closed, we have $z^* \in \mathcal P$. Clearly, $\pi_{\mathcal J}(z^*) = z^*\in \mathcal P$. Hence, $z^* \in \pi_{\mathcal J}(\mathcal P)$. This shows that $\pi_{\mathcal J}(\mathcal P)$ is closed.
Note that the above result may fail when $\mathcal P$ is not CP admissible, even if it is closed and convex. For example, consider $\mathcal P=\{x=(x_1, x_2) \, | \, x_2 \ge \frac{1}{x_1}, \ x_1 > 0 \} \subset \mathbb R^2$. Clearly, $\mathcal P$ is closed and convex but not CP admissible. Letting ${\mathcal J}=\{1\}$, we see that $\pi_{\mathcal J}(\mathcal P) = \{ (x_1, 0) \, | \, x_1 \in (0, \infty) \}$ and thus is not closed.
The following result gives a complete characterization of a closed, convex and CP admissible cone. Particularly, it shows that a closed, convex and CP admissible cone is a Cartesian product of Euclidean spaces and nonnegative or nonpositive orthants.
\[prop:CP\_admissible\_cone\] Let $\mathcal C$ be a closed convex cone in $\mathbb R^N$. Then $\mathcal C$ is CP admissible if and only if there exist four disjoint index subsets $\mathcal I_1$, $\mathcal I_+$, $\mathcal I_-$, and $\mathcal I_0$ (some of which can be empty) whose union is $\{1, \ldots, N\}$ such that $\mathcal C = \mathbb R^{{\mathcal I}_1}+ (\mathbb R_+)^{{\mathcal I}_+} + (\mathbb R_-)^{{\mathcal I}_-} + \{0\}^{{\mathcal I}_0}$ or equivalently $\mathcal C = \mathbb R_{{\mathcal I}_1} \times (\mathbb R_+)_{{\mathcal I}_+} \times (\mathbb R_-)_{{\mathcal I}_-} \times \{ 0 \}_{{\mathcal I}_0}$.
“If”. Suppose $\mathcal C = \mathbb R^{{\mathcal I}_1}+ \mathbb R^{{\mathcal I}_+}_+ + \mathbb R^{{\mathcal I}_-}_- + \{0\}^{{\mathcal I}_0}$, where the four index sets $\mathcal I_1$, $\mathcal I_+$, $\mathcal I_-$, and $\mathcal I_0$ form a disjoint union of $\{1, \ldots, N\}$. It is easy to see that $\mathcal C$ is closed and convex and that $\mathbb R^{{\mathcal I}_1}$, $\mathbb R^{{\mathcal I}_+}_+$, $\mathbb R^{{\mathcal I}_-}_-$ and $\{0\}^{{\mathcal I}_0}$ are all CP admissible. By (ii) of Proposition \[prop:properties\_CP\_admissible\], $\mathcal C$ is also CP admissible.
“Only If”. Let $\mathcal C$ be a closed convex cone which is CP admissible. For an arbitrary index $i\in \{1, \ldots, N\}$, let $\pi_i(\mathcal C):=\{ \pi_i(x) \, | \, x \in \mathcal C\} \subseteq \mathbb R^N$ and $[\pi_i(\mathcal C)]_i:=\{ \big(\pi_i(x))_i \, | \, x \in \mathcal C\} \subseteq \mathbb R$. Since $\mathcal C$ is a closed convex cone, it is easy to show via a similar argument for Lemma \[lem:CP\_adm\_closedness\] that $[\pi_i(\mathcal C)]_i$ is a closed convex cone in $\mathbb R$. This implies that $[\pi_i(\mathcal C)]_i$ equals either one of the following (polyhedral) cones in $\mathbb R$: $\mathbb R$, $\mathbb R_+$, $\mathbb R_-$, or $\{ 0 \}$. Define the index sets $\mathcal I_1:=\{ i \, | \, [\pi_i(\mathcal C)]_i = \mathbb R\}$, $\mathcal I_+:=\{ i \, | \, [\pi_i(\mathcal C)]_i = \mathbb R_+\}$, $\mathcal I_-:=\{ i \, | \, [\pi_i(\mathcal C)]_i = \mathbb R_-\}$, and $\mathcal I_0:=\{ i \, | \, [\pi_i(\mathcal C)]_i = \{ 0 \} \}$. Clearly, these index sets form a disjoint union of $\{1, \ldots, N\}$. Furthermore, since $\mathcal C$ is CP admissible, we have $\mathbb R^{{\mathcal I}_1} \subseteq \mathcal C$, $\mathbb R^{{\mathcal I}_+}_+ \subseteq \mathcal C$, $\mathbb R^{{\mathcal I}_-}_- \subseteq \mathcal C$, and $\{ 0 \}^{{\mathcal I}_0} \subseteq \mathcal C$. Since $\mathcal C$ is a convex cone, $ \mathbb R^{{\mathcal I}_1}+ (\mathbb R_+)^{{\mathcal I}_+} + (\mathbb R_-)^{{\mathcal I}_-} + \{0\}^{{\mathcal I}_0} \subseteq \mathcal C$. Conversely, for any $x \in \mathcal C$, it follows from the definition of $[\pi_i(\mathcal C)]_i$ and the disjoint property of the index sets ${\mathcal I}_1, {\mathcal I}_+, {\mathcal I}_-$ and ${\mathcal I}_0$ that $x \in \mathbb R^{{\mathcal I}_1}+ (\mathbb R_+)^{{\mathcal I}_+} + (\mathbb R_-)^{{\mathcal I}_-} + \{0\}^{{\mathcal I}_0}$. This shows that $\mathcal C = \mathbb R^{{\mathcal I}_1}+ (\mathbb R_+)^{{\mathcal I}_+} + (\mathbb R_-)^{{\mathcal I}_-} + \{0\}^{{\mathcal I}_0}$.
The next proposition presents a decomposition of a closed, convex and CP admissible set.
\[prop:CP\_adm\_decomposition\] Let ${\mathcal P}\subseteq \mathbb R^N$ be closed, convex and CP admissible. Then ${\mathcal P}= \mathcal W + \mathcal K$, where $\mathcal W\subseteq {\mathcal P}$ is a compact, convex and CP admissible set, and $\mathcal K\subseteq {\mathcal P}$ is a closed, convex and CP admissible cone.
For a given closed, convex and CP admissible set $\mathcal P$, we first construct a compact, convex and CP admissible set $\mathcal W$ contained in $\mathcal P$. It follows from the similar argument for Lemma \[lem:CP\_adm\_closedness\] and Proposition \[prop:CP\_admissible\_cone\] that for each $i \in \{1, \ldots, N\}$, $[\pi_{i} ({\mathcal P})]_i$ is a closed convex set in $\mathbb R$ which contains $0$ . Hence, each $[\pi_{i} ({\mathcal P})]_i$ must be in one of the following forms: $\mathbb R$, $[a_i, \infty)$ with $a_i \le 0$, $(-\infty, b_i]$ with $b_i \ge 0$, and $[a_i, b_i]$ with $a_i\le 0 \le b_i$, where in the last case, $a_i=b_i=0$ if $a_i=b_i$. These four forms respectively correspond to an unbounded set without lower and upper bounds, an unbounded set that is bounded from below, an unbounded set that is bounded frow above, and a bounded set. Define the following disjoint index sets whose union is $\{1, \ldots, N\}$: $$\begin{aligned}
{\mathcal I}_1 & \, := \, \{ i \, | \, [\pi_{i} ({\mathcal P})]_i = \mathbb R\}, \qquad \qquad \quad {\mathcal I}_+ \, := \, \{ i \, | \, [\pi_{i} ({\mathcal P})]_i \mbox{ is unbounded but bounded from below } \}, \\
{\mathcal I}_0 & \, := \, \{ i \, | \, [\pi_{i} ({\mathcal P})]_i \mbox{ is bounded} \}, \qquad
{\mathcal I}_ - \, := \, \{ i \, | \, [\pi_{i} ({\mathcal P})]_i \mbox{ is unbounded but bounded from above } \}.
\end{aligned}$$ Define the closed convex cone $\mathcal K:=\mathbb R^{{\mathcal I}_1}+ (\mathbb R_+)^{{\mathcal I}_+} + (\mathbb R_-)^{{\mathcal I}_-} + \{0\}^{{\mathcal I}_0}$. Since ${\mathcal P}$ is CP admissible and convex, we have $\mathcal K \subseteq {\mathcal P}$. Further, $\mathcal K$ is CP admissible in view of Proposition \[prop:CP\_admissible\_cone\]. Moreover, define the set $$\label{eqn:def_W_set}
\mathcal W \, := \, {\mathcal P}\cap \, \underbrace{ \Big \{ \, x =(x_{{\mathcal I}_1}, x_{{\mathcal I}_+}, x_{{\mathcal I}_-}, x_{{\mathcal I}_0} ) \, | \, x_{{\mathcal I}_1} =0, \ x_{{\mathcal I}_+} \le 0, \ x_{{\mathcal I}_-} \ge 0 \, \Big\} }_{:= \, \mathcal C}.$$ Clearly, $\mathcal W \subseteq {\mathcal P}$. Since the set $\mathcal C$ defined in (\[eqn:def\_W\_set\]) is closed and convex, $\mathcal W$ is also closed and convex. We show next that $\mathcal W$ is bounded and CP admissible. To proved the boundedness of $\mathcal W$, recall that (i) for each $i\in {\mathcal I}_+$, $[\pi_{i} ({\mathcal P})]_i=[a_i, \infty)$ for some $a_i \le 0$; (ii) for each $i\in {\mathcal I}_-$, $[\pi_{i} ({\mathcal P})]_i=(-\infty, b_i]$ for some $b_i \ge 0$; and (iii) for each $i \in {\mathcal I}_0$, $[\pi_{i} ({\mathcal P})]_i=[a_i, b_i]$ for some $a_i \le 0 \le b_i$. Hence, $\pi_i(\mathcal W)=\{0\}$ for each $i\in {\mathcal I}_1$, $\pi_i (\mathcal W) \in [a_i, 0]$ for each $i\in {\mathcal I}_+$, $\pi_i (\mathcal W) \in [0, b_i]$ for each $i\in {\mathcal I}_-$, and $\pi_i (\mathcal W) \in [a_i, b_i]$ for each $i\in {\mathcal I}_0$. Therefore, for each $x \in \mathcal W$, we have $\| x \|_1 = \| x_{{\mathcal I}_+}\|_1 + \| x_{{\mathcal I}_-}\|_1 + \| x_{{\mathcal I}_0}\|_1\le \sum_{i\in {\mathcal I}_+}|a_i| + \sum_{i\in {\mathcal I}_-}|b_i| +\sum_{i\in {\mathcal I}_0} \max(|a_i|, b_i)$. This shows that $\mathcal W$ is bounded and thus compact. Lastly, it is easy to see that the set $\mathcal C$ defined in (\[eqn:def\_W\_set\]) is CP admissible. Since ${\mathcal P}$ is CP admissible, by statement (i) of Proposition \[prop:properties\_CP\_admissible\], $\mathcal W$ is also CP admissible.
We show that $\mathcal P = \mathcal W + \mathcal K$ as follows. We first show that $\mathcal W + \mathcal K \subseteq \mathcal P$. Consider an arbitrary $z \in \mathcal W + \mathcal K$, i.e., $z=x+y$ with $x \in \mathcal W$ and $y \in \mathcal K$. Since $\mathcal W$ and $\mathcal K$ are both contained in the convex set ${\mathcal P}$ and since $\mathcal K$ is a cone, we see that for any $\lambda \in [0, 1)$, $$\lambda x + y = \lambda x + (1-\lambda) \frac{y}{1-\lambda} \in \mathcal P.$$ Furthermore, since ${\mathcal P}$ is closed, $x+y=\lim_{\lambda \uparrow 1} \big (\lambda x + y \big) \in \mathcal P$. This shows that $z \in {\mathcal P}$ and thus $\mathcal W + \mathcal K \subseteq {\mathcal P}$. We finally show that ${\mathcal P}\subseteq \mathcal W + \mathcal K$. Toward this end, consider an arbitrary $z =(z_1, \ldots, z_N)^T \in {\mathcal P}$, and define the vectors $x=(x_1, \ldots, x_N)^T$ and $y=(y_1, \ldots, y_N)^T$ as follows: $$x_i \, := \, \left\{ \begin{array}{lcc} 0 & \mbox{ if } \ \ i \in {\mathcal I}_1 \\ -(z_i)_- & \mbox{ if } \ \ i \in {\mathcal I}_+ \\ (z_i)_+ & \mbox{ if } \ \ i \in {\mathcal I}_- \\ z_i & \mbox{ if } \ \ i \in {\mathcal I}_0 \end{array} \right., \qquad \quad
y_i \, := \, \left\{ \begin{array}{lcc} z_i & \mbox{ if } \ \ i \in {\mathcal I}_1 \\ (z_i)_+ & \mbox{ if } \ \ i \in {\mathcal I}_+ \\ -(z_i)_- & \mbox{ if } \ \ i \in {\mathcal I}_- \\ 0 & \mbox{ if } \ \ i \in {\mathcal I}_0 \end{array} \right..$$ Clearly, $z=x+y$, $y \in \mathcal K$, and $x \in \mathcal C$, where $\mathcal C$ is defined in (\[eqn:def\_W\_set\]). Moreover, letting the index set ${\mathcal J}:= \{ i \in {\mathcal I}_+ \, | \, z_i < 0 \} \cup \{ i \in {\mathcal I}_- \, | \, z_i > 0 \} \cup {\mathcal I}_0$, we have $x=\pi_{\mathcal J}(z)$. Since ${\mathcal P}$ is CP admissible, it follows from Lemma \[lem:CP\_adm\_geometry\] that $x \in {\mathcal P}$, leading to $x\in \mathcal W$. This shows that $z \in \mathcal W + \mathcal K$, and thus ${\mathcal P}\subseteq \mathcal W + \mathcal K$.
The above proposition shows that $\mathcal K$ is the asymptotic cone (or recession cone) of $\mathcal P$. Furthermore, by using this proposition, we show the existence of an optimal solution of the underlying minimization problem given in Line 7 of Algorithm \[algo:constrained\_MP\] for an arbitrary index set ${\mathcal J}$ as follows.
\[coro:sol\_existence\_CP\_adm\] Let $\mathcal P \subseteq \mathbb R^N$ be a closed, convex and CP admissible set. Then for any matrix $A \in \mathbb R^{m \times N}$, any index set $\mathcal J \subseteq \{1, \ldots, N\}$, and any $y \in \mathbb R^m$, $\min_{w \in {\mathcal P}, { \mbox{supp} }(w) \subseteq {\mathcal J}} \| A w - y\|^2_2$ attains an optimal solution.
We first show that $A \mathcal P$ is a closed set for any matrix $A \in \mathbb R^{m \times N}$. It follows from Proposition \[prop:CP\_adm\_decomposition\] that $A {\mathcal P}= A \mathcal W + A \mathcal K$, where $\mathcal W$ is compact and $\mathcal K$ is a polyhedral cone. Note that $A \mathcal W$ is compact, and $A \mathcal K$ is a polyhedral cone and thus is closed. This implies that $A {\mathcal P}$ is closed. The desired result thus follows readily from statement (i) of Lemma \[lemma:sol\_existence\].
In what follows, we let $\mbox{cone}(\mathcal U)$ denote the conic hull of a nonempty set $\mathcal U$ in $\mathbb R^N$, i.e., $\mbox{cone}(\mathcal U)$ is the collection of all nonnegative combinations of finitely many vectors in $\mathcal U$.
\[prop:conic\_hull\] Let ${\mathcal P}$ be a closed, convex and CP admissible set in $\mathbb R^N$. Then $\mbox{cone}({\mathcal P})=\{ \lambda x \, | \, \lambda \ge 0, x \in {\mathcal P}\}$, and $\mbox{cone}({\mathcal P})$ is a closed, convex and CP admissible cone.
Since ${\mathcal P}$ is a convex set, it follows from a standard argument in convex analysis, e.g., [@Rockafellar_book70 Corollary 2.6.3], that $\mbox{cone}({\mathcal P})=\{ \lambda x \, | \, \lambda \ge 0, x \in {\mathcal P}\}$. Define the disjoint index sets whose union is $\{1, \ldots, N\}$: $$\begin{aligned}
{\mathcal L}_1 & \, := \, \{ i \, | \ \mbox{$0$ is in the interior of } [\pi_{i} ({\mathcal P})]_i \}, \qquad \qquad
{\mathcal L}_0 \, := \, \{ i \, | \ [\pi_{i} ({\mathcal P})]_i =\{ 0 \} \}, \notag \\
{\mathcal L}_+ & \, := \, \{ i \, | \ \inf [\pi_{i} ({\mathcal P})]_i =0, \, \mbox{ and $[\pi_{i} ({\mathcal P})]_i$ contains a positive number } \}, \label{eqn:conic_hll_indices} \\
{\mathcal L}_ - & \, := \, \{ i \, | \ \sup [\pi_{i} ({\mathcal P})]_i =0, \, \mbox{ and $[\pi_{i} ({\mathcal P})]_i$ contains a negative number } \}. \notag
\end{aligned}$$ Let $\mathcal C:=\mathbb R^{{\mathcal L}_1}+ (\mathbb R_+)^{{\mathcal L}_+} + (\mathbb R_-)^{{\mathcal L}_-} + \{0\}^{{\mathcal L}_0}$. In view of Proposition \[prop:CP\_admissible\_cone\], $\mathcal C$ is a closed, convex and CP admissible cone. In what follows, we show that $\mathcal C=\mbox{cone}({\mathcal P}) $ in two steps.
\(i) We first show that $\mbox{cone}({\mathcal P}) \subseteq \mathcal C$. For a given $x \in {\mathcal P}$, we write it as $x=(x_{{\mathcal L}_1}, x_{{\mathcal L}_0}, x_{{\mathcal L}_+}, x_{{\mathcal L}_-})$. Hence, $x=\pi_{{\mathcal L}_1}(x) + \pi_{{\mathcal L}_+}(x) + \pi_{{\mathcal L}_-}(x) + \pi_{{\mathcal L}_0}(x)$, where $\pi_{{\mathcal L}_1}(x) \in \mathbb R^{{\mathcal L}_1}$, $\pi_{{\mathcal L}_+}(x) \in (\mathbb R_+)^{{\mathcal L}_+}$, $\pi_{{\mathcal L}_-}(x) \in (\mathbb R_-)^{{\mathcal L}_-}$, and $\pi_{{\mathcal L}_0}(x) = 0 \in \{0\}^{{\mathcal L}_0}$. By the definition of $\mathcal C$, we have that $x \in \mathcal C$. Therefore, ${\mathcal P}\subseteq \mathcal C$. Since $\mbox{cone}({\mathcal P})$ is the smallest convex cone containing ${\mathcal P}$, we have $\mbox{cone}({\mathcal P}) \subseteq \mathcal C$. (ii) We next show that $\mathcal C \subseteq \mbox{cone}({\mathcal P})$. Consider a vector $x \in \mathbb R^{{\mathcal L}_1}$, where $x=(x_{{\mathcal L}_1}, x_{{\mathcal L}^c_1})=(x_{{\mathcal L}_1}, 0)$. By the definition of the index set $\mathcal L_1$ given in (\[eqn:conic\_hll\_indices\]), we see that there exists a sufficiently small positive number $\lambda$ such that $\lambda x_i \in [\pi_i({\mathcal P})]_i$ for each $i \in {\mathcal L}_1$. Let $v=(v_{{\mathcal L}_1}, v_{{\mathcal L}^c_1})$ with $v_{{\mathcal L}_1} := \lambda x_{{\mathcal L}_1}$ and $v_{{\mathcal L}^c_1}:=0$. Hence, $v\in \pi_{{\mathcal L}_1}({\mathcal P})$. Since ${\mathcal P}$ is CP admissible, $\pi_{{\mathcal L}_1}({\mathcal P}) \subseteq {\mathcal P}$ such that $v \in {\mathcal P}$. In view of $x = (1/\lambda) v$ and $\mbox{cone}({\mathcal P})=\{ \lambda x \, | \, \lambda \ge 0, x \in {\mathcal P}\}$, we deduce that $x \in \mbox{cone}({\mathcal P})$. Therefore, $\mathbb R^{{\mathcal L}_1} \subseteq \mbox{cone}({\mathcal P})$. It follows from a similar argument that $\mathbb R^{{\mathcal L}_+}_+\subseteq \mbox{cone}({\mathcal P})$, $\mathbb R^{{\mathcal L}_-}_-\subseteq \mbox{cone}({\mathcal P})$, and $\{0\}^{{\mathcal L}_0} \subseteq \mbox{cone}({\mathcal P})$. Since $\mbox{cone}({\mathcal P})$ is convex, we see that $\mathbb R^{{\mathcal L}_1}+ \mathbb R^{{\mathcal L}_+}_+ + \mathbb R^{{\mathcal L}_-}_- + \{0\}^{{\mathcal L}_0} \subseteq \mbox{cone}({\mathcal P})$. Hence, $\mathcal C \subseteq \mbox{cone}({\mathcal P})$.
Consequently, $\mathcal C = \mbox{cone}({\mathcal P})$. Finally, since $\mathcal C$ is closed and CP admissible, so is $\mbox{cone}({\mathcal P})$.
Note that if ${\mathcal P}$ is not CP admissible (even though closed and convex), its conic hull may [*not*]{} be closed in general. An example is the closed unit $\ell_2$-ball in $\mathbb R^N$ centered at $\mathbf e_1 \in \mathbb R^N$.
\[def:irreducible\_CP\_set\] A closed, convex and CP admissible set ${\mathcal P}$ is [*irreducible*]{} if the index set $\{ i \, | \, [\pi_{i} ({\mathcal P})]_i =\{ 0 \} \}$ is the empty set.
In light of Proposition \[prop:conic\_hull\], it is easy to see that a closed, convex and CP admissible set ${\mathcal P}$ is irreducible if and only if $\mbox{cone}({\mathcal P})$ is irreducible.
The above development shows that the class of CP admissible sets enjoy favorable properties indicated at the beginning of this section. For example, each CP admissible set contains sufficiently many sparse vectors due to the CP admissible property. Moreover, $\mathbb R^N$, $\mathbb R^N_+$ and their alikes belong to the class of CP admissible sets. In what follows, we show an additional important implication of CP admissible sets in Proposition \[prop:negative\_second\_term\], which is crucial to the development of sufficient conditions for uniform exact recovery in Section \[sect:suff\_cond\_exact\_recovery\]. To this end, we first present a technical result on the support of vectors.
\[lem:support\_result\] Let $u, v \in \mathbb R^N$ and ${\mathcal J}\subseteq \{1, \ldots, N\}$ be such that ${ \mbox{supp} }(v) \subseteq {\mathcal J}\subseteq { \mbox{supp} }(u)$. Then ${ \mbox{supp} }(u-v)\setminus {\mathcal J}\, = \, { \mbox{supp} }(u) \setminus {\mathcal J}$.
We show ${ \mbox{supp} }(u-v) \subseteq { \mbox{supp} }(u)$ first. Let $i \in { \mbox{supp} }(u-v)$. Hence, $u_i - v_i \ne 0$. We claim that $u_i \ne 0$, because otherwise, $u_i=0$ and $v_i \ne 0$, which implies $i\in { \mbox{supp} }(v) \subseteq { \mbox{supp} }(u)$, yielding a contradiction. Hence, ${ \mbox{supp} }(u-v) \subseteq { \mbox{supp} }(u)$. This leads to ${ \mbox{supp} }(u-v)\setminus {\mathcal J}\subseteq { \mbox{supp} }(u)\setminus {\mathcal J}$. Conversely, for any $i \in { \mbox{supp} }(u)\setminus {\mathcal J}$, we have $v_i =0$ (due to ${ \mbox{supp} }(v) \subseteq {\mathcal J}$) so that $(u-v)_i = u_i \ne 0$. Hence, $i \in { \mbox{supp} }(u-v)$. Since $i\notin{\mathcal J}$, we have $i \in { \mbox{supp} }(u-v)\setminus {\mathcal J}$. Therefore, ${ \mbox{supp} }(u)\setminus {\mathcal J}\subseteq { \mbox{supp} }(u-v)\setminus {\mathcal J}$. As a result, ${ \mbox{supp} }(u-v)\setminus {\mathcal J}= { \mbox{supp} }(u)\setminus {\mathcal J}$.
\[prop:negative\_second\_term\] Let $\mathcal P$ be a closed, convex and CP admissible set in $\mathbb R^N$. Given a matrix $A \in \mathbb R^{m\times N}$, a vector $0\ne u \in \Sigma_K \cap \mathcal P$, and any index set ${\mathcal J}\subset { \mbox{supp} }(u)$, let $v$ be an arbitrary solution to $\mathbf Q: \, \min_{ w \in \mathcal P, \, { \mbox{supp} }(w) \subseteq {\mathcal J}} \, \| A (w - u)\|^2_2 $. Then the following hold: $$\sum_{j \in { \mbox{supp} }(u-v)\cap{\mathcal J}} \langle A(u-v), A_{\bullet j} \rangle \cdot (u-v)_j \, \le \, 0,$$ and $$\|A (u-v)\|^2_2 \, \le \, \sum_{j \in { \mbox{supp} }(u)\setminus{\mathcal J}} \langle A(u-v), A_{\bullet j} \rangle \cdot (u-v)_j.$$
Note that such an optimal solution $v$ exists due to Corollary \[coro:sol\_existence\_CP\_adm\]. Define the convex function $g(z):= \| A_{\bullet {\mathcal J}} z - A u \|^2_2$ with $z\in \mathbb R^{|{\mathcal J}|}$, and the constraint set $\mathcal W := \{ z \, | \, (z, 0) \in \pi_{\mathcal J}(\mathcal P) \}$. It follows from Lemma \[lem:CP\_adm\_closedness\] that $\pi_{\mathcal J}(\mathcal P)$ is closed. Since ${\mathcal P}$ is convex, so is $\pi_{\mathcal J}(\mathcal P)$. Hence, $\pi_{\mathcal J}(\mathcal P)$ is closed and convex. This shows that $\mathcal W$ is also a closed convex set. Moreover, the underlying optimization problem $\mathbf Q$ can be equivalently formulated as the convex optimization problem: $\displaystyle \min_{ z \in \mathcal W } g(z)$. Therefore, the optimal solution $v=(v_{\mathcal J}, 0)$ satisfies the necessary and sufficient optimality condition given by the following variational inequality: $\langle \nabla g(v_{\mathcal J}), z - v_{\mathcal J}\rangle \ge 0$ for all $z \in \mathcal W$. Since $\mathcal P$ is CP admissible, we have $(u_{\mathcal J}, 0) \in \mathcal P$ so that $u_{\mathcal J}\in \mathcal W$. In view of $\nabla g(v_{\mathcal J}) = A^T_{\bullet {\mathcal J}}(A_{\bullet {\mathcal J}} v_{\mathcal J}- A u)=A^T_{\bullet {\mathcal J}} (Av-Au)$, we have $$0 \, \le \, \langle A^T_{\bullet {\mathcal J}}(A_{\bullet {\mathcal J}} \, v_{\mathcal J}- A u), u_{\mathcal J}- v_{\mathcal J}\rangle = \langle A v - A u, A_{\bullet {\mathcal J}} (u - v)_{\mathcal J}\rangle.$$ This implies that $\langle A (u -v), A_{\bullet {\mathcal J}} (u - v)_{\mathcal J}\rangle \le 0$. Consequently, we obtain $$\begin{aligned}
\lefteqn{ \sum_{j \in { \mbox{supp} }(u-v)\cap{\mathcal J}} \langle A(u-v), A_{\bullet j} \rangle \cdot (u-v)_j} \notag \\ [5pt]
& = & \sum_{j \in { \mbox{supp} }(u-v)\cap{\mathcal J}} \langle A(u-v), A_{\bullet j} \rangle \cdot (u-v)_j
+ \sum_{j \in [{ \mbox{supp} }(u-v)]^c\cap{\mathcal J}} \langle A(u-v), A_{\bullet j} \rangle \cdot (u-v)_j \notag \\ [5pt]
& = & \sum_{j \in {\mathcal J}} \langle A(u-v), A_{\bullet j} \rangle \cdot (u-v)_j \, = \, \langle A(u-v), A_{\bullet {\mathcal J}} (u - v)_{\mathcal J}\rangle \notag \\ [5pt]
& \, \le \, & 0. \notag $$ Furthermore, we have $$\begin{aligned}
\| A(u - v) \|^2_2 & = & \sum^N_{j=1} \langle A(u-v), A_{\bullet j} (u-v)_j \rangle = \sum_{j \in { \mbox{supp} }(u-v)} \langle A(u-v), A_{\bullet j} \rangle \cdot (u-v)_j \\
& = & \sum_{j \in { \mbox{supp} }(u-v)\setminus{\mathcal J}} \langle A(u-v), A_{\bullet j} \rangle \cdot (u-v)_j + \sum_{j \in { \mbox{supp} }(u-v)\cap{\mathcal J}} \langle A(u-v), A_{\bullet j} \rangle \cdot (u-v)_j \\
& \le & \sum_{j \in { \mbox{supp} }(u-v)\setminus{\mathcal J}} \langle A(u-v), A_{\bullet j} \rangle \cdot (u-v)_j \\
& = & \sum_{j \in { \mbox{supp} }(u)\setminus{\mathcal J}} \langle A(u-v), A_{\bullet j} \rangle \cdot (u-v)_j,\end{aligned}$$ where the last equation follows from Lemma \[lem:support\_result\].
Exact Vector Recovery on Closed, Convex, CP Admissible Cones for a Fixed Support via Constrained Matching Pursuit {#sect:exact_vector_recovery}
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This section is focused on the exact vector recovery on closed, convex and CP admissible cones for a fixed support. By Proposition \[prop:CP\_admissible\_cone\], such a cone is a Cartesian product of copies of $\mathbb R$, $\mathbb R_+$ and $\mathbb R_-$, which includes $\mathbb R^N$ and $\mathbb R^N_{+}$. It is shown in Section \[sect:suff\_cond\_exact\_recovery\] that closed, convex and CP admissible cones play an important role in characterizing exact recovery, even for general closed, convex and CP admissible sets (cf. Section \[subsect:suff\_cond\_noncone\]). We first introduce the definition of exact vector recovery.
Let a matrix $A\in \mathbb R^{m\times N}$ and a constraint set ${\mathcal P}$ be given. For a fixed vector $z \in \Sigma_K \cap \mathcal P$, we say that [*the exact vector recovery*]{} of $z$ is achieved from $y=A z$ via Algorithm \[algo:constrained\_MP\] if (i) the exact support recovery of $z$ is achieved, and (ii) along [*any*]{} sequence $\big( (x^k, j^*_k, {\mathcal J}_k) \big)_{k \in \mathbb N}$ for the given $z$, once ${\mathcal J}_s={ \mbox{supp} }(z)$ is reached, then the minimization problem given by Line 7 of Algorithm \[algo:constrained\_MP\] has a [*unique*]{} solution $x^s=z$. If the exact vector recovery of each $z \in \Sigma_K \cap \, \mathcal P$ is achieved, then we call [*the exact vector recovery on $\Sigma_K \cap \mathcal P$*]{} (or simply the exact vector recovery) is achieved via Algorithm \[algo:constrained\_MP\]. We also say that a matrix $A$ achieves the exact vector (resp. support) recovery on ${\mathcal P}$ if the exact vector (resp. support) recovery on $\Sigma_K \cap \mathcal P$ is achieved using $A$. Besides, for a fixed index set ${\mathcal S}$, we say that [*the exact vector recovery on ${\mathcal P}$ for ${\mathcal S}$*]{} is achieved if the exact vector recovery of any $z \in {\mathcal P}$ with ${ \mbox{supp} }(z)={\mathcal S}$ is achieved.
Revisit of Exact Vector Recovery on $\mathbb R^N$ for a Fixed Support via OMP: A Counterexample to a Necessary Exact Recovery Condition in the Literature {#subsect:OMP_counterexample}
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When the sparse recovery problem (\[eqn:constrained\_L0\]) is constraint free, i.e., ${\mathcal P}=\mathbb R^N$, the constrained matching pursuit scheme given by Algorithm \[algo:constrained\_MP\] reduces to the OMP [@PRKrishnaprasad_Asilomar93]. The OMP has been extensively studied in the signal processing and compressed sensing literature, and many results have been developed for support or vector recovery using the OMP [@FoucartRauhut_book2013; @MoS_TIT12]. In particular, “necessary” and sufficient conditions are established in [@FoucartRauhut_book2013 Proposition 3.5] for exact vector recovery via the OMP for a fixed support; the same “necessary” and sufficient conditions are also given by Tropp [@Tropp_ITI04 Theorems 3.1 and 3.10]. For the sake of completeness and the ease of the subsequent discussions, we present the real version of [@FoucartRauhut_book2013 Proposition 3.5] as follows, i.e., $A \in \mathbb R^{m\times N}$, $y \in \mathbb R^m$, and $x \in \mathbb R^N$, using slightly modified wording.
[@FoucartRauhut_book2013 Proposition 3.5] \[prop:Foucart\_prop3.5\] Given a matrix $A \in \mathbb R^{m\times N}$ with unit columns, every nonzero vector $x \in \mathbb R^N$ supported on a given index set ${\mathcal S}$ of size $s$ (i.e., ${ \mbox{supp} }(x)={\mathcal S}$ and $|{ \mbox{supp} }(x)|=s$) is recovered from $y=A x$ after at most $s$ iterations of OMP if and only if the following two conditions hold:
- The matrix $A_{\bullet {\mathcal S}}$ is injective (i.e., $A_{\bullet {\mathcal S}}$ has full column rank), and
- $$\label{eqn:condition_Prop_3.5}
\max_{j \in {\mathcal S}} \big| (A^T A z)_j \big | \, > \, \max_{j \in {\mathcal S}^c} \big |(A^T A z)_j \big|, \qquad \quad \forall \ 0\ne z \in \mathbb R^{N} \ \mbox{ with } \ { \mbox{supp} }(z) \subseteq {\mathcal S}.$$
Further, under condition (i), condition (\[eqn:condition\_Prop\_3.5\]) holds if and only if $$\label{eqn:exact_recovery_condition}
\big\| (A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}} )^{-1} A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}^c} \big\|_1 < 1,$$ where $\|\cdot \|_1$ denotes the matrix 1-norm.
The “proof” of this proposition can be found on page 68 of the well received monograph [@FoucartRauhut_book2013] by Foucart and Rauhut, and its equivalent condition (\[eqn:exact\_recovery\_condition\]) in term of the matrix 1-norm follows from [@FoucartRauhut_book2013 Remark 3.6]. Also see a similar sufficiency proof in [@Tropp_ITI04 Theorem 3.1] and a “necessity” proof in [@Tropp_ITI04 Theorem 3.10], where condition (\[eqn:exact\_recovery\_condition\]) is referred to as the [*exact recovery condition*]{} coined by Tropp in [@Tropp_ITI04]. Clearly, conditions (i) and (ii) are sufficient for the exact vector recovery. Further, condition (i) is necessary for the exact vector recovery. However, we find that condition (ii) only [*partially*]{} holds for the necessity of the exact vector recovery. Specifically, condition (ii) is necessary when the index set ${\mathcal S}$ satisfies $|{\mathcal S}|=1$ or $|{\mathcal S}|=2$; when $|{\mathcal S}|=3$, we construct a nontrivial counterexample (i.e., a matrix $A$) such that any nonzero vector $x \in \mathbb R^N$ with ${ \mbox{supp} }(x)={\mathcal S}$ is exactly recovered via the OMP using the matrix $A$ but this $A$ does not satisfy (\[eqn:condition\_Prop\_3.5\]) or its equivalence (\[eqn:exact\_recovery\_condition\]).
The construction of our counterexample is motivated by an unsuccessful attempt to justify the following implication, which is the last key step given in the necessity proof for [@FoucartRauhut_book2013 Proposition 3.5]: $$\begin{aligned}
\left[ \ \max_{j \in {\mathcal S}} \big| (A^T A z)_j \big | \, > \, \max_{j \in {\mathcal S}^c} \big |(A^T A z)_j \big|, \quad \forall \ 0 \ne z \in \mathbb R^{N} \ \mbox{ with } \ { \mbox{supp} }(z) = {\mathcal S}\ \right] \, \Longrightarrow \notag \\
\left[ \ \max_{j \in {\mathcal S}} \big| (A^T A z)_j \big | \, > \, \max_{j \in {\mathcal S}^c} \big |(A^T A z)_j \big|, \quad \forall \ 0\ne z \in \mathbb R^{N} \ \mbox{ with } \ { \mbox{supp} }(z) \subseteq {\mathcal S}\ \right ], \label{eqn:necessity_implication}\end{aligned}$$ where we assume that the exact vector recovery is achieved and $A_{\bullet {\mathcal S}}$ has full column rank. Note that the hypothesis of the implication given by (\[eqn:necessity\_implication\]) holds since it follows from the first step of the OMP using $A$. To elaborate an underlying reason for the failure of this implication, we define the function $q(z):= \max_{j \in {\mathcal S}} | (A^T A z)_j | - \max_{j \in {\mathcal S}^c} |(A^T A z)_j |$ for $z \in \mathbb R^N$ and the set $\mathcal R:=\{ z \in \mathbb R^N \, | \, z \ne 0, \ { \mbox{supp} }(z) = {\mathcal S}\}$. Clearly, $q(\cdot)$ is continuous. Further, any nonzero ${\widetilde}z \in \mathbb R^N$ with ${ \mbox{supp} }({\widetilde}z) \subset {\mathcal S}$ is on the boundary of $\mathcal R$ such that there exists a sequence $(z_k)$ in $\mathcal R$ converging to ${\widetilde}z$. Hence, the sequence $(q(z_k))$ converges to $q({\widetilde}z)$, where each $q(z_k)>0$ in view of the hypothesis of the implication (\[eqn:necessity\_implication\]). However, one can only conclude that $q({\widetilde}z) \ge 0$ instead $q({\widetilde}z)>0$. The counterexample we construct shows that when $|{\mathcal S}|=3$, there exists a matrix $A$ achieving the exact vector recovery via the OMP but the corresponding $q({\widetilde}z)=0$ for some $0\ne {\widetilde}z \in \mathbb R^N$ with ${ \mbox{supp} }({\widetilde}z) \subset {\mathcal S}$; see Remark \[remark:counter\_example\] for details. This example invalidates the implication (\[eqn:necessity\_implication\]).
A similar argument also explains the failure of Tropp’s necessity proof in [@Tropp_ITI04 Theorem 3.10]. In fact, the (nonzero) signal ${\mathbf s}_{bad}$ constructed in that proof is shown to satisfy $\rho({\mathbf s}_{bad})\ge 1$, which is equivalent to $q({\mathbf s}_{bad}) \le 0$. However, if ${ \mbox{supp} }({\mathbf s}_{bad})$ is a proper subset of the index set $\Lambda_{opt}$, which is equivalent to the index set ${\mathcal S}$ defined above, then the argument based on the first step of the OMP used in the proof for [@Tropp_ITI04 Theorem 3.10] becomes invalid. In fact, the counterexample we construct shows that when $|{\mathcal S}|=3$, there exists a matrix $A$ achieving the exact vector recovery via the OMP but a nonzero ${\widetilde}z$ with ${ \mbox{supp} }({\widetilde}z) \subset {\mathcal S}$ exists such that the corresponding $q({\widetilde}z)=0$ or equivalently $\rho({\widetilde}z) =1$. See Remark \[remark:counter\_example\] for details. [^2]
We introduce more assumptions and notation through the rest of the development in this section. Consider a matrix $A \in \mathbb R^{m\times N}$ with unit columns, i.e., $\|A_{\bullet i}\|_2=1$ for each $i=1, \ldots, N$. Define $\vartheta_{ij}:=\langle A_{\bullet i}, A_{\bullet j} \rangle$ for $i, j \in \{1, \ldots, N\}$, and for each $i$, define the function $$\label{eqn:g_i_def}
g_i(z) \, := \, \big | \langle A_{\bullet i}, A z \rangle \big | \, = \, \Big | \sum^N_{j=1} \vartheta_{ij} z_j \Big |, \qquad \quad \forall \, z =(z_1, \ldots, z_N)^T \in \mathbb R^N.$$
### Positive Necessity Results and Their Implications
This subsection presents certain cases where condition (\[eqn:exact\_recovery\_condition\]) (or equivalently (\[eqn:condition\_Prop\_3.5\])) is indeed necessary for the exact vector recovery for a given support ${\mathcal S}$. The first result shows that [@FoucartRauhut_book2013 Proposition 3.5] (or Proposition \[prop:Foucart\_prop3.5\] of the present paper) holds when the index set ${\mathcal S}$ is of size 1 or 2.
\[thm:nonconstrained\_S2\] For a matrix $A \in \mathbb R^{m\times N}$ with unit columns and an index set ${\mathcal S}$ with $|{\mathcal S}|=1$ or $|{\mathcal S}|=2$, the exact vector recovery of every nonzero vector $x \in \mathbb R^N$ with ${ \mbox{supp} }(x)={\mathcal S}$ is achieved from $y=A x$ via the OMP if and only if the conditions (i) and (ii) in Proposition \[prop:Foucart\_prop3.5\] hold.
In light of the prior discussions and the argument for [@FoucartRauhut_book2013 Proposition 3.5], we only need to show that the implication (\[eqn:necessity\_implication\]) holds when $A$ achieves the exact vector recovery via the OMP and $A_{\bullet {\mathcal S}}$ has full column rank. The case of $|{\mathcal S}|=1$ is trivial, and we focus on the case of $|{\mathcal S}|=2$ as follows. Without loss of generality, let ${\mathcal S}=\{ 1, 2\}$. In view of $g_i$’s defined in (\[eqn:g\_i\_def\]), it suffices to show that if $\max( g_1(z), g_2(z)) > \max_{i \ge 3} g_i (z), \forall \, z \mbox{ with } { \mbox{supp} }(z)=\{1, 2\}$, then $ \max (g_1(z), g_2(z)) > \max_{i \ge 3} g_i (z), \forall \, z$ with ${ \mbox{supp} }(z)=\{1\}$ or ${ \mbox{supp} }(z)=\{2 \}$. Since $A_{\bullet {\mathcal S}}$ has full column rank, the $2\times 2$ matrix $
A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}} = \begin{bmatrix} 1 & \vartheta_{12} \\ \vartheta_{12} & 1 \end{bmatrix}
$ is positive definite. Hence, $|\vartheta_{12}| <1$. For any $z$ with ${ \mbox{supp} }(z)=\{1\}$, we have $\max (g_1(z), g_2(z)) = \max( |z_1|, |\vartheta_{12} z_1|)= g_1(z)> g_2(z)$ because $z_1 \ne 0$ and $|\vartheta_{12}| <1$. Similarly, $\max (g_1(z), g_2(z)) = g_2(z)> g_1(z)$ when ${ \mbox{supp} }(z)=\{2\}$.
In what follows, we consider an arbitrary $z^*$ with ${ \mbox{supp} }(z^*)=\{1\}$ first. Note that $g_j(z^*)=|\vartheta_{j1} z^*_1|$ for each $j$, where $z^*_1\ne 0$. Since $z^*$ is on the boundary of $\mathcal R:=\{ z \in \mathbb R^N \, | \, { \mbox{supp} }(z)=\{1,2\} \}$ on which $\max( g_1(z), g_2(z)) > \max_{i \ge 3} g_i (z)$, we deduce via the continuity of $g_i$’s that $g_1(z^*)=\max (g_1(z^*), g_2(z^*)) \ge g_i(z^*)$ for each $i \ge 3$. We show next that $g_1(z^*)> g_i (z^*)$ for all $i \ge 3$ by contradiction. Suppose, in contrast, $g_1(z^*) = g_i(z^*)$ for some $i \ge 3$, i.e., $|z^*_1| = | \vartheta_{i 1} z^*_1|=\gamma$. For any $v \in \mathbb R^N$ with ${ \mbox{supp} }(v)=\{1, 2\}$ and $\| v\|_2>0$ sufficiently small, $\max (g_1(z^*+v), g_2(z^*+v) ) = g_1(z^*+v)$ due to $g_1(z^*)> g_2(z^*)$, and $z^*+v \in \mathcal R$ so that $g_1(z^*+v) > g_i(z^*+v)$. Therefore, we have $$\label{eqn:R^N_S=1,2}
|z^*_1 + p^T v_{{\mathcal S}} | \, > \, | \vartheta_{i 1} z^*_1 + q^T v_{{\mathcal S}}|,$$ where $p=(1, \vartheta_{12})^T$, $q=(\vartheta_{i1}, \vartheta_{i2})^T$, and $v_{\mathcal S}=(v_1, v_2)^T \in \mathbb R^2$. Letting $\gamma:=|z^*_1|>0$, we obtain four possible cases from $|z^*_1| = | \vartheta_{i 1} z^*_1|$: (i) $(z^*_1, \vartheta_{i 1} z^*_1)=(\gamma, \gamma)$; (ii) $(z^*_1, \vartheta_{i 1} z^*_1)=(\gamma, -\gamma)$; (iii) $(z^*_1, \vartheta_{i 1} z^*_1)=(-\gamma, \gamma)$; and (iv) $(z^*_1, \vartheta_{i 1} z^*_1)=(-\gamma, -\gamma)$. In each of these cases, it follows from (\[eqn:R\^N\_S=1,2\]) that $( {{\rm sgn}}(z^*_1) \cdot p - {{\rm sgn}}(\vartheta_{i 1} z^*_1) \cdot q)^T v_{{\mathcal S}}>0$ for all $\| v_{\mathcal S}\|>0$ sufficiently small, where ${{\rm sgn}}(\cdot)$ is the signum function. In view of ${ \mbox{supp} }(v_{\mathcal S})={ \mbox{supp} }(-v_{\mathcal S})$, we have $( {{\rm sgn}}(z^*_1) \cdot p - {{\rm sgn}}(\vartheta_{i 1} z^*_1) \cdot q)^T v_{{\mathcal S}}>0$ and $( {{\rm sgn}}(z^*_1) \cdot p - {{\rm sgn}}(\vartheta_{i 1} z^*_1) \cdot q)^T (-v_{{\mathcal S}})>0$ for all $\| v_{\mathcal S}\|_2>0$ sufficiently small. This yields a contradiction. Hence, $\max (g_1(z^*), g_2(z^*))> g_i (z^*)$ for all $i \ge 3$ when ${ \mbox{supp} }(z^*)=\{ 1\}$. The case of ${ \mbox{supp} }(z^*)=\{ 2\}$ also follows by interchanging the roles of $g_1$ and $g_2$. Consequently, the implication (\[eqn:necessity\_implication\]) holds, which leads to condition (ii) in Proposition \[prop:Foucart\_prop3.5\].
By leveraging the necessary and sufficient recovery conditions in Theorem \[thm:nonconstrained\_S2\] for a fixed support of size 2, we show that condition $(\mathbf H)$ is necessary for the exact vector or support recovery on $\Sigma_2$.
\[coro:condition\_H’\_necessary\_RN\_S2\] Let $A \in \mathbb R^{m\times N}$ have unit columns. Then $A$ achieves the exact vector recovery on $\Sigma_2$ if and only if (i) condition $(\mathbf H)$ holds on $\Sigma_2$, and (ii) any two distinct columns of $A$ are linearly independent.
“If”. In view of Proposition \[prop:condition\_H\_suppt\_recovery\], condition $(\mathbf H)$ yields the exact support recovery on $\Sigma_2$. Besides, condition (ii) guarantees that the unique $x^2$ equals $z$ for any $z \in \Sigma_2$ with $|{ \mbox{supp} }(z)|=2$. It also ensures that the unique $x^1=z$ for any $z \in \Sigma_2$ with $|{ \mbox{supp} }(z)|=1$. This yields the exact vector recovery on $\Sigma_2$.
“Only if”. Suppose $A$ achieves the exact vector recovery on $\Sigma_2$. Clearly, condition (ii) is necessary as shown before. To show that condition (i) is also necessary, consider a vector $z \in \Sigma_2$ with $|{ \mbox{supp} }(z)|=2$. Without loss of generality, we assume that ${ \mbox{supp} }(z)=\{ 1, 2\}$. Since $A$ achieves the exact vector recovery on $\Sigma_2$, it must achieve the exact support recovery for the fixed support ${\mathcal S}=\{1, 2 \}$. Hence it follows from Theorem \[thm:nonconstrained\_S2\] that $\| (A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}} )^{-1} A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}^c} \|_1 < 1$, which is equivalent to $$\label{eqn:condtion_H'_necessary_RN_S2}
1-\vartheta^2_{12} \, > \, \max_{j\in {\mathcal S}^c} \big( \, |\vartheta_{j1}- \vartheta_{j2} \vartheta_{12}| + |\vartheta_{j2}- \vartheta_{j1} \vartheta_{12}| \, \big).$$ Consider the three proper subsets of ${ \mbox{supp} }(z)=\{ 1, 2\}$, i.e., ${\mathcal J}=\emptyset$, ${\mathcal J}=\{1 \}$, and ${\mathcal J}=\{2 \}$. When ${\mathcal J}=\emptyset $, the inequality (\[eqn:condition\_H’\]) holds for $u=z$ and $v=0$ in light of $\max_{j \in {\mathcal S}} \big| (A^T A z)_j \big | \, > \, \max_{j \in {\mathcal S}^c} \big |(A^T A z)_j \big|$ obtained from the first step of the OMP. Moreover, we have either $|z_1 + \vartheta_{12} z_2| \ge |\vartheta_{12} z_1 + z_2|$ or $|z_1 + \vartheta_{12} z_2| \le |\vartheta_{12} z_1 + z_2|$. For the former case, we deduce from the exact support recovery of $z$ via the OMP that $j^*_1=1$ and ${\mathcal J}_1=\{ 1 \}$ such that $x^1=(A^T_{\bullet 1} A z) \mathbf e_1$ is the unique optimal solution to $\min_{{ \mbox{supp} }(w) \subseteq {\mathcal J}_1} \| A(z - w) \|^2_2$. Hence, by Corollary \[coro:Nec\_Suf\_suppt\_recovery\_RN\_RN+\], the exact support recovery shows that $f^*_2(z, x^1)< \min_{j \in {\mathcal S}^c} f^*_j(z, x^1)$, leading to the inequality (\[eqn:condition\_H’\]) for $u=z$ and $v=x^1$ when ${\mathcal J}=\{ 1 \}$. We then consider ${\mathcal J}=\{2\}$. In this case, the unique optimal solution $v^*$ to $\min_{{ \mbox{supp} }(w) \subseteq {\mathcal J}} \| A(z - w) \|^2_2$ is given by $v^*=(A^T_{\bullet 2} A z) \mathbf e_2 = (\vartheta_{12} z_1 + z_2) \mathbf e_2$. Therefore, $A^T_{\bullet j} A (z - v^*) = (\vartheta_{j1} - \vartheta_{j2} \vartheta_{12} ) z_1$ for any $j$. We thus have $|A^T_{\bullet 1} A (z-v^*)|= |1 - \vartheta^2_{12}| \cdot |z_1|$ and $|A^T_{\bullet j} A (z-v^*)|= |\vartheta_{j1} - \vartheta_{12} \vartheta_{j2}| \cdot |z_1|$, where $z_1 \ne 0$. Noting that $f^*_1(z, v^*) < \min_{j\in {\mathcal S}^c} f^*_j(z, v^*)$ if and only if $|A^T_{\bullet 1} A (z-v^*)| > \max_{j\in {\mathcal S}^c} |A^T_{\bullet j} A (z-v^*)|$, we deduce via the above results and (\[eqn:condtion\_H’\_necessary\_RN\_S2\]) that $f^*_1(z, v^*) < \min_{j\in {\mathcal S}^c} f^*_j(z, v^*)$, leading to the inequality (\[eqn:condition\_H’\]) for $u=z$ and $v=v^*$ when ${\mathcal J}=\{ 2 \}$. The other case where $|z_1 + \vartheta_{12} z_2| \le |\vartheta_{12} z_1 + z_2|$ can be established in a similar way. Further, for any $u\in \Sigma_2$ with $|{ \mbox{supp} }(u)|=1$ and ${\mathcal J}=\emptyset$, the inequality (\[eqn:condition\_H’\]) also holds. Thus condition $(\mathbf H)$ holds on $\Sigma_2$.
The next result shows that even though condition (\[eqn:exact\_recovery\_condition\]) (or equivalently (\[eqn:condition\_Prop\_3.5\])) may fail to be necessary, it is necessary for [*almost all*]{} the matrices achieving the exact vector recovery associated with a fixed support ${\mathcal S}$. This result also illustrates the challenge of constructing a counterexample. Toward this end, let $\mathcal U$ be the set of all matrices in $\mathbb R^{m\times N}$ with unit columns, i.e., $
\mathcal U \, := \, \big\{ A \in \mathbb R^{m\times N} \ | \, \|A_{\bullet i} \|_2=1, \ \forall \ i=1, \ldots, N \big\}.
$ Note that $\mathcal U$ is the Cartesian product of $N$ copies of unit $\ell_2$-spheres in $\mathbb R^m$. Hence, $\mathcal U$ is a compact manifold of dimension $(m-1)N$, and it attains a (finite) positive measure with a Lebesgue measure $\mu$ on $\mathcal U$. For a fixed index set ${\mathcal S}$, define the set $$\mathcal D \, := \, \big\{ A \in \mathcal U \, | \, \mbox{ $A$ achieves the exact vector recovery for the given support ${\mathcal S}$ } \big \}.$$ Recall that for any $A \in \mathcal D$, $A_{\bullet {\mathcal S}}$ has full column rank.
Let the set $\mathcal D' :=\{ A \in \mathcal D \, | \, \| (A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}} )^{-1} A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}^c} \|_1 = 1\}$, and $\mu$ be a Lebesgue measure on $\mathcal U$. Then $\mu(\mathcal D)>0$ and $\mu(\mathcal D')=0$.
For a given matrix $A\in \mathcal D$, we recall the function $q(z):= \max_{j \in {\mathcal S}} | (A^T A z)_j | - \max_{j \in {\mathcal S}^c} |(A^T A z)_j |$ for $z \in \mathbb R^N$ and the set $\mathcal R:=\{ z \in \mathbb R^N \, | \, z \ne 0, \ { \mbox{supp} }(z) = {\mathcal S}\}$ given below (\[eqn:necessity\_implication\]). Since $A$ achieves the exact vector recovery for the given support ${\mathcal S}$, we have $q(z)>0$ for all $z \in \mathcal R$. Moreover, it follows from the discussioins below (\[eqn:necessity\_implication\]) that $q({\widetilde}z) \ge 0$ for any nonzero ${\widetilde}z \in \mathbb R^N$ with ${ \mbox{supp} }({\widetilde}z) \subset {\mathcal S}$. By a similar argument for [@FoucartRauhut_book2013 Remark 3.6], we have $\| (A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}} )^{-1} A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}^c} \|_1 \le 1$ for any $A \in \mathcal D$.
Define the set $\mathcal D'' :=\{ A \in \mathcal D \, | \, \| (A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}} )^{-1} A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}^c} \|_1 < 1\}$. In view of the above argument, we see that $\mathcal D$ is the disjoint union of $\mathcal D'$ and $\mathcal D''$. Since $\mathcal D''$ is a (relatively) open subset in $\mathcal U$, we deduce that $\mu(\mathcal D'')>0$. Therefore, $\mu(\mathcal D) \ge \mu(\mathcal D'')>0$. Moreover, define $$\begin{aligned}
{\widetilde}{\mathcal D} & := \Big\{ \, A \in \mathcal U \, | \, \mbox{ $A_{\bullet {\mathcal S}}$ has full column rank, and $\| (A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}} )^{-1} A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}^c} \|_1 = 1$} \Big\}, \\
\mathcal W_j & := \Big\{ \, A \in \mathcal U \, | \, \mbox{ $A_{\bullet {\mathcal S}}$ has full column rank, and $\big\| \big[(A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}} )^{-1} A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}^c}\big]_{\bullet j} \big\|_1 = 1$} \Big\}, \quad j=1, \ldots, |{\mathcal S}^c|.
\end{aligned}$$ Hence, $\mathcal D' \subseteq {\widetilde}{\mathcal D} \subset \bigcup^{|{\mathcal S}^c|}_{j=1} \mathcal W_j$. Let $\mathbf a\in \mathbb R^{mN}$ be the vectorization of $A \in \mathbb R^{m\times N}$, i.e., $\mathbf a$ is generated by stacking the columns of $A$ on top of one another. For each fixed $j$, $\big\| \big[(A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}} )^{-1} A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}^c}\big]_{\bullet j} \big\|_1 = 1$ holds if and only if the piecewise polynomial function $G_j(\mathbf a)=0$, where $G_j(\mathbf a):= \sum^{|{\mathcal S}|}_{k=1} | G_{j, k}(\mathbf a) | - G_{j, k+1}(\mathbf a)$, and each $G_{j, k}(\cdot):\mathbb R^{mN} \rightarrow \mathbb R$ is a polynomial function. In view of this result, it is easy to verify that $\mathcal W_j$ is a subset of a finite union of the sets of the following form: $\big\{ \, A \in \mathcal U \, | \, \mbox{ $A_{\bullet {\mathcal S}}$ has full column rank, and $H_s(\mathbf a)=0$} \big\}$, where $H_s(\cdot)$ is a (nonzero) polynomial function. Clearly, each set of this form is a lower dimensional sub-manifold of $\mathcal U$ and thus is of zero measure. Thus $\mu(\mathcal W_j)=0$ for each $j$, and we thus have $\mu(\mathcal D')=0$.
### Construction of a Counterexample for a Fixed Support of Size 3 {#subsect:counterexample_S=3}
In this subsection, we construct a nontrivial counterexample to show that condition (\[eqn:exact\_recovery\_condition\]) (or equivalently (\[eqn:condition\_Prop\_3.5\])) fails to be necessary. The main result is given by the following theorem.
\[thm:counterexample\] For an index set ${\mathcal S}$ with $|{\mathcal S}|=3$, there exists an $A \in \mathbb R^{4\times 4}$ with unit columns such that $A$ achieves exact vector recovery for the fixed support ${\mathcal S}$ via the OMP, $A_{\bullet {\mathcal S}}$ has full column rank, and $\big\| (A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}} )^{-1} A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}^c} \big\|_1 = 1$.
To construct such a matrix $A$ indicated in the above theorem, we first present some preliminary results. Without loss of generality, let ${\mathcal S}=\{1, 2, 3\}$ and ${\mathcal S}^c=\{ 4 \}$. In view of the function $g_i$’s defined in (\[eqn:g\_i\_def\]), we introduce the following functions for $i=1, \ldots, 4$: $${\widehat}g_i(v) \, : = \, \big | h^T_i v |, \ \forall \ v \in \mathbb R^3, \quad \mbox{ where } \quad h_i := \big( \vartheta_{i1}, \vartheta_{i2}, \vartheta_{i3} )^T \in \mathbb R^3,$$ where we recall that $\vartheta_{ij}=\langle A_{\bullet i}, A_{\bullet j} \rangle$ for $i, j\in \{1, \ldots, 4\}$. Hence, $\max_{j \in {\mathcal S}} \big| (A^T A z)_j \big | \, > \, \max_{j \in {\mathcal S}^c} \big |(A^T A z)_j \big|$ for all $0 \ne z \in \mathbb R^{N}$ with ${ \mbox{supp} }(z) = {\mathcal S}$ if and only if the following holds: $$(\mathbf P): \quad \max_{i=1, 2, 3} {\widehat}g_i(v) \, > \, {\widehat}g_4(v), \quad \forall \ v=(v_1, v_2, v_3)^T\in \mathbb R^3 \ \mbox{ with } v_1\cdot v_2\cdot v_3 \ne 0.$$ Since each ${\widehat}g_i$ and $\max_{i=1, 2, 3} {\widehat}g_i(v)$ are convex piecewise affine functions [@MouShen_COCV18], it is not surprising that the feasibility of ($\mathbf P$) can be characterized by that of certain linear inequalities. The following lemma gives a necessary and sufficient condition for ($\mathbf P$) in term of the feasibility of some linear inequalities.
\[lem:feasbility\_sufficiency\] Let the matrix $H:=\begin{bmatrix} h_4 + h_1 & h_4-h_1 & h_4 + h_2 & h_4-h_2 & h_4 + h_3 & h_4-h_3 \end{bmatrix} \in \mathbb R^{3\times 6}$. Then $\max_{i=1, 2, 3} {\widehat}g_i(v) > {\widehat}g_4(v)$ for all $v=(v_1, v_2, v_3)^T\in \mathbb R^3$ with $v_1\cdot v_2\cdot v_3 \ne 0$ holds if and only if for each $\sigma :=(\sigma_1, \sigma_2, \sigma_3)\in \{ (\pm 1, \pm 1, \pm 1)\}$, there exist vectors $0\ne u \ge 0$ and $w \ge 0$ such that $ u + D_\sigma H w =0$, where the diagonal matrix $D_\sigma:=\mbox{diag}( \sigma_1, \sigma_2, \sigma_3) \in \mathbb R^{3\times 3}$.
Note that ($\mathbf P$) fails if and only if there exists ${\widehat}v \in \mathbb R^3$ with ${\widehat}v_1\cdot {\widehat}v_2\cdot {\widehat}v_3 \ne 0$ such that ${\widehat}g_i({\widehat}v) \le {\widehat}g_4({\widehat}v)$ for each $i=1,2, 3$. We claim that the latter statement holds if and only if there exists $v^*\in \mathbb R^3$ with $v^*_1\cdot v^*_2\cdot v^*_3 \ne 0$ such that $| h^T_i v^* | \le h^T_4 v^*$ for each $i=1, 2, 3$. The “if” part is obvious since $h^T_4 v^* \le |h^T_4 v^*|={\widehat}g_4(v^*)$. To show the “only if” part, we let $v^*= {{\rm sgn}}( h^T_4 {\widehat}v) \cdot {\widehat}v$, where ${\widehat}v$ satisfies the specified conditions. In view of ${\widehat}g_i(v^*)=|h^T_i v^*|=|h^T_i {\widehat}v|$ for $i=1, 2, 3$, $h^T_4 v^*= |h^T_4 {\widehat}v|={\widehat}g_4({\widehat}v)$, and each $v^*_i \ne 0$, we conclude that the desired result holds. This completes the proof of the claim. By using the above claim, we see that ($\mathbf P$) fails if and only if there exists $v^*\in \mathbb R^3$ with $v^*_1\cdot v^*_2\cdot v^*_3 \ne 0$ such that $| h^T_i v^* | \le h^T_4 v^*$ for each $i=1, 2, 3$, where the latter is further equivalent to $\pm h^T_i v^* \le h^T_4 v^*$ for each $i=1, 2, 3$ or equivalently $H^T v^* \ge 0$. Therefore, ($\mathbf P$) fails if and only if there exist $\sigma \in \{ (\pm 1, \pm 1, \pm 1)\}$ and ${\widetilde}v \in \mathbb R^3_{++}$ (i.e., ${\widetilde}v_i>0$ for each $i=1, 2, 3$) such that $H^T D_\sigma {\widetilde}v \ge 0$. By virtue of the Motzkin’s Transposition Theorem, we see that for a fixed $\sigma$, the linear inequality system $(D_\sigma H)^T {\widetilde}v \ge 0, {\widetilde}v >0$ has a solution ${\widetilde}v$ if and only if the linear inequality system $u + D_\sigma H w =0, 0\ne u \ge 0$ and $w \ge 0$ has no solution $(u, w)$. In other words, ($\mathbf P$) fails if and only if there exist $\sigma \in \{ (\pm 1, \pm 1, \pm 1)\}$ such that the linear inequality system $u + D_\sigma H w =0$, $0\ne u \ge 0$, and $w \ge 0$ has no solution. As a result, ($\mathbf P$) holds if and only if for any $\sigma \in \{ (\pm 1, \pm 1, \pm 1)\}$, there exist vectors $0\ne u \ge 0$ and $w \ge 0$ such that $ u + D_\sigma H w =0$.
By making use of the above preliminary results, we prove Theorem \[thm:counterexample\] as follows.
Consider the matrix $$\label{eqn:A_matrix_counterexample}
\displaystyle A \, = \, \begin{bmatrix} 1 & -\frac{1}{3} & -\frac{1}{3} & \frac{1}{3} \\
0 &\frac{2\sqrt{2}}{3} & -\frac{\sqrt{2}}{3} & \frac{\sqrt{2}}{3} \\
0 & 0 & \frac{\sqrt{6}}{3} & -\frac{\sqrt{6}}{12} \\
0 & 0 & 0 & \frac{\sqrt{10}}{4}\end{bmatrix} \in \mathbb R^{4\times 4}.$$ Recall that ${\mathcal S}=\{1, 2, 3\}$ and ${\mathcal S}^c=\{4\}$. It is easy to verify that $A$ is invertible with unit columns (i.e., $\|A_{\bullet i}\|_2=1$ for each $i$), $A_{\bullet{\mathcal S}}$ has full column rank, and $$\label{eqn:vartheta_value}
\vartheta_{12}=\vartheta_{21}=\vartheta_{13}=\vartheta_{31}=\vartheta_{23}=\vartheta_{32}=-\frac{1}{3},
\quad \vartheta_{41}=\vartheta_{42}=\frac{1}{3}, \quad \vartheta_{43} = -\frac{1}{2}.$$ Hence, $A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}}=\begin{bmatrix} h_1 & h_2 & h_3\end{bmatrix} \in \mathbb R^{3\times 3}$ and $A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}^c} = h_4$, where $$h_1 = \begin{bmatrix} 1 \\ -\frac{1}{3} \\ -\frac{1}{3} \end{bmatrix}, \quad
h_2 = \begin{bmatrix} -\frac{1}{3} \\ 1 \\ -\frac{1}{3} \end{bmatrix}, \quad
h_3 = \begin{bmatrix} -\frac{1}{3} \\ -\frac{1}{3} \\ 1 \end{bmatrix}, \quad
h_4 = \begin{bmatrix} \frac{1}{3} \\ \frac{1}{3} \\ -\frac{1}{2} \end{bmatrix}.$$ Furthermore, $$\Big( A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}} \Big)^{-1} = \begin{bmatrix} h_1 & h_2 & h_3\end{bmatrix}^{-1} = \begin{bmatrix} 1 & -\frac{1}{3} & -\frac{1}{3} \\ -\frac{1}{3} & 1 & -\frac{1}{3} \\ -\frac{1}{3} & -\frac{1}{3} & 1 \end{bmatrix}^{-1} = \frac{3}{4} \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix}$$ such that $\big\| (A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}} )^{-1} A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}^c}\big\|_1 = \frac{3}{8}+\frac{3}{8}+\frac{1}{4}= 1$. The rest of the proof consists of two parts: the first part shows that $\max_{j \in {\mathcal S}} \big| (A^T A z)_j \big| > \max_{j \in {\mathcal S}^c} \big |(A^T A z)_j \big|$ for all $z$ with ${ \mbox{supp} }(z) = {\mathcal S}$, and the second part shows that $A$ achieves the exact vector recovery for the index set ${\mathcal S}$.
We first show the following claim: $$\label{eqn:claim_I}
\mbox{ Claim I: $\quad \max_{j \in {\mathcal S}} \big| (A^T A z)_j \big| > \max_{j \in {\mathcal S}^c} \big |(A^T A z)_j \big|$ \ for all $z$ with ${ \mbox{supp} }(z) = {\mathcal S}$}.$$ In view of Lemma \[lem:feasbility\_sufficiency\], we only need to show that for each $\sigma \in \{ (\pm 1, \pm 1, \pm 1)\}$, there exist vectors $0\ne u \ge 0$ and $w \ge 0$ such that $ u + D_\sigma H w =0$, where the matrix $$H = \begin{bmatrix} h_4 + h_1 & h_4-h_1 & h_4 + h_2 & h_4-h_2 & h_4 + h_3 & h_4-h_3 \end{bmatrix} = \begin{bmatrix} \frac{4}{3} & -\frac{2}{3} & 0 & \frac{2}{3} & 0 & \frac{2}{3} \\
0 & \frac{2}{3} & \frac{4}{3} & -\frac{2}{3} & 0 & \frac{2}{3} \\
-\frac{5}{6} & -\frac{1}{6} & -\frac{5}{6} & -\frac{1}{6} & \frac{1}{2} & -\frac{3}{2} \end{bmatrix}.$$ Toward this end, we give a specific solution $(u, w)$ to the above linear inequality system for each $\sigma$:
- $\sigma=(1, 1, 1)$. A solution is given by $u=-(H_{\bullet 2} + H_{\bullet 4})=(0, 0, \frac{1}{3})^T$ and $w=\mathbf e_2 + \mathbf e_4$;
- $\sigma=(1, 1, -1)$. A solution is given by $u=H_{\bullet 5}=(0, 0, \frac{1}{2})^T$ and $w=\mathbf e_5$;
- $\sigma=(1, -1, 1)$. A solution is given by $u=(\frac{2}{3}, \frac{2}{3}, \frac{1}{6})^T$ and $w=\mathbf e_2$;
- $\sigma=(-1, 1, 1)$. A solution is given by $u=(\frac{2}{3}, \frac{2}{3}, \frac{1}{6})^T$ and $w=\mathbf e_4$;
- $\sigma=(1, -1, -1)$. A solution is given by $u=H_{\bullet 5}=(0, 0, \frac{1}{2})^T$ and $w=\mathbf e_5$;
- $\sigma=(-1, 1, -1)$. A solution is given by $u=H_{\bullet 5}=(0, 0, \frac{1}{2})^T$ and $w=\mathbf e_5$;
- $\sigma=(-1, -1, 1)$. A solution is given by $u=(\frac{3}{4}, 0, \frac{5}{6})^T$ and $w=\mathbf e_1$;
- $\sigma=(-1, 1, -1)$. A solution is given by $u=H_{\bullet 5}=(0, 0, \frac{1}{2})^T$ and $w=\mathbf e_5$.
Hence, Claim I holds in light of Lemma \[lem:feasbility\_sufficiency\].
We show next that the matrix $A$ achieves the exact vector recovery via the OMP for the given index set ${\mathcal S}=\{ 1, 2, 3\}$. Let $z$ be an arbitrary vector in $\mathbb R^4$ with ${ \mbox{supp} }(z)={\mathcal S}$, and $y=A z = A_{\bullet {\mathcal S}} z_{\mathcal S}$. Consider the following three steps of the OMP: $\bullet$ Step 1: Since $x^0=0$ and $y=A z$, it follows from (\[eqn:claim\_I\]) that $\max_{i=1, 2, 3} | A^T_{\bullet i} A (z - x^0)| > | A^T_{\bullet 4} A( z - x^0)|$. Hence, by Corollary \[coro:Nec\_Suf\_suppt\_recovery\_RN\_RN+\], $j^*_1\in {\mathcal S}= \{1, 2, 3\}$ and ${\mathcal J}_1=\{ j^*_1\}$. Also, $x^1:={\operatornamewithlimits{\arg\min}}_{{ \mbox{supp} }(w) \subseteq {\mathcal J}_{1} } \, \| y - A w\|^2_2$ is given by $x^1 = (A^T_{\bullet j^*_1} A_{\bullet {\mathcal S}} z_{{\mathcal S}}) \cdot \mathbf e_{j^*_1}$. Note that $x^1_{j^*_1} \ne 0$ in view of Proposition \[prop:index\_set\]. $\bullet$ Step 2: We first prove the following claim: for any $j_1 \in {\mathcal S}= \{1, 2, 3\}$ and $u = (A^T_{\bullet j_1} A z) \cdot \mathbf e_{j_1} \in \mathbb R^4$, $\max_{i=1, 2, 3} | A^T_{\bullet i} A ( z- u) | > | A^T_{\bullet 4} A ( z - u) |$.
For any $j_1\in \{1, 2, 3\}$ and its corresponding $u$, let $v:=z - u$. Note that $v_i=z_i \ne 0$ for each $i\in {\mathcal S}\setminus \{ j_1 \}$. Therefore, if $z_{j_1} \ne A^T_{\bullet j_1} A z$, then ${ \mbox{supp} }(v)={\mathcal S}$ so that the claim holds by virtue of (\[eqn:claim\_I\]). To handle the case where $z_{j_1} = A^T_{\bullet j_1} A z$, we consider $j_1=1$ first. Since $A^T_{\bullet 1} A z = A^T_{\bullet 1} A_{\bullet {\mathcal S}} z_{\mathcal S}= h^T_1 z_{\mathcal S}= z_1 - \frac{1}{3} z_2 - \frac{1}{3}z_3$, we must have $z_3=-z_2 \ne 0$. Therefore, $v_{\mathcal S}= z_{\mathcal S}- u_{\mathcal S}= z_2 \cdot (0, 1, -1)^T$. It follows from ${\widehat}g_i(v_{\mathcal S})=| h^T_i v_{\mathcal S}|$ and $h_i$’s given before that ${\widehat}g_1(v_{\mathcal S})= 0$, ${\widehat}g_2(v_{\mathcal S})= {\widehat}g_3(v_{\mathcal S})= \frac{4}{3}|z_2|$, and ${\widehat}g_4(v_{\mathcal S})=\frac{5}{6}|z_2|$. Consequently, $\max_{i=1, 2, 3} | A^T_{\bullet i} A( z- u) | =\max_{i=1,2,3} {\widehat}g_i(v_{\mathcal S}) > {\widehat}g_4(v_{\mathcal S})=| A^T_{\bullet 4} A (z - u) |$. Due to the symmetry of the matrix $A$, it can be shown via a similar argument that the above result also holds for $z_{j_1} = A^T_{\bullet j_1} A z$ with $j_1=2$ or $j_1=3$. This completes the proof of the claim.
By the above claim, we see that $\max_{i=1, 2, 3} | A^T_{\bullet i} A (z- x^1) | > | A^T_{\bullet 4} A (z - x^1) |$ for the vector $x^1$ obtained from Step 1. Therefore, $j^*_2 \in {\mathcal S}$ and $j^*_2 \ne j^*_1$ in view of Lemma \[lem:exact\_suppt\_recovery\_index\]. Hence, ${\mathcal J}_2=\{j^*_1, j^*_2\}$, and $x^2:={\operatornamewithlimits{\arg\min}}_{{ \mbox{supp} }(w) \subseteq {\mathcal J}_{2} } \, \| y - A w\|^2_2$ is given by $x^2_{{\mathcal J}_2}= \big( A^T_{\bullet {\mathcal J}_2} A_{\bullet {\mathcal J}_2} \big)^{-1} A^T_{\bullet {\mathcal J}_2} A_{\bullet {\mathcal S}} z_{\mathcal S}$, and $x^2_i=0$ for $i\notin {\mathcal J}_2$.
$\bullet$ Step 3: Note that for any index set ${\mathcal I}\in \big\{ \{1, 2\}, \{1, 3\}, \{2, 3\}\big\}$, it follows from a direct calculation on the matrix $A$ that $$w = \big( A^T_{\bullet {\mathcal I}} A_{\bullet {\mathcal I}} \big)^{-1} A^T_{\bullet {\mathcal I}} A_{\bullet {\mathcal S}} z_{\mathcal S}= \begin{bmatrix} z_s - \frac{1}{2} z_p \\ z_t -\frac{1}{2} z_p \end{bmatrix},$$ where $s, t \in {\mathcal I}$ with $s<t$, and $p\in {\mathcal S}\setminus {\mathcal I}$. Hence, (i) if ${\mathcal J}_2=\{1, 2\}$, then $(z-x^2)_{\mathcal S}= z_3\cdot (\frac{1}{2}, \frac{1}{2}, 1)^T$; (ii) if ${\mathcal J}_2=\{1, 3\}$, then $(z-x^2)_{\mathcal S}= z_2\cdot (\frac{1}{2}, 1, \frac{1}{2})^T$; and (iii) if ${\mathcal J}_3=\{2, 3\}$, then $(z-x^2)_{\mathcal S}= z_1\cdot (1, \frac{1}{2}, \frac{1}{2})^T$. Therefore, for the vector $x^2$ obtained from Step 2, we have ${ \mbox{supp} }(z-x^2)={\mathcal S}$. It follows from (\[eqn:claim\_I\]) that $\max_{i=1, 2, 3} | A^T_{\bullet i} A (z- x^2) | > | A^T_{\bullet 4} A (z - x^2) |$. This shows that $j^*_3 \in {\mathcal S}$ with $j^*_3 \notin {\mathcal J}_2$. Hence, ${\mathcal J}_3={\mathcal J}_2\cup\{j^*_3\}={\mathcal S}$. Since $A_{\bullet {\mathcal S}}$ has full column rank, we see that $x^3:={\operatornamewithlimits{\arg\min}}_{{ \mbox{supp} }(w) \subseteq {\mathcal J}_{3} } \, \| y - A w\|^2_2$ satisfies $x^3=z$. This shows that $z$ is uniquely recovered via the OMP using the matrix $A$.
\[remark:counter\_example\] We make a few remarks about the counterexample constructed above.
- It is easy to verify that for the given matrix $A$ in (\[eqn:A\_matrix\_counterexample\]), when $v=\alpha \cdot (1, 1, 0)^T$ for any $0 \ne \alpha \in \mathbb R$, ${\widehat}g_1(v)={\widehat}g_2(v)= {\widehat}g_3(v)= {\widehat}g_4(v)= \frac{2}{3}|\alpha|$. Hence, $\max_{i=1,2,3} {\widehat}g_i(v) = {\widehat}g_4(v)$. Letting $z=(z_{\mathcal S}, z_{{\mathcal S}^c})\in \mathbb R^4$ with $z_{\mathcal S}=v$ and $z_{{\mathcal S}^c}=0$, we have $\max_{i=1, 2, 3} g_i(z) = g_4(z)$, leading to a counterexample to the implication (\[eqn:necessity\_implication\]) used in the necessity proof for [@FoucartRauhut_book2013 Proposition 3.5]. Besides, letting ${\widetilde}m=|{\mathcal S}|=3$, since $A$ is invertible, all the ${\widetilde}m$-term representations are unique, and the condition $\| (A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}} )^{-1} A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}^c} \|_1 = 1$ implies the failure of the “Exact Recovery Condition” defined in Tropp’s paper [@Tropp_ITI04] (i.e., $\| (A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}} )^{-1} A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}^c} \|_1 < 1$). However, any $z$ with ${ \mbox{supp} }(z)={\mathcal S}$ can be exactly recovered via the OMP, yielding a counterexample to [@Tropp_ITI04 Theorem 3.10].
- There are multiple $4\times 4$ real matrices satisfying the conditions specified in Theorem \[thm:counterexample\] as long as their columns are unit and the inner products of their distinct columns defined by $\vartheta_{ij}$ equal to the values given in (\[eqn:vartheta\_value\]). In particular, for the matrix $A$ given in (\[eqn:A\_matrix\_counterexample\]) and any orthogonal matrix $P \in \mathbb R^{4 \times 4}$, $PA$ also satisfies the conditions in Theorem \[thm:counterexample\].
The counterexample constructed in the previous theorem can be extended to one with a larger size.
Suppose an index set ${\mathcal S}\subseteq\{ 1, \ldots, N\}$ is of size 3, i.e., $|{\mathcal S}|=3$. Then for any $m \ge 4$ and $N \ge 4$, there exists a matrix ${\widehat}A \in \mathbb R^{m\times N}$ with unit columns such that ${\widehat}A$ achieves the exact vector recovery for the fixed support ${\mathcal S}$ via the OMP, ${\widehat}A_{\bullet {\mathcal S}}$ has full column rank, and $\big\| ({\widehat}A^T_{\bullet {\mathcal S}} {\widehat}A_{\bullet {\mathcal S}} )^{-1} {\widehat}A^T_{\bullet {\mathcal S}} {\widehat}A_{\bullet {\mathcal S}^c} \big\|_1 = 1$.
Without loss of generality, let ${\mathcal S}=\{1, 2, 3\}$. For any $N \ge 4$, define the matrix $B \in \mathbb R^{4\times N}$ as $
B := \begin{bmatrix} A & B_{\bullet 5} & \cdots & \cdots & B_{\bullet N} \end{bmatrix},
$ where the matrix $A$ is given in (\[eqn:A\_matrix\_counterexample\]), and $B_{\bullet k}=\pm A_{\bullet 4}$ for each $k \ge 5$. Then let $
{\widehat}A := \begin{bmatrix} B \\ 0_{(m-4)\times N} \end{bmatrix} \in \mathbb R^{m\times N}.
$ Straightforward calculations show that ${\widehat}A$ satisfies the desired properties by observing that almost all the required properties of ${\widehat}A$ rely on $\langle {\widehat}A_{\bullet i}, {\widehat}A_{\bullet j}\rangle$’s, which are defined by $\vartheta_{ij}$’s or $h_i$’s of the matrix $A$.
Exact Vector Recovery on the Nonnegative Orthant $\mathbb R^N_+$ for a Fixed Support
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We consider the exact vector recovery on the nonnegative orthant $\mathbb R^N_+$ for a fixed support ${\mathcal S}$ using constrained matching pursuit. Without loss of generality, we assume that the matrix $A \in \mathbb R^{m\times N}$ has unit columns, i.e., $\|A_{\bullet i}\|_2=1$ for each $i=1, \ldots, N$. A necessary condition is given as follows.
\[lem:nonnegative orthant\_Nec01\] Given a matrix $A \in \mathbb R^{m\times N}$ with unit columns and an index set ${\mathcal S}$ of size $s$, the exact vector recovery of every nonzero vector $x \in \mathbb R^N_+$ with ${ \mbox{supp} }(x)={\mathcal S}$ is achieved via constrained matching pursuit only if $A_{\bullet {\mathcal S}}$ has full column rank.
Assume, in contrast, that $A_{\bullet {\mathcal S}}$ does not have full column rank. Let $r:=|{\mathcal S}|$. Then there exist a nonzero vector $v \in \mathbb R^{r}$ such that $A_{\bullet {\mathcal S}} v =0$. For a given nonzero $x \ge 0$ with ${ \mbox{supp} }(x)={\mathcal S}$, suppose at the $r$th step, the exact support of $x$ is recovered from $y=A x$ via constrained matching pursuit. It follows from Algorithm \[algo:constrained\_MP\] that one need to solve the constrained minimization problem $
{\bf Q}: \ \min_{w \in \mathbb R^{r}_+} \| A_{\bullet {\mathcal S}} w - y \|^2_2,
$ where $y=A_{\bullet {\mathcal S}} x_{\mathcal S}$, to recover $x_{\mathcal S}$. Since $x_{\mathcal S}>0$ and $v \ne 0$, there exists a small positive constant $\varepsilon$ such that $x_{\mathcal S}+ \varepsilon v >0$. Noting that $A_{\mathcal S}(x_{\mathcal S}+ \varepsilon v)= A_{\mathcal S}x_{\mathcal S}= y$, we see that $x_{\mathcal S}+ \varepsilon v$ is a solution to the minimization problem ${\bf Q}$. Hence, ${\bf Q}$ has multiple optimal solutions which can be different from the desired solution $x_{\mathcal S}$. This leads to a contradiction. Consequently, $A_{\bullet {\mathcal S}}$ has full column rank.
In light of statement (ii) of Corollary \[coro:Nec\_Suf\_suppt\_recovery\_RN\_RN+\] for $x^0=0$, we easily obtain another necessary condition for the exact support recovery (and thus exact vector recovery) of any $z \in \mathbb R^N_+$ with ${ \mbox{supp} }(z)={\mathcal S}$: $$\max_{j \in { \mbox{supp} }(z)} (A^T_{\bullet j} A z )_+ \, > \, \max_{j\in [{ \mbox{supp} }(z)]^c} ( A^T_{\bullet j} A z )_+, \qquad \forall \ z \in \mathbb R^N_+ \ \mbox{ with } { \mbox{supp} }(z)={\mathcal S},$$ which is equivalent to $\| (A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}} v )_+ \|_\infty > \| (A^T_{\bullet {\mathcal S}^c} A_{\bullet {\mathcal S}} v )_+ \|_\infty$ for all $v \in \mathbb R^{|{\mathcal S}|}_{++}$.
### Necessary and Sufficient Conditions for Exact Vector Recovery for a Fixed Support of Size 2
We derive necessary and sufficient conditions for exact vector recovery on $\mathbb R^N_+$ for a given support ${\mathcal S}$ with $|{\mathcal S}|=2$. Recall that $\vartheta_{ij}:=\langle A_{\bullet i}, A_{\bullet j} \rangle$ for $i, j \in \{1, \ldots, N\}$. Besides, the following lemma is needed.
\[lem:positive\_sign\] Let $M \in \mathbb R^{m\times m}$ be a positive definite matrix. Then for any $z \in \mathbb R^m$ with $z > 0$, there exists $i\in \{1, \ldots, m\}$ such that $(M z)_i>0$.
Suppose, in contrast, that there exists $z>0$ such that $M z \le 0$. Since $z>0$, we have $z^T Mz \le 0$. As $M$ is positive definite, we deduce that $z^T M z =0$ so that $z =0$. This yields a contradiction.
\[thm:nonnegative constraint\_S2\] Given a matrix $A \in \mathbb R^{m\times N}$ with unit columns and the index set ${\mathcal S}=\{1, 2\}$, every nonzero vector $x \in \mathbb R^N_+$ with ${ \mbox{supp} }(x)={\mathcal S}$ is recovered from $y=A x$ via constrained matching pursuit if and only if the following conditions hold:
- $A_{\bullet {\mathcal S}}$ has full column rank or equivalently $|\vartheta_{12}|<1$;
- $ \max \big( (z_1 + \vartheta_{12} z_2)_+, \, (\vartheta_{12} z_1 + z_2)_+ \big) > \max_{j \in {\mathcal S}^c} \big(\vartheta_{j1} z_1 + \vartheta_{j2} z_2 \big)_+, \ \forall \, (z_1, z_2)^T \in \mathbb R^2_{++}$;
- $1-\vartheta^2_{12} \, > \, \max_{j\in {\mathcal S}^c} \big( \, (\vartheta_{j2}- \vartheta_{12}\vartheta_{j1})_+, \ (\vartheta_{j1}- \vartheta_{12}\vartheta_{j2})_+ \, \big)$.
“Only if”. Clearly, the condition that $A_{\bullet {\mathcal S}}$ has full column rank is necessary for the exact vector recovery in view of Lemma \[lem:nonnegative orthant\_Nec01\]. Since $A_{\bullet {\mathcal S}}$ has full column rank if and only if $A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}} = \begin{bmatrix} 1 & \vartheta_{12} \\ \vartheta_{12} & 1 \end{bmatrix}$ is positive definite, we see that $A_{\bullet {\mathcal S}}$ has full column rank if and only if $|\vartheta_{12}|<1$. For an arbitrary $z \in \mathbb R^N$ with $z_{\mathcal S}=(z_1, z_2)>0$, let $y= A z = A_{\bullet {\mathcal S}} z_{\mathcal S}$. At Step 1, since $x^0=0$, it follows from statement (ii) of Corollary \[coro:Nec\_Suf\_suppt\_recovery\_RN\_RN+\] that any $j^*_1 \in {\mathcal S}$ if and only if $
\max_{j\in {\mathcal S}} \langle A_{\bullet j}, A z \rangle_+ > \max_{j\in {\mathcal S}^c} \langle A_{\bullet j}, A z \rangle_+.
$ This leads to condition (ii), in light of $\langle A_{\bullet 1}, A z \rangle_+ = (z_1 + \vartheta_{12} z_2)_+$, $\langle A_{\bullet 2}, A z \rangle_+ = (\vartheta_{12}z_1 + z_2)_+$, and $\langle A_{\bullet j}, A z \rangle_+ = (\vartheta_{j1} z_1 + \vartheta_{j2} z_2 )_+$. Since $\begin{bmatrix} 1 & \vartheta_{12} \\ \vartheta_{12} & 1 \end{bmatrix}$ is positive definite, it follows from Lemma \[lem:positive\_sign\] that for any $(z_1, z_2)>0$, at least one of $\vartheta_{12}z_1 + z_2$ and $\vartheta_{j1} z_1 + \vartheta_{j2} z_2$ is positive. Further, in view of $|\vartheta_{12}|<1$ and the fact that for $a, b \in \mathbb R$, $b_+>a_+$ if and only if $b>0$ and $b>a$, it is easy to verify that for any $(z_1, z_2)>0$, (a) $(z_1 + \vartheta_{12} z_2)_+ > (\vartheta_{12} z_1 + z_2)_+$ if and only if $z_1>z_2$; (b) $(z_1 + \vartheta_{12} z_2)_+ < (\vartheta_{12} z_1 + z_2)_+$ if and only if $z_1 < z_2$; and (c) $(z_1 + \vartheta_{12} z_2)_+ = (\vartheta_{12} z_1 + z_2)_+>0$ if and only if $z_1 =z_2$. Hence, we have that $j^*_1=1$ if $z_1>z_2>0$, $j^*_1=2$ if $z_2>z_1>0$, and $j^*_1\in\{1, 2\}$ if $z_1=z_2>0$. Moreover, ${\mathcal J}_1=\{ j^*_1\}$, and $x^1:={\operatornamewithlimits{\arg\min}}_{w \ge 0, { \mbox{supp} }(w) \subseteq {\mathcal J}_{1} } \, \| A_{\bullet {\mathcal S}} z_{\mathcal S}- A w\|^2_2$ is given by $x^1= \langle A_{\bullet {\mathcal S}} z_{{\mathcal S}}, A_{\bullet j^*_1} \rangle_+ \cdot \mathbf e_{j^*_1}$, where $\langle A_{\bullet {\mathcal S}} z_{{\mathcal S}}, A_{\bullet j^*_1} \rangle_+ >0$ by Proposition \[prop:index\_set\]. In what follows, we consider $j^*_1=1$ corresponding to $z_1\ge z_2>0$ first. In this case, $x^1= (z_1 + \vartheta_{12} z_2) \cdot \mathbf e_1$. Hence, $(z - x^1)_{\mathcal S}= (-\vartheta_{12}, 1)^T \cdot z_2$. It follows from statement (ii) of Corollary \[coro:Nec\_Suf\_suppt\_recovery\_RN\_RN+\] that a necessary and sufficient condition to select $j^*_2=2$ at Step 2 is $$\label{eqn:Step2_Nec}
\langle A(z - x^1), A_{\bullet 2} \rangle_+ \, > \, \max_{j \in {\mathcal S}^c} \langle A(z - x^1), A_{\bullet j} \rangle_+,$$ where $\langle A(z - x^1), A_{\bullet 2} \rangle_+= (1-\vartheta^2_{12}) \cdot z_2$ and $\langle A(z - x^1), A_{\bullet j} \rangle_+ = (\vartheta_{j2}- \vartheta_{12}\vartheta_{j1})_+ \cdot z_2$ for each $j\in {\mathcal S}^c$. Hence, when $z_1\ge z_2>0$, an equivalent condition for (\[eqn:Step2\_Nec\]) is $1-\vartheta^2_{12} > \max_{j \in {\mathcal S}^c} (\vartheta_{j2}- \vartheta_{12}\vartheta_{j1})_+$. When $j^*_1=2$ corresponding to $z_2 \ge z_1>0$, we deduce via a similar argument that a necessary and sufficient condition for $j^*_2=1$ at Step 2 is $1-\vartheta^2_{12} > \max_{j \in {\mathcal S}^c} (\vartheta_{j1}- \vartheta_{12}\vartheta_{j2})_+$. This gives rise to condition (iii).
“If”. As indicated in the “only if” part, condition (ii) is sufficient for $j^*_1 \in {\mathcal S}$ at Step 1, and condition (iii) is sufficient for $j^*_2 \in {\mathcal S}\setminus \{ j^*_1\}$ at Step 2. Hence, under conditions (ii) and (iii), the exact support ${\mathcal S}$ is recovered from $y=A z$ in two steps for any $z\in \mathbb R^N$ with $z_{\mathcal S}>0$, i.e., ${\mathcal J}_2={\mathcal S}$. Note that the optimality condition for $x^2:={\operatornamewithlimits{\arg\min}}_{w \ge 0, { \mbox{supp} }(w) \subseteq {\mathcal J}_{2} } \, \| A_{\bullet {\mathcal S}} z_{\mathcal S}- A w\|^2_2$ is given by the linear complementarity problem (LCP): $0 \le x^2_{\mathcal S}\perp A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}} (x^2_{{\mathcal S}} - z_{\mathcal S}) \ge 0$. Since $A_{\bullet {\mathcal S}}$ has full column rank, $A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}}$ is positive definite such that the LCP has a unique solution $x^2_{\mathcal S}= z_{\mathcal S}$ or equivalently $x^2=z$. This shows that the exact vector recovery is achieved for any $z\in \mathbb R^N$ with $z_{\mathcal S}>0$ under conditions (i)-(iii).
Applying the necessary and sufficient conditions given in Theorem \[thm:nonnegative constraint\_S2\], it is shown in the next corollary that condition $(\mathbf H)$ is necessary for the exact vector or support recovery on $\Sigma_2 \cap \mathbb R^N_+$.
\[coro:condition\_H’\_necessary\_RN+\_S2\] Let $A \in \mathbb R^{m\times N}$ be a matrix with unit columns. Then the exact vector recovery on $\Sigma_2 \cap \, \mathbb R^N_+$ is achieved if and only if (i) condition $(\mathbf H)$ holds on $\Sigma_2 \cap \, \mathbb R^N_+$, and (ii) any two distinct columns of $A$ are linearly independent.
The “if” part is similar to that given in the proof of Corollary \[coro:condition\_H’\_necessary\_RN\_S2\]. For the “only if” part, let $A$ achieve the exact vector recovery on $\Sigma_2 \cap \, \mathbb R^N_+$. Clearly, condition (ii) is necessary in light of Lemma \[lem:nonnegative orthant\_Nec01\]. To show that condition (i) is necessary, we consider an arbitrary $z \in \Sigma_2 \cap \, \mathbb R^N_+$ with ${ \mbox{supp} }(z)=\{1, 2\}:={\mathcal S}$. Hence, $A$ achieves the exact support recovery for the fixed support ${\mathcal S}$. Therefore, conditions (ii) and (iii) of Theorem \[thm:nonnegative constraint\_S2\] hold. Consider the three proper subsets of ${\mathcal S}$, i.e., ${\mathcal J}=\emptyset$, ${\mathcal J}=\{1\}$, and ${\mathcal J}=\{2 \}$. When ${\mathcal J}=\emptyset$, we see that the inequality (\[eqn:condition\_H’\]) holds for $u=z$ and $v=0$ in light of statement (ii) of Corollary \[coro:Nec\_Suf\_suppt\_recovery\_RN\_RN+\] and conditions (ii) of Theorem \[thm:nonnegative constraint\_S2\]. Furthermore, we have either $(z_1 + \vartheta_{12} z_2)_+ \ge (\vartheta_{12} z_1 + z_2)_+$ or $(z_1 + \vartheta_{12} z_2)_+ \le (\vartheta_{12} z_1 + z_2)_+$. For the former case, we deduce from Algorithm \[algo:constrained\_MP\] that $j^*_1=1$ and ${\mathcal J}_1=\{ 1 \}$ such that $x^1=(A^T_{\bullet 1} A z)_+ \mathbf e_1$ is the unique optimal solution to $\min_{w\ge 0, { \mbox{supp} }(w) \subseteq {\mathcal J}_1} \| A(z - w) \|^2_2$. Hence, the exact support recovery of $z$ shows that $f^*_2(z, x^1)< \min_{j \in {\mathcal S}^c} f^*_j(z, x^1)$, yielding (\[eqn:condition\_H’\]) for $u=z$ and $v=x^1$ when ${\mathcal J}=\{ 1 \}$. We then consider ${\mathcal J}=\{2\}$. Similarly, the unique optimal solution $v^*$ to $\min_{w \ge 0, { \mbox{supp} }(w) \subseteq {\mathcal J}} \| A(z - w) \|^2_2$ is given by $v^*=(A^T_{\bullet 2} A z)_+ \mathbf e_2 = (\vartheta_{12} z_1 + z_2)_+ \mathbf e_2$. Consider two sub-cases:
- $(\vartheta_{12} z_1 + z_2)_+ \le 0$. In this case, $v^*=0$ such that $z-v^*=z$. Hence, $(A^T_{\bullet 1} A (z - v^*))_+=(z_1+\vartheta_{12} z_2)_+$ and $(A^T_{\bullet j} A (z-v^*))_+= (\vartheta_{j1}z_1 + \vartheta_{j2} z_2)_+$ for $j \in {\mathcal S}^c$. Since $ \max \big( (z_1 + \vartheta_{12} z_2)_+, \, (\vartheta_{12} z_1 + z_2)_+ \big) = (z_1 + \vartheta_{12} z_2)_+$, we deduce via condition (ii) of Theorem \[thm:nonnegative constraint\_S2\] that $f^*_1(z, v^*) < \min_{j\in {\mathcal S}^c} f^*_j(z, v^*)$, yielding the inequality (\[eqn:condition\_H’\]) for $u=z$ and $v=v^*$ when ${\mathcal J}=\{ 2 \}$.
- $(\vartheta_{12} z_1 + z_2)_+ \ge 0$. In this case, $z-v^*= (1, -\vartheta_{12}) z_1$ such that $(A^T_{\bullet 1} A (z - v^*))_+=(1 - \vartheta^2_{12}) \cdot z_1$ and $(A^T_{\bullet j} A (z-v^*))_+= (\vartheta_{j1} - \vartheta_{12} \vartheta_{j2})_+ \cdot z_1$ for $j \in {\mathcal S}^c$, where $z_1>0$. By condition (iii) of Theorem \[thm:nonnegative constraint\_S2\] that $f^*_1(z, v^*) < \min_{j\in {\mathcal S}^c} f^*_j(z, v^*)$, yielding (\[eqn:condition\_H’\]) for $u=z$ and $v=v^*$ when ${\mathcal J}=\{ 2 \}$.
The other case where $(z_1 + \vartheta_{12} z_2)_+ \le (\vartheta_{12} z_1 + z_2)_+$ can be established similarly. In addition, for any $u\in \Sigma_2\cap \mathbb R^N_+$ with $|{ \mbox{supp} }(u)|=1$ and ${\mathcal J}=\emptyset$, (\[eqn:condition\_H’\]) also holds. Hence, condition $(\mathbf H)$ holds on $\Sigma_2 \cap \, \mathbb R^N_+$.
### Necessary and Sufficient Conditions for Exact Vector Recovery for a Fixed Support of Size 3
We first present some preliminary results. Given a (possibly non-square) matrix $$M = \begin{bmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{bmatrix},$$ where $M_{ij}$’s are submatrices of $M$ with $M_{11}$ being invertible, the Schur complement of $M_{11}$ in $M$, denoted by $M/M_{11}$, is given by $M/M_{11}:= M_{22} - M_{21} M^{-1}_{11} M_{12}$. When $M$ is square, the Schur determinant formula says that $\det (M/M_{11}) = \det M / \det M_{11}$ [@CPStone_book92 Proposition 2.3.5]. Particularly, when $M$ is positive definite, any of its Schur complement is also positive definite.
\[lem:R+\_optimal\_solution\] Given a matrix $A\in \mathbb R^{m\times N}$ and an index set ${\mathcal S}$ such that $A_{\bullet {\mathcal S}}$ has full column rank, let the matrix $M:=A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}}$. For a nonempty index set ${\mathcal J}\subset {\mathcal S}$ and $z \in \mathbb R^N_+$ with ${ \mbox{supp} }(z)={\mathcal S}$, let $x^*$ be the unique solution to $\min_{w \ge 0, { \mbox{supp} }(w) \subseteq {\mathcal J}} \| A (z-w) \|^2_2$ whose support is given by ${\mathcal J}^*$, i.e., ${ \mbox{supp} }(x^*)={\mathcal J}^*$. Define the index set ${\mathcal I}:={\mathcal S}\setminus {\mathcal J}^*$. Then $A^T_{\bullet {\mathcal J}^*}A (z- x^*)=0$, and $$A^T_{\bullet {\mathcal I}}A (z- x^*) = \big( M/M_{{\mathcal J}^* {\mathcal J}^*} \big) \cdot z_{{\mathcal I}}, \quad
A^T_{\bullet {\mathcal S}^c}A (z- x^*) = A^T_{\bullet {\mathcal S}^c} \big[ I - A_{\bullet {\mathcal J}^*} \big( A^T_{\bullet {\mathcal J}^*} A_{\bullet {\mathcal J}^*} \big)^{-1} A^T_{\bullet {\mathcal J}^*} \big] A_{\bullet {\mathcal I}} \cdot z_{{\mathcal I}}.$$ Moreover, $\max_{j\in{\mathcal I}} [A^T_{\bullet j}A(z - x^*)]_+=\max_{j\in{\mathcal S}\setminus {\mathcal J}} [A^T_{\bullet j}A(z - x^*)]_+>0$.
Since $x^*$ is the unique optimal solution to $\min_{w \ge 0, { \mbox{supp} }(w) \subseteq {\mathcal J}} \| A (z- w) \|^2_2$, we have $x^*=(x^*_{\mathcal J}, 0)$, where $x^*_{\mathcal J}$ is the solution to $\min_{u \ge 0} \| A_{\bullet {\mathcal J}} u - A_{\bullet {\mathcal S}} z_{\mathcal S}\|^2$, and $A_{\bullet {\mathcal J}}$ has full column rank. Therefore, $x^*_{\mathcal J}$ is a solution to the linear complementarity problem: $0 \le u \perp A^T_{\bullet {\mathcal J}} A_{\bullet {\mathcal J}} u - A^T_{\bullet {\mathcal J}} A_{\bullet {\mathcal S}} z_{\mathcal S}\ge 0$. In view of ${ \mbox{supp} }(x^*)={\mathcal J}^* \subseteq {\mathcal J}$, we deduce that $x^*_{{\mathcal J}^*} = \big( A^T_{\bullet {\mathcal J}^*} A_{\bullet {\mathcal J}^*} )^{-1} A^T_{\bullet {\mathcal J}^*} A_{\bullet {\mathcal S}} z_{\mathcal S}> 0$. Using ${\mathcal I}= {\mathcal S}\setminus {\mathcal J}^*$ and $A_{\bullet {\mathcal S}} z_{\mathcal S}= A_{\bullet {\mathcal J}^*} z_{{\mathcal J}^*} + A_{\bullet {\mathcal I}} z_{\mathcal I}$, we further have $x^*_{{\mathcal J}^*} = z_{{\mathcal J}^*} + \big( A^T_{\bullet {\mathcal J}^*} A_{\bullet {\mathcal J}^*} )^{-1} A^T_{\bullet {\mathcal J}^*} A_{\bullet {\mathcal I}} z_{\mathcal I}$. Hence, $$A (z - x^*) = A_{\bullet {\mathcal S}} (z_{\mathcal S}- x^*_{\mathcal S}) = A_{\bullet {\mathcal J}^*} \big( z_{{\mathcal J}^*} - x^*_{{\mathcal J}^*} \big) + A_{\bullet {\mathcal I}} z_{\mathcal I}= \big[ -A_{\bullet {\mathcal J}^*} \big( A^T_{\bullet {\mathcal J}^*} A_{\bullet {\mathcal J}^*} )^{-1} A^T_{\bullet {\mathcal J}^*} A_{\bullet {\mathcal I}} + A_{\bullet{\mathcal I}} \big] z_{\mathcal I}.
$$ Direct calculations yield $A^T_{\bullet {\mathcal I}}A (z- x^*) = \big[ A^T_{\bullet {\mathcal I}} A_{\bullet{\mathcal I}} - A^T_{\bullet{\mathcal I}}A_{\bullet {\mathcal J}^*} \big( A^T_{\bullet {\mathcal J}^*} A_{\bullet {\mathcal J}^*} )^{-1} A^T_{\bullet {\mathcal J}^*} A_{\bullet {\mathcal I}} \big] z_{\mathcal I}= \big( M/M_{{\mathcal J}^* {\mathcal J}^*} \big) \cdot z_{{\mathcal I}}$; the other equation also follow readily.
Since $M/M_{{\mathcal J}^* {\mathcal J}^*}$ is positive definite and $z_{\mathcal I}>0$, it follows from Lemma \[lem:positive\_sign\] and the expression for $A^T_{\bullet {\mathcal I}}A (z- x^*)$ derived above that there exists an index $j\in {\mathcal I}$ such that $A^T_{\bullet j}A(z - x^*)>0$. Hence, $\max_{j\in{\mathcal I}} [A^T_{\bullet j}A(z - x^*)]_+>0$. Furthermore, since ${\mathcal I}={\mathcal S}\setminus {\mathcal J}^*$ and ${\mathcal J}^* \subseteq {\mathcal J}\subset {\mathcal S}$, we have ${\mathcal I}= ({\mathcal S}\setminus {\mathcal J}) \cup ({\mathcal J}\setminus {\mathcal J}^*)$. However, it follows from the linear complementarity condition for $x^*_{\mathcal J}$ that $A^T_{\bullet {\mathcal J}}A( x^* - z) = A^T_{\bullet {\mathcal J}} A_{\bullet {\mathcal J}} x^*_{\mathcal J}- A^T_{\bullet {\mathcal J}} A_{\bullet {\mathcal S}} z_{\mathcal S}\ge 0$, which implies that $A^T_{\bullet {\mathcal J}}A(z- x^* ) \le 0$ or equivalently $[A^T_{\bullet j}A(z- x^* )]_+ =0$ for all $j \in {\mathcal J}$. Therefore, $\max_{j\in{\mathcal I}} [A^T_{\bullet j}A(z - x^*)]_+=\max_{j\in{\mathcal S}\setminus {\mathcal J}} [A^T_{\bullet j}A(z - x^*)]_+$.
\[lem:R+\_S3\_Step2\] Let $U:= \begin{bmatrix} \alpha & \gamma \\ \gamma & \beta \end{bmatrix} \in \mathbb R^{2\times 2}$ be a positive definite matrix for real numbers $\alpha, \beta$ and $\gamma$. Define the set $\mathcal W :=\big\{ (u_1, u_2) \in \mathbb R^2_{++} \, | \, \big( \alpha u_1 + \gamma u_2 \big)_+ \ge \big( \gamma u_1 + \beta u_2 \big)_+ \big\}$. Then $\mathcal W$ is nonempty if and only if $\alpha > \gamma$. Furthermore, if $\mathcal W$ is nonempty, then $\{ u_2 \, | \, (u_1, u_2) \in \mathcal W \}=\mathbb R_{++}$.
Since $U$ is positive definite, we have $\alpha>0$, $\beta>0$, and $\alpha\beta > \gamma^2$. To show the “if” part, suppose $\alpha> \gamma$. Then for a fixed $u_1>0$, we have $\alpha u_1 > \gamma u_1$ and $\alpha u_1>0$. Therefore, for a sufficiently small $u_2>0$, it is easy to see that $( \alpha u_1 + \gamma u_2 )_+ \ge ( \gamma u_1 + \beta u_2 )_+$. This shows that $\mathcal W$ is nonempty. To prove the “only if” part, suppose $\mathcal W$ is nonempty but $\alpha \le \gamma$. Note that this implies that $\gamma>0$. Since $\alpha \cdot \beta > \gamma^2$ (due to the positive definiteness of $U$), we have $
\displaystyle \beta > \frac{\gamma}{\alpha} \cdot \gamma \ge \gamma.
$ Therefore, $\beta>\gamma\ge \alpha>0$. Hence, for any $(u_1, u_2)>0$, we have $\alpha u_1 \le \gamma u_1$ and $\gamma u_2 < \beta u_2$ such that $
0 < \alpha u_1 + \gamma u_2 < \gamma u_1 + \beta u_2.
$ This implies that $\mathcal W$ is empty, yielding a contradiction. Finally, when $\mathcal W$ is nonempty, we see, in view of $\alpha > \gamma$ proven above, that for any $u_2>0$, there exists a sufficiently large $u_1>0$ such that $\alpha u_1 + \gamma u_2>0$ and $\alpha u_1 + \gamma u_2> \gamma u_1 + \beta u_2$. This shows that $( \alpha u_1 + \gamma u_2 )_+ > ( \gamma u_1 + \beta u_2 )_+$. Hence, $\{ u_2 \, | \, (u_1, u_2) \in \mathcal W \}=\mathbb R_{++}$.
\[thm:nonnegative constraint\_S3\] Given a matrix $A \in \mathbb R^{m\times N}$ with unit columns and the index set ${\mathcal S}=\{1, 2, 3\}$, let $M:=A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}}$. Then every nonzero vector $x \in \mathbb R^N_+$ with ${ \mbox{supp} }(x)={\mathcal S}$ is recovered from $y=A x$ via constrained matching pursuit if and only if each of the following conditions holds:
- $A_{\bullet {\mathcal S}}$ has full column rank;
- $\big\| (A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}} u )_+ \big\|_\infty > \big\| (A^T_{\bullet {\mathcal S}^c} A_{\bullet {\mathcal S}} u )_+ \big\|_\infty$ for all $u \in \mathbb R^3_{++}$;
- For any ${\mathcal J}\in \{ \{1\}, \{2\}, \{3\}\}$, $\big\| (M/M_{{\mathcal J}{\mathcal J}} \, v )_+ \big\|_\infty > \big\| (A^T_{\bullet {\mathcal S}^c}[I- A^T_{\bullet {\mathcal J}} A_{\bullet {\mathcal J}}] A_{{\mathcal S}\setminus{\mathcal J}}v )_+ \big\|_\infty$ for all $v \in \mathbb R^2_{++}$;
- All the following implications hold: $$\begin{aligned}
\Big[ 1-\vartheta^2_{12} > \min(\Delta_{13}, \Delta_{23}) \Big] & \ \Longrightarrow \ \Big[ \det M > \max_{i \in {\mathcal S}^c} \big( \vartheta_{i3} (1-\vartheta^2_{12}) - \vartheta_{i1} \Delta_{13} - \vartheta_{i2}\Delta_{23} \big)_+ \Big], \\
\Big[ 1-\vartheta^2_{13} > \min(\Delta_{12}, \Delta_{23}) \Big] & \ \Longrightarrow \ \Big[ \det M > \max_{i \in {\mathcal S}^c} \big( \vartheta_{i2} (1-\vartheta^2_{13}) - \vartheta_{i1} \Delta_{12} - \vartheta_{i3}\Delta_{23} \big)_+ \Big], \\
\Big[ 1-\vartheta^2_{23} > \min(\Delta_{12}, \Delta_{13}) \Big] & \ \Longrightarrow \ \Big[ \det M > \max_{i \in {\mathcal S}^c} \big( \vartheta_{i1} (1-\vartheta^2_{23}) - \vartheta_{i2} \Delta_{12} - \vartheta_{i3}\Delta_{13} \big)_+ \Big],
\end{aligned}$$ where $\Delta_{12}:=\vartheta_{12}-\vartheta_{13}\vartheta_{23}$, $\Delta_{13}:=\vartheta_{13}-\vartheta_{12}\vartheta_{23}$, and $\Delta_{23}:=\vartheta_{23} - \vartheta_{12}\vartheta_{13}$.
\[remark:R+3\_conditions\] We comment on the above conditions before presenting a proof:
- Since the matrix $M = \begin{bmatrix} 1 & \vartheta_{12} & \vartheta_{13} \\ \vartheta_{12} & 1 & \vartheta_{23} \\ \vartheta_{13} & \vartheta_{23} & 1 \end{bmatrix}$, its determinant $\det M = 1 + 2\vartheta_{12} \vartheta_{13} \vartheta_{23} - \vartheta^2_{12} - \vartheta^2_{13}-\vartheta^2_{23}$.
- If the hypothesis of an implication in condition (iv) fails, then that implication holds even when the conclusion statement is false. Hence, that implication is vacuously true and can be neglected.
- Since each Schur complement of $M:=A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}}$ is positive definite, we have $(1-\vartheta^2_{13})(1-\vartheta^2_{23}) \ge \Delta^2_{12}$, $(1-\vartheta^2_{12})(1-\vartheta^2_{23}) \ge \Delta^2_{13}$, and $(1-\vartheta^2_{12})(1-\vartheta^2_{13}) \ge \Delta^2_{23}$. By virtue of these inequalities, it is easy to verify that at least two hypotheses of the three implications in condition (iv) must hold.
“If”. Suppose conditions (i)-(iv) hold. Fix an arbitrary $z=(z_{\mathcal S}, 0) \in \mathbb R^N_+$ with $z_{\mathcal S}=(z_1, z_2, z_3)\in \mathbb R^3_{++}$, and let $y = A z = A_{\bullet S} z_{\mathcal S}$. Consider the following three steps of Algorithm \[algo:constrained\_MP\]:
$\bullet$ Step 1: Let $x^0=0$. Since $y=A_{\bullet {\mathcal S}} z_{\mathcal S}$, it follows from condition (ii) that $\max_{i=1, 2, 3} ( A^T_{\bullet i} A_{\bullet {\mathcal S}} z_{\mathcal S})_+ > \max_{j \in {\mathcal S}^c} (A^T_{\bullet j} A_{\bullet {\mathcal S}} z_{\mathcal S})_+$. Hence, it follows from Algorithm \[algo:constrained\_MP\] that $j^*_1\in {\mathcal S}= \{1, 2, 3\}$, and the index set ${\mathcal J}_1=\{ j^*_1\}$. Further, $x^1:={\operatornamewithlimits{\arg\min}}_{x \ge 0, { \mbox{supp} }(x) \subseteq {\mathcal J}_{1} } \, \| y - A x\|^2_2$ is given by $x^1 = \langle A_{\bullet j^*_1}, A_{\bullet {\mathcal S}} z_{{\mathcal S}} \rangle_+ \mathbf e_{j^*_1}$, where $\langle A_{\bullet j^*_1}, A_{\bullet {\mathcal S}} z_{{\mathcal S}} \rangle_+ > 0$ in view of Proposition \[prop:index\_set\].
$\bullet$ Step 2: By observing that $x^1$ is the optimal solution obtained from Step 1 with ${ \mbox{supp} }(x^1)={\mathcal J}_1=\{ j^*_1\}$, it follows from Lemma \[lem:R+\_optimal\_solution\] and $A^T_{\bullet {\mathcal J}_1} A_{\bullet {\mathcal J}_1}=1$ that by letting the index set ${\mathcal I}:= {\mathcal S}\setminus {\mathcal J}_1$, $$\begin{aligned}
\max_{i\in {\mathcal S}} \big( A^T_{\bullet i} A(z - x^1) \big)_+ & \, = \ \big \| \big( A^T_{\bullet {\mathcal S}} A(z - x^1) \big)_+ \big\|_\infty \, = \, \big\| \big( (M/M_{{\mathcal J}_1 {\mathcal J}_1}) \cdot z_{\mathcal I}\big)_+ \big\|_\infty, \\
\max_{j\in {\mathcal S}^c} \big( A^T_{\bullet j} A(z - x^1) \big)_+ & \, = \ \big \| \big( A^T_{\bullet {\mathcal S}^c} A(z - x^1) \big)_+ \big\|_\infty \, = \, \big\| (A^T_{\bullet {\mathcal S}^c}[I- A_{\bullet {\mathcal J}_1} A^T_{\bullet {\mathcal J}_1}] A_{{\mathcal I}} \cdot z_{\mathcal I})_+ \big\|_\infty.\end{aligned}$$ Noting that the Schur complement $M/M_{{\mathcal J}_1 {\mathcal J}_1}$ is positive definite and $z_{\mathcal I}>0$, we deduce via Lemma \[lem:positive\_sign\] that $\big\| \big( (M/M_{{\mathcal J}_1 {\mathcal J}_1}) \cdot z_{\mathcal I}\big)_+ \big\|_\infty>0$. By $z_{\mathcal I}>0$ and condition (iii), we have $ \max_{i\in {\mathcal S}} \big( A^T_{\bullet i} A(z - x^1) \big)_+ > \max_{j\in {\mathcal S}^c} \big( A^T_{\bullet j} A(z - x^1) \big)_+$. In light of Algorithm \[algo:constrained\_MP\], we see that $j^*_2:= {\operatornamewithlimits{\arg\max}}_{i\in {\mathcal S}} \big( A^T_{\bullet i} A(z - x^1) \big)_+$ satisfies $j^*_2 \in {\mathcal I}$, and ${\mathcal J}_2=\{ j^*_1, j^*_2\} \subset {\mathcal S}$ with $j^*_1 \ne j^*_2$. Moreover, let $x^2$ be the unique optimal solution to $\min_{w \ge 0, { \mbox{supp} }(w)\subseteq {\mathcal J}_2} \| y - A w \|^2$. Then it follows from Proposition \[prop:index\_set\] that ${ \mbox{supp} }(x^2)={\mathcal J}_2$.
$\bullet$ Step 3: Let the index $j_3$ be such that $\{ j_3 \}={\mathcal S}\setminus {\mathcal J}_2$. Note that $\{ j^*_2, j_3\}={\mathcal I}$. Hence, the Schur complement $U:=M/M_{{\mathcal J}_1 {\mathcal J}_1}$ is one of the following $2\times 2$ positive definite matrices: $$\label{eqn:U_matrices}
U^1: = \begin{bmatrix} 1 - \vartheta^2_{12} & \Delta_{23} \\ \Delta_{23} & 1 - \vartheta^2_{13} \end{bmatrix}, \quad
U^2: = \begin{bmatrix} 1 - \vartheta^2_{12} & \Delta_{13} \\ \Delta_{13} & 1 - \vartheta^2_{23} \end{bmatrix}, \quad
U^3: = \begin{bmatrix} 1 - \vartheta^2_{13} & \Delta_{12} \\ \Delta_{12} & 1 - \vartheta^2_{23} \end{bmatrix},$$ where $\Delta_{ij}$’s are defined in condition (iv), $U^1= M/M_{11}$, $U^2= M/M_{22}$, and $U^3= M/M_{33}$. Hence, $U=\begin{bmatrix} \alpha & \gamma \\ \gamma & \beta \end{bmatrix}$ is positive definite, where $\alpha, \beta \in \{ 1 - \vartheta^2_{12}, 1-\vartheta^2_{13}, 1-\vartheta^2_{23} \}$ with $\alpha \ne \beta$, and $\gamma \in \{ \Delta_{12}, \Delta_{13}, \Delta_{23} \}$. Furthermore, either $(U_{1\bullet} z_{\mathcal I})_+\ge (U_{2\bullet} z_{\mathcal I})_+$ or $(U_{2\bullet} z_{\mathcal I})_+\ge (U_{1\bullet} z_{\mathcal I})_+$, where $U_{i \bullet}$ denotes the $i$th row of $U$. Since $z_{\mathcal I}>0$, it follows from Lemma \[lem:R+\_S3\_Step2\] that either $\alpha > \gamma$ or $\beta > \gamma$. We first consider the case where $\alpha > \gamma$. In this case, $\alpha= 1- \vartheta^2_{j^*_1, j^*_2}$, $\beta = 1 - \vartheta^2_{j^*_1, j_3}$, and $\gamma=\Delta_{j^*_2, j_3}$. In light of the implications given by condition (iv), we have that $$\label{eqn:R+_step_3}
\det M \, > \, \max_{i \in {\mathcal S}^c} \big( \vartheta_{i, j_3} (1-\vartheta^2_{j^*_1, j^*_2}) - \vartheta_{i, j^*_1} \Delta_{j^*_1, j_3} - \vartheta_{i, j^*_2}\Delta_{j^*_2, j_3} \big)_+.$$ Additionally, since $x^2$ is the unique solution to $\min_{w \ge 0, { \mbox{supp} }(w) \subseteq {\mathcal J}_2} \| y - A w \|^2$ with ${ \mbox{supp} }(x^2)={\mathcal J}_2$, it follows from Lemma \[lem:R+\_optimal\_solution\] that by letting ${\widetilde}{\mathcal I}:= {\mathcal S}\setminus {\mathcal J}_2=\{ j_3\}$, $$\begin{aligned}
\max_{i \in {\mathcal S}} \big( A^T_{\bullet i} A(z- x ^2) \big)_+ & \, = \, \big \| \big( (M/M_{{\mathcal J}_2{\mathcal J}_2}) \cdot z_{{\widetilde}{\mathcal I}} \big)_+ \big\|_\infty, \\
\max_{i\in {\mathcal S}^c} \big( A^T_{\bullet i} A(z - x^2) \big)_+ & \, = \, \big\| (A^T_{\bullet {\mathcal S}^c}[I- A^T_{\bullet {\mathcal J}_2} ( A_{\bullet {\mathcal J}_2} A_{\bullet {\mathcal J}_2} )^{-1} A^T_{\bullet {\mathcal J}_2}] A_{{\widetilde}{\mathcal I}} \cdot z_{{\widetilde}{\mathcal I}} )_+ \big\|_\infty.\end{aligned}$$ Note that $z_{{\widetilde}{\mathcal I}}$ and $M/M_{{\mathcal J}_2{\mathcal J}_2}$ are positive scalars. It follows from the Schur determinant formula that $M/M_{{\mathcal J}_2{\mathcal J}_2} = \det (M/M_{{\mathcal J}_2{\mathcal J}_2} )=\det M/\det(M_{{\mathcal J}_2{\mathcal J}_2})$. Thus $\max_{i \in {\mathcal S}} \big( A^T_{\bullet i} A(z- x ^2) \big)_+ = \det M/\det(M_{{\mathcal J}_2{\mathcal J}_2}) \cdot z_{{\widetilde}{\mathcal I}}$. Further, direct calculations show that $( A_{\bullet {\mathcal J}_2} A_{\bullet {\mathcal J}_2} )^{-1} A^T_{\bullet {\mathcal J}_2} A_{{\widetilde}{\mathcal I}} = ( \Delta_{j^*_1, j_3}, \Delta_{j^*_2, j_3})^T/\det(M_{{\mathcal J}_2 {\mathcal J}_2})$. In view of this result and $\det(M_{{\mathcal J}_2{\mathcal J}_2})= 1-\vartheta^2_{j^*_1, j^*_2}$, we have, for each $i \in {\mathcal S}^c$, $$\big( (A^T_{\bullet i}[I- A^T_{\bullet {\mathcal J}_2} ( A_{\bullet {\mathcal J}_2} A_{\bullet {\mathcal J}_2} )^{-1} A^T_{\bullet {\mathcal J}_2}] A_{{\widetilde}{\mathcal I}} \cdot z_{{\widetilde}{\mathcal I}} \big)_+ = \frac{z_{{\widetilde}{\mathcal I}}}{\det( M_{{\mathcal J}_2{\mathcal J}_2})} \Big( \vartheta_{i, j_3} (1-\vartheta^2_{j^*_1, j^*_2}) - \vartheta_{i, j^*_1} \Delta_{j^*_1, j_3} - \vartheta_{i, j^*_2}\Delta_{j^*_2, j_3} \Big)_+.$$ These results and the inequality (\[eqn:R+\_step\_3\]) imply that $\max_{i \in {\mathcal S}} \big( A^T_{\bullet i} A(z- x ^2) \big)_+ > \max_{i\in {\mathcal S}^c} \big( A^T_{\bullet i} A(z - x^2) \big)_+$. The other case where $\beta>\gamma$ can be established by the similar argument. Therefore, following Algorithm \[algo:constrained\_MP\], $j^*_3:= {\operatornamewithlimits{\arg\max}}_{i\in {\mathcal S}} \big( A^T_{\bullet i} A(z - x^2) \big)_+$ satisfies $j^*_3 = j_3$. This yields ${\mathcal J}_3= {\mathcal S}$. Since $A_{\bullet {\mathcal S}}$ has full column rank, the exact vector recovery is achieved.
“Only if”. Suppose every nonzero vector $x \in \mathbb R^N_+$ with ${ \mbox{supp} }(x)={\mathcal S}$ is recovered from $y=A x$ via constrained matching pursuit for a given matrix $A \in \mathbb R^{m\times N}$ and the index set ${\mathcal S}=\{1, 2, 3\}$. It follows from Lemma \[lem:nonnegative orthant\_Nec01\] that condition (i) must hold. Besides, by setting $x^0=0$, we see via Corollary \[coro:Nec\_Suf\_suppt\_recovery\_RN\_RN+\] that $\max_{i \in {\mathcal S}} (A^T_{\bullet i} A_{\bullet S} z_{\mathcal S})_+ > \max_{j \in {\mathcal S}^c} (A^T_{\bullet j} A_{\bullet S} z_{\mathcal S})_+$ holds for all $z_{\mathcal S}\in \mathbb R^3_{++}$. This yields condition (ii).
For each $p\in {\mathcal S}$, define the set $\mathcal W_p :=\{ z_{\mathcal S}\in \mathbb R^3_{++} \, | \, (A^T_{\bullet p} A_{\bullet S} z_{\mathcal S})_+ = \max_{i \in {\mathcal S}} (A^T_{\bullet i} A_{\bullet S} z_{\mathcal S})_+ \}$. Clearly, $\mathbb R^3_{++} = \mathcal W_1 \cup \mathcal W_2 \cup \mathcal W_3$. Since the matrix $M:=A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}}$ given by (c) of Remark \[remark:R+3\_conditions\] is positive definite, we observe $|\vartheta_{ij}| <1$ for any $i \ne j$. Based on this observation, it is easy to show that for any given $(z_2, z_3)>0$, there exists a sufficiently large $z_1>0$ such that $(z_1, z_2, z_3) \in \mathcal W_1$. Hence, $\mathcal W_1$ is nonempty and $\{ (z_2, z_3) \, | \, z_{\mathcal S}=(z_1, z_2, z_3) \in \mathcal W_1 \}=\mathbb R^2_{++}$. By a similar argument, we deduce that $\mathcal W_2$ and $\mathcal W_3$ are nonempty and $\{ (z_1, z_3) \, | \, z_{\mathcal S}=(z_1, z_2, z_3) \in \mathcal W_2 \}=\mathbb R^2_{++}$ and $\{ (z_1, z_2) \, | \, z_{\mathcal S}=(z_1, z_2, z_3) \in \mathcal W_3 \}=\mathbb R^2_{++}$. Since $\mathbb R^3_{++} = \mathcal W_1 \cup \mathcal W_2 \cup \mathcal W_3$, $z_{\mathcal S}$ belongs to one of $\mathcal W_i$’s for any $z_{\mathcal S}\in \mathbb R^3_{++}$. For each $p \in {\mathcal S}$, it follows from Algorithm \[algo:constrained\_MP\] that for any $z \in \mathcal W_p$, the corresponding unique $x^1= (A^T_{\bullet p} A_{\bullet S} z_{\mathcal S})_+ \mathbf e_p$, where $(A^T_{\bullet p} A_{\bullet S} z_{\mathcal S})_+>0$. Moreover, we must have $\max_{i \in {\mathcal S}} \big( A^T_{\bullet i} A (z - x^1) \big)_+> \max_{j \in {\mathcal S}^c} \big( A^T_{\bullet j} A (z - x^1) \big)_+$. This condition, as shown at Step 2 of the “if” part, is equivalent to $
\big\| (M/M_{{\mathcal J}_1{\mathcal J}_1} \, z_{\mathcal I})_+ \big\|_\infty > \big\| (A^T_{\bullet {\mathcal S}^c}[I- A^T_{\bullet {\mathcal J}_1} A_{\bullet {\mathcal J}_1}] A_{{\mathcal I}} z_{\mathcal I})_+ \big\|_\infty,
$ where ${\mathcal J}_1=\{ p \}$ and ${\mathcal I}= {\mathcal S}\setminus {\mathcal J}_1$. Since $\{ z_{\mathcal I}\, | \, z_{\mathcal S}\in \mathcal W_p\}= \mathbb R^2_{++}$ as shown before, we obtain condition (iii).
To establish condition (iv), we first show the following claim: if $1-\vartheta^2_{12}>\min(\Delta_{13}, \Delta_{23})$ holds true, then there exists $z \in \mathbb R^N_+$ with ${ \mbox{supp} }(z)={\mathcal S}$ such that when $y=A z$, Algorithm \[algo:constrained\_MP\] give rises to ${\mathcal J}_2=\{ 1, 2\}$. To prove this claim, it is noted that $1-\vartheta^2_{12}>\min(\Delta_{13}, \Delta_{23})$ is equivalent to $1-\vartheta^2_{12} > \Delta_{23}$ or $1-\vartheta^2_{12} > \Delta_{13}$. For the former case, i.e., $1-\vartheta^2_{12} > \Delta_{23}$, it follows from Lemma \[lem:R+\_S3\_Step2\] and $U^1=M/M_{11}$ given in (\[eqn:U\_matrices\]) that there exists $v:=(v_1, v_2)^T \in \mathbb R^2_{++}$ such that $(U^1_{1\bullet} v )_+ \ge (U^1_{2\bullet} v)_+$. Further, as shown previously, there exists a sufficiently large $v_0>0$ such that ${\widetilde}z=({\widetilde}z_{\mathcal S}, 0)$ with ${\widetilde}z_{\mathcal S}:=({\widetilde}z_1, {\widetilde}z_2, {\widetilde}z_3)=(v_0, v_1, v_2)$ satisfies ${\widetilde}z \in \mathcal W_1$. This implies via Lemma \[lem:R+\_optimal\_solution\] and the argument for Step 1 of the “if” part that when $y= A {\widetilde}z$, Algorithm \[algo:constrained\_MP\] give rises to $(j^*_1, j^*_2)=(1, 2)$ and ${\mathcal J}_2=\{ 1, 2\}$. The similar argument can be used to show that if $1-\vartheta^2_{12} > \Delta_{13}$ holds, then there exists $z \in \mathbb R^N_+$ with ${ \mbox{supp} }(z)={\mathcal S}$ such that when $y=A z$, Algorithm \[algo:constrained\_MP\] give rises to $(j^*_1, j^*_2)=(2, 1)$ and ${\mathcal J}_2= \{1, 2\}$. The above proof can be extended to show that if $1-\vartheta^2_{13}>\min(\Delta_{12}, \Delta_{23})$ (respectively $1-\vartheta^2_{13}>\min(\Delta_{12}, \Delta_{23})$) holds, then there exists $z \in \mathbb R^N_+$ with ${ \mbox{supp} }(z)={\mathcal S}$ such that when $y=A z$, Algorithm \[algo:constrained\_MP\] yields ${\mathcal J}_2=\{ 1, 3 \}$ (respectively ${\mathcal J}_2=\{ 2, 3 \}$).
As indicated in Remark \[remark:R+3\_conditions\], if the hypothesis of an implication in condition (iv) is false, then that implication holds true vacuously. Now consider an implication in condition (iv) whose hypothesis holds true. Then there exists $z \in \mathbb R^N_+$ with ${ \mbox{supp} }(z)={\mathcal S}$ such that Algorithm \[algo:constrained\_MP\] yields ${\mathcal J}_2:=\{ j^*_1, j^*_2\}$ from $y=A z$. Hence, the corresponding $x^2$ obtained from $y=Az$ via Algorithm \[algo:constrained\_MP\] satisfies ${ \mbox{supp} }(x^2)={\mathcal J}_2$. Since the exact support recovery implies that $\max_{i \in {\mathcal S}} (A^T_{\bullet i} A(z- x^2))_+>\max_{j \in {\mathcal S}^c} (A^T_{\bullet i} A(z- x^2))_+$, we deduce, in view of ${ \mbox{supp} }(x^2)={\mathcal J}_2$, Lemma \[lem:R+\_optimal\_solution\] and the argument for Step 3 of the “if” part, that $$\frac{\det M}{\det(M_{{\mathcal J}_2{\mathcal J}_2}) } \cdot z_{{\widetilde}{\mathcal I}} \, > \, \frac{z_{{\widetilde}{\mathcal I}}}{\det( M_{{\mathcal J}_2{\mathcal J}_2})} \Big( \vartheta_{i, j_3} (1-\vartheta^2_{j^*_1, j^*_2}) - \vartheta_{i, j^*_1} \Delta_{j^*_1, j_3} - \vartheta_{i, j^*_2}\Delta_{j^*_2, j_3} \Big)_+,$$ where ${\widetilde}{\mathcal I}=\{j_3\}={\mathcal S}\setminus {\mathcal J}_2$, $z_{{\widetilde}{\mathcal I}}\in \mathbb R_{++}$, and $\det(M_{{\mathcal J}_2{\mathcal J}_2}) =1-\vartheta^2_{j^*_1, j^*_2}$. This yields condition (iv).
### Sufficient Conditions for Exact Vector Recovery on $\mathbb R^N_+$ for a Fixed Support
When a given support ${\mathcal S}$ is of size greater than or equal to 4, necessary [*and*]{} sufficient conditions are difficult to obtain due to increasing complexities. Hence, we seek neat sufficient conditions in this subsection.
\[thm:R+\_sufficient\_cond\] Given a matrix $A \in \mathbb R^{m\times N}$ with unit columns and the index set ${\mathcal S}\subset \{1, \ldots, N\}$, let $M:=A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}}$. Then every nonzero vector $z \in \mathbb R^N_+$ with ${ \mbox{supp} }(z)={\mathcal S}$ is recovered from $y=A z$ via constrained matching pursuit if the following conditions hold:
- $A_{\bullet {\mathcal S}}$ has full column rank or equivalently $M$ is positive definite; and
- For any (possibly empty) index set ${\mathcal J}\subset {\mathcal S}$, $$\label{eqn:R+_suff_cond}
\big\| (M/M_{{\mathcal J}{\mathcal J}} \, x )_+ \big\|_\infty \, > \, \big\| \big(A^T_{\bullet {\mathcal S}^c}[I- A^T_{\bullet {\mathcal J}} ( A^T_{\bullet {\mathcal J}} A_{\bullet {\mathcal J}} )^{-1} A_{\bullet {\mathcal J}}] A_{\bullet{\mathcal S}\setminus{\mathcal J}} \, x \big)_+ \big\|_\infty, \quad \ \forall \ x \in \mathbb R^{|{\mathcal S}\setminus{\mathcal J}|}_{++},$$ where $M/M_{{\mathcal J}{\mathcal J}}$ is the Schur complement of $M_{{\mathcal J}{\mathcal J}}$ in $M$.
Due to condition (i), it suffices to show the exact support recovery of each $z\in \mathbb R^N_+$ with ${ \mbox{supp} }(z)={\mathcal S}$ via Algorithm \[algo:constrained\_MP\] from $y=A z$. Toward this end, we see via a similar argument for Corollary \[coro:Nec\_Suf\_suppt\_recovery\_RN\_RN+\] that condition $(\mathbf H)$ given by (\[eqn:condition\_H’\]) holds if for any $0\ne u \in \mathbb R^N_+$ with ${ \mbox{supp} }(u)={\mathcal S}$, any index set ${\mathcal J}\subset {\mathcal S}$, and the (unique) optimal solution $v={\operatornamewithlimits{\arg\min}}_{w \ge 0, { \mbox{supp} }(w)\subseteq {\mathcal J}} \| A (u - w )\|^2_2$, the following holds: $$\max_{i\in {\mathcal S}} \big( A^T_{\bullet i} A(u - v) \big)_+ = \max_{i\in {\mathcal S}\setminus {\mathcal J}} \big( A^T_{\bullet i} A(u - v) \big)_+ \, > \, \max_{j \in {\mathcal S}^c} \big( A^T_{\bullet j} A(u - v) \big)_+,$$ where the first equation follows from Lemma \[lem:exact\_suppt\_recovery\_index\]. Let ${\mathcal J}^*:={ \mbox{supp} }(v)$. Hence, ${\mathcal J}^* \subseteq {\mathcal J}\subset {\mathcal S}$. Since $v$ is the optimal solution to $\min_{w \ge 0, { \mbox{supp} }(w)\subseteq {\mathcal J}} \| A (u - w )\|^2_2$, we deduce via Lemma \[lem:R+\_optimal\_solution\] that $$\begin{aligned}
\max_{i\in {\mathcal S}} \big( A^T_{\bullet i} A(u - v) \big)_+ & \, = \ \big \| \big( A^T_{\bullet {\mathcal S}} A(u - v) \big)_+ \big\|_\infty \, = \, \big\| \big( (M/M_{{\mathcal J}^* {\mathcal J}^*}) \cdot u_{\mathcal I}\big)_+ \big\|_\infty, \\
\max_{j\in {\mathcal S}^c} \big( A^T_{\bullet j} A(u - v) \big)_+ & \, = \ \big \| \big( A^T_{\bullet {\mathcal S}^c} A(u - v) \big)_+ \big\|_\infty \, = \, \big \| \big(A^T_{\bullet {\mathcal S}^c}[I- A^T_{\bullet {\mathcal J}^*} ( A^T_{\bullet {\mathcal J}^*} A_{\bullet {\mathcal J}^*} )^{-1} A_{\bullet {\mathcal J}^*}] A_{\bullet{\mathcal I}} \cdot u_{\mathcal I}\big)_+ \big\|_\infty,\end{aligned}$$ where ${\mathcal I}:= {\mathcal S}\setminus {\mathcal J}^*$ is nonempty. Since $u_{\mathcal I}>0$, we see that $\max_{i\in {\mathcal S}} \big( A^T_{\bullet i} A(u - v) \big)_+ \, > \, \max_{j \in {\mathcal S}^c} \big( A^T_{\bullet j} A(u - v) \big)_+$ holds under condition (ii). This leads to the desired result.
In what follows, we develop conditions to verify the inequality given in (\[eqn:R+\_suff\_cond\]), which leads to a numerical scheme to check (\[eqn:R+\_suff\_cond\]). Fix an index set ${\mathcal J}\subset {\mathcal S}$, and let $r:=|{\mathcal S}\setminus {\mathcal J}|$. Further, let $M/M_{{\mathcal J}{\mathcal J}}=[ p_1, \cdots, p_r]$, and $E:=\big(A^T_{\bullet {\mathcal S}^c}[I- A^T_{\bullet {\mathcal J}} ( A^T_{\bullet {\mathcal J}} A_{\bullet {\mathcal J}} )^{-1} A_{\bullet {\mathcal J}}] A_{\bullet{\mathcal S}\setminus{\mathcal J}} \big)^T=[ q_1, \cdots, q_{|{\mathcal S}^c|} ]$, namely, $p_i \in \mathbb R^r$ is the $i$th column of $M/M_{{\mathcal J}{\mathcal J}}$ and $q_j \in \mathbb R^r$ is the $j$th column of $E$.
\[lem:verification\_cond\] The inequality (\[eqn:R+\_suff\_cond\]) for a fixed index set ${\mathcal J}\subset {\mathcal S}$ holds if and only if for each $q_j\in \mathbb R^r$, there exist $w\in \mathbb R^r_+$ and $0 \ne (w', \beta) \in \mathbb R^r_+ \times \mathbb R_+$ such that $[ p_1 - q_j, p_2-q_j, \cdots, p_r - q_j] w = w'+ \beta \cdot q_j$.
Since the Schur complement $M/M_{{\mathcal J}{\mathcal J}}$ is symmetric, it is easy to see that the inequality (\[eqn:R+\_suff\_cond\]) fails if and only if there exists $v>0$ such that $\max_{i=1, \ldots, r} ( p^T_i v)_+ \le (q^T_j v)_+$ for some $j$. In view of Lemma \[lem:positive\_sign\], we deduce that $\max_{i=1, \ldots, r} ( p^T_i v)_+>0$ such that $q^T_j v >0$ for this $j$. Hence, the inequality system $\max_{i=1, \ldots, r} ( p^T_i v)_+ \le (q^T_j v)_+, v >0$ is equivalent to the following linear inequality system: $$\mbox{(I)}: \quad v>0, \ \ q^T_j v >0, \ \ q^T_j v \ge p^T_i v, \ \ \forall \ i=1, \ldots, r.$$ By Motzkin’s Transposition Theorem, (I) has no solution if and only if there exist $w\in \mathbb R^r_+$ and $0 \ne (w', \beta) \in \mathbb R^r_+ \times \mathbb R_+$ such that $[ p_1 - q_j, p_2-q_j, \cdots, p_r - q_j] w = w'+ \beta \cdot q_j$, yielding the desired result.
The condition derived in the above lemma can be effectively verified via a linear program for the given matrices $M/M_{{\mathcal J}{\mathcal J}}$ and $E$.
Exact Vector Recovery on $\mathbb R^{N_1} \times \mathbb R^{N_2}_+$ for a Fixed Support
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In this subsection, we briefly discuss an extension of the preceding exact vector recovery results to a Cartesian product of copies of $\mathbb R$ and $\mathbb R_+$. Let ${\mathcal I}_1$ and ${\mathcal I}_+$ be two nonempty index subsets that form a disjoint union of $\{1, \ldots, N\}$. Consider the constraint set ${\mathcal P}= \mathbb R_{{\mathcal I}_1} \times (\mathbb R_+)_{{\mathcal I}_+}$. The following preliminary result can be easily extended from Corollary \[coro:Nec\_Suf\_suppt\_recovery\_RN\_RN+\] and Lemma \[lem:nonnegative orthant\_Nec01\]; its proof is thus omitted.
\[lem:preliminary\_results\_RR+\] Let $A\in \mathbb R^{m\times N}$ be a matrix with unit columns, and ${\mathcal P}=\mathbb R_{{\mathcal I}_1} \times (\mathbb R_+)_{{\mathcal I}_+}$. The following hold:
- Let $0 \ne z \in \Sigma_K \cap {\mathcal P}$ with $|{ \mbox{supp} }(z)|=r$. Then the exact support recovery of $z$ is achieved if and only if for any sequence $\big( (x^k, j^*_k, {\mathcal J}_k) \big)_{k \in \mathbb N}$ generated by Algorithm \[algo:constrained\_MP\] with $y=A z$, $$\begin{aligned}
& \max\Big( \, \max_{j \in ({ \mbox{supp} }(z)\setminus {\mathcal J}_k) \cap {\mathcal I}_1} |A^T_{\bullet j} A(z - x^k)|, \ \max_{j \in ({ \mbox{supp} }(z)\setminus {\mathcal J}_k) \cap {\mathcal I}_+} [A^T_{\bullet j} A(z - x^k) ]_+ \, \Big) \\
& \, > \ \max\Big( \, \max_{j\in [{ \mbox{supp} }(z)]^c\cap {\mathcal I}_1} | A^T_{\bullet j} A(z - x^k) |, \ \max_{j\in [{ \mbox{supp} }(z)]^c\cap {\mathcal I}_+} [ A^T_{\bullet j} A(z - x^k) ]_+ \Big), \quad \forall \ k=0, 1, \ldots, r-1.
\end{aligned}$$
- Let ${\mathcal S}$ be a nonempty index subset of $\{1, \ldots, N\}$. The exact vector recovery of every vector $x \in {\mathcal P}$ with ${ \mbox{supp} }(x)={\mathcal S}$ is achieved via constrained matching pursuit only if $A_{\bullet {\mathcal S}}$ has full column rank.
The next result characterizes the exact vector recovery on ${\mathcal P}$ for a given support ${\mathcal S}$ of size 2.
Given a matrix $A \in \mathbb R^{m\times N}$ with unit columns and the index set ${\mathcal S}=\{1, 2\}$ with $1 \in {\mathcal I}_1$ and $2\in {\mathcal I}_+$, every vector $x \in {\mathcal P}=\mathbb R_{{\mathcal I}_1} \times (\mathbb R_+)_{{\mathcal I}_+}$ with ${ \mbox{supp} }(x)={\mathcal S}$ is recovered from $y=A x$ via constrained matching pursuit if and only if the following conditions hold:
- $A_{\bullet {\mathcal S}}$ has full column rank or equivalently $|\vartheta_{12}|<1$;
- $\displaystyle \max \big( |z_1 + \vartheta_{12} z_2|, \, (\vartheta_{12} z_1 + z_2)_+ \big) > \max\Big( \max_{j \in {\mathcal S}^c\cap {\mathcal I}_1} |\vartheta_{j1} z_1 + \vartheta_{j2} z_2 \big|, \, \max_{j \in {\mathcal S}^c\cap {\mathcal I}_+} \big(\vartheta_{j1} z_1 + \vartheta_{j2} z_2 \big)_+ \Big)$,\
$\forall \, (z_1, z_2)^T \in \big(\mathbb R \setminus \{ 0 \} \big) \times \mathbb R_{++}$;
- $1-\vartheta^2_{12} \, > \,
\max\Big( \max_{j\in {\mathcal S}^c\cap {\mathcal I}_1} \, |\vartheta_{j2}- \vartheta_{12}\vartheta_{j1}|, \ \max_{j \in {\mathcal S}^c \cap {\mathcal I}_+}(\vartheta_{j2}- \vartheta_{12}\vartheta_{j1})_+, \ \max_{j\in {\mathcal S}^c} | \vartheta_{j1}- \vartheta_{12}\vartheta_{j2} | \, \Big)$.
“Only if”. Suppose the exact vector recovery is achieved for any $x \in {\mathcal P}$ with ${ \mbox{supp} }(x)={\mathcal S}$. Condition (i) follows from statement (ii) of Lemma \[lem:preliminary\_results\_RR+\], and condition (ii) follows from Step 1 of Algorithm \[algo:constrained\_MP\] and statement (i) of Lemma \[lem:preliminary\_results\_RR+\] with $x^0=0$ and ${\mathcal J}_0=\emptyset$. To establish condition (iii), we first notice via $|\vartheta_{12}|<1$ that for any $z\in {\mathcal P}$ with ${ \mbox{supp} }(z)={\mathcal S}$, i.e., $z_1 \ne 0$ and $z_2>0$, $|z_1 + \vartheta_{12} z_2 | \ge (\vartheta_{12} z_1 + z_2)_+$ if and only if $|z_1| \ge z_2 >0$, and $|z_1 + \vartheta_{12} z_2 | \le (\vartheta_{12} z_1 + z_2)_+$ if and only if $z_2 \ge |z_1|>0$. When the former holds, i.e., $|z_1|\ge z_2>0$, we have $j^*_1=1$ and $x^1=(z_1+ \vartheta_{12} z_2) \cdot \mathbf e_1$. Hence, $A^T_{\bullet j} A(z - x^1)= (\vartheta_{j2} - \vartheta_{j1} \vartheta_{12}) z_2$. Using Step 2 of Algorithm \[algo:constrained\_MP\] and statement (i) of Lemma \[lem:preliminary\_results\_RR+\] with ${\mathcal J}_1=\{1 \}$, it is easy to obtain $1-\vartheta^2_{12} > \max\big( \max_{j\in {\mathcal S}^c\cap {\mathcal I}_1} \, |\vartheta_{j2}- \vartheta_{12}\vartheta_{j1}|, \ \max_{j \in {\mathcal S}^c \cap {\mathcal I}_+}(\vartheta_{j2}- \vartheta_{12}\vartheta_{j1})_+\big)$. We next consider the case where $z_2 \ge |z_1|>0$. In this case, $j^*_1=2$ such that $x^1=(\vartheta_{12} z_1 + z_2)_+ \cdot \mathbf e_2$, where $\vartheta_{12} z_1 + z_2 > 0$. Hence, $A^T_{\bullet j} A(z - x^1)= (\vartheta_{j1} - \vartheta_{j2} \vartheta_{12}) z_1$. Applying Step 2 of Algorithm \[algo:constrained\_MP\] and statement (i) of Lemma \[lem:preliminary\_results\_RR+\] with ${\mathcal J}_1=\{2 \}$, we have that $$(1-\vartheta^2_{12}) |z_1| > \max\big( \max_{j\in {\mathcal S}^c\cap {\mathcal I}_1} \, |(\vartheta_{j1}- \vartheta_{12}\vartheta_{j2}) z_1|, \ \max_{j \in {\mathcal S}^c \cap {\mathcal I}_+}[(\vartheta_{j1}- \vartheta_{12}\vartheta_{j2}) z_1 ]_+\big).$$ It is easy to show that $(1-\vartheta^2_{12}) |z_1| > \max_{j \in {\mathcal S}^c \cap {\mathcal I}_+}[(\vartheta_{j1}- \vartheta_{12}\vartheta_{j2}) z_1 ]_+$ for any $z_1 \ne 0$ if and only if $1-\vartheta^2_{12} > \max_{j \in {\mathcal S}^c \cap {\mathcal I}_+}|\vartheta_{j1}- \vartheta_{12}\vartheta_{j2}|$. This yields $1-\vartheta^2_{12}> \max_{j\in {\mathcal S}^c} | \vartheta_{j1}- \vartheta_{12}\vartheta_{j2} |$, and condition (iii).
“If”. This part can be shown in a similar way by reversing the previous argument.
Necessary and sufficient conditions for the exact vector recovery on ${\mathcal P}$ for a given support ${\mathcal S}$ of size 3 can be established via a similar argument for Theorem \[thm:nonnegative constraint\_S3\]. Instead doing this, we provide a sufficient condition for a given support of arbitrary size. To simplify notation, we define the following function $F_{{\mathcal I}, {\mathcal J}}: \mathbb R_{\mathcal I}\times (\mathbb R_+)_{\mathcal J}\rightarrow \mathbb R$ for given index sets ${\mathcal I}$ and ${\mathcal J}$: $F_{{\mathcal I}, {\mathcal J}}( v ):=\max\big( \max_{i\in {\mathcal I}} |v_i|, \max_{i\in {\mathcal J}} (v_i)_+\big)$.
\[thm:RR+\_sufficient\_cond\] Given a matrix $A \in \mathbb R^{m\times N}$ with unit columns and the index set ${\mathcal S}\subset \{1, \ldots, N\}$, let $M:=A^T_{\bullet {\mathcal S}} A_{\bullet {\mathcal S}}$, ${\mathcal S}_1:={\mathcal S}\cap {\mathcal I}_1$, and ${\mathcal S}_+:={\mathcal S}\cap {\mathcal I}_+$. Then every vector $z \in {\mathcal P}$ with ${ \mbox{supp} }(z)={\mathcal S}$ is recovered from $y=A z$ via constrained matching pursuit if the following conditions hold:
- $A_{\bullet {\mathcal S}}$ has full column rank or equivalently $M$ is positive definite; and
- For any (possibly empty) index sets ${\mathcal L}_1 \subset {\mathcal S}_1$ and ${\mathcal L}_+ \subset {\mathcal S}_+$, letting ${\widetilde}{\mathcal L}:={\mathcal L}_1 \cup {\mathcal L}_+$, $$\begin{aligned}
& F_{{\mathcal S}_1\setminus {\mathcal L}_1, \, {\mathcal S}_+\setminus {\mathcal L}_+}\left( M/M_{{\widetilde}{\mathcal L}{\widetilde}{\mathcal L}} \begin{pmatrix} v_{{\mathcal S}_1 \setminus {\mathcal L}_1} \\ v_{{\mathcal S}_+ \setminus {\mathcal L}_+} \end{pmatrix} \right) \notag \\
& > \
F_{{\mathcal S}^c\cap {\mathcal I}_1, \, {\mathcal S}^c\cap {\mathcal I}_+}\left( \big(A^T_{\bullet {\mathcal S}^c}[I- A^T_{\bullet {\widetilde}{\mathcal L}} ( A^T_{\bullet {\widetilde}{\mathcal L}} A_{\bullet {\widetilde}{\mathcal L}} )^{-1} A_{\bullet {\widetilde}{\mathcal L}}] A_{\bullet{\mathcal S}\setminus{\widetilde}{\mathcal L}}
\begin{pmatrix} v_{{\mathcal S}_1 \setminus {\mathcal L}_1} \\ v_{{\mathcal S}_+ \setminus {\mathcal L}_+} \end{pmatrix} \right)
\end{aligned}$$ for all $v_{{\mathcal S}_+ \setminus {\mathcal L}_+} >0$ and all $v_{{\mathcal S}_1 \setminus {\mathcal L}_1}$ whose each element is nonzero.
Let ${\mathcal J}\subset {\mathcal S}$ be a nonempty index set. Since ${\mathcal P}=\mathbb R_{{\mathcal I}_1} \times (\mathbb R_+)_{{\mathcal I}_+}$ is a closed convex cone, it follows from the discussions at the end of Section \[sect:Constrained\_MP\] that the necessary and sufficient optimality condition for an optimal solution $x^*=(x^*_{\mathcal J}, 0)$ of the underlying minimization problem $\min_{w \in {\mathcal P}, { \mbox{supp} }(w) \subseteq {\mathcal J}} \| A w - A z \|^2_2$ is given by: $\mathcal C \in x^*_{\mathcal J}\perp A^T_{\bullet {\mathcal J}}( A_{\bullet {\mathcal J}} x^*_{\mathcal J}- A z ) \in \mathcal C^*$, where $z \in {\mathcal P}$ is such that ${ \mbox{supp} }(z)={\mathcal S}$, the convex cone $\mathcal C:=\{ w_{\mathcal J}\, | \, (w_{{\mathcal J}}, 0) \in {\mathcal P}\}= \mathbb R_{{\mathcal I}_1 \cap {\mathcal J}} \times (\mathbb R_+)_{{\mathcal I}_+\cap {\mathcal J}}$ and the dual cone $\mathcal C^*$ is given by $\mathcal C^*=\{ 0 \}\times (\mathbb R_+)_{{\mathcal I}_+\cap {\mathcal J}}$. Hence, we have that $$A^T_{\bullet {\mathcal I}_1 \cap {\mathcal J}} ( A_{\bullet {\mathcal J}} x^*_{\mathcal J}- A z ) = A^T_{\bullet {\mathcal I}_1 \cap {\mathcal J}} A(x^*-z)=0,$$ where $({\mathcal I}_1 \cap {\mathcal J}) \subset {\mathcal S}_1$, and $$0 \le x^*_{{\mathcal I}_+ \cap {\mathcal J}} \perp A^T_{\bullet {\mathcal I}_+ \cap {\mathcal J}} ( A_{\bullet {\mathcal J}} x^*_{\mathcal J}- A z ) \ge 0,$$ where $x^*_{\mathcal J}=(x^*_{{\mathcal I}_1 \cap {\mathcal J}}, x^*_{{\mathcal I}_+ \cap {\mathcal J}})$ with $x^*_{{\mathcal I}_+ \cap {\mathcal J}} \ge 0$. Let the index set $\mathcal L_+:=\{ i \in {\mathcal I}_+ \cap {\mathcal J}\, | \, x^*_i>0\}$. Thus $\mathcal L_+ \subset {\mathcal S}_+$ and $A^T_{\bullet \mathcal L_+} A(x^*-z)=0$. Set ${\mathcal L}_1:={\mathcal I}_1 \cap {\mathcal J}$, and ${\widetilde}{\mathcal L}:={\mathcal L}_1 \cup {\mathcal L}_+$. Hence, ${\mathcal L}_1$ and ${\mathcal L}_+$ are disjoint subsets of ${\mathcal S}$ with $A^T_{\bullet {\widetilde}{\mathcal L}} A(z - x^*)=0$. Further, $x^*_{{\mathcal S}\setminus {\widetilde}{\mathcal L}}=0$. Hence, $ A^T_{\bullet {\mathcal S}\setminus {\widetilde}{\mathcal L}} A (z - x^*) = M/M_{{\widetilde}{\mathcal L}{\widetilde}{\mathcal L}} (z - x^*)_{{\mathcal S}\setminus {\widetilde}{\mathcal L}} = M/M_{{\widetilde}{\mathcal L}{\widetilde}{\mathcal L}} \, z_{{\mathcal S}\setminus {\widetilde}{\mathcal L}}$, and $A^T_{\bullet {\mathcal S}^c} A (z - x^*) = A^T_{\bullet {\mathcal S}^c}[I- A^T_{\bullet {\widetilde}{\mathcal L}} ( A^T_{\bullet {\widetilde}{\mathcal L}} A_{\bullet {\widetilde}{\mathcal L}} )^{-1} A_{\bullet {\widetilde}{\mathcal L}}] A_{\bullet{\mathcal S}\setminus{\widetilde}{\mathcal L}} \, z_{{\mathcal S}\setminus {\widetilde}{\mathcal L}}$. Since ${\mathcal S}$ is a disjoint union of ${\mathcal S}_1$ and ${\mathcal S}_+$, $z_{{\mathcal S}\setminus {\widetilde}{\mathcal L}} = (z_{{\mathcal S}_1\setminus {\mathcal L}_1}, z_{{\mathcal S}_+\setminus {\mathcal L}_+})$, where $z_{{\mathcal S}_+\setminus {\mathcal L}_+}>0$ and each element of $z_{{\mathcal S}_1\setminus {\mathcal L}_1}$ is nonzero. Further, $$\max\Big( \max_{j \in {\mathcal S}_1\setminus {\mathcal J}} |A^T_{\bullet j} A (z -x^*)|, \max_{j\in {\mathcal S}_+ \setminus {\mathcal J}} [ A^T_{\bullet j} A (z -x^*) ]_+\Big) \, = \, F_{{\mathcal S}_1\setminus {\mathcal L}_1, \, {\mathcal S}_+\setminus {\mathcal L}_+}\left( M/M_{{\widetilde}{\mathcal L}{\widetilde}{\mathcal L}} \, z_{{\mathcal S}\setminus {\widetilde}{\mathcal L}} \right),$$ and $$\begin{aligned}
&\max\Big( \max_{j \in {\mathcal S}^c\cap {\mathcal I}_1} |A^T_{\bullet j} A (z -x^*)|, \max_{j\in {\mathcal S}^c \cap {\mathcal I}_+} [ A^T_{\bullet j} A (z -x^*) ]_+\Big) \\
& \, = \ F_{{\mathcal S}^c\cap {\mathcal I}_1, \, {\mathcal S}^c\cap {\mathcal I}_+}\left( \big(A^T_{\bullet {\mathcal S}^c}[I- A^T_{\bullet {\widetilde}{\mathcal L}} ( A^T_{\bullet {\widetilde}{\mathcal L}} A_{\bullet {\widetilde}{\mathcal L}} )^{-1} A_{\bullet {\widetilde}{\mathcal L}}] A_{\bullet{\mathcal S}\setminus{\widetilde}{\mathcal L}}\, z_{{\mathcal S}\setminus {\widetilde}{\mathcal L}} \right).
\end{aligned}$$ Consequently, under the condition (ii), condition $(\mathbf H)$ holds, leading to the exact vector recovery.
Sufficient Conditions for Uniform Exact Recovery on Convex, CP Admissible Sets via Constrained Matching Pursuit {#sect:suff_cond_exact_recovery}
===============================================================================================================
In this section, we derive sufficient conditions for uniform exact support and vector recovery via constrained matching pursuit using the restricted isometry-like and restricted orthogonality-like constants. For this purpose, we introduce the following constants.
\[def:RIP\_MC\_constants\] For a given (possible non-CP admissible) set ${\mathcal P}$, a matrix $A \in \mathbb R^{m\times N}$, and disjoin index sets ${\mathcal S}_1, {\mathcal S}_+, {\mathcal S}_-$ whose union is $\{1, \ldots, N\}$, we say that
- A real number $\delta$ is [*of Property RI on ${\mathcal P}$*]{} if $0<\delta <1$ and $(1-\delta) \cdot \| u - v \|^2_2 \le \| A(u-v) \|^2_2 $ for all $u, v\in \Sigma_K \cap {\mathcal P}$ with ${ \mbox{supp} }(v) \subset { \mbox{supp} }(u)$;
- A real number $\theta$ is [*of Property RO on ${\mathcal P}$ corresponding to ${\mathcal S}_1, {\mathcal S}_+, {\mathcal S}_-$*]{} if $\theta>0$ and for all $u, v\in \Sigma_K \cap {\mathcal P}$ with ${ \mbox{supp} }(v) \subset { \mbox{supp} }(u)$, the following holds: $$\begin{aligned}
\lefteqn{
\max\Big( \max_{j \in [{ \mbox{supp} }(u)]^c \cap {\mathcal S}_1} |\langle A(u-v), A_{\bullet j} \rangle|, \
\max_{j \in [{ \mbox{supp} }(u)]^c \cap {\mathcal S}_+} \langle A(u-v), A_{\bullet j} \rangle_+,
} \qquad \qquad \qquad \qquad \\
& \max_{j \in [{ \mbox{supp} }(u)]^c \cap {\mathcal S}_-} \langle A(u-v), A_{\bullet j} \rangle_- \Big ) \ \le \ \theta \cdot \|u-v\|_2.
\end{aligned}$$
We also denote these two constants by $\delta_{K, {\mathcal P}}$ and $\theta_{K, {\mathcal P}}$ respectively to emphasize their dependence on ${\mathcal P}$.
When ${\mathcal P}=\mathbb R^N$, the constant $\delta_{K, {\mathcal P}}$ resembles the restricted isometry constant, and the constant $\theta_{K, {\mathcal P}}$ is closely related to the $(K, 1)$-restricted orthogonality constant [@FoucartRauhut_book2013 Definition 6.4].
Cone Case {#subsect:suff_cond_cone}
---------
We first consider the case where $\mathcal P$ is an irreducible, closed, convex and CP admissible cone; see Definition \[def:irreducible\_CP\_set\] for the irreducibility. It follows from Proposition \[prop:CP\_admissible\_cone\] that ${\mathcal P}$ is a Cartesian product of copies of $\mathbb R, \mathbb R_+$ and $\mathbb R_-$, i.e., ${\mathcal P}= \mathbb R_{{\mathcal I}_1} \times (\mathbb R_+)_{{\mathcal I}_+} \times (\mathbb R_-)_{{\mathcal I}_-}$, where ${\mathcal I}_1, {\mathcal I}_+$ and ${\mathcal I}_-$ form a disjoint union of $\{1, \ldots, N\}$. The following theorem gives a sufficient condition for condition $(\mathbf H)$ on ${\mathcal P}$, and thus for the exact support recovery on ${\mathcal P}$, in terms of the constants $\theta_{K, {\mathcal P}}$ and $\theta_{K, {\mathcal P}}$ introduced in Definition \[def:RIP\_MC\_constants\].
\[thm:sufficency\_cone\_case\] Let ${\mathcal P}= \mathbb R_{{\mathcal I}_1} \times (\mathbb R_+)_{{\mathcal I}_+} \times (\mathbb R_-)_{{\mathcal I}_-}$, where the index sets ${\mathcal I}_1, {\mathcal I}_+$ and ${\mathcal I}_-$ form a disjoint union of $\{1, \ldots, N\}$, and let $A \in \mathbb R^{m\times N}$ be a matrix with unit columns. Suppose there exist constants $\delta_{K, {\mathcal P}}$ of Property RI on ${\mathcal P}$ and $\theta_{K, {\mathcal P}}$ of Property RO on ${\mathcal P}$ corresponding to ${\mathcal I}_1, {\mathcal I}_+$ and ${\mathcal I}_-$ such that $$\label{eqn:sufficient_condition_cone}
1- \delta_{K, {\mathcal P}} \, > \, \sqrt{K} \cdot \theta_{K, {\mathcal P}}.$$ Then condition $(\mathbf H)$ given by (\[eqn:condition\_H’\]) holds on ${\mathcal P}$.
Given any $0 \ne u \in \Sigma_K \cap \, {\mathcal P}$ and any index set $\mathcal J \subset { \mbox{supp} }(u)$, let $v$ be an arbitrary optimal solution to $\min_{w \in \mathcal P, \ { \mbox{supp} }(w)\subseteq \mathcal J} \| A (u - w) \|^2_2$. Hence, for each $j\in { \mbox{supp} }(u)\setminus {\mathcal J}$, either $j \in {\mathcal I}_1$, $j \in {\mathcal I}_+$ or $j \in {\mathcal I}_-$. For any $j \in [{ \mbox{supp} }(u) \setminus {\mathcal J}] \cap {\mathcal I}_1$, we have $\mathbb I_j(v) = \mathbb R$, where $\mathbb I_j(v)$ is defined in (\[eqn:interval\_j\]). For any $j\in [{ \mbox{supp} }(u) \setminus {\mathcal J}] \cap {\mathcal I}_+$, it follows from $j \notin {\mathcal J}$ that $v_j=0$ and $\mathbb I_j(v) = \mathbb R_+$. Similarly, for any $j\in [{ \mbox{supp} }(u) \setminus {\mathcal J}] \cap {\mathcal I}_-$, we have $\mathbb I_j(v) = \mathbb R_-$. Further, in light of $\|A_{\bullet j}\|_2=1, \forall \, j$ and the expressions for $f^*_j(u,v)$ given below (\[eqn:f\*\_j\_convex\]), we have $$\begin{aligned}
f^*_j(u,v) & = & \| A(u-v)\|^2_2 - |\langle A(u-v), A_{\bullet j} \rangle|^2, \qquad \forall \ j \in [{ \mbox{supp} }(u)\setminus {\mathcal J}] \cap {\mathcal I}_1, \\
f^*_j(u,v) & = & \| A(u-v)\|^2_2 - [\langle A(u-v), A_{\bullet j} \rangle_+]^2, \quad \ \forall \ j \in [{ \mbox{supp} }(u)\setminus {\mathcal J}] \cap {\mathcal I}_+, \\
f^*_j(u,v) & = & \| A(u-v)\|^2_2 - [\langle A(u-v), A_{\bullet j} \rangle_-]^2, \quad \ \forall \ j \in [{ \mbox{supp} }(u)\setminus {\mathcal J}] \cap {\mathcal I}_-.
\end{aligned}$$ Define the following quantities: $$\begin{aligned}
\Gamma_1 & := & \max\Big( \, \max_{j \in [ { \mbox{supp} }(u)\setminus \mathcal J] \cap {\mathcal I}_1 } |\langle A(u-v), A_{\bullet j} \rangle|, \ \max_{j \in [ { \mbox{supp} }(u)\setminus \mathcal J] \cap {\mathcal I}_+ } \langle A(u-v), A_{\bullet j} \rangle_+, \notag \\ [5pt]
& & \qquad \quad \max_{j \in [ { \mbox{supp} }(u)\setminus \mathcal J] \cap {\mathcal I}_- } \langle A(u-v), A_{\bullet j} \rangle_- \, \Big), \label{eqn:cone_equivalent_H'_condition} \\ [5pt]
\Gamma_2 & := & \max\Big( \, \max_{j \in [{ \mbox{supp} }(u)]^c \cap {\mathcal I}_1 } |\langle A(u-v), A_{\bullet j} \rangle|, \ \max_{j \in [{ \mbox{supp} }(u)]^c \cap {\mathcal I}_+ } \langle A(u-v), A_{\bullet j} \rangle_+, \notag \\ [5pt]
& & \qquad \quad \, \max_{j \in [{ \mbox{supp} }(u)]^c \cap {\mathcal I}_- } \langle A(u-v), A_{\bullet j} \rangle_- \, \Big). \notag\end{aligned}$$ Note that if $\Gamma_1 > \Gamma_2$, then $\min_{j \in { \mbox{supp} }(u)\setminus \mathcal J} f^*_{j}(u, v) \, < \, \min_{j \in [{ \mbox{supp} }(u)]^c} f^*_{j}(u, v)$ such that condition $(\mathbf H)$ given by (\[eqn:condition\_H’\]) holds. Hence, it suffices to show that $\Gamma_1 > \Gamma_2$ as follows.
By virtue of the definition of the constant $\theta_{K, {\mathcal P}}$ corresponding to ${\mathcal I}_1, {\mathcal I}_+$ and ${\mathcal I}_-$, we deduce that $\Gamma_2 \le \theta_{K, {\mathcal P}} \cdot \| u - v \|_2$. Besides, in view of Proposition \[prop:negative\_second\_term\] and the definition of $\Gamma_1$ in (\[eqn:cone\_equivalent\_H’\_condition\]), we have $$\begin{aligned}
\lefteqn{ \| A (u-v)\|^2_2 \ \ \le \ \sum_{j \in { \mbox{supp} }(u)\setminus{\mathcal J}} \langle A(u-v), A_{\bullet j} \rangle \cdot (u-v)_j } \\ & = & \sum_{j \in [{ \mbox{supp} }(u)\setminus{\mathcal J}] \cap {\mathcal I}_1 } \langle A(u-v), A_{\bullet j} \rangle \cdot (u-v)_j + \sum_{j \in [{ \mbox{supp} }(u)\setminus{\mathcal J}] \cap {\mathcal I}_+ } \langle A(u-v), A_{\bullet j} \rangle \cdot (u-v)_j \\ [5pt]
& & \qquad \quad + \sum_{j \in [{ \mbox{supp} }(u)\setminus{\mathcal J}] \cap {\mathcal I}_- } \langle A(u-v), A_{\bullet j} \rangle \cdot (u-v)_j \\ & \le & \sum_{j \in [{ \mbox{supp} }(u)\setminus{\mathcal J}] \cap {\mathcal I}_1 } |\langle A(u-v), A_{\bullet j} \rangle| \cdot |(u-v)_j| + \sum_{j \in [{ \mbox{supp} }(u)\setminus{\mathcal J}] \cap {\mathcal I}_+ } \langle A(u-v), A_{\bullet j} \rangle_+ \cdot (u-v)_j \\ & & \qquad \quad + \sum_{j \in [{ \mbox{supp} }(u)\setminus{\mathcal J}] \cap {\mathcal I}_- } \langle A(u-v), A_{\bullet j} \rangle_- \cdot |(u-v)_j| \\ & \le & \Gamma_1 \cdot \big \|(u-v)_{{ \mbox{supp} }(u)\setminus {\mathcal J}} \big \|_1 \ \le \ \Gamma_1 \cdot \sqrt{|{ \mbox{supp} }(u)\setminus {\mathcal J}|} \cdot \|u-v \|_2, \\
& \le & \Gamma_1 \cdot \sqrt{K} \cdot \|u-v \|_2,\end{aligned}$$ where the second inequality follows from the fact that $u_j>0=v_j$ for each $j \in [{ \mbox{supp} }(u)\setminus{\mathcal J}] \cap {\mathcal I}_+$. Therefore, by the definition of the constant $\delta_{K, {\mathcal P}}$, we have $$(1- \delta_{K, {\mathcal P}} ) \cdot \|u-v\|^2_2 \, \le \, \|A (u-v)\|^2_2 \, \le \, \Gamma_1 \cdot \sqrt{K} \cdot \|u-v \|_2.$$ Since ${ \mbox{supp} }(v) \subset { \mbox{supp} }(u)$, we have $\| u-v \|_2>0$. This further implies that $ [(1- \delta_{K,{\mathcal P}})/\sqrt{K}] \cdot \| u-v\|_2 \le \Gamma_1$. Using $\Gamma_2 \le \theta_{K, {\mathcal P}} \cdot \| u - v \|_2 $ and the assumption that $1- \delta_{K, {\mathcal P}} > \sqrt{K} \cdot \theta_{K, {\mathcal P}}$ given in (\[eqn:sufficient\_condition\_cone\]), we obtain $\Gamma_1 > \Gamma_2$. As a result, condition $(\mathbf H)$ holds.
Since $\theta_{K, {\mathcal P}}$ and $\theta_{K, {\mathcal P}}$ may be difficult to find numerically due to the conditions such as ${ \mbox{supp} }(v) \subset { \mbox{supp} }(u)$ in their definitions, it is desired that similar constants independent of the above mentioned conditions can be used. This leads to the following quantities.
\[def:delta\_theta\_hat\] Let a matrix $A \in \mathbb R^{m\times N}$ with unit columns and the index sets ${\mathcal I}_1, {\mathcal I}_+$ and ${\mathcal I}_-$ which form a disjoint union of $\{1, \ldots, N\}$ be given.
- The constant ${\widehat}\delta_K \in (0, 1)$ is such that $(1-{\widehat}\delta_K) \cdot \| x \|^2_2 \le \| Ax \|^2_2 $ for all $x \in \Sigma_K$;
- The constant ${\widehat}\theta_K>0$ corresponding to the index set ${\mathcal I}_1, {\mathcal I}_+$ and ${\mathcal I}_-$ is such that for any $x \in \Sigma_K$, $$\max \Big( \, \max_{ j \in {\mathcal I}_1} |\langle Ax, A_{\bullet j} \rangle|, \ \max_{ j \in {\mathcal I}_+} \langle Ax, A_{\bullet j} \rangle_+, \ \max_{ j \in {\mathcal I}_-} \langle Ax, A_{\bullet j} \rangle_- \, \Big) \, \le \, {\widehat}\theta_K \cdot \|x\|_2.$$
To emphasize the dependence of the above constants on $A$ (when ${\mathcal I}_1, {\mathcal I}_+$ and ${\mathcal I}_-$ are fixed), we also write them as ${\widehat}\delta_K(A)$ and ${\widehat}\theta_K(A)$, respectively.
Based on Definition \[def:delta\_theta\_hat\], it is easy to see that ${\widehat}\delta_K$ is of Property RI and ${\widehat}\theta_K$ is of Property RO, both on ${\mathcal P}$. Hence, by Theorem \[thm:sufficency\_cone\_case\], we obtain the following corollary immediately; its proof is omitted.
\[coro:sufficiency\_uniform\_constants\] For a given matrix $A \in \mathbb R^{m\times N}$ with unit columns and a closed, convex, and CP admissible cone ${\mathcal P}$ defined by the index sets ${\mathcal I}_1, {\mathcal I}_+$ and ${\mathcal I}_-$, if there exist positive constants ${\widehat}\delta_K$ and ${\widehat}\theta_K$ given by Definition \[def:delta\_theta\_hat\] such that $ 1-{\widehat}\delta_K> \sqrt{K} \cdot {\widehat}\theta_K$, then condition $(\mathbf H)$ given by (\[eqn:condition\_H’\]) holds.
In what follows, we discuss the constants ${\widehat}\delta_K$ and ${\widehat}\theta_K$ subject to perturbations of $A$.
\[prop:A\_perturbation\] Let a matrix $A^\diamond \in \mathbb R^{m\times N}$ be such that there exist constants ${\widehat}\delta_K(A^\diamond) \in (0, 1)$ and ${\widehat}\theta_{K}(A^\diamond)>0$ satisfying $1- {\widehat}\delta_K(A^\diamond)> \sqrt{K} \cdot {\widehat}\theta_{K}(A^\diamond)$. Then there exists a constant $\eta>0$ such that for any $A $ with $\| A - A^\diamond \|_2 < \eta$, there exist constants ${\widehat}\delta_K(A)>0$ and ${\widehat}\theta_{K}(A)>0$ satisfying the conditions given by Definition \[def:delta\_theta\_hat\] such that $1- {\widehat}\delta_K(A)> \sqrt{K} \cdot {\widehat}\theta_{K}(A)$.
For the given matrix $A^\diamond$ and the positive constants ${\widehat}\delta_K(A^\diamond)$ and ${\widehat}\theta_{K}(A^\diamond)$, it suffices to show that for any $\varepsilon>0$, there exist constants $\eta'>0$ and $\eta''>0$ such that (i) for each $A$ with $\| A - A^\diamond\|_2 < \eta'$, there exists a constant ${\widehat}\delta_K(A)>0$ satisfying condition (i) of Definition \[def:delta\_theta\_hat\] such that $|{\widehat}\delta_K(A)- {\widehat}\delta_K(A^\diamond)| < \varepsilon$; and (ii) for each $A$ with $\| A - A^\diamond\|_2 < \eta''$, there exists a constant ${\widehat}\theta_K(A)>0$ satisfying condition (ii) of Definition \[def:delta\_theta\_hat\] such that $|{\widehat}\theta_K(A)- {\widehat}\theta_K(A^\diamond)| < \varepsilon$. To show the existence of $\eta'$, we use the inequality $\big| \|Ax\|_2 - \| A^\diamond x\|_2 \big| \le \| A - A^\diamond \|_2 \cdot \|x \|_2$ for any $A$ and $x$ [@ShenMousavi_SIOPT18 Proposition 5.3]. Hence, for all $A$ in the neighborhood $\mathcal U$ of $A^\diamond$ given by $\mathcal U=\{ A \, | \, \|A - A^\diamond\|_2<\alpha\}$ for some $\alpha>0$, we have $\big| \|A x \|^2_2 - \| A^\diamond x \|^2_2 \big | = \big| \|A x \|_2 - \| A^\diamond x \|_2 \big | \cdot ( \|A x \|_2 + \| A^\diamond x \|_2) \le \| A - A^\diamond\|_2 \cdot \|x\|_2 \cdot (2\| A^\diamond\|_2 + \alpha) \cdot \| x\|_2 \le c' \cdot \| A - A^\diamond\|_2 \cdot \| x \|^2_2$ for all $x$, where $c' :=2\| A^\diamond\|_2 + \alpha>0$. Hence, $\| A x \|^2_2 \ge \| A^\diamond x \|^2_2 - c' \cdot \| A - A^\diamond\|_2 \cdot \| x \|^2_2 \ge [1-{\widehat}\delta_K(A^\diamond) - c'\cdot \| A - A^\diamond\|_2] \cdot \| x \|^2_2$ for all $x$. Letting ${\widehat}\delta_K(A):= {\widehat}\delta_K(A^\diamond) + c' \cdot \| A - A^\diamond \|_2$, we can obtain a positive constant $\eta'$ with $0<\eta'<\min(\varepsilon/c', \alpha)$ such that for each $A$ with $\| A - A^\diamond\|_2 < \eta'$, $|{\widehat}\delta_K(A)- {\widehat}\delta_K(A^\diamond)| < \varepsilon$.
To show the existence of $\eta''$, define the function $h_j$ for a fixed index $j$ and a matrix $A$: $$h_j(A, x) \, := \, \left\{ \begin{array}{lll} |\langle Ax, A_{\bullet j} \rangle|, & \mbox{ if } \ j \in {\mathcal I}_1; \\ \langle Ax, A_{\bullet j} \rangle_+, & \mbox{ if } \ j \in {\mathcal I}_+; \\
\langle Ax, A_{\bullet j} \rangle_-, & \mbox{ if } \ j \in {\mathcal I}_-. \end{array} \right.$$ Using the fact that $|x_+- y_+| \le |x -y |$ and $|x_- - y_-| \le |x - y|$ for any $x, y \in \mathbb R$, we have, for each $j$, $$\begin{aligned}
| h_j(A, x) - h_j(A^\diamond, x)| & \le & |\langle Ax, A_{\bullet j} \rangle -\langle A^\diamond x, A^\diamond_{\bullet j} \rangle | \\
& = & \Big| \langle A^\diamond x, (A-A^\diamond)_{\bullet j} \rangle + \langle (A-A^\diamond) x, A^\diamond_{\bullet j} \rangle + \langle (A-A^\diamond) x, (A-A^\diamond)_{\bullet j} \rangle \Big| \\
& \le & | \langle A^\diamond x, (A-A^\diamond) \mathbf e_{j} \rangle| + |\langle (A-A^\diamond) x, A^\diamond \mathbf e_{j} \rangle| + | \langle (A-A^\diamond) x, (A-A^\diamond) \mathbf e_{j} \rangle | \\
& \le & \| A - A^\diamond \|_2 \cdot [ 2 \|A^\diamond\|_2 + \| A - A^\diamond \|_2 \big] \cdot \| x \|_2,\end{aligned}$$ where the last inequality follows from Cauchy-Schwarz inequality and $\| \mathbf e_j \|_2=1$. Therefore, for all $A$ in the neighborhood $\mathcal U$ of $A^\diamond$ given by $\mathcal U=\{ A \, | \, \|A - A^\diamond\|_2<\beta\}$ for some $\beta>0$, we obtain the constant $c:= 2 \|A^\diamond\|_2 +\beta>0$ such that for each $j$, $h_j(A, x) \le h_j(A^\diamond, x) + c \cdot \| A - A^\diamond \|_2 \cdot \| x \|_2$. In view of $$\max_j h_j(A, x) \, = \, \max \Big( \, \max_{ j \in {\mathcal I}_1} |\langle Ax, A_{\bullet j} \rangle|, \ \max_{ j \in {\mathcal I}_+} \langle Ax, A_{\bullet j} \rangle_+, \ \max_{ j \in {\mathcal I}_-} \langle Ax, A_{\bullet j} \rangle_- \, \Big),$$ we further have $$\begin{aligned}
\max_j h_j(A, x) & \le & \max_j h_j(A^\diamond, x) + c \cdot \| A - A^\diamond \|_2 \cdot \| x \|_2 \, \le \, {\widehat}\theta_K (A^\diamond) \cdot \| x \|_2 + c \cdot \| A - A^\diamond \|_2 \cdot \| x \|_2 \\
& \le & \big[ {\widehat}\theta_K(A^\diamond) + c \cdot \| A - A^\diamond \|_2 \big] \cdot \| x \|_2.\end{aligned}$$ By letting ${\widehat}\theta_K(A):= {\widehat}\theta_K(A^\diamond) + c \cdot \| A - A^\diamond \|_2$, it is easy to obtain a positive constant $\eta''$ with $0<\eta''<\min(\varepsilon/c, \beta)$ such that for each $A$ with $\| A - A^\diamond\|_2 < \eta''$, $|{\widehat}\theta_K(A)- {\widehat}\theta_K(A^\diamond)| < \varepsilon$.
\[remark:A\_perturb\] The above proposition shows that for fixed index sets ${\mathcal I}_1, {\mathcal I}_+$ and ${\mathcal I}_-$, $\mathcal A := \{ A \in \mathbb R^{m\times N} \, | \, 1- {\widehat}\delta_K(A)> \sqrt{K} \cdot {\widehat}\theta_{K}(A) \}$ is an open set in the matrix space $\mathbb R^{m \times N}$. Since the set of matrices of completely full rank, i.e., $A \in \mathbb R^{m\times N}$ is such that every $m\times m$ submatrix of $A$ is invertible [@ShenMousavi_SIOPT18], is open and dense in the matrix space $\mathbb R^{m\times N}$, we conclude that for any $A \in \mathcal A$ and an arbitrarily small $\varepsilon>0$, there exists a matrix $A' \in \mathcal A$ of complete full rank such that $\| A' - A \|< \varepsilon$. An advantage of using the matrix $A'$ is that it leads to a unique $x^k$ in each step (cf. Lemma \[lemma:sol\_existence\]) and thus gives rise to the exact vector recovery, provided that the sparsity level $K \le m$.
Non-cone Case {#subsect:suff_cond_noncone}
-------------
In this subsection, we consider the case where an irreducible, convex and CP admissible set ${\mathcal P}$ is not a cone. We exploit the positive homogeneous property of the functions used to characterize the two constants $\delta_{K, {\mathcal P}}$ and $\theta_{K, {\mathcal P}}$ in Definition \[def:RIP\_MC\_constants\] and obtain sufficient conditions for exact support recovery on ${\mathcal P}$. Toward this end, we recall that a function $f:\mathbb R^N \rightarrow \mathbb R$ is positively homogeneous of degree $p \in \mathbb N$ if for any $\lambda \ge 0$, $f(\lambda x) = \lambda^p \cdot f(x)$ for all $x \in \mathbb R^N$. We start from a technical lemma.
\[lem:positive\_homo\_equivalence\] Let $\mathcal P$ be a convex set in $\mathbb R^N$ containing the zero vector, $g:\mathbb R^N \times \mathbb R^N \rightarrow \mathbb R$ be a positively homogeneous function of degree $p$, and the set $\mathcal K \subset \mathbb R^N\times \mathbb R^N $ be such that $(0, 0) \in \mathcal K $ and $\mathcal K =\lambda \mathcal K $ for any $\lambda > 0$. Then $g(u, v) \le 0$ holds for all $(u, v) \in (\mathcal P \times \mathcal P) \cap \mathcal K $ if and only if $g(x, y) \le 0$ holds for all $(x, y) \in (\mbox{cone}(\mathcal P) \times \mbox{cone}(\mathcal P) ) \cap \mathcal K$.
Since $\mathcal P$ is a subset of $\mbox{cone}(\mathcal P)$, the “if” part holds trivially. To show the “only if” part, suppose $g(u, v) \le 0$ holds for all $(u, v) \in (\mathcal P \times \mathcal P) \cap \mathcal K $. Since $\mathcal P$ is convex, we have $\mbox{cone}(\mathcal P)=\{ \lambda x \, | \, x \in \mathcal P, \lambda \ge 0 \}$ [@Rockafellar_book70 Corollary 2.6.3]. Hence, for any $(x, y) \in (\mbox{cone}(\mathcal P) \times \mbox{cone}(\mathcal P) ) \cap \mathcal K$, there exist (possibly distinct) real numbers $\alpha, \beta \in \mathbb R_+$ and $(u, v) \in (\mathcal P \times \mathcal P) \cap \mathcal K$ such that $x= \alpha u$ and $y = \beta v$. We claim that there exist a pair $({\widehat}u, {\widehat}v) \in (\mathcal P \times \mathcal P) \cap \mathcal K$ and a positive constant $\lambda$ such that $(x, y) =\lambda \cdot ({\widehat}u, {\widehat}v)$. We show this claim for four possible cases as follows:
- $x=y=0$. Then we choose ${\widehat}u= {\widehat}v=0$ and any $\lambda>0$, using the fact that $0 \in {\mathcal P}$ and $(0, 0) \in \mathcal K$.
- $x\ne 0$ and $y=0$. This implies that $\alpha$ must be positive. Since $(x, y)=(\alpha u, 0) \in \mathcal K$ and $\mathcal K =\lambda \mathcal K$ for any $\lambda>0$, we have $(u, 0) = (1/\alpha)(x, y) \in \mathcal K$. Further, since ${\mathcal P}$ contains the zero vector, we have $(u, 0)\in (\mathcal P \times \mathcal P) \cap \mathcal K$. Therefore, by letting $({\widehat}u, {\widehat}v)=(u, 0)$ and $\lambda =\alpha>0$, the desired result holds.
- $x= 0$ and $y \ne 0$. This follows readily by interchanging the roles of $x$ and $y$ in case (b).
- $x \ne 0$ and $y \ne 0$. In this case, both $\alpha>0$ and $\beta>0$. Without loss of generality, we assume that $\alpha \ge \beta$. Since $0 < \beta/\alpha \le 1$ and ${\mathcal P}$ is a convex set containing the zero vector and $v$, we see that the vector ${\widehat}v:= ( \beta/\alpha) v$ belongs to ${\mathcal P}$. Hence, $y=\alpha {\widehat}v$ such that $(x, y)= \alpha ( u, {\widehat}v) \in \mathcal K$. Letting ${\widehat}u:=u$ and $\lambda=\alpha>0$, we have $({\widehat}u, {\widehat}v) \in (\mathcal P \times \mathcal P) \cap \mathcal K$ and $(x, y)=\lambda \cdot ({\widehat}u, {\widehat}v)$.
In light of the above claim, we deduce that for any $(x, y) \in (\mbox{cone}(\mathcal P) \times \mbox{cone}(\mathcal P) ) \cap \mathcal K$, $g(x, y) = g(\lambda({\widehat}u, {\widehat}v)) =
\lambda^p \cdot g({\widehat}u, {\widehat}v) \le 0$ by the positive homogeneity of $g(\cdot, \cdot)$.
\[prop:constant\_equivalence\] Let ${\mathcal P}$ be a convex set in $\mathbb R^N$ containing the zero vector, $A \in \mathbb R^{m\times N}$ be a matrix, and the index sets ${\mathcal S}_1, {\mathcal S}_+, {\mathcal S}_-$ form a disjoint union of $\{1, \ldots, N\}$. Then the following hold:
- A real number $\delta$ is of Property RI on ${\mathcal P}$ if and only if it is of Property RI on $\mbox{cone}({\mathcal P})$;
- A real number $\theta$ is of Property RO on ${\mathcal P}$ corresponding to ${\mathcal S}_1, {\mathcal S}_+, {\mathcal S}_-$ if and only if it is of Property RO on $\mbox{cone}({\mathcal P})$ corresponding to ${\mathcal S}_1, {\mathcal S}_+, {\mathcal S}_-$.
Define the set $\mathcal K:=\{(0, 0)\} \cup \{ (u, v) \in \Sigma_K \times \Sigma_K \, | \, { \mbox{supp} }(v) \subset { \mbox{supp} }(u) \} \subset \mathbb R^N \times \mathbb R^N$. It is easy to verify that $(0, 0) \in \mathcal K$, and $\mathcal K = \lambda \mathcal K$ for any positive number $\lambda$. For any fixed real numbers $\delta \in (0, 1)$ and $\theta>0$, define the functions $g_\delta:\mathbb R^N \times \mathbb R^N \rightarrow \mathbb R$ and $h_\theta:\mathbb R^N \times \mathbb R^N \rightarrow \mathbb R$: $$\begin{aligned}
g_\delta(u, v) & := & (1-\delta) \|u-v\|^2_2 - \| A(u-v)\|^2_2, \\
h_\theta(u, v) & := & \max \Big( \, \max_{ j \in [{ \mbox{supp} }(u)]^c \cap {\mathcal S}_1} |\langle A(u-v), A_{\bullet j} \rangle|, \ \max_{ j \in [{ \mbox{supp} }(u)]^c \cap{\mathcal S}_+} \langle A(u-v), A_{\bullet j} \rangle_+, \\
& & \quad \max_{ j \in [{ \mbox{supp} }(u)]^c \cap{\mathcal S}_-} \langle A(u-v), A_{\bullet j} \rangle_- \, \Big) - \theta \|(u-v)\|_2.
\end{aligned}$$ Clearly, $g_\delta$ is positively homogeneous of degree two. Furthermore, when $\lambda=0$, we see that for any $u$ and $v$ in $\mathbb R^N$, $h_\theta(\lambda u, \lambda v)=h_\theta(0, 0)=0=\lambda h_\theta(u, v)$. Besides, in view of ${ \mbox{supp} }(u)={ \mbox{supp} }(\lambda u)$ for any $\lambda>0$ and any $u$, we also have that $h_\theta(\lambda u, \lambda v)=\lambda h_\theta(u, v)$ for any $u, v$ and any $\lambda>0$. Therefore, $h_\theta$ is positively homogeneous of degree one. As a result, we obtain the following equivalent implications: $$\begin{aligned}
\Big[ \, \mbox{$\delta$ is of Property RI on ${\mathcal P}$} \, \Big] & \Longleftrightarrow & \Big[ \, g_\delta(u,v) \le 0, \ \forall \, (u, v) \in ({\mathcal P}\times {\mathcal P}) \cap \mathcal K \, \Big] \\
& \iff & \Big[ \, g_\delta(u,v) \le 0, \ \forall \, (u, v) \in (\mbox{cone}({\mathcal P}) \times \mbox{cone}({\mathcal P})) \cap \mathcal K \, \Big] \\
& \Longleftrightarrow & \Big[ \, \mbox{$\delta$ is of Property RI on $\mbox{cone}({\mathcal P})$} \, \Big],
\end{aligned}$$ where the first and last double implications follow from the definition of $\delta$ on ${\mathcal P}$ or $\mbox{cone}({\mathcal P})$ given by Definition \[def:RIP\_MC\_constants\], and the second double implication follow from Lemma \[lem:positive\_homo\_equivalence\]. Similarly, we can show that $\theta$ is of Property RO on ${\mathcal P}$ if and only if it is of Property RO on $\mbox{cone}({\mathcal P})$ using $h_\theta$.
By applying the above proposition and the conic hull of a closed, convex, and CP admissible set given by Proposition \[prop:conic\_hull\], we obtain sufficient conditions for exact support recovery in the following theorem.
\[thm:sufficency\_non\_cone\] Let $A \in \mathbb R^{m\times N}$ be a matrix with unit columns, and ${\mathcal P}$ be an irreducible, closed, convex, and CP admissible set in $\mathbb R^N$ whose conic hull is given by $\mbox{cone}({\mathcal P})=\mathbb R_{{\mathcal I}_1} \times (\mathbb R_+)_{{\mathcal I}_+}\times (\mathbb R_-)_{{\mathcal I}_-}$, where ${\mathcal I}_1, {\mathcal I}_+$ and ${\mathcal I}_-$ form a disjoint union of $\{1, \ldots, N\}$. Then condition $(\mathbf H)$ holds on ${\mathcal P}$ under either one of the following conditions:
- There exist constants $\delta_{K,\mbox{cone}({\mathcal P})}$ of Property RI and $\theta_{K, \mbox{cone}({\mathcal P})}$ of Property RO corresponding to ${\mathcal I}_1, {\mathcal I}_+$ and ${\mathcal I}_-$ such that $ 1-\delta_{K,\mbox{cone}({\mathcal P})}> \sqrt{K} \cdot \theta_{K,\mbox{cone}({\mathcal P})}$;
- There exist constants $\delta_{K,{\mathcal P}}$ of Property RI and $\theta_{K, {\mathcal P}}$ of Property RO corresponding to ${\mathcal I}_1, {\mathcal I}_+$ and ${\mathcal I}_-$ such that $ 1-\delta_{K,{\mathcal P}}> \sqrt{K} \cdot \theta_{K,{\mathcal P}}$.
\(i) It follows from Theorem \[thm:sufficency\_cone\_case\] that if $ 1-\delta_{K,\mbox{cone}({\mathcal P})}> \sqrt{K} \cdot \theta_{K,\mbox{cone}({\mathcal P})}$, then condition $(\mathbf H)$ holds on $\mbox{cone}({\mathcal P})$. Since ${\mathcal P}$ is a subset of $\mbox{cone}({\mathcal P})$, condition $(\mathbf H)$ also holds on ${\mathcal P}$.
\(ii) Suppose $ 1-\delta_{K,{\mathcal P}}> \sqrt{K} \cdot \theta_{K,{\mathcal P}}$ holds for the constants $\delta_{K,{\mathcal P}}$ of Property RI and $\theta_{K, {\mathcal P}}$ of Property RO corresponding to ${\mathcal I}_1, {\mathcal I}_+$ and ${\mathcal I}_-$. Since ${\mathcal P}$ is CP admissible, it contains the zero vector. Since ${\mathcal P}$ is also convex, we deduce via Proposition \[prop:constant\_equivalence\] that $\delta_{K,{\mathcal P}}$ is a constant of Property RI on $\mbox{cone}({\mathcal P})$ and $\theta_{K, {\mathcal P}}$ is a constant of Property RO on $\mbox{cone}({\mathcal P})$ corresponding to ${\mathcal I}_1, {\mathcal I}_+$ and ${\mathcal I}_-$. Therefore, by Theorem \[thm:sufficency\_cone\_case\], condition $(\mathbf H)$ holds on $\mbox{cone}({\mathcal P})$. It thus follows from statement (i) that condition $(\mathbf H)$ holds on ${\mathcal P}$.
Theorem \[thm:sufficency\_non\_cone\] gives a sufficient condition for $(\mathbf H)$ and thus exact support recovery on a closed, convex, and CP admissible set by leveraging its conic hull. Despite the simplicity of its proof, Theorem \[thm:sufficency\_non\_cone\] provides a potentially effective way to establish the exact support recovery for the following reasons. It is usually difficult to find and compute the constants $\delta_{K,{\mathcal P}}$ and $\theta_{K,{\mathcal P}}$ for a general closed, convex and CP admissible set ${\mathcal P}$. On the other hand, computing the constants $\delta_{k,\mbox{cone}({\mathcal P})}$ and $\theta_{k,\mbox{cone}({\mathcal P})}$ is easier, due to the simple structure of $\mbox{cone}({\mathcal P})$ as illustrated in Propositions \[prop:CP\_admissible\_cone\] and \[prop:conic\_hull\]. Note that the conditions $ 1-\delta_{K,{\mathcal P}}> \sqrt{K} \cdot \theta_{K,{\mathcal P}}$ and $ 1-\delta_{K,\mbox{cone}({\mathcal P})}> \sqrt{K} \cdot \theta_{K,\mbox{cone}({\mathcal P})}$ are equivalent in view of Proposition \[prop:constant\_equivalence\]. Hence, the latter condition in term of $\mbox{cone}({\mathcal P})$ does not lead to conservativeness.
Theorem \[thm:sufficency\_non\_cone\] can be extended to a non-CP admissible set as long as the closure of its conic hull is CP admissible. This is shown in the following corollary.
\[coro:non\_CP\_admissible\] Let $A \in \mathbb R^{m \times N}$ and ${\mathcal P}$ be a closed convex set containing the zero vector. Suppose the closure of $\mbox{cone}(P)$, denoted by $\mathcal C$, is CP admissible. Then the following hold:
- If there exist constants $\delta_{K,\mathcal C}$ of Property RI on $\mathcal C$ and $\theta_{K, \mathcal C}$ of Property RO on $\mathcal C$ such that $ 1-\delta_{K,\mathcal C}> \sqrt{K} \cdot \theta_{K,\mathcal C}$, then condition $(\mathbf H)$ holds on ${\mathcal P}$.
- If $K\le m$ and the constants ${\widehat}\delta_K(A)$ and ${\widehat}\theta_K(A)$ corresponding to the index sets for $\mathcal C$ given in Definition \[def:delta\_theta\_hat\] are such that $1-{\widehat}\delta_{K}(A) > \sqrt{K} \cdot {\widehat}\theta_{K}(A)$, then there exists a matrix $A' \in \mathbb R^{m\times N}$ sufficiently close to $A$ such that the exact vector recovery on $\Sigma_K \cap \mathcal P$ is achieved using $A'$.
\(i) Let $\mbox{cl}(\cdot)$ denote the closure of a set. It follows directly from the fact that $\mathcal P \subseteq \mathcal C:=\mbox{cl}(\mbox{cone}({\mathcal P}))$ and the similar argument for statement (i) of Theorem \[thm:sufficency\_non\_cone\].
\(ii) This result follows from Corollary \[coro:sufficiency\_uniform\_constants\], statement (i), Proposition \[prop:A\_perturbation\], and Remark \[remark:A\_perturb\].
For illustration, consider the set ${\mathcal P}:=\{ x \in \mathbb R^N \, | \, \| x - \mathbf e_1 \|_2 \le 1\}$, which is a convex set containing the zero vector. As indicated right after the proof of Proposition \[prop:conic\_hull\], ${\mathcal P}$ is not CP admissible but the closure of its conic hull is given by $\mathcal C=\mathbb R_+ \times \mathbb R^{N-1}$ and is thus CP admissible. Suppose $ 1-\delta_{K,\mathcal C}> \sqrt{K} \cdot \theta_{K,\mathcal C}$. Then by Corollary \[coro:non\_CP\_admissible\], condition $(\mathbf H)$ holds on ${\mathcal P}$. Another example is a convex set whose interior contains the zero vector. In this case, the closure of its conic hull is $\mathbb R^N$ for which a similar sufficient condition in terms of $\delta_{K,\mathcal C}$ and $\theta_{K, \mathcal C}$ with $\mathcal C=\mathbb R^N$ can be established.
It is interesting to ask whether the sufficient condition $1-\delta_{K,{\mathcal P}}> \sqrt{K} \cdot \theta_{K,{\mathcal P}}$ derived in Theorem \[thm:sufficency\_non\_cone\] for condition $(\mathbf H)$ can be improved using similar techniques for the cone case given in the proof of Theorem \[thm:sufficency\_cone\_case\]. In spite of many tries, our efforts show that these techniques do not yield better (i.e., less restrictive) sufficient conditions in terms of $\delta_{K,{\mathcal P}}$ and $\theta_{K,{\mathcal P}}$. Although this finding does not rule out the possibility of the existence of better sufficient conditions in terms of $\delta_{K,{\mathcal P}}$ and $\theta_{K,{\mathcal P}}$ because it only gives certain sufficient conditions, it demonstrates a potential difficulty of further improving the obtained sufficient conditions using the same line of ideas given in the proof of Theorem \[thm:sufficency\_cone\_case\]. It also justifies the importance of the sufficient conditions in terms of $\delta_{K,\mbox{cone}({\mathcal P})}$ and $\theta_{K,\mbox{cone}({\mathcal P})}$ in Theorem \[thm:sufficency\_non\_cone\].
Conclusions {#sect:conclusion}
===========
This paper studies the exact support and vector recovery on a constraint set via constrained matching pursuit. We show the exact recovery critically relies on a constraint set, and introduce the class of CP admissible sets. Rich properties of these sets are exploited, and various exact recovery conditions are developed for convex CP admissible cones or sets. Future research includes the exact recovery of constrained sparse vectors subject to noise and errors via constrained matching pursuit.
[**Acknowledgements.**]{} The authors would like to thank Dr. Joel A. Tropp for a helpful discussion on the counterexample given in Section \[subsect:OMP\_counterexample\].
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[^1]: Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, MD 21250, U.S.A. Emails: shenj@umbc.edu and smousav1@umbc.edu.
[^2]: In a private communication, Dr. Joel A. Tropp pointed out to the authors that this issue may be related to the borderline case indicated in Footnote 2 in his paper [@Tropp_ITI04].
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We have obtained deep $B$ and $R$-band CCD photometry for five dwarf elliptical galaxies that were previously identified on Schmidt films covering the region of the Centaurus A (CenA) group. From a Fourier analysis of the $R$-band CCD images we determined the surface brightness fluctuation (SBF) magnitude $\bar{m}_R$ for each stellar system. All magnitudes are similar and, given the small colour spread, suggest that these low surface brightness galaxies lie approximately at the same distance, regardless of the assumed SBF zero point. Long-slit spectra have been acquired to derive redshifts for two of the dwarfs, ESO269-066 and ESO384-016. The velocities, $v_\odot=784$and $v_\odot=561$, respectively, identify them unambiguously as CenA group members. An age (H$\delta_{\mbox{A}}$) – metallicity (C$_2 \lambda$4668) analysis of the spectra reveals an underlying old and metal-poor stellar population in both cases. Combining photometric and spectroscopic results we find strong evidence that indeed all dwarf galaxies are CenA group members.
Based on Cepheid, TRGB, and PNLF distances published for the two main CenA group galaxies NGC5128 and NGC5253, we adopted a mean group distance of 3.96Mpc to calibrate the apparent fluctuation magnitudes. The resulting [*absolute*]{} SBF magnitudes $\bar{M}_R$ of the dEs correlate with the dereddened colours $(B-R)_0$ as predicted by Worthey’s stellar synthesis models using the theoretical isochrones of Bertelli and collaborators. This good agreement allows a calibration of the SBF method for dwarf ellipticals in the colour range $0.8<(B-R)_0<1.5$. However, two branches of stellar populations appear in the $\bar{M}_R$-colour plane, and care has to be taken to decide which branch applies to a given observed dwarf. For dwarfs with $(B-R)_0<1$ there is very little colour dependence ($\bar{M}_R \approx -1.2$), in accord with our previous SBF analysis of faint, blue Sculptor group dEs. For red dwarfs, $(B-R)_0>1.2$, the $\bar{M}_R$-colour relation is steep, and accurate colours are needed to achieve SBF distances with an uncertainty of only 10%. One of the dwarfs, ESO219-010, is located slightly behind the core of the CenA group at $\sim$4.8Mpc, while the remaining four recover the mean group distance of 3.96 Mpc that was put into the calibration. The depth of the group is only 0.5Mpc which identifies the CenA group as a spatially well isolated galaxy aggregate, in contrast to the nearby Sculptor group.
author:
- 'H. Jerjen'
- 'K.C. Freeman'
- 'B. Binggeli'
title: Testing the Surface Brightness Fluctuations Method for Dwarf Elliptical Galaxies in the Centaurus A Group
---
INTRODUCTION
============
Dwarf elliptical (dE) galaxies are the most numerous type of galaxies in the nearby universe. Their existence is tightly correlated with the effective galaxy density of the environment, i.e. dEs reside predominantly in cluster cores (Binggeli et al. 1985; Ferguson 1989; Ferguson & Sandage 1990; Jerjen & Dressler 1997) or accompany giant galaxies as satellites in the field (Binggeli, Tarenghi & Sandage 1990). This naturally renders dEs as ideal objects to estimate distances to all sorts of galaxy aggregates. However, little advantage could be taken of the dE clustering properties to date, mainly because accurate distance measurements to dEs still rely on the resolution of the galaxy light into stars, such as any method based on the colour-magnitude diagram (CMD, e.g. Smecker-Hane et al. 1994; DaCosta et al. 1996; Stetson 1997), or the tip of the red giant branch (TRGB, Harris et al. 1998; Grebel & Guhathakurta 1999). This requirement makes observations expensive, restricting the practicability essentially to local systems.
An alternative, very promising distance estimator for dEs is based on “surface brightness fluctuations” (SBF). The theoretical framework of the SBF method was introduced and discussed by Tonry & Schneider (1988). The idea is to quantify the pixel-to-pixel brightness variation on a CCD image of a gas-free stellar system due to the varying numbers of stars within each pixel. At a constant surface brightness (star density) this variation is then inversely proportional to the distance of the galaxy. In practice, one has to measure the so-called apparent fluctuation magnitude $\bar{m}_X$ in the photometric passband $X$. This quantity is the ratio of the second and first moments of the underlying stellar luminosity function. If sufficient information is known about the stellar content of a galaxy type, the absolute fluctuation magnitude $\bar{M}_X$ can be predicted from stellar synthesis models or empirically determined from calibrators, thus yielding a distance modulus of the galaxy.
So far SBF applications concentrated on high-surface brightness giant ellipticals (Tonry, Ajhar & Luppino 1989, 1990; Tonry & Schechter 1990; Pahre & Mould 1994; Thomson et al. 1997; Tonry et al. 1997), bulges of spiral galaxies (Tonry 1991; Luppino & Tonry 1993), and globular clusters (Ajhar & Tonry 1994). The first attempt to obtain SBF distances for [*dwarf*]{} ellipticals was made by Bothun et al. (1991), followed only recently by our study of nearby dEs (Jerjen, Freeman, & Binggeli 1998, hereafter JFB98) located in the outskirts of the Local Group (LG). In the latter study, the $R$-band SBF distances for five dEs in the Sculptor group were measured to an internal accuracy of 5–10%. The difficulty with the SBF method for dwarfs is the inherent difference in the stellar populations of giant and dwarf ellipticals, preventing the adoption of existing calibration values from previous work on giants. To bypass the lack of empirical results, we based the SBF distances on the zero point predictions of stellar synthesis models (Worthey 1994). As it turned out, the calibrating $\bar{M}_R$ is fairly insensitive to the star formation history for these very faint and blue Sculptor dEs. The resulting distances showed that the dwarf galaxies closely follow the spatial distribution of the main Sculptor group galaxies, which are spread over a distance range between 1.7Mpc and 4.4Mpc.
However, this large line-of-sight extension of the Sculptor group is clearly not ideal for the empirical calibration of the SBF method. To improve this situation, we continue our SBF project here with a study of five dE galaxies identified in CentaurusA (Cen A) group region. There are two major advantages of working in the CenA group rather than the Sculptor group. Firstly, the CenA group is physically and spatially much better defined than the Sculptor group. The Sculptor aggregate is, in fact, an unbound and almost freely expanding “cloud” of galaxies (see JFB98), while CenA is certainly a rich and well concentrated group with a small distance dispersion of the member galaxies. Secondly, the CenA group’s mean distance is well established via Cepheids and tip of the red giant branch magnitudes measured for the two main group galaxies, NGC5128 (= CentaurusA) and NGC5253.
Owing to their greater mean distances, the CenA group dwarf Es are also intrinsically brighter on average than the Sculptor dwarfs. Thus in this paper we also extend the calibration of the SBF method to brighter magnitudes. As it turns out, for brighter dEs the calibration is somewhat more difficult. Brighter dwarfs are on average also [*redder*]{} than faint ones, and for red objects, according to the stellar synthesis models, the calibrating $\bar{M}_R$ depends rather strongly on the colour. Moreover, in a certain colour range the calibration is not unique. However, we show that by a proper treatment of the $\bar{M}_R$ - colour relation(s) and with reliable colour data, accurate SBF distances can be derived also for brighter dEs.
The paper is organized as follows. In §2 we describe the observations and data reduction procedures. This includes the spectroscopy of two of the five sample dEs, for which radial velocities as well as ages and metallicities are derived. §3 is the SBF analysis resulting in the apparent fluctuation magnitudes of the galaxies. In §4 these fluctuation magnitudes are then calibrated by means of stellar synthesis models which are critically discussed in terms of the $\bar{M}_R$ - colour dependence. Summary and conclusions are given in §5.
OBSERVATIONS AND DATA REDUCTION
===============================
As reported elsewhere (Jerjen, Binggeli, & Freeman 2000, hereafter JBF00), 13 low surface brightness dwarf galaxies with dE morphology were identified in the region of the CenA group. They have been detected as part of a visual inspection of $\approx 50$ fine-grain IIIa-J emulsion SRC Schmidt films covering the group’s area: $12^h 30^m < \mbox{R.A.(1950)} < 15^h,\, -20^\circ
< \mbox{Decl.(1950)} < -50^\circ$. For the present SBF study we selected a subsample of five dEs. These galaxies have sufficiently large isophotal diameters providing enough independent measure points for the SBF analysis and their profiles are mostly unaffected by the disturbing light of bright nearby stars. In Figure \[fig9\] we show the distribution of the sample dEs on the sky.
Imaging
-------
The photometric data for the five dwarf galaxies were taken with the 2.3m ANU telescope at Siding Spring Observatory in the dark time period 1997 April 9–11. The CCD camera was mounted at the Nasmyth focus B. On the first night the detector was a 2k$\times$1k EEV thick CCD with 22$\mu$m pixels and a pixel scale of 0.55$''$. On the remaining two nights we used a 2k$\times$800 SITe thinned CCD, AR coated, 15$\mu$m pixels, and a pixel scale of 0.375$''$. The CCD gain was set to 1$e^-$ ADU$^{-1}$ for all exposures. The field of view is mechanically restricted to a $6'.7$ diameter circle.
All three nights were photometric and seeing ranged from 1.1 to 1.4$''$ FWHM. A series of 4–5 $R$-band images of 600–720sec duration were taken for each galaxy yielding a total exposure time of 2400–3600sec per object. Such long integration times are required to allow the SBF signal from a low-surface brightness galaxy to grow and surpass the high level of sky shot noise present in the power spectrum. This noise level continuously increases when working with redder photometric passbands and is rather high in $I$ and $K$, the favourite filters for SBF studies on giant ellipticals. Even though the fluctuation signal also gets stronger this only partially compensates for the brighter sky, thus the Cousins $R$-band appears to be the best compromise for the work on dwarf ellipticals. Furthermore, with $R$-band images we avoid fringing effects which occur with thinned CCDs beyond 7000Å and which can severely affect the weak fluctuation signal.
The $R$-band observations of a galaxy were supplemented with a series of $6\times$600sec $B$ band images to get colour information. Besides the science images we took high signal-to-noise twilight flats every dusk and dawn period and photometric standards in the E regions (Graham 1982) throughout the nights.
The CCD data were reduced with standard IRAF procedures. The bias was subtracted from all of the images, and we used the combined twilight flats to flatten the science images. On each frame we determined the sky level from 10–15 selected star-free regions well away from the galaxies. The rms scatter showed that all images were flat to $0.1-0.2$% over the full frame. Each set of sky-subtracted images of a galaxy was registered and median combined to generate a composite image. The calibration of these master images was carried out with aperture photometry of the standard stars. Uncertainty in the photometric zero point was measured at 0.02 mag.
Figures \[fig1\] and \[fig2\] show the $R$-band images of the five dEs. Among early-type dwarf galaxies there is some morphological variety (see Sandage & Binggeli 1984). At the bright end of the dE luminosity function, there is a variant that was named dS0, because it is often distinguished by a S0-like, two-component structure (see also Binggeli & Cameron 1991). ESO384-016 is clearly of this type, ESO269-066 arguably so. Bright dwarf ellipticals also host quite often an unresolved, centrally located star-like object, which is possibly a massive globular cluster formed in or fallen into the core region of the galaxy. ESO219-010 and ESO269-066 are the galaxies that show qualitative evidence for such a nucleus. However, a star projected onto the galaxy centre by chance cannot be ruled out in either case; thus a “:” goes with the classifications.
In Table \[tbl-1\] we list the fundamental parameters of the dwarf galaxies: the morphological type (col. \[2\]), coordinates (cols.\[3-4\]), the total $B$ and $R$ magnitudes (cols. \[5-6\]), the effective radius $r_{eff,R}$ containing half of the total light (col. \[7\]), and the mean effective surface brightness $\langle
\mu \rangle_{eff,R}$ (col. \[8\]). A full account of the observations including a discussion of the radial surface brightness profiles, colour gradients, and structure parameters of the galaxies is given in JBF00. The dereddened overall $(B-R)^T$ colours (col. \[10\]) were derived from cols.\[5-6\] and corrected for foreground extinction using the IRAS/DIRBE maps of dust IR emission (Schlegel et al. 1998) and the ratio $A_B: A_R : E(B-V)=4.315: 2.673: 1$. According to these authors the extinction estimates (col. \[9\]) have an accuracy of $16$%. The observed colour range $1.1< (B-R)_0^T < 1.5$ (col. \[10\]) covered by our CenA group candidates is redder than for the Sculptor dEs (JFB98) and again is typical for dEs whose $B - R$ values range from 0.5 to 2.3 (Evans et al. 1990).
Spectroscopy
------------
The two sample galaxies with the highest surface brightness were selected for a spectroscopic follow-up to estimate their ages and metal abundances and to measure their redshifts. We obtained two spectra of ESO269-066, on April 19, 1996 and two spectra of ESO384-016 on February 2, 1997, at the Nasmyth A focus of the 2.3m ANU telescope using a 600 line grating at the blue side of the double-beam spectrograph. Each exposure was of 2000sec duration. The detector was a SITe 1752$\times$532 thinned CCD with an across dispersion of $0.91''$pixel$^{-1}$ and a spectral resolution of 1.1Å pixel$^{-1}$. The grating angle was set to cover the wavelength range $3500$Å$-5500$Å. We acquired the data using a long slit ($2''\times 6'.7$) under good seeing conditions. The slit was positioned at the galaxy centre and aligned along the major axis.
We reduced the CCD data in a straightforward manner with standard IRAF procedures. After the bias level was removed, each image was flatfielded and sky line subtracted. Wavelength calibration was performed using emission line spectra of a Ne-Ar lamp observed immediately after each galaxy spectrum. The combined galaxy spectra are plotted in Figure \[fig3\]. Both dwarfs are pure absorption line systems with prominent Balmer lines (H$_\beta$ $\lambda 4861$, H$_\gamma$ $\lambda 4340$, H$_\delta$ $\lambda 4101$) and a H and K doublet ($\lambda 3968$, $\lambda 3934$). All lines are slightly stronger in the spectrum of ESO384-016.
The age and metallicity of the dwarf galaxies were estimated (L. A. Jones 1999, private communication) by comparing the observed index strengths of the lines C$_2\lambda$4668 (Worthey 1994) and H$\delta_{\mbox{A}}$ at $\lambda 4102.9$ (Jones & Worthey 1995) with the index strengths computed from single-burst population models. The C$_2\lambda$4668 line is a blend from C$_2$, Mg, Fe, and other elements and is considered as a very good metallicity indicator with a 2–3 times better orthogonal separation of age and metallicity than the most commonly used Mg$_2$ index (Jones & Worthey 1995). On the other hand, H$\delta_{\mbox{A}}$ is relatively metallicity-insensitive, thus useful in determining age. Furthermore, H$\delta_{\mbox{A}}$ is less affected by emission from ionised hydrogen than for instance H$\beta$ and less diluted by light from giant stars.
In Figure \[fig4\] we show the location of the dwarf elliptical galaxies in the H$\delta_{\mbox{A}}$ versus C$_2$4668 (age–metallicity) diagram. Both galaxies appear in the old, metal-poor part of the diagram where crowding in the parameter space leads to poor resolution. ESO384-016 seems to be slightly more metal-rich and younger relative to ESO269-066. However, a quantification is impossible due to large uncertainties. Also, it has to be borne in mind that the assumed single-burst model need not be correct.
To determine the redshifts of the dwarfs, their spectra were continuum-subtracted and cross-correlated with the spectrum of the velocity standard HD176047, a K1III giant. A heliocentric velocity of v$_\odot=561(\pm32)$ was measured for ESO384-016, which is in good agreement with velocities observed for well-known CenA group galaxies such as NGC5128 (v$_\odot$=562) or NGC5236 (v$_\odot$=516). For ESO269-066 we measured v$_\odot$=784($\pm$31) which lies at the high end of the velocity distribution of the CenA group. The most complete catalogues of CenA group galaxies (Côté et al. 1997, hereafter C97; Banks et al. 1999) list three galaxies with comparable velocities: SGC1259.6-1659 (v$_\odot$=732), DDO161 (v$_\odot$=747), and AM1331-451 (v$_\odot$=831). A fourth galaxy, AM1318-444 ($13^h$ $21^m$ $47^s\!.1$, $-45^\circ$ $03'$ $49''$ J2000.0) with v$_\odot$=741, was reported just recently (JBF00). Assuming D=4Mpc, the small distance and velocity differences between ESO269-066 and AM1331-451 ($250$kpc, $\Delta$v=47) or AM1318-444 ($100$kpc, $\Delta$v=$-43$) may indicate a small subclump within the group (see Figure \[fig9\]).
Overall, the observed redshifts provide clear evidence that ESO269-066 and ESO384-016 are indeed CenA group members. This picture is further qualitatively supported by the facts that (i) the CenA group is well isolated in velocity space (Tully & Fisher 1987; C97) i.e. has no overlap in velocity space with a nearby background galaxy aggregate, and that (ii) dwarf ellipticals found in low density regions are almost exclusively close companions of massive galaxies (Binggeli, Tarenghi & Sandage 1990; Ferguson & Binggeli 1994; JFB98). A quantitative confirmation of these kinematic distance estimates is of course provided by the following SBF analysis.
SBF ANALYSIS
============
The deep $R$-band CCD master images of our galaxies were prepared according to the procedure described in TS88 and JFB98. Point sources were identified with the DAOPHOT routines (Stetson 1987) and the affected CCD areas were replaced by nearby uncontaminated patches of the galaxy from the same surface brightness range. The isophotes of the cleaned galaxy were modelled with the ISOPHOT package within STSDAS by fitting ellipses with variable radius, ellipticity, and position angle. The mean galaxy brightness distribution was subtracted from the master image. The residual image was then divided by the square root of the mean profile to produce the fluctuation image. This last step normalizes the strength of the pixel-to-pixel variation over the galaxy’s surface due to differences in surface brightness (stellar density).
On each fluctuation image, two subimages (F1 and F2) were selected within the 25.5 $R$magarcsec$^{-2}$ isophote of the galaxy for the SBF analysis. The only exception was ESO219-010 where superimposed foreground stars and the relative small size of the galaxy allowed us to work with only one subimage. The size of a subframe was between $40\times 40$ and $70\times 70$ pixels. Under the given seeing conditions this area corresponds to 900–4000 independent points carrying the SBF fluctuation signal. The overlap of subimages was kept minimal ($<5$%) to get two independent distance measurements for a galaxy. Objects previously identified with DAOPHOT on a subimage were replaced by a randomly selected area from the region outside of the subimage but in the same isophotal range. The fraction of pixels patched in this way was less than 5% of the total subimage area. All cleaned fluctuation images were Fourier-transformed and the azimuthally averaged power spectrum calculated (Figures \[fig5\] and \[fig6\]). From well isolated bright stars on the master images we derived the power spectrum of the PSF. We then fitted a linear combination of the power spectrum of the PSF and a constant to the the galaxy power spectrum, $$\mbox{PS}_{\mbox{gal}}(k)=P_0\cdot \mbox{PS}_{\mbox{PSF}}(k) + P_1.$$ Low spatial frequencies ($k<4$) which can be affected by the galaxy subtraction were excluded from the fit. The results are listed in Table \[tbl-2\]. From the amplitude $P_0$ (col. \[7\]) we calculated the apparent fluctuation magnitude (col. \[9\]) with the formula $\bar{m}_R=m_1-2.5\cdot\log(P_0/t)$. Thereby, the quantity $m_1$ (col. \[2\]) is the magnitude of a star yielding 1 ADU per second on the CCD and $t$ (col. \[3\]) is the integration time of a single exposure. $P_1$ (col. \[8\]) is the scale-free (white-noise) component in the power spectrum which is proportional to the ratio of the sky (col. \[6\]) and the galaxy surface brightness (col. \[5\]) averaged over the subimage. To keep the white-noise level low it is thus crucial to restrict the SBF analysis to the high-surface brightness parts of the galaxy. The combination of $n$ exposures reduced the white-noise level by a factor of $\sqrt{n}$.
The overall error of the apparent fluctuation magnitude $\bar{m}_R^0$ is dominated by the power spectrum fitting error which contributes 3–10%. Applying the SBF method to subimages of relatively high surface brightness ($\bar{g}$, col. \[5\]) helped to keep the error which is due to the uncertainty in the sky determination (col. \[6\]) small, i.e. 3% or less. Errors from PSF normalisation and the shape variation of the stellar PSF over the CCD area are equally small (1–3%). In Column 7 of Table \[tbl-2\] we list the $P_0$ values with the combined error from all these sources. Assuming a photometric accuracy of $\Delta m_1=0.02$mag (col. \[2\]) and adopting a 16% error for foreground extinction (Schlegel et al. 1997), the formal internal uncertainty in $\bar{m}_R^0$ is between 0.05 to 0.14mag (col. \[11\]). We like to stress that the two narrow but distinct magnitude ranges covered by $m_1$ are due to the use of two different CCD detectors with different zero points (see section 2.1).
CALIBRATION OF FLUCTUATION MAGNITUDES AND DISTANCES
===================================================
Theoretical stellar populations
-------------------------------
First let us recall that most faint dEs are not single-burst populations like globular clusters (Da Costa 1997). A rather diverse and complex set of star formation histories (SFH) is observed among the local dwarf spheroidals. Their stellar populations range from old (Ursa Minor) and mainly old (e.g. Tucana, LeoII) through intermediate-age episodic SFH (e.g. Carina, LeoI) to intermediate-age continuous SFH (e.g. Fornax). Phoenix and LSG3 are classified as dE/Im, because they show similarities to both dwarf spheroidals and dwarf irregulars. These systems are dominated by an old metal-poor population with no evidence for [*major*]{} star formation activities after the initial episode 8-10 Gyr ago. However, both systems have a minor population of young stars, with ages of about 150 Myr, which makes these galaxies resemble dwarf irregulars.
To explore the age- and metallicity-dependency of $\bar{M}_R$ for dE-like stellar populations we employed Worthey’s on-line model interpolation engine [^1]. We considered separately the two offered sets of isochrones: (1) the amalgamation of the stellar evolutionary isochrones of VandenBerg and collaborators and the Revised Yale Isochrones (Green et al. 1987; hereafter RYI) as described in Worthey (1994), and (2) the Padova isochrone library (Bertelli et al. 1994; hereafter PI).
We computed $\bar{M}_R$ and $(B-R)$ values for the following 2-component stellar populations. The age of the main population was set to 17, 12, or 8 Gyr and its metallicity set in the range $-2.0$[^2], $-1.9$, ...,$-1.0$, $-0.5$, $-0.25$, $0$. The second population, 5 Gyr old and of solar metallicity, contributed in weight to the whole population at the 0 ($\sim$ pure case), 10, 20 or 30% level. In all cases we assumed a Salpeter IMF. The results of the models are illustrated in Figure \[fig7\] where we have plotted $\bar{M}_R$ versus $(B-R)$. The models are separated according to the underlying isochrone library. In each case, two distinct branches are visible. However, only one branch is well defined, while the other, though recognisable, is only sparsely populated. We will call the steep linear and shallow quadratic component the red and blue branch, respectively, for reasons given below.
In order to understand the significant differences between the predictions of the two sets of isochrones for $(B-R)<1.3$ one has to recall that $\bar{M}_R$ corresponds to the luminosity-weighted luminosity of the underlying stellar population. This quantity is roughly the luminosity of a giant star, thus critically depends on the post red giant branch (RGB) evolution. While the theoretical isochrones of Bertelli et al. (1994) are the most complete set available with a full RGB evolution implemented, this part of the stellar evolution is missing in the RYI. Worthey assumed in his RYI-based models the horizontal branch to remain in a red clump near the giant branch, which is incorrect for metal-poor populations. Because of these limitations we will focus in the following on the results of the PI based models.
A few words on the interpretation of the PI colour-fluctuation magnitude diagram. Da Costa (1997) pointed out that (i) the colour of the red giant branch is largely independent of age, but is strongly dependent on metallicity (his Figure1) and (ii) that for a fixed metallicity the colour of the horizontal branch has a discontinuity from blue for the oldest populations ($\sim 15$Gyr) to red at younger ages ($\leq 10$Gyr, his Figure4). The latter effect is more marked for lower metallicities. These two trends describe very accurately the behaviour of $\bar{M}_R$ above and below $(B-R)=-1.3$ (or \[Fe/H\]$\sim-1.0$) as we illustrate in Figure \[fig8\]. In the models, only the very old (17Gyr) and metal-poor (\[Fe/H\]$<-1.0$) populations are located on the blue branch. With an increasing fraction of younger stars the total population gets redder and the model values reach quickly the red branch.
To quantify the location of the two branches, we deduced two analytical expressions for the locus of theoretical values by least-squares fitting to the model data: $\bar{M}_R=6.09\cdot (B-R)_0-8.78$ for the red, and $\bar{M}_R=1.89\cdot
[(B-R)_0-0.77]^2-1.23$ for the blue branch, respectively.
Observed stellar populations
----------------------------
The observed $\bar{m}_R^0$ magnitudes (col. \[11\], Table \[tbl-2\]) occupy a small range with $\sigma=0.3$mag which, given the small colour range, implies roughly the same distance for our sample galaxies independently from any assumption on the zero point of the SBF distance estimator. Furthermore, we found quantitative evidence in §2.2 that two of the dEs are CenA group members according to their redshifts. This immediately suggests that indeed [*all*]{} sample dEs are CenA group galaxies. Based on this assumption we can investigate the empirical calibration of the SBF method for dEs.
In Table \[tbl-4\] we compiled accurate distances of two main CenA group galaxies from the literature. Three distances are published for NGC5128: Harris et al. (1999) measured $D=3.9(\pm0.3)$Mpc based on the tip of the red giant branch (TRGB) magnitude. A distance from the planetary nebulae luminosity function (PNLF) of $D=3.9(\pm0.3)$Mpc was given by Hui et al. (1993), and $D=3.6(\pm 0.2)$Mpc was obtained by Tonry & Schechter (1990) with the SBF method for giant ellipticals. The two latter values have been adjusted to the presently favoured LG distance scale set by $(m - M)_0$(M31) = 24.5 (van den Bergh 1995; Fernley et al. 1998; Harris 1999). A mean value of $D=3.8(\pm0.1)$Mpc \[(m–M)=$27.90\pm 0.05$\] is taken as the true distance of NGC5128. The second galaxy is NGC5253. Saha et al. (1995) reported a distance modulus of (m–M)$=28.08(\pm 0.10)$ derived from Cepheids. NGC5128 and NGC5253 are the only CenA group galaxies with accurate distances to date, which makes the estimation of the group distance difficult in principal. However, their redshifts (col. \[2\], Table \[tbl-4\]) are close to the mean group velocity at v$_\odot$=551(C97) and we adopt their average distance modulus (m–M)$=27.99(\pm0.17)$, or $D$ = 3.96 Mpc, for the whole CenA group.
We note that more distance determinations are available for these two galaxies (e.g. Della Valle & Melnick 1992; Shopbell, Bland-Hawthorn & Malin 1993; Soria et al. 1996) as well as for other group members (e.g. De Vaucouleurs 1979). However, these results have been superseded by newer data, or are less secure.
Using the independent CenA group distance we converted the fluctuation magnitude of each SBF field (col. \[4\], Table \[tbl-3\]) into an absolute magnitude. The results are shown in Figure \[fig7\] whereby the colour (col. \[3\], Table \[tbl-3\]) was determined for each SBF field individually because integrated colours as listed in col. \[10\] of Table \[tbl-1\] are not reliable due to observed colour gradients (see JBF00). The estimated error in the local $(B-R)_0$ is composed of the sky uncertainty in the $B$ and $R$-band images, the photometric error, and an extinction error of 16% (Schlegel et al. 1998).
Theory vs. observations and the final calibration
-------------------------------------------------
A comparison between observations and theoretical models based on PI (Figure \[fig7\], [*right panel*]{}) reveals a remarkable agreement. All dEs (except ESO219-010 which will be discussed separately below) are close to the locus of theoretical values. Each branch exhibited by the models is populated with the data of at least one sample galaxy. This confirms the previous assumption that the dEs are CenA group members, and that the value of the adopted CenA group distance modulus of 27.99 is approximately correct for the dwarfs. It is important to note that the assignment of an individual galaxy to one of the two model branches may be less problematic than it first appears. As we have data for two (ideally more) SBF fields per galaxy with a significant colour difference, we can decide on the basis of colour slope. For instance, between the two fields of ESO384-016, we measured $\Delta \bar{M}_R=0.08(\pm0.08)$mag and $\Delta (B-R)_0=0.20(\pm0.06)$mag. This clearly indicates that ESO384-016 belongs on the blue branch. Due to smaller differences in colour the situation for ESO269-066, AM1339-445, and AM1343-452 is less clear. One could argue for each of them that it is actually a member of the blue branch and that the offset from the model values reflects the relative distance from the assumed group centre. However, this explanation appears highly unlikely as all three galaxies would by chance have to fall at the right place on the red branch.
We prefer to interpret the results in Figure \[fig7\] [*right panel*]{} as a convincing demonstration how well theoretical models of mainly old, metal-poor stellar populations can predict the colour dependence of $\bar{M}_R$. The systematic offset between the dE data and the analytical expressions that describe the two branches (see above) is $-0.03$mag with a scatter of $\sigma=0.16$mag. This scatter accounts for genuine differences between models and observations as well as for the depth effect among the four dEs. Taking into account the zero point from the empirical data and the shape of the relation from the models we can formulate a semi-empirical calibration of the SBF method as distance indicator for dEs:
$$\begin{aligned}
\bar{M}_R = 6.09\cdot (B-R)_0-8.81\end{aligned}$$
for the red, steeply rising branch in the colour range $1.10<(B-R)_0<1.50$, and
$$\begin{aligned}
\bar{M}_R = 1.89\cdot [(B-R)_0-0.77]^2-1.26\end{aligned}$$
for the blue branch in the range $0.80<(B-R)_0<1.35$. The decision which of the formulas is appropriate for a galaxy remains subject to the observed trend of $\bar{m}_R$ as a function of $(B-R)$ colour.
We note that for fairly blue old galaxies, say with $(B-R)_0<1.0$, there is almost no colour dependence of $\bar{M}_R$. For $(B-R)_0$ = 1.0 and 0.8, equation (3) gives $\bar{M}_R$ = $-1.16$ and $-1.26$, respectively. This is in accord with $\bar{M}_R \approx$ $-1.15$ adopted in our previous paper (JFB98) for the blueish Sculptor group dwarf spheroidals.
Resulting distances
-------------------
We are now in a position to derive individual distances for the dwarfs. First, by using formulas (2) and (3) we have calculated the calibration constant for each SBF field, as listed in col. \[6\] of Table \[tbl-3\]. In col. \[5\] of that table we also give the “observed” $\bar{M}_R$ based on $m-M$ = 27.99, as plotted in Figure \[fig7\]. The errors in the calibrating $\bar{M}_R$ are simply the propagated colour errors given in col. \[3\] of Table \[tbl-3\]. ESO219-010 and 384-016 have been assigned to the blue branch (equation 3); the three other dwarfs to the main, red branch (equation 2). The errors in $\bar{M}_R$ are accordingly much smaller for the former than the latter, as the red branch is quite steep. [*Assuming no error in the models and in the assignment to them*]{}, we now take any difference between $\bar{M}_R$(obs) and $\bar{M}_R$(calib) as a real difference in the distance between the object and the adopted mean group distance of $(m-M)$ = 27.99. The individual distance moduli for the dwarf fields, listed in col. \[7\] of Table \[tbl-3\], follow then simply by combining $\bar{m}^0_R$ (col. \[4\]) and $\bar{M}_R$(calib) (col. \[6\]). Finally, the corresponding distances for the fields are averaged for each galaxy (where two fields are available), giving the error-weighted mean distances in Mpc listed in col. \[8\] of Table \[tbl-3\]. The distance errors vary between 3 and 9% for the two blue branch dwarfs, and 13 to 19% for the three red branch dwarfs.
The distance of ESO219-010, of 4.79 Mpc, is significantly larger than the distances of the remaining four dwarfs (if assigned to the red branch in Figure \[fig7\], the distance would even be 7.3 Mpc). While its CenA group membership is not questioned, this galaxy is likely lying outside the core region of the group, and it should be excluded for a calculation of the mean group distance. The four remaing dEs have a mean distance of 4.00 Mpc (m$-$M = 28.01), which by chance exactly coincides with the pre-assumed group distance inferred from the well-determined distances of NGC5128 and NGC5253. The small 1$\sigma$ distance dispersion of 0.20 Mpc \[or $\sigma (\rm m-M)$ = 0.10 mag\] is not only a confirmation of the previous assumption that the CenA group is well defined and concentrated, but again shows that the SBF method for dwarf ellipticals works quite well in this case. In Figure \[fig9\] we show the sky distribution of the CenA group main members and the positions of the five dEs. We note that ESO219-010 lies near the edge of this distribution, again suggesting that this is an outer member of the group. It is tempting to associate ESO384-016 with the galaxy pair NGC5253/5236, and to see ESO269-066 as a companion of AM1331-451. However, more distance and velocity information for the remaing galaxies is needed to unambiguously identify any subclumps, if present, in the CenA group.
SUMMARY AND CONCLUSIONS
=======================
This is the second study on the surface brightness fluctuation method as useful distance indicator for low surface brightness dwarf elliptical galaxies. In a first paper (JFB98) we have applied the SBF method to dwarf spheroidal members of the nearby Sculptor group. The Sculptor dwarfs (with one exception) are very faint ($M_B>-11$) and blue \[$(B-R)_0^T<1.1$\]. As with LG dwarf spheroidals (daCosta et al. 1996), many of them show signs of recent star formation. This forced us to look, by means of stellar synthesis models provided by Worthey (1994), for any variation of the calibrating $\bar{M}_R$ with age and metallicity of the underlying stellar population. Very fortunately, [*for the faint, blue, and metal-poor*]{} Sculptor dwarfs this variation turned out to be minimal, and a preliminary calibration constant of $\bar{M}_R \approx -1.15$ was adopted.
The present paper was intended to improve the SBF calibration and to extend it to brighter magnitudes. We have carried out $B$ and $R$ CCD photometry of five dE galaxies, AM1339-445, AM1343-452, ESO219-010, ESO269-066, and ESO384-016, suspected members of the CentaurusA group. The SBF analysis of their $R$-band images revealed similar fluctuation magnitudes suggesting the same approximate distance to the Local Group (given their similar colour). The missing link for their association to the CenA group is provided by the spectroscopic redshifts we measured for ESO269-066 and ESO384-016. Both dwarfs have heliocentric velocities in good agreement with velocities found for main CenA group members.
For the calibration of the measured apparent fluctuation magnitudes we employed a mean distance modulus for the CenA group of 3.96 Mpc ($m-M$ = 27.99 mag) based on reliable (Cepheid, TRGB, PNLF) distances to NGC5128 and NGC5253. Again we had to study the dependence of the resulting absolute fluctuation magnitudes $\bar{M}_R$ on age and metallicity, i.e. in practice on the $(B-R)_0$ colour of the underlying stellar population, by drawing on the on-line tool provided by Worthey (see footnote 1).
Here now comes the difference from the Sculptor dwarfs in our first paper. The CenA group dwarfs are more distant and hence on average brighter than the Sculptor dwarfs. As a result, owing to a general colour-luminosity relation (e.g., Ferguson 1994, Secker et al. 1997), the CenA dEs are also significantly redder, lying in the colour range $1.0<(B-R)_0<1.3$. For these red colours the stellar population synthesis models show a strong dependence of $\bar{M}_R$ on $(B-R)_0$. Moreover, two branches are populated by the models in the $\bar{M}$-colour plane. One branch is more populated by models based on Revised Yale Isochrones, the other by models based on Padova Isochrones (Bertelli et al. 1994). We have argued that the Revised Yale Isochrones are unreliable for the purpose of predicting $\bar{M}_R$ magnitudes for old, metal-poor, and sufficiently red stellar populations due to the lack of a full post red giant evolution. That the present dEs are dominated by old, metal-poor stars is indeed confirmed by the absorption line spectra of ESO269-066 and ESO384-016.
Aside from theoretical considerations, it was found that the fluctuation magnitudes, when forced to a mean distance of 3.96 Mpc, are surprisingly well matched by the model curves. An exception is ESO219-010 which seems to be slightly more distant than the core of the CenA group. We also found that the assignment of a particular galaxy to one of the two branches is fairly unambigous, especially if more than one subfield per galaxy is analysed. In this way, by assuming that the scatter around the model curves, given by equations (2) and (3) is entirely due to a dispersion in the individual distances (depth effect), we achieved SBF distances with an accuray of 3 to 9% for the two dwarfs belonging to the “blue” branch, and 13 to 19% for the three dwarfs belonging to the “red” branch. The distance dispersion of the dwarfs (again excepting ESO219-010) is quite small, with a depth of ca. 0.5 Mpc, confirming the isolation and compactness of the CenA group.
As regards the application of the SBF method to dEs in general, the situation for red dwarfs \[$(B-R)_0>1.3$\] is certainly much less favourable than for blue ones. As future applications will tend to reach greater distances, the objects will be brighter and hence redder. The slope of the $\bar{M}$-colour relation in the $R$-band is very steep (the slope is 6.09, see equation 2), such that accurate colours are prerequisite for a proper SBF analysis. In different photometric bands the $\bar{M}$-colour relation is somewhat less, but still considerably steep (e.g. $4.5\pm0.25$ in $\bar{M}_I$ and $V-I$, see Tonry et al. 1997). Given the easy access and convenience of optical $R$-band imaging and the avoidance of fringing problems (as compared to the $I$-band) it is probably no great gain to change the photometric band. As we have shown in this paper, by working with several subfields per galaxy it should be feasible to achieve accurate SBF distances in the $R$-band (with an error of less than, or of the order of 10%) also for bright and red dwarf ellipticals.
We thank Lewis Jones for interesting discussions and for providing Figure \[fig4\]. H.J. and B.B. are grateful to the [*Swiss National Science Foundation*]{} for financial support. H.J. thanks further the Freiwillige Akademische Gesellschaft der Universität Basel and the Janggen-Pöhn Stiftung for their partial financial support. This project made extensively use of Worthey’s World Wide Web model interpolation engine and made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, Caltech, under contract with the National Aeronautics and Space Administration.
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=6.5cm
=6.5cm =6.5cm\
=6.5cm =6.5cm
=8.0cm
[cccccccccc]{} ESO219-010& dE,N &12h56m09.6s & $-$50d08m38s & 16.42 & 14.73 & 17.4 & 22.92 & 0.96 & 1.32\
ESO269-066&dE,N: &13h13m09.5s & $-$44d52m56s & 14.59 & 12.96 & 39.7 & 22.95 & 0.40 & 1.48\
AM1339-445&dE &13h42m05.8s & $-$45d12m21s & 16.32 & 14.76 & 23.8 & 23.63 & 0.48 & 1.38\
AM1343-452&dE &13h46m17.8s & $-$45d41m05s & 17.57 & 16.02 & 14.7 & 23.85 & 0.52 & 1.35\
ESO384-016&dS0/Im &13h57m01.2s & $-$35d19m59s & 15.11 & 13.90 & 19.2 & 22.31 & 0.32 & 1.09\
[ccccccccccc]{} ESO219-010 & 24.58 & 5$\times$ 600 & 1.4 & 242 & 2572(3)& 26.3(2.7) & 4.5(0.05) & 27.98(0.10) & 0.59(0.09) & 27.39(0.14)\
ESO269-066 F1 & 24.56 & 5$\times$ 600 & 1.3 & 473 & 2120(2)& 59.4(3.3) & 1.8(0.08) & 27.07(0.06) & 0.25(0.04) & 26.82(0.07)\
F2 & & & & 447 & & 56.9(3.8) & 2.0(0.08) & 27.11(0.07) & & 26.86(0.08)\
AM1339-445 F1 & 24.03 & 4$\times$ 600 & 1.2 & 223 & 3566(7)& 39.4(2.4) & 7.9(0.10) & 26.99(0.06) & 0.30(0.05) & 26.69(0.08)\
F2 & & & & 204 & & 45.4(2.9) & 7.8(0.14) & 26.83(0.07) & & 26.53(0.09)\
AM1343-452 F1 & 24.58 & 5$\times$ 720 & 1.4 & 162 & 2896(4)& 60.2(5.0) & 6.8(0.13) & 27.27(0.09) & 0.32(0.05) & 26.95(0.10)\
F2 & & & & 226 & & 57.6(3.3) & 4.3(0.09) & 27.32(0.06) & & 27.00(0.08)\
ESO384-016 F1 & 24.03 & 5$\times$ 600 & 1.1 & 428 & 3290(6)& 31.5(1.0) & 4.9(0.07) & 27.23(0.04) & 0.20(0.03) & 27.03(0.05)\
F2 & & & & 205 & & 29.4(1.2) & 6.2(0.08) & 27.31(0.05) & & 27.11(0.06)\
[cccccccc]{} ESO219-010 & F1 & 1.13(0.08) & 27.39(0.14) & $-$0.60 & $-$1.01(0.11) & 28.40(0.18) & 4.79(0.43)\
& F1 & 1.24(0.06) & 26.82(0.07) & $-$1.17 & $-$1.26(0.36) & 28.08(0.37) &\
\[0cm\]\[0cm\][ESO269-066]{} & F2 & 1.26(0.06) & 26.86(0.08) & $-$1.13 & $-$1.14(0.36) & 28.00(0.37) & \[0cm\]\[0cm\][4.05(0.53)]{}\
& F1 & 1.28(0.06) & 26.69(0.08) & $-$1.30 & $-$1.01(0.36) & 27.70(0.37) &\
\[0cm\]\[0cm\][AM1339-445]{} & F2 & 1.19(0.06) & 26.53(0.09) & $-$1.46 & $-$1.56(0.36) & 28.09(0.37) & \[0cm\]\[0cm\][3.75(0.49)]{}\
& F1 & 1.31(0.12) & 26.95(0.10) & $-$1.04 & $-$0.83(0.73) & 27.78(0.74) &\
\[0cm\]\[0cm\][AM1343-452]{} & F2 & 1.27(0.07) & 27.00(0.08) & $-$0.99 & $-$1.08(0.42) & 28.08(0.43) & \[0cm\]\[0cm\][3.97(0.74)]{}\
& F1 & 1.01(0.04) & 27.03(0.05) & $-$0.96 & $-$1.15(0.04) & 28.18(0.06) &\
\[0cm\]\[0cm\][ESO384-016]{} & F2 & 1.21(0.05) & 27.11(0.06) & $-$0.88 & $-$0.89(0.08) & 28.00(0.10) & \[0cm\]\[0cm\][4.23(0.11)]{}\
[ccccc]{} NGC5128 & 562 & 27.98($\pm 0.15$)& TRGB& Harris et al. 1999\
& & 27.97($\pm 0.14$)& PNLF& Hui et al. 1993\
& & 27.78($\pm 0.10$)& SBF& Tonry & Schechter 1990\
NGC5253 & 404 & 28.08($\pm 0.10$)& Cepheids& Saha et al. 1995\
[^1]: http://199.120.161.183:80/$\sim$worthey/dial/dial\_a\_model.html
[^2]: The model limits for the Bertelli isochrones are $-1.3$ for 17Gyr and $-1.7$ for 12 and 8Gyrs
|
---
abstract: 'Point defects in the binary group-IV monochalcogenide monolayers of SnS, SnSe, GeS, GeSe are investigated using density-functional-theory calculations. Several stable configurations are found for oxygen defects, however we give evidence that these materials are less prone to oxidation than phosphorene, with which monochalcogenides are isoelectronic and share the same orthorhombic structure. Concurrent oxygen defects are expected to be vacancies and substitutional oxygen. We show that it is energetically favorable oxygen be incorporated into the layers substituting for a chalcogen ($\rm O_{S/Se}$ defects), and different from most of the other defects investigated, this defect preserves the electronic structure of the material. Thus, we suggest that annealing treatments can be useful for the treatment of functional materials where loss mechanisms due to the presence of defects are undesirable.'
author:
- 'Lídia C. Gomes'
- 'A. Carvalho'
- 'A. H. Castro Neto'
bibliography:
- 'Bibliography.bib'
title: 'Vacancies and oxidation of 2D group-IV monochalcogenides'
---
Introduction
============
Layered group-IV monochalcalgenides has become an important group of materials within the ever-growing family of two-dimensional crystals. Among the binary IV-VI compounds, SnS, SnSe, GeS and GeSe form a sub-group with orthorhombic structure belonging to the space group $D^{16}_{2h}$. Even though bulk structural, electronic and optical properties of these materials have been investigated since the 70’s [@PhysRevB.16.1616; @nc.39.709; @krist.148.295; @PhysRevB.41.5227], more recently their photovoltaic properties have been gaining considerable attention due to the increasing demand for efficient energy conversion technologies [@am402550s; @ashley.arxiv.1507]. The optimal band gap for photovoltaic solar cells of the naturally occurring bulk SnS, also known as herzenbergite [@PhysRevB.70.235114; @PhysRevB.92.085406], boosted experimental and theoretical research on this material in the last years.
Additional interest in group-IV monochalcogenides arose with the advances in experimental techniques of production and manipulation of low-dimensional materials, paving an avenue for research in the 2D field. While the most studied 2D materials are hexagonal, as graphene, phosphorene layers are orthorhombic,[@PhysRevLett.114.046801] and therefore became a paradigm of anisotropy in 2D. Anisotropy has important consequences, for example the enhanced thermoelectric effect arising from the fact that the preferential axes for heat and electronic conduction are orthogonal.[@fei-NL-14-288] Monolayer group-IV monochalcogenides are isoelectronic with phosphorene and share the same structure, but a lower symmetry, and therefore are expected to show large spin-orbit splitting [@PhysRevB.92.085406] piezoelectricity and high ionic dielectric screening [@PhysRevB.92.214103], all of them absent in phosphorene. Experimental progress in growth and exfoliation has already resulted in the isolation of bilayers, and the isolation of monolayer is expected.[@nn303745e; @am402550s; @jacs.5b08236]
In this article, we reveal yet another aspect in which group-IV monochalcogenides are more promising than phosphorene: their resistance to oxidation. In fact one of the hindrances to the research and use of phosphorene is its tendency to oxidize.[@PhysRevLett.114.046801; @PhysRevB.91.085407] Exposed to air, few-layer samples degrade in less than one hour,[@doganov-arXiv:1412.1274] eventually producing phosphorus oxide and phosphoric acid. Thus, phosphorene devices require immediate encapsulation in order to maintain their I-V characteristics.[@avsar-arXiv:1412.1191; @zhu-NL-15-1883] However, group-IV chalcogenides have stronger bonds and therefore are expected to be less prone to oxidation. In this article, we use first principles calculations to investigate point defects in group-IV monochalcogenide monolayers. Section \[O-defects\] is dedicated to the study of chemisorbed oxygen defects. Subsequently, in Section \[V-O-defects\] we study the effects of intrinsic vacancy defects and substitutional oxygen. Conclusions are presented in Section \[conclusions\].
Methods
=======
We use first-principles calculations based on density-functional theory to obtain the electronic and structural properties of oxidized monolayer monochalcogenides. We employ a first-principles approach based on Kohn-Sham density functional theory (KS-DFT) [@PhysRev.140.A1133], as implemented in the [Quantum ESPRESSO]{} code. [@Giannozzi2009]. The exchange correlation energy was described by the generalized gradient approximation (GGA) using the PBE [@PhysRevLett.77.3865] functional. Interactions between valence electrons and ionic cores are described by Troullier-Martins pseudopotentials [@PhysRevB.43.1993]. The Kohn-Sham orbitals were expanded in a plane-wave basis with a cutoff energy of 70 Ry, and for the charge density, a cutoff of 280 Ry was used. The Brillouin-zone (BZ) was sampled using a $\Gamma$-centered 10$\times$10$\times$1 grid following the scheme proposed by Monkhorst and Pack [@PhysRevB.13.5188].
We used periodic boundary conditions along the three dimensions. The layers are placed in the $x$-$y$ plane, with the $y$ axis parallel to the puckering direction, where atoms are arranged in a zigzag shape. Along the perpendicular $x$ axis, the atoms form an armchair configuration. In direction perpendicular to the layers, we used vacuum regions of 10 Å between adjacent images. Convergence tests with greater vacuum thickness were performed, and the values used are enough to avoid spurious interaction between neighbouring images.
The isolated defects were modeled using 3$\times$3 supercells ($M_{18}C_{18}$, with $M$=Sn,Ge and $C$=S,Se). The unitcells with an adsorbed oxygen atom have, therefore, $M_{18}C_{18}$O chemical composition, while for vacancies and O substitutional the concentrations are $M_{18(17)}C_{17(18)}$ and $M_{18(17)}C_{17(18)}$O, respectively.
Results
=======
Oxygen defects {#O-defects}
--------------
### Crystal Structure and Energetics
Group-IV monochalcogenides are isoelectronic with phosphorus, and their monolayer form assumes a corrugated structure very similar to phosphorene, with all atoms three-fold coordinated. The presence of two atomic species lowers the symmetry, and thus the bulk structure belongs to space group $Pnma-D^{16}_{2h}$, while black phosphorus is $Pnma-D^{18}_{2h}$. In the monolayer form, they also lose inversion symmetry in the perpendicular direction of the layers, which places them in the $Pn2_{1}m-C_{2\nu}^{7}$ space group.
Oxygen atoms can be adsorbed at different positions. As an initial step, we consider five configurations for isolated oxygen defects, derived from the models for oxygen defects in phosphorene considered in Ref. [@PhysRevLett.114.046801]. The models can be divided into dangling, horizontal bridge and interstitial bridge configurations, and are shown in the first column in Fig. \[O-MC\].
In dangling oxygen configurations the oxygen atom is bonded to only one lattice atom, borrowing two electrons from one of the lone pairs of $C$ or $M$. Thus, in group-IV monochalcogenides there are two distinct dangling bond configurations, unlike in phosphorene where there is only one. These models are labeled O-$M$($C$)-$MC$, where $M$($C$) = Sn, Ge (S, Se) indicates the species oxygen is bonded to. The two dangling oxygen structures are shown in Figs. \[O-MC\] (a) and (b).
We also consider two horizontal bridge configurations consisting of one oxygen positioned mid-way between two Sn(Ge) atoms of either two neighbor zigzag chains (i.e. along the armchair direction), or the same zigzag chain, along the zigzag direction. These models are labeled O-hb-ac-$MC$ (Fig. \[O-MC\](c)) and O-hb-zz-$MC$ (Fig. \[O-MC\](d)), respectively. In the bridge-type configurations the oxygen forms two single bonds to its nearest neighbors. In the interstitial configuration, named O-ib-$MC$, the oxygen is initially placed at a bond center ie. between a $M$ and $C$ atoms, as shown in Fig. \[O-MC\](e).
For all defects, the five initial structures (first column in Fig. \[O-MC\]) give rise to the respective optimized structures presented in columns 2 to 5 for SnS, SnSe, GeS and GeSe. The four materials show some variation in the final structures. After optimization, most defects result in significant distortion of the $MC$ structure in their neighborhood. Exceptions are dangling oxygen defects bonded to Ge in GeS and GeSe, for which the lattice remains little changed.
In addition, not all structures are stable for all four materials. Take as an example the SnS oxygen defects. In the horizontal bridge configuration O-hb-ac-SnS, oxygen atoms are initially bonded to two Sn atoms at equal distances O$-$Sn $\simeq$ 2.3 Å. After optimization, oxygen pulls one of the Sn atoms to $h^{top}_{out}$= 0.80 Å above the monolayer plane and along $x$ direction, towards to the second atom bonded to O. This is exactly the same structure adopted by the dangling oxygen O-Sn-SnS after optimization, as can be seen in Fig. \[O-MC\]. The dangling oxygen and bridge oxygen configurations, O-S-SnS and O-hb-zz-SnS, assume also very similar structures after optimization, with the lattices less affected by the introduction of the oxygen atoms. In this case, the sulfur in the O-S bond is pulled into the layer by about $h_{in}$= 0.37 Å while the tin atom directly below it is pushed $h^{\rm bot}_{\rm out}$= 0.30 Å out of the layer surface. For SnSe, the initial O-$M$ and O-hb-ac models also adopt the same final structures, as well O-$C$ and O-hb-zz. The main effect of the interstitial oxygen in the O-ib model is to push the chalcogen atom bonded to it slightly out of the monolayer plane.
[m[1.8cm]{} m[1.2cm]{} m[1.2cm]{} m[1.2cm]{} m[1.4cm]{}]{} & $h^{top}_{out}$ & $h_{in}$ & $h^{bot}_{out}$ & $h^{top}_{out}-ib$\
SnS & 0.80 & 0.37 & 0.30 & 0.70\
SnSe & 1.52 & 0.68 & 0.57 & 0.61\
GeS & 1.33 & 0.40 & 0.37 & 0.56\
GeSe & 1.60 & 0.74 & 0.62 & 0.51\
Along with an analysis of the structural changes, we investigate the energetic stability of the oxidized materials. The binding energy $E_b$ per oxygen atom is defined as:
$$E_b = E_{l+\rm O} - \left(E_{l} + N_{\rm O}\mu_{\rm O} \right)
\label{eb}$$
where $ E_{l+\rm O}$ is the total energy of the defective layers, $E_{l}$ is the energy of the pristine layers (without the oxygen atom), $\mu_{\rm O}$ is the chemical potential of oxygen and $N_{\rm O}$ is the number of oxygen atoms per unit cell, which we have chosen as $N_{\rm O}$=1 in this work. A natural choice for $\mu_{\rm O}$, is the chemical potential of $\rm O_2$ molecules as the oxygen source, from which $\mu_{\rm O}$ is obtained by $E_{\rm O_2}$/2, where $E_{\rm O_2}$ is the total energy of an $\rm O_2$ molecule. Defined as in Eq. \[eb\], a negative $E_b$ indicates that the defect formation is energetically favorable (exothermic reaction). The calculated $E_{b}$ for all materials and defect models are presented in Table \[eb-tab\].
Both Sn chalcogenides have low oxygen binding energies, of about $-$0.7 eV, while Ge chalcogenides have oxygen binding energies around $-$1.2-1.4 eV. This indicates that the latter are the most susceptible to oxidation. Still, even in this case the absorption energy is lower than in phosphorene, where the binding energy for the dangling configuration is found to be $-2.08$ eV using a similar method.
Overall, comparing the energies of the different defects across the four materials (Table \[eb-tab\]), we find that the lowest energy configuration is always the horizontal-bridge oxygen along the armchair direction, O-hb-ac.
[m[2.4cm]{} m[1.4cm]{} m[1.4cm]{} m[1.4cm]{} m[1.4cm]{}]{} & SnS & SnSe & GeS & GeSe\
O-$M$ & -0.75 & -0.75 & -0.83 & -0.69\
O-$C$ & -0.74 & -0.51 & -0.64 & -0.42\
O-hb-ac & -0.74$^{*}$ & -0.75$^{*}$ & -1.47 & -1.22\
O-hb-zz & -0.74$^{**}$ & -0.51$^{**}$& -0.77 & -0.42\
O-ic & -0.62 & -0.56 & -0.63 & -0.46\
### Electronic Properties
Fig. \[bands-O\] shows the electronic bandstructures for all oxygen defect configurations. The total density of states (DOS) and the contribution of each atomic specie to the electronic states, i.e., the projected density of states (PDOS), are also presented. The electronic properties of monolayer monochalcogenides are appreciably affected by the introduction of oxygen. The characteristic valleys observed for the pristine structures [@PhysRevB.92.085406] are strongly modified when they are exposed to oxygen. For all materials, the oxygen states hybridize with those of the $MC$ atoms.
Let us first concentrate on the low energy horizontal bridge O-hb-ac-$MC$ defect configuration (Fig. \[bands-O\](c)). For SnS, SnSe and GeSe, empty level states are introduced in the gap at 75 meV, 178 meV and 148 meV, respectively, from the conduction band minima, localized along the $\Gamma$-Y lines of the Brillouin zone (BZ). In GeS, an occupied state is formed 134 meV above the top of the valence band, while an empty state is calculated at $\approx$ 25 meV below the lowest conduction band. An analysis of the PDOS shows that the acceptor and donor bands are mainly formed by the states of the $MC$ atoms, and only a minor contribution from the oxygen atom (less than $\sim$ 8% in SnS). The horizontal bridge defects O-hb-zz-$MC$, in contrast, do not introduce gap states and are, therefore, electrically neutral for all materials.
The electronic bands, total and partial density of states for the monolayers with dangling oxygen configurations are presented in Figs. \[bands-O\] (a) and (b). When the oxygen is bonded to a chalcogen atom (O-$C$ structures), no gap states are introduced. On the other hand, when oxygen is bonded to a group-IV atom (O-$M$ structures), the conduction band is perturbed and acceptor levels are introduced. The gap state is localized mainly on the Sn or Ge atoms directly bonded to the oxygen. Fig. \[1-O-Sn-SnS.rho\] shows the charge density contribution of such defective in-gap state in SnS.
For the interstitial bridge defects, O-ib-$MC$, the top of the valence band is less affected, if compared to the other defect models, while the conduction band is still strongly modified in comparison to the pristine layers. In particular, for GeSe an acceptor state is formed a few meV bellow the conduction band minimum.
Activation energies can be estimated by the Marker Method (MM), as detailed in Ref. [@marker]. This method allows us to estimate energy levels comparing ionization energies of defective systems in different charge states, referent to a known [*marker*]{} system. A natural choice as reference system is the pristine (undefective) material, when other appropriate markers, for which reliable experimental data exists, are not available. With these considerations, its shown [@marker] that the electrical levels of an unknown system can be defined with respect to a known marker system by means of their total energies. We need to emphasize that the MM is most reliable for comparison between chemically (and structurally) similar systems. In the case of 2D materials modeled with periodic boundary conditions, the marker method is an efficient way to cancel, in a good approximation, the energy resulting from the spurious electrostatic interaction between neighbouring cells[@PhysRevB.89.081406].
For the models discussed up to now, we focus on calculate activation energies for the O-$M$ and low energy O-hb-ac systems, which present in-gap defective acceptor bands for all the monochalcogenides investigated here. The activation energies of the defective states are calculated from the differences in electron affinities: $$I_D - I_m = [E_D(0) - E_D(q)] - [E_m(0) - E_m(q)]
\label{ae}$$ where $E(q)$ is the energy of the supercell in charge state $q=\lbrace-,+\rbrace$, and the sub-indices $m$($D$) refer to pristine (defective) systems. Charged systems with an extra electron (q = $-$1) or a missing electron (q = +1) are considered for the pristine and defective monolayers with acceptor or donor states, respectively. The total energies used in Eq. \[ae\] are computed from GGA-PBE exchange-correlation functional. The calculated results are summarized in Table \[ae-tab\].
All acceptor levels are deep, lying in the upper half of the bandgap. The O-hb-ac defect in GeS presents the shallower state, with activation energy of 50 meV. GeS is also the only material which presents a donor defective state. Such state is deep, with an activation energy of 100 meV.
[m[2cm]{} m[1.2cm]{} m[1.2cm]{} m[1.2cm]{} m[1.2cm]{} m[0.8cm]{}]{} & SnS & SnSe & GeS & GeSe &\
O-$M$ & 0.17 & 0.31 & 0.11 & 0.22 & ($-$/0)\
O-hb-ac & 0.17 & 0.31 & 0.05 & 0.33 & ($-$/0)\
& - & - & 0.10 & - & (0/+)\
V$_{C}$ & 0.21 & - & 0.07 & - & (0/+)\
![(Color online) Charge density contribution to the defective state of the O-Sn-SnS model. The charge density distributions for the other compounds (SnSe, GeS and GeSe) are very similar and are not shown for simplicity.[]{data-label="1-O-Sn-SnS.rho"}](./1-O-Sn-SnSe.rho.eps)
Vacancies and Substitutional Oxygen {#V-O-defects}
-----------------------------------
Besides the oxygen defects discussed in the previous section, intrinsic defects is another class of dominant defects present in 2D materials. Bulk SnS, for instance, is characterized by an intrinsic p-type conductivity due to typical acceptor states formed by Sn vacancies ($\rm V_{Sn}$) [@apl.100.032104]. S vacancies ($\rm V_{S}$) can also be formed under appropriate Sn-rich conditions as well substitutional oxygen at S sites ($\rm O_{S}$). Experimental studies also indicate the presence of vacancies in single-crystal GeSe nanosheets, and discuss the role of the resulting defective states in the photoresponse of this material [@am402550s]. In this section, we investigate intrinsic defects in the monolayers of group-IV monochalcogenides. We discuss the energetic of the systems with introduction of four different types of defects: two vacancies, V$_{M}$ and V$_{C}$, and two substitutional oxygen defects, O$_{M}$ and O$_{C}$.
Formation energies ($E_f$) are calculated using $$E_f = E_{\rm def} - \left(N_{M}\mu_{M} + N_{C}\mu_{C} + N_{\rm O}\mu_{\rm O} \right)
\label{eb2}$$ where $E_{def}$ is the total energy of the defective structure, $\mu_{i}$ and $N_{i}$ are the chemical potential and number of atoms of $i$ type. The chemical potentials for $M$ and $C$ ($\mu_{M}$ and $\mu_{C}$, respectively) for Ge and Sn-rich conditions are taken from the diamond structure of these elements. The chemical potentials for Se and S-rich environments are calculated using the molecular crystal (R$\overline{3}$ phase) Se$_6$ and the S$_8$ molecule [@chemsci.7.1082]. The formation energy ($E_{f}$) interval is presented in Fig. \[vac-subst\], where, as in the previous section, our definition of formation energy yields negative values for exoenergetic processes.
![(Color online) Formation energies $E_{f}$ (eV) for vacancies and substitutional oxygen defects for $M$-rich ($M$ = Sn, Ge) and $C$-rich ($C$ = S, Se) conditions. All the vacancies present positive formation energies, an indication that their formation process is endothermic. On the other side, oxygen substitutional at the chalcogen sites ($\rm O_{S}$ and $\rm O_{Se}$ defects), are energetically favorable to occur in these systems, given their remarkable negative $E_{b}$ values.[]{data-label="vac-subst"}](./EbXmu.eps)
The first marked result is the indication of stability of all O-substitutional defects at the chalcogen sites, given by the negative values of the $\rm O_{S}$ and $\rm O_{Se}$ defects. In addition to the $\rm O_{S/Se}$, the only defect model that presents negative formation energy is the $\rm O_{Ge}$ in GeSe, under Se-rich condition. All vacancies, as well the remainder substitutional oxygen $\rm O_{Sn/Ge}$, are endothermic processes and, at least in the growth environments considered here, should not be favorable to occur in the single-layer of these materials.
Formation energies of vacancies and oxygen substituting for Sn/Ge are positive, however the formation energies of oxygen replacing S/Se are negative. Thus, the reaction $\rm V_{S/Se}+\frac{1}{2} O_{2}\rightarrow O_{S/Se}$ is energetically favorable.
### Electronic properties
Inspection of the electronic structure of the different types of vacancy defects indicates that Sn/Ge vacancies remove valence states, acting as shallow acceptors and displacing the Fermi level below the valence band top, as shown in Fig. \[bands-V-O\](a). In the case of chalcogen vacancies, perturbed valence and conduction bands appear in the selenides. A similar perturbed state has been predicted for bulk GeSe with the same type of defect [@am402550s]. For the sulphides, S vacancies introduce deep donor states, localized at Sn and Ge atoms in the vicinity of the missing S atom (Fig. \[bands-V-O\]b). Activation energies of the $\rm V_{S}$ states are presented in Table \[ae-tab\] and show that the defect state is shallower in GeS (70 meV) than in SnS (210 meV).
$\rm O_{Sn/Ge}$ is also shallow acceptor in all cases except GeS, where it introduces a deep state instead (Fig.\[bands-V-O\](c)). This deep state is highly localized on the oxygen atom and on the S atom bonded to it, as shown by the charge density distribution of the defect state, plotted in Fig. \[o-ge-ges-rho\].
![(Color online) Isosurface of charge density for the defect band introduced in the $\rm O_{Ge}$ substitutional defect in GeS. The plot shows the localized nature of the bands in the region around the oxygen and first S and Ge neighbouring atoms.[]{data-label="o-ge-ges-rho"}](./O-Ge-GeS-rho-iso0.002.eps)
In contrast, substitutional $\rm O_{S/Se}$ defects are isoelectronic with the pristine structure and do not introduce gap states for most of the compounds, as shown in Fig. \[bands-V-O\]d, leaving the bandstructure mostly unaffected.
Given the lack of experimental results on the properties of monolayer monochalcogenides, we establish a comparison with bulk and few-layer material. In Ref. [@am402550s], for instance, photoresponse analysis of single-crystal GeSe nanosheets shows a slow decay time, which is attributed to defect states created in the samples upon chemical process or light and heat application in the fabrication process. Indeed, a previous first-principles study of defects in bulk GeSe, shows the presence of middle gap states for vacancies or interstitial atoms. The energetic preference of oxygen to be incorporated into the layers, occupying the chalcogen sites (S and Se atoms), at the two limits of chemical potentials has also been reported by a previous study of surface passivation of bulk SnS [@jap.115.173702].
Conclusions
===========
Point defects in group-IV monochalcogenides monolayers - SnS, SnSe, GeS and GeSe - are investigated using first principles density functional theory calculations. Energetic and structural analysis of five different models for chemisorbed oxygen atoms, reveals a better resistance of these materials to oxidation if compared to their isostructural partner, phosphorene. Amongst all monochalcogenides, GeS is the most prone to oxidation, as it presents larger binding energies for four of the five models investigated.
Electronic structure calculations show that the most stable oxygen configurations have deep acceptor states, and so do the chalcogen vacancies. However, oxygen substitution leads to neutral defects which preserve the electronic structure of the pristine material. Substitutional oxygen forms spontaneously at the chalcogen sites in the presence of chalcogen vacancies and oxygen. This indicate that annealing/laser healing of vacancy defects will be effective in removing gap states in group-IV monochalcogenides, as was found for TMDCs.[@junpeng]
In contrast, Sn/Ge vacancies are shallow acceptors, and therefore are expected to confer $p$-type character to chalcogen-rich material. In this case, annealing in oxygen is not expected to be an effective passivation technique.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the National Research Foundation, Prime Minister Office, Singapore, under its Medium Sized Centre Programme and CRP award “Novel 2D materials with tailored properties: beyond graphene" (Grant number R-144-000-295-281). The first-principles calculations were carried out on the GRC high-performance computing facilities.
\[Bibliography\]
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Introduction {#sec-intro}
============
Inflation generates adiabatic density perturbations that can seed the formation of structure in the Universe. They arise from quantum fluctuations in the field that drives inflation and are stretched to astrophysical size by the enormous growth of the scale factor during inflation [@scalar]. The magnitude of these perturbations was recognized early on to be important in constraining inflationary models. The nearly scale-invariant value for the scalar spectral index, $n\approx 1$, is considered to be one of the three principal predictions of inflation, and the deviation of $n$ from unity is an important probe of the underlying dynamics of inflation [@Turner_10things].
The advantage of scale-invariant primordial density perturbations was first spelled out nearly three decades ago [@Harrison; @Zeldovich]: any other spectrum, in the absence of a long-wavelength or short-wavelength cutoff, will have excessively large perturbations on small scales or large scales.[^1] Even though inflation provided the first realization of such a spectrum, long before inflation many cosmologists considered the scale-invariant spectrum to be the only sensible one. For this reason, the inflationary prediction of a deviation from scale invariance – even if small – becomes all the more important.
One of the pioneering papers on inflationary fluctuations [@bst] emphasized that the fluctuations were not precisely scale-invariant; the first quantitative discussion followed a year later [@Steinh_Turner]. The COBE DMR detection of CBR anisotropy awakened the inflationary community to the testability of the inflationary density-perturbation prediction. The connection between $(n-1)$ and the underlying inflationary potential was pointed out soon thereafter [@davisetal; @Liddle_Lyth], and the possibility of reconstructing the inflationary potential from measurements of CBR anisotropy began being discussed [@recon]. It is now quite clear that the degree of deviation from scalar invariance is an important test and probe of inflation.
Particular inflationary potentials and the values of $n$ they predict have been widely discussed in literature (see e.g., Refs. [@Turner93; @Lyth96]). Lyth and Riotto [@Lyth96], for example, remark that many inflationary potentials can be written in the form $V(\phi )=V_0(1\pm \mu \phi^p)$ (in the interval relevant for inflation), and conclude that virtually all potentials of this form give $0.84<n<0.98$ or $1.04<n<1.16$ (also see Ref. [@Steinh_Turner]). Experimental limits on $n$, derived from CBR anisotropy measurements, are not yet very stringent, $0.7<n<1.2$ [@White_etal; @Tegmark]. Even the stronger bound claimed by Bond and Jaffe [@bond_etal], $n=0.95\pm 0.06$, falls far short of the potential of future CBR experiments (e.g., the MAP and Planck satellites), $\sigma_n \sim 0.01$ [@Eisenstein_etal].
The purpose of our paper is to discuss the general issue of the deviation from scale invariance, and to explain why scale invariance is a generic feature of inflation. In so doing, we will take a very agnostic approach to models. In view of our lack of knowledge about physics of the scalar sector and of the inflationary-energy scale, this seems justified. As we show, the slow-roll conditions necessary for inflation are closely related to the possible deviation from scale invariance. To illustrate what must be done to achieve significant deviation from scale invariance, we discuss models based upon smooth potentials where $n$ is much smaller than and much larger than unity.
Why inflationary perturbations are nearly scale invariant {#sec-ngg1}
=========================================================
The equations governing inflation are well known [@kt] $$\begin{aligned}
\ddot\phi + 3H\dot\phi + V^\prime (\phi ) & = & 0 \\
H^2 \equiv \left({\dot a\over a}\right)^2
& = & {8\pi \over 3{m_{\scriptscriptstyle{PL}}}^2}\,\left[V(\phi )
+ {1\over 2}\dot\phi^2\right] \\
N \equiv \ln (a_f/a_i)
& = & \int_{\phi_i}^{\phi_f}Hdt \\
\delta_H^2 (k) & \simeq & V^3/V^{\prime 2} \propto k^{n-1},\end{aligned}$$ where $a(t)$ is the cosmic scale factor, derivatives with respect to the field $\phi$ are denoted by prime, and derivatives with respect to time by overdot. The quantity $\delta_H$ is the post-inflation horizon-crossing amplitude of the density perturbation, which, if the perturbations are not precisely scale invariant is a function of comoving wavenumber $k$. (The dimensionless amplitude $\delta_H$ also corresponds to the dimensionless amplitude of the fluctuations in the gravitational potential.)
In computing the density perturbations, the value of the potential and its first derivative are evaluated when the scale $k$ crossed outside the horizon during inflation. Because both $V$ and $V^\prime$ can vary, $\delta_H^2 \propto k^{n-1}$ in general depends upon scale; exact scale-invariance corresponds to $n=1$. For most models, $\delta_H^2$ is not a true power law, but rather $n$ varies slowly with scale, typically $|dn/d\ln k|\leq 10^{-3}$ [@Kosowsky_Turner]; in fact, both $n$ and $dn/d\ln k$ are measurable cosmological parameters and can provide important information about the potential.
In the slow-roll approximation the $\ddot\phi$ term is neglected in the equation of motion for $\phi$ and the kinetic term is neglected in the Friedmann equation [@Steinh_Turner; @kt]: $$\begin{aligned}
\dot\phi & \simeq & {V^\prime \over 3H} \\
N & \simeq & -{8\pi\over {m_{\scriptscriptstyle{PL}}}} \int_{\phi_i}^{\phi_f}
{d\phi \over x(\phi )}. \\\end{aligned}$$ The power-law index $n$ is given by [@Liddle_Lyth; @Turner93] $$(n-1) = -\frac{x_{60}^2}{8 \pi} +
\frac{{m_{\scriptscriptstyle{PL}}}x_{60}'}{4 \pi}, \label{n-1}\\$$ where $x(\phi ) \equiv {m_{\scriptscriptstyle{PL}}}V^\prime (\phi ) /V(\phi )$ measures the steepness of the potential and $x^\prime =dx/d\phi$ measures the change in steepness. (Higher-order corrections are discussed and the next correction is given in Ref. [@LT].) The subscript “60” indicates that these parameters are evaluated roughly 60 e-folds before the end of inflation, when the scales relevant for structure formation crossed outside the horizon.
Deviation from scale invariance is a generic prediction since the inflationary potential cannot be absolutely flat, and it is controlled by the steepness and the change in steepness of the potential. Significant deviation from scale invariance requires a steep potential or one whose steepness changes rapidly. Further, Eq. (\[n-1\]) immediately hints that it is easier to make models with a “red spectrum” ($n < 1$), than with a “blue spectrum” ($n > 1$), because the first term in Eq. (\[n-1\]) is manifestly negative, while the second term can be of either sign. In addition, $x_{60}^2/8 \pi$ is usually larger in absolute value than ${m_{\scriptscriptstyle{PL}}}x_{60}'/4\pi$.
The two conditions on the potential needed to ensure the validity of the slow-roll approximation are (see e.g., Refs. [@Steinh_Turner; @kt]): $$\begin{aligned}
{m_{\scriptscriptstyle{PL}}}V^{\prime}/V = x & \la & \sqrt{48\pi} \\
{m_{\scriptscriptstyle{PL}}}^2 V^{\prime\prime}/V &\la & 24\pi.\end{aligned}$$ Note that the first slow-roll condition constrains the first term in the expression for $(n-1)$, and the second slow-roll condition constrains the second term since, ${m_{\scriptscriptstyle{PL}}}x^\prime = {m_{\scriptscriptstyle{PL}}}^2 V^{\prime\prime}/V
-x^2$.
A model that can give $n$ significantly less than 1 is power-law inflation [@Abbott_Wise; @Lucchin_Mattarese] (there are other models too [@Steinh_Turner; @Freese_etal]). It also illustrates the tension between sufficient inflation and large deviation from scale invariance. The potential for power-law inflation is exponential, $$V=V_0 \exp(-\beta \phi/{m_{\scriptscriptstyle{PL}}}),$$ the scale factor of the Universe evolves according to a power law $$a(t) \propto t^{16 \pi/\beta^2} \equiv t^p
\qquad \mbox{with} \qquad p\equiv 16 \pi/\beta^2,$$ and $$\dot\phi = \sqrt{p\over 4\pi}{{m_{\scriptscriptstyle{PL}}}\over t}.$$ Further, $n$ can be calculated exactly in the case of power-law inflation [@Lyth_Stewart_92] $$(n-1) = {2\over 1-p}\ \rightarrow\ -{2\over p}\ \ {\rm (slow-roll\ limit)}.$$
For this potential $x = -\beta$, $x'=0$ (constant steepness), and the slow-roll constraint implies $|\beta| \la 7$, or $p \ga 1$. This is not very constraining as $p>1$ is required for the superluminal expansion necessary for inflation [@htw]. The quantitative requirement of sufficient inflation to solve the horizon problem and a safe return to a radiation-dominated Universe before big-bang nucleosynthesis (reheat temperature $T_{\rm RH} \gg 1\,$MeV and reheat age $t_{\rm RH} \ll 1\,$sec) and baryogenesis ($T_{\rm RH}
> 1\,$TeV and $t_{\rm RH} < 10^{-12}\,$sec) restricts $p$ more seriously.
In particular, the amount of inflation is depends upon when inflation ends: $$N = -{8\pi \over {m_{\scriptscriptstyle{PL}}}}\int^{\phi_f}_{\phi_i}
\,{d\phi \over x(\phi )} = p\ln (H_i/H_f),$$ where $H_i = p/t_i$ and $H_f=p/t_f$. The number of e-folds $N$ required to solve the horizon problem (i.e., expand a Hubble-sized patch at the beginning of inflation to comoving size larger than the present Hubble volume) is approximately 60, but depends upon $H_i$ and $H_f$ if $p$ is not $\gg 1$ (see e.g., Ref. [@kt]): $$N > 74 + \ln (H_i/H_f) +{1\over 2}\ln (H_f/{m_{\scriptscriptstyle{PL}}}).$$
Bringing everything together, the constraint to $p$ is $$p > 1 + {74\over \ln (H_i /H_f)} + {1\over 2} {\ln (H_f/{m_{\scriptscriptstyle{PL}}})
\over \ln (H_i/H_f)}.$$ Based upon the gravity-wave contribution to CBR anisotropy $H_i$ must be less than about $10^{-5}{m_{\scriptscriptstyle{PL}}}$ and the baryogenesis constraint implies $H_f \ga (1\,{\rm TeV})^2/{m_{\scriptscriptstyle{PL}}}\sim
10^{-32}{m_{\scriptscriptstyle{PL}}}$. Since reheating is not expected to be very efficient and baryogenesis may require a temperature much greater than $1\,$TeV (if it involves GUT, rather than electroweak, physics), we can safely say that $H_f \gg 10^{-32}{m_{\scriptscriptstyle{PL}}}$. Thus, sufficient inflation and safe return to a radiation-dominated Universe before baryogenesis requires: $$\begin{aligned}
p & \gg & 2 \\
(1-n) & \ll & 2.\end{aligned}$$ Even insisting that $H_f \ga (10^{13}\GeV )^2/{m_{\scriptscriptstyle{PL}}}$, a typical inflation scale, only leads to $p\ga 5$ and $n\ga 0.5$, which is still a large deviation from scale invariance.
While the exponential potential allows a very large deviation from $n=1$, it illustrates the tension between achieving sufficient inflation and large deviation from scale invariance: because $(1-n) = 2/(p-1)$, large deviation from scale invariance implies a slow, prolonged inflation, $\ln (t_f/t_i) \simeq N(1-n)/2$, with the change in the inflaton field being many times the Planck mass, $\Delta \phi \simeq N
\sqrt{(1-n)/(8\pi)}\, {m_{\scriptscriptstyle{PL}}}\gg {m_{\scriptscriptstyle{PL}}}$. Other models also exhibit this tension: For example, for the potential $V(\phi ) = V_0-m^2\phi^2/2 +
\lambda\phi^4/4$, the lower limit to $n$ is set by the condition of sufficient inflation [@Steinh_Turner].
Achieving $n$ significantly greater 1 provides a different challenge since the first term in the equation for $(n-1)$ is negative and the work must be done by the change-in-steepness term, ${m_{\scriptscriptstyle{PL}}}x^\prime /4\pi$. To see the difficulty of doing so, let us assume that we can expand the slow-roll parameter $x(\phi)$ around a point $\phi_*$ in the slow-roll region: $$x(\phi)\approx x_{*}+x_{*}'(\phi-\phi_{*}). \label{Taylor}$$ This expression holds for potentials whose steepness does not change much in the slow-roll region. $N$ can now be evaluated explicitly: $$\begin{aligned}
N = -\frac{8 \pi}{{m_{\scriptscriptstyle{PL}}}}\int_{\phi_i}^{\phi_f} \frac{d\phi}{x(\phi)}
= \frac{8 \pi}{x_{60}'{m_{\scriptscriptstyle{PL}}}}\ln{\left ( \frac{x_i}{x_f}\right )}, \label{N}\end{aligned}$$ where $x_i$ and $x_f$ are understood to have been evaluated according to expression (\[Taylor\]). Combining expressions (\[N\]) and (\[n-1\]), we get $$n-1=\frac{2}{N}\ln{\left (\frac{x_i}{x_f}\right)} - \frac{x_{60}^2}{8 \pi},
\label{hard}$$ and the difficulty of obtaining large $n-1$ is now more transparent. For example, to get $n\approx 1.5$ with $N\geq 60$ we need $\ln(x_i/x_f)
> 15$ – more, if $x_{60}^2/8 \pi$ is not negligible. Not only does such a large change seem unnatural, but it probably invalidates the expansion in Eq. (\[Taylor\]).
Note, Eq. (\[hard\]) (and others below) make it appear that $(n-1)$ depends directly upon the amount of inflation. This is not really the case, because $N$ is the number of e-folds that occur during the time $x$ evolves from $x_i$ to $x_f$. In relating $(n-1)$ to properties of the potential it is probably most useful to set $N=60$, and further to expand $x(\phi)$ around $\phi_{60}$, the era relevant to creating our present Hubble volume. Therefore, we choose $\phi_i=\phi_{*}=\phi_{60}$.
Now further specialize to the case where $x_{60}^2/8\pi \ll |{m_{\scriptscriptstyle{PL}}}x_{60}^\prime |/4\pi$ and $|x_{60}|\gg
|x_{60}'\Delta\phi|$, where $\Delta \phi = \phi_f-\phi_i$. Here we have explicitly assumed that the change in the steepness of the potential is small. It now follows that $$\begin{aligned}
N & \simeq & {8\pi \over {m_{\scriptscriptstyle{PL}}}}{\left | \Delta \phi \over
x_{60}\right |} \\
(n-1) & \simeq & {2\over N}
{\left |\Delta \phi \over x_{60}\right |} \,
x_{60}^\prime < {2\over N}\end{aligned}$$ (note that $\Delta \phi$ and $x_{60}$ are of opposite sign). Thus, we get a very strong constraint on $n$ in this case, $(n-1)< 0.04$, and learn that to achieve $n$ significantly greater than unity, the scalar field must change by much more than ${m_{\scriptscriptstyle{PL}}}$.
One well-known class of inflationary models that gives $n\geq 1$ is hybrid inflation [@Linde]; in the slow-roll region, $ V(\phi )
\simeq V_0(1+ \mu \phi^2)$. In these models, $$\begin{aligned}
N & \simeq & {4\pi \over \mu \,{m_{\scriptscriptstyle{PL}}}^2} \ln (\phi_i/\phi_f) \\
(n-1) & \simeq & {{m_{\scriptscriptstyle{PL}}}x^\prime \over 4\pi} = {\mu\, {m_{\scriptscriptstyle{PL}}}^2\over 2\pi}
\simeq {2\over N} \ln (\phi_i/\phi_f).\end{aligned}$$ Thus, $n$ significantly larger than 1 can be achieved, albeit at the expense of an exponentially long roll, $\phi_i/\phi_f=\exp [N(n-1)/2]$. However, $\phi_f$ may not be arbitrarily small here – in fact, the smallest value it can take in the semi-classical approximation is equal to the magnitude of quantum fluctuations of the field, $H/2\pi$ (this is further discussed in the next section). This constraint, in combination with the other constraints, limits the maximum value of $n$ in hybrid inflation scenarios to $n\leq 1.2$ [@Lyth96].
To end, as well as summarize, this discussion, let us rewrite Eq. (\[hard\]) by expressing $x_{60}^2/8\pi$ in terms of $N$ and $\Delta \phi$ by assuming that $x(\phi )$ doesn’t change too much: $$(n-1) \simeq {2\over N}\ln (x_i/x_f)
-{8\pi \over N^2}\left({\Delta\phi \over {m_{\scriptscriptstyle{PL}}}}\right)^2.
\label{theessence}$$ As this equation illustrates, unless $\Delta \phi /{m_{\scriptscriptstyle{PL}}}$ is large or the steepness changes significantly, $|n-1|\la 2/N
\approx 0.04$. This is certainly borne out by inflationary model building: with a few notable exceptions all models predict $|n-1|\leq 0.1$ [@Lyth96].
Models with very blue spectra
=============================
Constraints
-----------
The conditions for successful inflation were spelled out a decade ago [@Steinh_Turner; @kt]. The [*règles de jeu*]{} are:
$\bullet$ Slow-roll conditions must be satisfied.
$\bullet$ Sufficient number of e-folds to solve the horizon problem ($N\ga 60$).
$\bullet$ Density perturbations of the correct amplitude $$\delta_H \sim V_{60}^{3/2}/V_{60}^\prime \sim 10^{-5}.$$
$\bullet$ The distance that $\phi$ rolls in a Hubble time must exceed the size of quantum fluctuations, otherwise the semi-classical approximation breaks down $${\dot{\phi}}H^{-1} \gg H/2\pi \qquad \Rightarrow \qquad V^\prime
\gg V^{3/2}/{m_{\scriptscriptstyle{PL}}}^3,$$ which is automatically satisfied if the density perturbations are small. Additionally, no aspect of inflation should hinge upon $\phi_i$ or $\phi_f$ being smaller than $H/2\pi$, the size of the quantum fluctuations.
$\bullet$ “Graceful exit” from inflation. The potential should have a stable minimum with zero energy around which the field oscillates at the stage of reheating. The reheat temperature must be sufficiently high to safely return the Universe to a radiation-dominated phase in time for baryogenesis and BBN.
$\bullet$ No overproduction of undesired relics such as magnetic monopoles, gravitinos, or other nonrelativistic particles.
There are additional constraints that the potential should obey in order to give $n\gg 1$:
\(a) ${m_{\scriptscriptstyle{PL}}}x_{60}'/4\pi$ has to be large and positive, while $x_{60}^2/8\pi$ should be negligible[^2]. Therefore $ |x_{60}|\la
O(1)$ and ${m_{\scriptscriptstyle{PL}}}x_{60}^\prime\simeq 4 \pi(n-1)$. In other words, at 60 e-folds before the end of inflation the potential should be nearly flat and starting to slope upwards.
\(b) To obtain 60 e-folds of inflation, the potential should be nearly flat in some region during inflation. However, the potential must not become too flat, since then density perturbations diverge ($\delta_H \propto 1/V^\prime$). Therefore, the potential should have a point of approximate inflection where $V'(\phi)$ is small but not zero.
Example 1
---------
A potential with the characteristics just mentioned is $$V=V_0 + M^4 \left [\sinh\left (\frac{\phi-\phi_1}{f}\right )+ e^{-
{\textstyle}{\frac{\phi}{g}}}\right ],$$ where $M$, $f$ $g$ and $\phi_1$ are constants with dimension of mass. The plot of the potential, with the parameters calculated below, is shown in the top panel of Fig. 1. The hyperbolic sine was invoked to satisfy requirements (a) and (b), while the exponential was used to produce a stable minimum.
We make the following assumptions to make the analysis simpler (later justified by our choice of parameters below):
1\) $V_0$ dominates the potential in the slow-roll region, $$V_0 \gg M^4 \sinh \left (\frac{\phi-\phi_1}{f}\right )
\quad \mbox{for} \quad \phi_i>\phi > \phi_f.$$
2\) $f \gg g\;$ so that the factor $\exp(-\phi/g)$ can be completely ignored in the slow-roll region.
3\) $(\phi-\phi_1)/f$ is at least of the order of a few for $\phi_i > \phi > \phi_f$, so that $\sinh [(\phi-\phi_1)/f] \gg 1$.
4\) For simplicity we take $\phi_i=\phi_{60}$.\
In terms of the dimensionless parameter $K \equiv {\displaystyle}{\frac{M^4 {m_{\scriptscriptstyle{PL}}}}{V_0 f}}$, $$\begin{aligned}
x &\simeq& K \cosh\left (\frac{\phi-\phi_1}{f}\right ) , \\
x'&\simeq& \frac{K}{f} \sinh\left (\frac{\phi-\phi_1}{f}\right )
-\frac{K^2}{{m_{\scriptscriptstyle{PL}}}} \cosh^2\left (\frac{\phi-\phi_1}{f}\right ) .\end{aligned}$$ The condition that $x_{60}\la {\cal O}(1)$ becomes $$K \cosh \left (\frac{\phi_{60} -\phi_1}{f}\right )
\la {\cal O}(1), \label{x60_negl}$$ and the end of inflation occurs one of the slow-roll conditions breaks down; in this case ${m_{\scriptscriptstyle{PL}}}^2 V^{\prime\prime}/V \simeq 24 \pi$, or $$\frac{{m_{\scriptscriptstyle{PL}}}K}{f} \sinh\left (\frac{\phi_f-\phi_1}{f}\right )
\simeq {24\pi}.
\label{infl_ends}$$ We can now write $$(n-1)\simeq
\frac{{m_{\scriptscriptstyle{PL}}}K}{4\pi f}\sinh \left( \frac{\phi_{60}-\phi_1}{f} \right) .
\label{n-1_mypot}$$
That inflation produces density perturbations of the correct magnitude implies $$\sqrt{V_0}\approx 4.3 \cdot10^{-6} x_{60}\, {m_{\scriptscriptstyle{PL}}}^2. \label{V0}$$ The expression for the number of e-folds can be calculated analytically. Introducing $\alpha=(\phi-\phi_1)/f $, we have: $$\begin{aligned}
N &=& -\frac{8 \pi}{{m_{\scriptscriptstyle{PL}}}}\int_{\phi_i}^{\phi_f} \frac{d
\phi}{x(\phi)} \nonumber\\
&=& -\frac{8 \pi f}{K {m_{\scriptscriptstyle{PL}}}} \left. \tan^{-1}[ \sinh(\alpha)]
\: \right |_{\alpha_i}^{\alpha_f}
\approx \frac{8 \pi^2 f}{K {m_{\scriptscriptstyle{PL}}}}. \ \label{N_mypot}\end{aligned}$$ In the last equality we used the fact that both $\alpha_i$ and $|\alpha_f |$ are at least of the order of a few, so that $\tan^{-1}[\sinh(\alpha_i)]\approx -\tan^{-1}[\sinh(\alpha_f)]\approx
\pi/2$. This assumption will also be fully justified with our choice of parameters below.
Finally, the potential should have a stable minimum (with $V=0$) at some $\phi=\phi_R$. This implies that $V(\phi_R)=0$ and $V'(\phi_R)=0$.
Before proceeding, we must specify $n$. We choose, somewhat arbitrarily, $n=2$. Of course, for such a large $n$ we should include terms beyond the lowest order, complicating the analysis. But we are not looking for accuracy – if $n=2$ is obtainable to first order, then one can certainly say that $n \gg 1$ is obtainable. (In fact, for the two potentials chosen, the second-order correction decreases $n-1$ only slightly.)
We now have to choose parameters $V_0$, $M$, $f$, $g$, $\phi_1$, $\phi_{60}$, $\phi_f$ and $\phi_R$ to satisfy Conditions (\[x60\_negl\] - \[N\_mypot\]), as well as $V(\phi_R)=0$ and $V'(\phi_R)=0$. The choice of these parameters is by no means unique, however. Here is such a set:
To verify our analytic results we integrated the equation of motion for $\phi$ numerically and computed the spectrum of density perturbations. We did so neglecting the $\ddot{\phi}$ in the equation of motion for $\phi$ and the kinetic energy of the field (slow-roll approximation) and taking both these quantities into account. The result is that $N_{\rm slow\ roll} = 57.3$ and $N_{\rm exact}= 57.9$. Thus, the field really rolls as predicted by analytic methods ($N\approx 60$), and the slow-roll approximation holds well for this potential.
The numerical results for the spectrum of density perturbations did contain a surprise, shown in Fig. 2. While this potential achieved large $n$, slightly smaller than 2, over a few e-folds $n$ falls to a smaller value[^3]. Indeed, even restricting the spectrum to astrophysically interesting scales, $1\,$Mpc to $10^4\,$Mpc, the spectrum is not a good power law, $| dn/d\ln k| \sim 0.3$, and is reminiscent of the “designer spectra” with special features constructed in Ref. [@bbs]. The reason is simple: in achieving $x^\prime
\sim 1$ an even larger value of $x^{\prime\prime}$ was attained.
Example 2
---------
Is there anything special about the hyperbolic sine? Not really – for example, a potential of the form “$\phi + \phi^3$” also works. Consider the potential $$V=V_0 + M^4 \left [\left (\frac{\phi-\phi_1}{f}\right )+
\left (\frac{\phi-\phi_1}{f}\right )^3 + e^{-{\textstyle}{\frac{\phi}{g}}}\right].$$ Again, we assume that $V_0$ dominates during inflation, that $\phi_i=\phi_{60}$ and that $\exp(-\phi/g)$ can be ignored in the inflationary region. To evaluate $N$, we further assume that $|(\phi_{60}-\phi_1)/f| \gtrsim 1$ and $|(\phi_f-\phi_1)/f| \gtrsim
1$. All of these assumptions are justified by the choice of parameters below.
The analysis of the inflationary constraints is similar. We conclude that large $n$ (here $n= 2$) is possible, with the following parameters:
This potential is shown in the bottom panel of Fig. 1. Numerical integration of the equation of motion shows that our “60 e-folds” is actually $N_{\rm slow roll}=55.0$ and $N_{\rm exact}=56.0$. Further, just as with the hyperbolic sine potential, $n\sim 2$ is achieved, but the spectrum of perturbations is not a good power law. Both potentials achieve a large change in steepness by having inflation occur near an approximate inflection point; however, the derivative of the change in steepness is also large, and $n$ varies significantly. The change in $n$ can be mitigated at the expense of a smaller value of $n$; see Fig. 2.
Conclusions {#concl}
===========
The deviation of inflationary density perturbations from exact scale invariance ($n=1$) is controlled by the steepness of the potential and the change in steepness, cf. Eq. (\[n-1\]). The steepness of the potential also controls the relationship between the amount of inflation and change in the field driving inflation, $N\sim 8\pi (\Delta \phi /{m_{\scriptscriptstyle{PL}}})/x$. A very “red spectrum” can be achieved at the expense of a steep potential and prolonged inflation ($t_f/t_i \gg 1$ and $\Delta \phi \gg {m_{\scriptscriptstyle{PL}}}$); the simplest example is power-law inflation. A very “blue spectrum” can be achieved at the expense of a large change in steepness near an inflection point in the potential and a poor power law. In both cases there appears to be a degree of unnaturalness.
The robustness of the inflationary prediction of that density perturbations are approximately scale-invariant is expressed by Eq. (\[theessence\]), $$(n-1) \simeq {2\over N}\ln (x_i/x_f)
-{8\pi \over N^2}\left({\Delta\phi \over {m_{\scriptscriptstyle{PL}}}}\right)^2.
\nonumber$$ Unless the change in steepness of the potential is large, $|\ln (x_i/x_f)| \gg 1$, or the duration of inflation is very long, $\Delta \phi \gg {m_{\scriptscriptstyle{PL}}}$, the deviation from scale invariance must be small, $|n-1|\la {\cal O}
(2/N) \sim 0.1$. Even for an extreme range in $n$, say from $n=0.5$ to $n\sim 1.5$, the variation of $\delta_H$ over astrophysically interesting scales, $\sim$1Mpc to $\sim 10^4\,$Mpc, is not especially large – a factor of $10$ or so – but is easily measurable.
Inflation also predicts a nearly scale-invariant spectrum of gravitational waves (tensor perturbations). The deviation from scale invariance is controlled solely by the first term in $(n-1)$ [@Turner93; @Liddle_Lyth], $n_T = -x_{60}^2/8\pi$. Thus, only a red spectrum is possible, with the same remarks applying as for density (scalar) perturbations with $n\ll 1$. In addition, the relative amplitude of the scalar and tensor perturbations is related to the deviation of the tensor perturbations from scale invariance, $T/S \simeq
-7n_T$ ($S$ and $T$ are respectively the scalar and tensor contributions to the variance of the quadrupole anisotropy of the CBR). Detection of the gravity-wave perturbations is an important, but very challenging, test of inflation; if, in addition, the spectral index of the tensor perturbations can be measured, it provides a consistency test of inflation [@Turner_GW].
Finally, measurements of the anisotropy of the CBR and of the power spectrum of inhomogeneity today which will be made over the next decade will probe the nature of the primeval density perturbations and determine $n$ precisely ($\sigma_n \sim 0.01$) [@Eisenstein_etal]. By so doing they will provide a key test of inflation and provide insight into the underlying dynamics. On the basis of our work here, as well as previous studies (see e.g., Ref. [@Lyth96]), one would expect $(n-1) \sim {\cal O}(0.1)$ or less, but not precisely zero. The determination that $|n-1| \ga {\cal O}(0.2)$, or for that matter $n=1$, would point to a handful of less generic potentials. The deviation of $n$ from unity is a key test of inflation and provides valuable information about the underlying potential [@recon].
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[^1]: Inflation provides a natural cutoff on comoving scales smaller than $\sim$1km, the horizon size at the end of inflation; perturbations on scales larger than the present horizon will not be important until long into the future. Thus, for inflation exact scale invariance is not necessary to avoid problems with excessively large perturbations.
[^2]: Of course, $x_{60}^2/8\pi$ is not required to be negligible, but it seems that it is even more difficult to get large $(n-1)$ without this assumption.
[^3]: Starting the roll higher on the potential will increase the highest $n$ achieved without violating any of the constraints. However, $n$ will fall to equally low values after a few e-folds as with the original $\phi_i$.
|
---
abstract: 'The behavior of a nanodevice based upon double-walled carbon nanotube oscillators driven by periodically applied thermal gradients (7 and 17 K/nm) is investigated by numerical calculations and classical molecular dynamics simulations. Our results indicate that thermophoresis can be effective to initiate the oscillator and that suitable heat pulses may provide an appropriate way to tune its behavior. Sustained regular oscillatory as well as chaotic motions were observed for the systems investigated in this work.'
author:
- 'V. R. Coluci'
- 'V. S. Timóteo'
- 'D. S. Galvão'
nocite: '[@tangney]'
title: Thermophoretically Driven Carbon Nanotube Oscillators
---
The remarkable properties of carbon nanotubes (CNTs) have allowed the proposition, design and exploitation of nanomechanical devices. One example is the CNT oscillator proposed by Zheng and Jiang [@zheng-prl] based upon the experimental realization of low-friction nanoscale linear bearings [@zettl]. Linear nanomotors based on coupled CNTs have also been experimentally realized [@nakayama1; @nakayama2]. CNT oscillators, in their simplest form, consist of a double-walled CNT where the inner tube can move inside the outer tube in an oscillatory low-friction motion of high frequency. This motion is induced and maintained by van der Waals interactions between the inner and the outer tubes. Many works have contributed to understand the characteristics of CNT oscillators such as their oscillation frequency dependence upon CNT parameters [@zheng-prb], stability [@vitor; @groove; @apl1], friction [@rivera; @servantie-1; @tangney], energy dissipation [@servantie-prl; @guo; @zhao; @popov1], and chaotic behavior [@vitor-caos]. Key issues regarding these devices are how to initialize them and how to sustain the motion in a controllable way. Different strategies have been proposed to start the oscillatory motion, e.g., by applying magnetic and electrical fields [@vitor; @popov2], by accelerating encapsulated charged elements located inside the inner tube [@ions], and by thermal expansion of encapsulated gases [@gas].
In this letter we propose an alternative way to start, maintain, and control the oscillatory motion, based on thermophoresis, i.e., the use of thermal gradients to induce mass transport. In this case, the thermal gradient is imposed on the outer tube by external agents such as electrical currents. This approach was inspired on the recent findings that demonstrated the possibility of producing nanoscale thermal motors based on CNTs [@rurali]. Thermophoretic mass transport through CNTs has been theoretically investigated for solid gold nanoparticles [@schoen] and water nanodroplets [@zambrano], but not yet for CNT-based oscillators.
The concept of the thermophoretically driven device is presented in Fig. 1. A CNT containing a “X”-like junction and a movable internal CNT is placed between two contacts. An electrical current $I$ can be injected through the contacts to create thermal gradients along the outer tube, with temperatures $T_1$ and $T_2$ ($T_1 > T_2$). The Joule heating will create a hotter region at the junction producing a heat flux along the $x$-direction. As will demonstrate below suitable choices of periodic electrical current pulses provide ways to start, tune and control the oscillatory behavior of the inner tube. It is important to mention that the configurations of tubes on the device of Fig. 1 are feasible to be obtained with current technological capabilities [@suspended-review].
In order to investigate the dynamics of the device shown in Fig. 1 we have used analytical models and molecular dynamics (MD) simulations. For the analytical part we considered a simplified model where the motion of the inner tube of mass $m$, represented by the center of mass coordinate $x$, is described by the following equation of a non-linear, damped, driven oscillator $$\displaystyle m\ddot{x} - F(x) + \gamma{\mathop{\mathrm{sgn}}}(\dot{x})|\dot{x}|^2 = \alpha \frac{dT}{dx}\; p(t).$$ where $F$ is the restoring force due to van der Waals interactions between inner and outer tubes, the damped term is associated with a friction proportional to $|\dot{x}|^2$ (Ref. 11), with a damped coefficient $\gamma$. The effect of the heating caused by the electrical current is translated into a driven “thermal” force which is proportional to the thermal gradient $dT/dx$. This model takes into account only the translational motion considering all tubes rigid, discarding rotational and vibrational modes. Despite the existence of these effects in real situations, the simplified analysis of the translational motion allows a fast estimative of the relative importance and contributions of geometrical (tube lengths and diameters, etc.) and dynamical aspects (temperature, damping constants, etc.) that can be obtained by numerical integration of Eq. (1). These analysis are combined with classical MD simulations based on the Brenner’s potential [@brenner] (explicitly taking into account rotations and vibrations) to validate Eq. (1) and to extract the relevant parameters ($\gamma$, $\alpha$, and $|F|$). Non-bonded interactions were described by a Lennard-Jones 6-12 potential with $\sigma = 3.37$ [Å]{} and $\epsilon = 4.2038$ meV [@mao; @lj]. The equations of motion were integrated with a third-order Nordisieck predictor-corrector algorithm [@nordsieck] using a time step of 0.5 fs. We considered the (10,10)@(15,15) double-walled CNT as a model for the oscillator, with lengths of 100 nm and 5 nm for the outer and inner tubes, respectively [@mov]. The atoms within 0.5 nm of the outer tube extremities were kept fixed during the simulations. To create the thermal gradient on the outer tube, the Berendsen’s thermostat [@berendsen] was applied to about 1100 atoms that were within 5 nm of each fixed extremity. We considered thermal gradients of 7 and 17 K/nm that are of the same order of magnitude that can be experimentally obtained [@zettl1]. Such large thermal gradient values also provided high signal-to-noise ratios in our MD simulations. Initially, we established a constant thermal gradient on the outer tube while the inner tube was separately equilibrated at 300 K. Then, we inserted the inner tube in the middle the outer tube and subtracted inner tube center of mass velocity from the velocity of its individual atoms. Fig. 2 (a) displays the time evolution of the center of mass of the inner tube and the resulting fitting curves, obtained with the following parameters: $|F|= 1.92 $ nN, $\gamma = 1.48~(2.24) $ pN ps$^2$/[Å]{}$^2$, and $\alpha = 8.08~(6.26) $ pN nm/K for $7~(17)$ K/nm. Despite the simplicity of the model the fitting is quite good. Discrepancies are present and are due to other effects not included in the model such as the tube rotations and vibrations. The imposed thermal gradient causes an acceleration of the inner tube initially at rest. The acceleration is larger for larger thermal gradients. When reaching the outer tube extremity, the inner tube experiences a restoring force which causes its retraction. Due to the presence of a constant thermal gradient, the inner tube is accelerated towards the extremity of the outer tube.
In order to simulate a temporal dependence of the driving force we used a periodic square wave-like pulse function $p(t)$ ($0\leq p(t)\leq 1$), which will be responsible to “turn on” (current turned on) and “turn off” (current turned off) the thermal gradient. The $dT/dx$ behavior as a function of $x$ is determined by the device geometry, electrical current pulse through $p(t)$, and by the thermal CNT conductivity values. Our MD simulations showed that a constant $dT/dx$ is obtained (after a transient period) for a large range of different temperatures [@mov]. These results also showed that thermal gradients can be a very effective way to initialize the oscillatory regime of the inner tube.
In order to obtain $x(t)$, Eq. (1) was then numerically integrated using the fourth-order Runge-Kutta algorithm with a fixed time step of 1 fs, with $x(0)=$ 1 [Å]{} and $\dot{x}(0)=$ 0. Two examples of possible regimes are shown in Fig. 2 (b). When the thermal force is time independent ($p(t)=1$, Fig. 2(b), curve 1), the inner tube is initially accelerated ($x(t)\propto t^2$) until reaching one of the extremity of the outer tube where it is subjected to the restoring force $F$. This force leads the inner tube to retract until some point where $F$ is compensated by the thermal force which sends it back to the extremity. This process continues back and forth until the inner tube is stopped. This is analog to the case of letting a steel ball to fall towards a rigid ground subject to the action of the gravitational and frictional forces. On the other hand, when a periodic pulse is used, the behavior of the inner tube is completely different (Fig. 2 (b), curve 2). We can see that the pulse can initialize and maintain the inner tube oscillations. The example presented in Fig. 2 (b) is for a tube movement that is not regular and does not show a well defined oscillatory period. By tuning the period of thermal pulses regular and periodic oscillations can also be obtained, as shown below.
For a fixed value of $dT/dx$, the overall behavior of the inner tube for different values of the driven period of $p(t)$ can be analyzed using the bifurcation diagram shown in Fig. 3. The diagram presents the values of $x$ collected after 30 ns in a 60 ns-integration at times that are in phase with the period of $p(t)$. As we can see the diagram exhibits a very complex structure. However, it is possible to determine values of the period that provide sustained (i.e., regular oscillatory) and chaotic motions. These values are illustrated by the vertical dashed lines in Fig. 3 and their corresponding temporal profile movements are shown in Fig. 4. For a period of 290 ps, the inner tube reaches, after a transient period, a sustainable oscillatory motion on one side of the outer tube with the oscillatory period matching the imposed one. On the other hand, for the driven period of 320 ps the inner tube also reaches a sustainable motion but now its displacement covers both (right and left) sides of the outer tube. However, in this case, the oscillatory period is twice the driven one. This result is similar to what is observed in several non-linear dynamical systems, where the period-doubling is an indication of a route to chaos [@caos]. Finally, for a driven period of 500 ps, the inner tube again reaches both sides of the outer tube but the motion is chaotic, characterized by the large spreading of $x$ values shown in Fig. 3.
The behavior of the CNT-based oscillator subject to periodically driven thermal forces results from an interplay between the thermal gradient and the lengths of the inner and outer tubes. The inner tube length significantly affects the bifurcation diagrams and, consequently, the necessary driven periods for regular motions. The thermal gradient greatly influences the overall behavior of the CNT oscillator since, for relatively high values of $dT/dx$, the acceleration of the inner tube can be so high that it could be ejected, destroying the nanodevice. For long outer tubes ($\sim$ 1 $\mu$m), the inner tube can reach a steady velocity state which can change the system dynamics, not allowing periodic motions that cover all the outer tube (Fig. 4 (b)) [@mov]. From the practical point of view, very long outer tubes will be strongly influenced by off-axis tube oscillations. Preliminary MD simulations of models for the nanodevice shown in Fig. 1 indicated high amplitude off-axis oscillations of the outer tube when its extremities are not clamped. On the other hand, clamped extremities (e.g., achieved by the use of additional CNT junctions along the outer tube) can prevent such vibrations and increase the stability of the nanodevice proposed here.
In summary, based on numerical calculations and molecular dynamics simulations we demonstrated that the use of thermal gradients can be an effective approach to initialize, control, and tune CNT-based oscillators. Financial support from the Brazilian agencies FAPESP (grant 2007/03923-1) and CNPq is acknowledged.
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[**Supplementary Information**]{}\
**Thermophoretically Driven Carbon Nanotube Oscillators**\
V. R. Coluci, V. S. Timóteo, and D. S. Galvão
Equation of Motion
==================
The motion of the inner tube of the CNT oscillator considering only translational motions (rigid tubes) is given by $$\displaystyle m\ddot{x} - F(x) + \gamma{\mathop{\mathrm{sgn}}}(\dot{x})|\dot{x}|^2 = \alpha \frac{dT}{dx}\; p(t)$$ where the restoring force is written as
$$\displaystyle F(x)=F_{vdW}\left\{\Theta(x+r)[1-\Theta(x-r)][\Theta(-x-\Delta)-\Theta(x-\Delta)]\right\}$$
with
$$\displaystyle \Theta(x)=\frac{1}{2}+\frac{x}{2\sqrt{x^2+\delta^2}}$$
($\delta = 0.1 $ [Å]{}) and $$\displaystyle r\equiv\frac{L+l}{2} \;\;\;\;\Delta\equiv\frac{L-l}{2},$$ where $F_{vdW}$ is the magnitude of the van der Waals restoring force, $L$ and $l$ are the outer and inner tube lengths, respectively. The form of $F(x)$ is shown in Fig. S1, for $L=100$ nm and $l=5$ nm.
The thermal gradient was chosen to be practically constant on the outer tube length based on molecular dynamics simulations (see below) according to the following expression
$$\displaystyle \frac{dT}{dx}(x)\equiv \left| \frac{dT}{dx}\right| \left[\frac{2}{1+\exp(-30x)} - 1\right].$$
Thus, the term $\alpha dT/dx$ represents the thermal force due to thermal vibrations of the outer tube caused by the externally imposed thermal gradient (Fig. S1). The regime of a constant thermal gradient is obtained after a transient period once the electrical current pulse is applied. We have analyzed the dynamics of the transient period using molecular dynamics simulations (see below).
The thermal gradient can be turned on or off accordingly to the presence or not of the electrical current flux through the supporting tube. The function $p(t)$ determines when the current would be on or off. We choose a square wave-like function to represent $p(t)$ with the form
$$\displaystyle p(t)= s(\bmod(t,P))$$
where
$$\displaystyle s(\tau)\equiv \frac{1}{[1+\exp[-\beta(\tau-t_1)]]\;[1+\exp[-\beta(-\tau+t_2)]]},$$
with $P$ being the period of $p(t)$. The modulo function $\bmod$ was used to provide the $p(t)$ periodicity. The parameters $\beta (=10 $ ps$^{-1})$, $t_1=P/6$, and $t_2=4t_1$ characterize $p(t)$. An example of $p(t)$ is depicted in Fig. 2 (B) for $P=400$ ps.
FIG. S1:
Carbon oscillator model
=======================
The model used in the molecular dynamics simulations is represented in Fig. S2. It is formed by a double-walled carbon nanotube with combination (10,10)@(15,15). Outer tube atoms that are fixed during the simulations are represented in blue, thermostated atoms in yellow ($T_1$) and red ($T_2$). Temperatures $T_1$ and $T_2$ were imposed at the beginning of the simulations.
FIG. S2:
We used this model to determine the temperature profile along the outer tube as a function of time once the thermal gradient is created. The analysis of the movement of the inner tube motion provided the values for the parameters $F_{vdw}$, $|dT/dx|$, $\gamma$, and $\alpha$.
Temperature Profile
===================
Fig. S3 presents the temperature profile of the outer tube for the system (10,10)@(15,15) for different times after a thermal gradient of 7 K/nm is imposed between the two extremities of the outer tube. As we can see an approximately linear variation of the temperature with $x$ is attained after about 50 ps, the period $P$ of $p(t)$ should be larger than 100 ps in order to keep the CNT oscillator operating in the constant thermal gradient regime.
FIG. S3:
Dependence upon tube lengths and thermal gradients
==================================================
As mentioned in the main text the behavior of the thermophoretically driven CNT oscillator depends upon both the inner tube length and the thermal gradient. In order to evaluate this dependence, we identified the cases where there is an oscillation (regular or chaotic) of the inner tube. In such cases the motion is limited within the outer tube. On the other hand, for some set of parameters, the inner tube was strongly accelerated and ejected off the outer tube. The evolution of the inner tube position was obtained by numerical integration of Eq. (1) for different values of the thermal gradient and driving period $P$. These results are only estimations since we used parameters for the case of inner tube length of 5 nm and thermal gradient of 7 K/nm ($F_{vdW}= 1.92 $ nN, $\gamma = 1.48 $ pN ps$^2$/[Å]{}$^2$, and $\alpha = 8.08 $ pN nm/K) to all other thermal gradient cases.
Figure S4 shows a phase diagram which indicates, for which combinations of the driving period and the thermal gradient $|dT/dx|$, the inner tube motion is limited. As one can see, the phase diagram exhibits a complicated structure. The longer the inner tube, the higher the thermal gradient necessary to completely eject the inner tube.
FIG. S4:
The overall behavior of the oscillator will also depend upon the outer tube length. The inner tube can reach a steady velocity state if the outer tube is long enough. For example, for the thermal gradient of 7 K/nm and an inner tube length of 5 nm, steady velocity state was observed for an outer tube length of 1 $\mu$m, as shown in Figure S5. For this case, the behavior of the device will be different as represented in the bifurcation diagram of Figure S6. The inner tube can now perform periodic motions only at one side of the outer tube. This is because the inner tube is not able to return to the middle of the outer tube for such large outer tube length. Furthermore, such large outer tube lengths may cause additional issues such as off-axis vibrations (if not clamped) which may make the device not viable. Modifying the thermal pulse (form, periods, etc) might provide a way to allow periodic motions covering all long outer tubes.
FIG. S5:
FIG. S6:
For low thermal gradients (below $\sim$ 1 K/nm), steady velocity state is expected to be achieved for even longer outer tubes. By reducing the temperature, a decreased in the friction is expected since nanotube vibrations (one of the causes of energy dissipation) are reduced. This is indeed observed in the $\gamma$ values of the fitted parameters for the cases of 7 and 17 K/nm. Thus, we expect a decreasing of the $\gamma$ value for lower thermal gradients than 7 K/nm.
Since we have not performed molecular dynamics simulations for low thermal gradients (below 1 K/nm), which provide the necessary parameters for Eq. (1), we can only estimate the behavior for such cases. The estimation is done by numerically integration Eq. (1) for different parameters of $\gamma$ and $\alpha$. The parameter $F_{vdw}$ will not change since we are not changing the double-walled carbon nanotube considered.
Keeping the same $\alpha$ value of the 7 K/nm case and using $\gamma = $ 0.5 pN ps$^2$/[Å]{}$^2$, we can see in Figure S7 that the steady velocity state will be achieved for even longer outer tubes for a thermal gradient of 0.1 K/nm. Thus, the value of 1 $\mu$m for the outer tube length can be considered as a lower limit where stead velocity state is expected, for thermal gradients below $\sim$ 10 K/nm.
FIG. S7:
|
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author:
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\
České vysoké učení technické v Praze, FJFI, Břehová 7, 115 19 Praha 1, Czech Republic\
E-mail:
- |
Boris Tomášik\
Univerzita Mateja Bela, Tajovského 40, 97401 Banská Bystrica, Slovakia\
České vysoké učení technické v Praze, FJFI, Břehová 7, 115 19 Praha 1, Czech Republic\
E-mail:
title: Evolution of higher moments of multiplicity distribution
---
Motivation
==========
The main motivation of this work is the observation that overall observed multiplicity of different types of particles from ultrarelativistic heavy-ion collisions agrees with the statistical model at temperatures above $160~\mathrm{MeV}$ [@Andronic:2017pug]. On the other hand, the phase transition temperature can be also determined from multiplicity fluctuations. Higher-order susceptibilities determined by lattice QCD are then compared with data on higher-order moments of the multiplicity distribution. The extracted temperature is usually lower than $160~\mathrm{MeV}$ [@Alba:2014eba].
The main aim of this work is to study the evolution of the multiplicity distribution in fireball that cools down after chemical freeze-out. We want to know the answer to the question: Can different moments of the multiplicity distribution be influenced by the temperature decrease differently? This is important, because then we could obtain different apparent temperatures from different moments.
For the description of the evolution of the multiplicity distribution we use a master equation. Particularly, we focus on the higher factorial moments from which all other kinds of moments, e.g. central moments or the coefficients of skewness and kurtosis, can be calculated. We first study in Section \[s:2\] the relaxation of factorial moments when the temperature is fixed. Then, in Section \[s:3\] the cooling scenario is investigated. In Section \[s:4\] we draw conclusions on the extracted apparent temperature and we summarise in Section \[s:5\].
Relaxation of factorial moments {#s:2}
===============================
We will consider a binary reversible process $a_1 a_2 \leftrightarrow b_1 b_2$. Here, none of the involved species are identical to each other and it is understood that $b$ particles carry a conserved charge while the $a$ particles do not. We will also assume that there is a sufficiently large pool of $a's$ which is basically untouched by this chemical process. For our study we shall investigate multiplicity distribution of species that conserve an abelian charge, e.g. strangeness.
The master equation [@1] for $P_n(\tau)$, the probability of finding $n$ pairs $b_1 b_2$, formulated in dimensionless time $\tau$ has the following form $$\frac{dP_n(\tau)}{d \tau} = \epsilon \left[ P_{n-1}(\tau) - P_n(\tau)\right] - \left[ n^2 P_n(\tau) - \left( n+1\right) ^2 P_{n+1}(\tau)\right],$$ where $n$ goes from 0 to $\infty$, and the constant $\epsilon$ is defined as $\epsilon = G \left\langle N_{a_1} \right\rangle \left\langle N_{a_2} \right\rangle /L $. Here, $ \left\langle N_{a_1} \right\rangle $, $ \left\langle N_{a_2} \right\rangle $ are (initial) averaged number of particles $a_1, a_2 $, and $G, L$ stand for the momentum-averaged cross-section of the gain process ($a_1 a_2 \rightarrow b_1 b_2$) and the loss process ($b_1 b_2 \rightarrow a_1 a_2$), respectively $$\label{eq:1}
G = \langle \sigma_G v \rangle \, , \qquad
L = \langle \sigma_L v \rangle\, .$$ The dimensionless time $\tau = t/\tau_0^c$ is formulated with the help of the relaxation time $\tau_{0} ^c = V/L$, where $V$ is the effective volume.
From the master equation we can derive the equilibrium distribution of the factorial moments. For this purpose, the master equation can be converted into a partial differential equation for the generating function [@1] $$\label{eq:2}
g(x, \tau) = \sum_{n=0}^{\infty} x^n P_n (\tau),$$ where $x$ is an auxiliary variable. If we multiply (\[eq:2\]) by $x^n$ and sum over $n$, we find that [@1] $$\frac{\partial g(x, \tau)}{\partial \tau}= (1-x)(xg''+g'-\epsilon g),$$ where $g' = \partial g/\partial x$. The generating function obeys the normalisation condition $$g(1, \tau) = \sum_{n=0}^{\infty} P_n (\tau) = 1.$$ The equilibrium solution, $g_{eq} (x)$, must not depend on time, thus it obeys the following equation $$x g_{eq} ^{''} + g_{eq} ^{'} - \epsilon g_{eq} = 0.$$ The solution that is regular at $x = 0$ is then given by $$g_{eq} (x) = \frac{I_0 (2 \sqrt{\epsilon x})}{I_0 (2 \sqrt{\epsilon})}.$$ Here, $I_0(x)$ is the Bessel function. From the derivatives of the generating function we can easily determine the factorial moments, since $\langle N!/(N-k)!\rangle = \partial^kg/\partial x^k$. (The first and second factorial moments have been calculated in [@1; @2].) $$\begin{aligned}
\label{eq:3}
\langle N \rangle_{eq} %F_{1, eq}
& = & \sqrt{\varepsilon}
\frac{I_1(2\sqrt{\varepsilon})}{I_0(2\sqrt{\varepsilon})} \\
\langle N(N-1) \rangle_{eq} %F_{2, eq}
& = & - \frac{1}{2} \sqrt{\varepsilon}
\frac{I_1(2\sqrt{\varepsilon})}{I_0(2\sqrt{\varepsilon})} + \frac{1}{2}\varepsilon
\frac{I_2(2\sqrt{\varepsilon}) + I_0(2\sqrt{\varepsilon})}{I_1(2\sqrt{\varepsilon})} \\
\left \langle \frac{ N!}{(N-3)!} \right \rangle_{eq}%F_{3, eq}
& = &
\frac{3}{4} \sqrt{\varepsilon}\frac{I_1(2\sqrt{\varepsilon})}{I_0(2\sqrt{\varepsilon})} \nonumber - \frac{3}{4} \varepsilon\left ( 1 + \frac{ I_2(2\sqrt{\varepsilon})}{I_0(2\sqrt{\varepsilon})}\right ) \\
&&+ \frac{1}{4} \varepsilon^{3/2}
\frac{I_3(2\sqrt{\varepsilon}) + 3I_1(2\sqrt{\varepsilon})}{I_0(2\sqrt{\varepsilon})}\\
\left \langle \frac{ N!}{(N-4)!} \right \rangle_{eq}%F_{4, eq}
& = & -\frac{15}{8} \sqrt{\varepsilon} \frac{I_1(2\sqrt{\varepsilon})}{I_0(2\sqrt{\varepsilon})} \nonumber + \frac{15}{8} \varepsilon \left (\frac{I_2(2\sqrt{\varepsilon})}{I_0(2\sqrt{\varepsilon})} +1\right) \\
&&-\frac{3}{4} \varepsilon^{3/2}
\frac{3I_1(2\sqrt{\varepsilon}) +I_3(2\sqrt{\varepsilon})}{I_0(2\sqrt{\varepsilon})}
+ \frac{1}{8} \varepsilon^2 \left ( 3 +
\frac{4I_2(2\sqrt{\varepsilon}) + I_4(2\sqrt{\varepsilon})}{I_0(2\sqrt{\varepsilon})}\right ) \, .
\end{aligned}$$
Now, we let the distribution of the multiplicities relax with the help of master equation. We calculate the evolution of *scaled* factorial moments, which are defined as $$\begin{aligned}
\label{eq:4}
F_2 & = & \frac{\langle N (N-1) \rangle}{\langle N \rangle^2}\\
F_3 & = & \frac{\left \langle N (N-1) (N-2)\right \rangle}{\langle N \rangle^3}\\
F_4 & = & \frac{\left \langle N (N-1) (N-2) (N-3)\right \rangle}{\langle N \rangle^4} \, .\end{aligned}$$
For numerical calculations, binomial initial conditions were used $$\begin{aligned}
\label{eq:5}
P_0(\tau = 0) & = & 1-N_0\\
P_1(\tau = 0) & = & N_0\\
P_n(\tau = 0) & = & 0\, \qquad \mbox{for}\phantom{m}n>1\, ,\end{aligned}$$ where $N_0 = 0.005$ $(N_0 = \left\langle N \right\rangle (\tau = 0))$.
The evolution of second to fourth scaled factorial moments divided by their equilibrium values is shown in Figure \[fig:1\].
![Time evolution of scaled factorial moments divided by their equilibrium values for constant temperature and $\epsilon = 0.1$.[]{data-label="fig:1"}](obrazek_1_new.eps){width="70.00000%"}
The value of the parameter $\epsilon$ has been set to 0.1. It is important to say here that we have obtained qualitatively similar results also with other sets of parameters. In Fig. \[fig:1\] we can see that relaxation time for all moments is the same. However, during relaxation higher moments depart further from their equilibrium values than the lower moments.
Higher moments in a cooling fireball {#s:3}
====================================
Realistic description of the fireball evolution must include decreasing temperature. If temperature changes, also the relaxation time will change. Thus one cannot use the dimensionless time because relaxation time was the typical scale in introducing the dimensionless time. Now we need to go back to real time in the master equation and calculate the creation and annihilation terms for each temperature. The master equation takes the form $$\label{eq:6}
\frac{dP_n(t)}{dt} = \frac{G}{V} \left\langle N_{a_1} \right\rangle \left\langle N_{a_2} \right\rangle \left[ P_{n-1} (t) - P_n (t) \right] -
\frac{L}{V} \left[ n^2 P_n (t) - (n+1)^2 P_{n+1} (t) \right] .$$ We shall study, how higher moments evolve in a scenario with a decreasing temperature. In our simulations we shall assume that the system is established in equilibrium at the hadronisation temperature $T = 165~\mathrm{MeV}$. The fireball then cools down further. We investigate, how does the distribution of multiplicities change.
To answer this question we have used a simple toy model in which the temperature, volume and density behave like in 1D longitudinally boost-invariant expansion (Bjorken scenario). The effective volume grows linearly $$\label{eq:7}
V(t) = V_0 \frac{t}{t_0},$$ the temperature drops according to $$\label{eq:8}
T^3 (t)= T_0^3 \frac{t_0}{t}$$ and the particle density drops like $$\label{eq:9}
\rho (t) = \rho_0 \frac{t_0}{t}.$$
In the calculations, we have set $V_0 = 125 ~\mathrm{fm^3}$, $T_0 = 165 ~\mathrm{MeV}$ and $\rho_0 = 0.08 ~\mathrm{fm^{-3}}$ for the initial state of the evolution. Motivated by the femtoscopic measurements we set the final time to $10 ~\mathrm{fm/c}$ and the final temperature to 100 MeV. This leads then to $t_0 = 2.2 ~\mathrm{fm/c}$.
For this calculation we have to choose the particular inelastic process. We have chosen the reaction system $\pi^+ + n \leftrightarrow K^+ + \Lambda^0$. For the moment we shall use a parametrisation of the cross-section [@3] $$\label{eq:10}
\sigma_{\pi N}^{\Lambda K} = \left \{
\begin{array}{lc}
0\, \mbox{fm}^2 & \sqrt{s} < \sqrt{s_0}\\
\frac{0.054 (\sqrt{s} - \sqrt{s_0})}{0.091}\, \mbox{fm}^2 & \sqrt{s_0}\le\sqrt{s}<\sqrt{s_0}+0.09\,\mbox{GeV}\\
\frac{0.0045}{\sqrt{s} - \sqrt{s_0}} \, \mbox{fm}^2 & \sqrt{s} \ge \sqrt{s_0}+0.09\,\mbox{GeV}
\end{array}
\right .$$ where $\sqrt{s_0}$ is the threshold energy of the reaction and the energies are given in $\mathrm{GeV}$. Since we will assume density-dependent mass of $\Lambda$, the threshold energy will also depend on the density.
We shall assume that the mass of $\Lambda$ hyperon depends on baryon density as $$\label{eq:11}
m(\rho) = -2.2 ~\mathrm{GeV \cdot fm^3} \cdot \rho + m_{\Lambda0}\, .$$ Hence, the hyperon mass becomes identical to that of the proton at the highest baryon density $\rho_0$ at which our calculations starts, and returns to the vacuum value $m_{\Lambda 0}$ if baryon density vanishes.
The scaled factorial moments for the cooling scenario are shown in Figure \[fig:2\].
![Scaled factorial moments for the gradual change of temperature. Solid lines: evolution of moments according to master equation. Dashed lines: equilibrium values at the given temperature.[]{data-label="fig:2"}](factorial_moments_en.eps){width="70.00000%"}
The freeze-out temperature {#s:4}
==========================
In Figure \[fig:3\] we demonstrate the potential danger in case of extraction of the (apparent) freeze-out temperature from the different moments.
![Apparent freeze-out temperature of factorial moments. Solid lines: evolution of moments according to master equation. Dashed lines: equilibrium values at the given temperature.[]{data-label="fig:3"}](factorial_moments_cary_en.eps){width="70.00000%"}
At the hadronisation temperature we set the moments to equilibrium values, then we let them evolve. Let us assume that the evolution is finished at $T = 100$ MeV. All moments are off-equilibrium, there. Auxiliary lines in Fig. \[fig:3\] demonstrate, how different values of the temperature would be obtained from different orders of the moments if they are interpreted as equilibrated.
In experimental data, more conveniently, the central moments are used. $$\begin{aligned}
\label{eq:12}
\mu_1 & = & \langle N \rangle = M\\
\mu_2 & = & \langle N^2 \rangle - \langle N\rangle^2 = \sigma^2\\
\mu_3 & = & \langle (N-\langle N\rangle )^3\rangle \\
\mu_4 & = & \langle (N-\langle N\rangle )^4\rangle.\end{aligned}$$ Often, one uses their combinations like the coefficient of skewness $$\label{eq:13}
S = \frac{\mu_3}{\mu_2^{3/2}}$$ or the coefficient of kurtosis $$\label{eq:14}
\kappa = \frac{\mu_4}{\mu_2^2} - 3.$$
We also look at the volume-independent ratios which are often measured. These are, e.g. $$\begin{aligned}
R_{32} & = & \frac{\mu_3}{\mu_2} = S\sigma\\
R_{42} & = & \frac{\mu_4}{\mu_2} - 3\mu_2 = \kappa\sigma^2.\end{aligned}$$
Results are plotted in Fig. \[fig:4\].
![Central moments, skewness, kurtosis and volume-independent ratios $S \sigma$ and $\kappa \sigma^2$ for the scenario with density-dependent mass of $\Lambda$ and the decreasing temperature. Thick solid lines: numerically calculated evolution, thin dotted lines: equilibrium values at the given temperature.[]{data-label="fig:4"}](obrazek_7_new2.eps){width="100.00000%"}
We can see that while the central moments are decreasing with the time evolution, the coefficients of skewness and kurtosis are increasing. Only slight changes are seen for the volume independent ratios $S \sigma$ and $\kappa \sigma^2$. So the extracted apparent temperature strongly depends on the chosen observable. In real collisions we have non-equilibrium evolution of the moments and it is very difficult to determine the unique freeze-out temperature from them.
Conclusion {#s:5}
==========
If equilibrium is broken, higher factorial moments of the multiplicity distribution depart further from their equilibrium values than the lower moments. Evolution of chemical reaction off equilibrium may show different temperatures for different orders of the (factorial or central) moments. We demonstrated this on the example of $ \pi^+ + n \leftrightarrow K^+ + \Lambda^0$. The behavior of the combination of the central moments depends on the combination of moments we choose. Caution is mandatory when we want to extract the freeze-out temperature from higher moments of the multiplicity distributions.
This work was supported by the grant 17-04505S of the Czech Science Foundation (GAČR).
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---
abstract: 'A set of $n$ boxes, located on the vertices of a hypergraph $G$, contain known but different rewards. A [*Searcher*]{} opens all the boxes in some hyperedge of $G$ with the objective of collecting the maximum possible total reward. Some of the boxes, however, are booby trapped. If the Searcher opens a booby trapped box, the search ends and she loses all her collected rewards. We assume the number $k$ of booby traps is known, and we model the problem as a zero-sum game between the maximizing Searcher and a minimizing [*Hider*]{}, where the Hider chooses $k$ boxes to booby trap and the Searcher opens all the boxes in some hyperedge. The payoff is the total reward collected by the Searcher. This model could reflect a military operation in which a drone gathers intelligence from guarded locations, and a booby trapped box being opened corresponds to the drone being destroyed or incapacitated. It could also model a machine scheduling problem, in which rewards are obtained from successfully processing jobs but the machine may crash. We solve the game when $G$ is a $1$-uniform hypergraph (the hyperedges are all singletons), so the Searcher can open just 1 box. When $G$ is the complete hypergraph (containing all possible hyperedges), we solve the game in a few cases: (1) same reward in each box, (2) $k=1$, and (3) $n=4$ and $k=2$. The solutions to these few cases indicate that a general simple, closed form solution to the game appears unlikely.'
author:
- 'Thomas Lidbetter[^1]'
- 'Kyle Y. Lin[^2]'
bibliography:
- '../bib/references.bib'
title: A Search Game on a Hypergraph with Booby Traps
---
[****Keywords:**** game theory; search games; discrete optimization]{}
Introduction {#sec:introduction}
============
We consider the following game between a Hider and a Searcher. There is a set $[n] \equiv \{1,\ldots,n\}$ of boxes, with box $i$ containing a reward of $r_i \ge 0$, for $i \in [n]$. We also make a standing assumption that, without loss of generality, $r_1 \geq \cdots \geq r_n$. The boxes are identified with the vertices of a hypergraph $G$. The Hider sets booby traps in $k$ of the boxes, where $1 \le k \le n-1$, so his strategy set is ${[n]^{(k)} \equiv \{H \subset [n]: |H|=k \}}$. The Searcher chooses a subset $S \subset [n]$ of boxes to search, where $S$ is the hyperedge of a hypergraph $G$ with vertices $V$ and hyperedges $E \subset 2^V$.
If the Hider plays $H$ and the Searcher plays $S$, the payoff $R(S,H)$ is given by $$R(S,H) =
\begin{cases}
r(S), & \text{if } H \cap S = \emptyset, \\
0, & \text{otherwise,}
\end{cases}$$ where $r(S) \equiv \sum_{i \in S} r_i$ is the sum of the rewards in $S$. In other words, the Searcher keeps the sum of all the rewards in the boxes she opens unless one or more of them is booby trapped, in which case, she gets nothing. If the Searcher uses a mixed strategy $p$ (that is, a probability distribution over subsets $S \subset [n]$) and the Hider uses a mixed strategy $q$ (a probability distribution over subsets $H \in [n]^{(k)}$), we write the expected payoff as $R(p,q)$. We also write $R(p, H)$ and $R(S, q)$ if one player uses a pure strategy while the other player uses a mixed strategy.
This game could be an appropriate model for a military scenario in which a drone is used to gather intelligence at several locations, and $r_i$ is the expected value of the intelligence gathered at location $i$. A known number $k$ of the locations are guarded, and flying the drone near these locations would result in its incapacitation. Alternatively, the Searcher may be collecting rewards in the form of stolen weapons or drugs from locations at which capture is possible, or the Searcher could be a burglar stealing valuable possessions from houses in a neighborhood, some of which are monitored by security cameras. The graph structure could correspond to geographical constraints. The case of the complete hypergraph, where $E=2^V$, corresponds to no constraints on the Searcher’s choice of subset. The case where $E$ is $1$-uniform, so that every hyperedge consists of a single vertex, corresponds to the Searcher being limited to searching only one location. If $E$ is $2$-uniform, so that $G$ is a graph, the Searcher must choose locations corresponding to the endpoints of an edge of the graph.
The game could also model a scheduling problem in which there are $n$ jobs with utilities $r_i$ which are obtained from a successful execution of job $i$. For example, jobs may correspond to computer programs. A total of $k$ of the programs are bugged, and each bug will crash the machine so that all data is lost. The objective is to find a subset of jobs to run that maximizes the worst-case expected utility, assuming Nature chooses which $k$ jobs are bugged.
This work lies in the field of [*search games*]{}, as discussed in [@AlpernGal], [@Gal2011], and [@Hohzaki]. Search games involving objects hidden in boxes have previously been considered in [@Lidbetter] and [@LidbetterLin]. In these works, the objective of the Searcher is to [*minimize*]{} a total cost of finding a given number of hidden objects. [@Agnetis] consider a machine scheduling problem in which rewards are collected from processing jobs and the machine may crash, similarly to our problem. But in their setting, each job will independently cause the machine to fail with a given probability.
Since this is a zero-sum game, it could be solved by standard linear programming methods, but this approach would be inefficient for large $k$, or if the hypergraph has a large number of hyperedges. In this work, we concentrate on two special cases of the game, with the aim of finding concise, closed-form solutions. We first solve the case where $G$ is a 1-uniform hypergraph in Section \[sec:null\]. In Section \[sec:complete\], we consider the complete hypergraph, and solve the game for three special cases: (1) same reward in each box, (2) $k=1$, and (3) $n=4$, $k=2$. We also give some general bounds, and make a conjecture on the form of the optimal solution. Finally, we offer concluding remarks in Section \[sec:conclusion\].
The game on a 1-uniform hypergraph {#sec:null}
==================================
We begin with the special case that $G$ is a $1$-uniform hypergraph, so that every hyperedge is a singleton (though every singleton may not be a hyperedge). In other words, the Searcher can open only 1 box, and her strategy set is simply some subset $A$ of the set $[n]$ of vertices. If the Searcher is restricted to boxes in $A$, then any Hider strategy that does not hide all $k$ booby traps in $A$ is (weakly) dominated by another Hider strategy that does. Hence, without loss of generality, we may assume that $G$ is the complete $1$-uniform hypergraph whose hyperedges are [*all*]{} the singletons. A mixed strategy for the Searcher is a probability vector $\mathbf x \in \mathbb R^n$ with $\sum_{j=1}^n x_j =1, x_j \ge 0$ for all $j$.
We first obtain a class of lower bounds on the value of the game, by defining a Searcher strategy for every subset of boxes.
\[lem:null\] For a subset $A \subseteq [n]$ of boxes with $|A| \geq k$, let the Searcher strategy $\mathbf x \equiv \mathbf x^A$ be given by $$x^A_j =
\begin{cases}
\lambda(A)/r_j, & \text{ if } j \in A, \\
0, & \text{ otherwise,}
\end{cases}$$ where $\lambda(A) = \left( \sum_{i \in A} 1/r_i \right)^{-1}$. The strategy $\mathbf{x}^A$ guarantees an expected payoff of at least $(|A|-k)\lambda(A)$.
*Proof.* The expected payoff of the Searcher strategy $\mathbf{x}^A$ against the Hider’s strategy $H$ is $$R(\mathbf{x}^A, H) = \sum_{j \in A-H} x^A_j \, r_j = |A- H| \, \lambda(A) \geq (|A|- |H|) \, \lambda(A) = (|A|- k) \, \lambda(A),$$ where the lower bound is obtained when $H \subseteq A$. $\Box$
Recall that $r_1 \geq r_2 \geq \cdots \geq r_n$. If the Searcher is restricted to choosing a strategy of the form described in Lemma \[lem:null\], for some $|A|=t \ge k$, then it is clear that the subset maximizing $(|A|- k) \, \lambda(A)$ is $[t] = \{1, 2, \ldots, t\}$. For $t=k, k+1, \ldots, n$, define $$V(t) \equiv (t-k) \lambda ([t]),
\label{eq:V(t)}$$ which is the expected payoff guaranteed by choosing $A = [t]$ in Lemma \[lem:null\]. Our main result is that, when $G$ is the complete 1-uniform hypergraph, the value of the game is $\max_{t=k,\ldots,n} V(t)$. For example, if $n=3$ and $k=1$ with $(r_1,r_2,r_3)=(10,10,1)$, then $V(1)=0$, $V(2) = 5$, and $V(3) = 5/3$, so the value of the game is $V(2)=5$ and the Searcher opens either box 1 or box 2 each with probability 0.5. Intuitively, if the rewards in different boxes are lopsided, then it is better for the Searcher to avoid those boxes with the lowest rewards altogether. We need a lemma before presenting the theorem.
For $t \geq k+1$, the two inequalities $V(t) \geq V(t-1)$ and $r_t \geq V(t)$ are equivalent, where $V(t)$ is defined in (\[eq:V(t)\]). \[le:r\_t\]
*Proof.* The first inequality is equivalent to $$\frac{t-k}{\frac{1}{r_1} + \cdots + \frac{1}{r_t}} \geq \frac{t-1-k}{\frac{1}{r_1} + \cdots + \frac{1}{r_{t-1}}}$$ Multiplying both denominators on both sides and cancelling common terms yields $$\frac{1}{r_1} + \cdots + \frac{1}{r_t} \geq (t-k) \frac{1}{r_t},$$ which is equivalent to $r_t \geq V(t)$, thus completing the proof. $\Box$
\[th:null\] Consider the search game with $n$ boxes and $k$ booby traps played on the complete $1$-uniform hypergraph. Define $$t^* \equiv \arg \max_{t = k,\ldots,n} V(t),$$ where $V(t)$ is defined in . The strategy $\mathbf x^{[t^*]}$ described in Lemma \[lem:null\] is optimal for the Searcher. For the Hider, any strategy that distributes the $k$ booby traps among the boxes in $[t^*]$ in such a way that box $j \in [t^*]$ contains a booby trap with probability $$y_j \equiv 1- \frac{V(t^*)}{r_j}$$ is optimal. The value of the game is $V(t^*)$.
*Proof.* By Lemma \[lem:null\], the Searcher guarantees an expected payoff at least $V(t^*)$ by using the strategy $\mathbf x^{[t^*]}$, so $V(t^*)$ is a lower bound for the value of the game.
To show that $V(t^*)$ is also an upper bound for the value of the game, first note that $t^* \ge k+1$, since $V(k) = 0 < V(k+1)$. In addition, by definition of $t^*$, we have that $V(t^*-1) \le V(t^*)$, which is equivalent to $V(t^*) \le r_{t^*}$ by Lemma \[le:r\_t\], so $y_j \in [0,1]$ for $j \in [t^*]$. One can also verify that $\sum_{j =1}^{t^*} y_j = k$.
If the Hider’s strategy has the property described in the theorem, then the expected payoff against any Searcher strategy $j \in [t^*]$ is $r_j (1-y_j) = V(t^*)$ and the expected payoff against any Searcher strategy $j \notin [t^*]$ is $r_j$. By definition of $t^*$, we have that $V(t^*) \ge V(t^*+1)$, which is equivalent to $V(t^* +1) \ge r_{t^*+1}$ from the proof in Lemma \[le:r\_t\]. Combining two inequalities yields that $V(t^*) \geq V(t^*+1) \geq r_{t^*+1}$. In other words, opening box $j \notin [t^*]$ results in payoff $r_j \leq r_{t^*+1} \leq V(t^*)$. Consequently, $V(t^*)$ is an upper bound for the value of the game, which completes the proof. $\Box$
There are many Hider strategies that will give rise to the property required in Theorem \[th:null\]; that is, the Hider distributes $k$ booby traps in $t^*$ boxes in such a way that box $j \in [t^*]$ contains a booby trap with probability $y_j \in [0,1]$, where $\sum_{j=1}^{t^*} y_j = k$. One way to implement such a Hider strategy can be found in Definition 2.1 in [@Gal-Casas]. Partition the interval $[0, k]$ into subintervals of lengths $y_1, \ldots, y_{t^*}$. Generate $\theta$ from the uniform distribution in $[0,1]$ and select the $k$ boxes corresponding to the $k$ subintervals containing the points $\theta, \theta+1, \ldots, \theta+(k-1)$. By construction, the Hider will choose exactly $k$ boxes to put booby traps, and box $i$ will contain a booby trap with probability $y_i$, for $i \in [t^*]$.
In the special case where all the rewards are equal, we have $V(t) = (t-k)/t$, which is maximized at $V(n) = (n-k)/n$. The Searcher’s optimal strategy is to open each box with probability $1/n$, and any Hider strategy that puts a booby trap in each box with the same probability $k/n$ is optimal—such as choosing every subset of $k$ boxes with probability $1/{n \choose k}$.
In another special case when $k=n-1$, we have $V(n-1)=0$, so the value of the game is $V(n) = (n- (n-1)) \lambda([n]) = ( \sum_{j =1}^n 1/r_j )^{-1}$. The Searcher’s optimal strategy is to open box $j$ with probability $\lambda([n])/ r_j$, for $j \in [n]$. The Hider’s optimal strategy needs to put a booby trap in box $j$ with probability $1- (n - (n-1)) \lambda([n])/ r_j$. Because the Hider has $n-1$ booby traps, the only strategy that meets this requirement is for the Hider to leave box $j$ free of booby trap with probability $\lambda([n])/ r_j$, for $j \in [n]$.
The game on the complete hypergraph {#sec:complete}
===================================
This section concerns the extreme case where $G$ is the complete (non-uniform) hypergraph, so that a Searcher strategy is any $S \subset [n]$. Note that if $k=n-1$, the Searcher should open only 1 box, so the structure of the hypergraph becomes irrelevant; the solution presented in Section \[sec:null\] is also optimal.
For the case of complete hypergraph, we present the solution to the three special cases: (1) equal rewards in each box; (2) $k=1$, and (3) $n=4$, $k=2$. We then give some general bounds on the value of the game, and make a conjecture on the optimal solution based on our findings.
The case with equal rewards {#sec:equal}
---------------------------
We begin our analysis with the special case where all the rewards are equal, which we set to $1$ without loss of generality.
\[thm:eq\] Consider the search game on the complete hypergraph with $r_j=1$ for $j \in [n]$, so this game is characterized by only the number of boxes $n$ and the number of booby traps $k$. The Hider’s optimal strategy is to choose some $H \in [n]^{(k)}$ uniformly at random. The Searcher’s optimal strategy is to open $m^* = \lceil \frac{n-k}{k+1} \rceil$ boxes at random. In particular, $m^*=1$ if $k \geq \frac{n-1}{2}$. The value of the game is given by $$U(n, k) \equiv \frac{{n-m^* \choose k} m^*}{{n \choose k}}.$$
*Proof.* By symmetry, it is optimal for the Hider to choose uniformly at random between all his pure strategies.
Because each box contains the same reward, the Searcher’s decision reduces to the number of boxes she opens. Write $F(m)$ for the expected reward when the Searcher opens $m$ boxes at random, and the booby trap is located in some arbitrary set of $k$ boxes. We calculate $F(m)$ by considering the Searcher’s $m$ boxes to be fixed and supposing that a randomly chosen set of $k$ boxes are booby trapped. The Searcher gets a reward of $m$ if none of the boxes she has chosen are booby trapped; otherwise she gets nothing. Hence, $$\begin{aligned}
F(m) = \frac{{n-m \choose k} m}{{n \choose k}}. \label{eq:F}\end{aligned}$$ The ratio $F(m+1)/F(m)$ is given by $$\frac{F(m+1)}{F(m)} = \frac{(n-k-m)(m+1)}{(n-m)m}.$$ Therefore, $F(m+1) \le F(m)$ if and only if $$m \ge \frac{n-k}{k+1}.$$ It follows that $F(m)$ is maximized at $m^* = \lceil (n-k)/(k+1) \rceil$. The value of the game is $F(m^*)$, as given in the statement of the theorem. $\Box$
Note that if $\theta \equiv k/n \le 1/2$ is held constant, and $n$ and $k$ tend to $\infty$, then the optimal search strategy in the limit is to open $m^*=\lceil (1-\theta)/\theta \rceil$ boxes, which is independent of $n$. The same result is obtained independently in Example 2.1c in [@ross] with a dynamic programming formulation. Writing $F(m)$ as $$F(m) = \left( \frac{n-k}{n} \right) \left( \frac{n-k-1}{n-1} \right) \cdots \left( \frac{n-k-m+1}{n-m+1} \right) m,$$ we can verify that the value of the game in the limit is $$\lim_{n \rightarrow \infty} F(m^*) =m^* (1-\theta)^{m^*}.$$ One can interpret $(1-\theta)^{m^*}$ as the probability that none of the $m^*$ boxes opened by the Hider contains a booby trap in the limit as $n \rightarrow \infty$.
Suppose now that $k$ is held constant and let $n \rightarrow \infty$. In the limit, the optimal number of boxes to open tends to infinity, and so does the value of the game. To calculate the proportion of the total reward $n$ the Searcher can obtain, we write out the probability ${n-m^* \choose k} / {n \choose k}$ from (\[eq:F\]) that none of the Searcher’s boxes are booby trapped as $$\prod_{i=0}^{k-1} \left( 1- \frac{m^*}{n-i} \right),
\label{eq:success_prob}$$ where $m^* = \lceil (n-k)/(k+1) \rceil$. Because $$\frac{n-k}{k+1} \leq m^* < \frac{n-k}{k+1} + 1 = \frac{n+1}{k+1},$$ the probability in satisfies the bounds $$\prod_{i=0}^{k-1} \left( 1- \frac{\frac{n+1}{n-i}}{k+1} \right) < \prod_{i=0}^{k-1} \left( 1- \frac{m^*}{n-i} \right) \leq \prod_{i=0}^{k-1} \left( 1- \frac{\frac{n-k}{n-i}}{k+1} \right).$$ Since the upper bound and the lower bound approach to the same limit as $n \rightarrow \infty$, we can conclude that $$\lim_{n \rightarrow \infty} \prod_{i=0}^{k-1} \left( 1- \frac{m^*}{n-i} \right) = \left(1 - \frac{1}{k+1} \right)^k.$$ Hence, in the limit as $n \rightarrow \infty$, the ratio of the value of the game to the total reward $n$ is $$\begin{aligned}
\lim_{n \rightarrow \infty} \frac{U(n,k)}{n} = \frac{1}{k+1} \left(1- \frac{1}{k+1} \right)^k. \label{eq1}\end{aligned}$$
The case with $k=1$ booby trap
------------------------------
We now consider the special case in which the Hider has only $k=1$ booby trap. Recall that a Searcher’s pure strategy is $S \subset [n]$. In order to present an optimal strategy for the Searcher, define $S^* \subset [n]$ to be a subset of boxes that minimizes $|r(S) - r(\bar{S})|$, where $\bar{S}$ denotes the complement of $S$.
We state and prove optimal strategies for the game in the case $k=1$. Let $R_0 = \sum_{i=1}^n r_i$.
\[theorem:k=1\] Consider the search game on the complete hypergraph with $k=1$. Let $S^* \subset [n]$ be a subset of boxes that minimizes $|r(S) - r(\bar{S})|$. It is optimal for the Searcher to choose $S^*$ with probability $$p(S^*) = \frac{r(\bar{S}^*)}{R_0};$$ otherwise choose $\bar{S}^*$. It is optimal for the Hider to put the booby trap in box $i$ with probability $q_i = r_i/R_0$, for $i \in [n]$. The value $V$ of the game is $$V = \frac{r(S^*) \, r(\bar{S}^*)}{R_0}.$$
*Proof.* Suppose the Searcher uses the strategy $p$ and that the booby trap is in some box $j$. If $j \in S^*$, the expected payoff is $$p(\bar{S}^*) \, r(\bar{S}^*) = \frac{r(S^*) \, r(\bar{S}^*)}{R_0}.$$ Similarly, if $j \in \bar{S}^*$, the expected payoff is the same. Therefore, $V \ge r(S^*) r(\bar{S}^*)/R_0$.
On the other hand, suppose the Hider uses the strategy $q$. If the Searcher opens some subset $S$ of boxes, then the expected payoff is $$r(S) \sum_{i \in \bar{S}} q_i = \frac{r(\bar{S}) \, r(S)}{R_0}.$$ The numerator in the preceding is equal to $$r(S) \, r(\bar{S}) = r(S)(R_0 - r(S)) = - \left(r(S) - \frac{R_0}{2} \right)^2 + \frac{R_0^2}{4} = -\frac{(r(S) - r(\bar{S}))^2}{4} + \frac{R_0^2}{4},$$ which is maximized by taking $S=S^*$ by definition of $S^*$. In other words, the Hider’s strategy $q$ guarantees that the expected payoff (for the Searcher) is at most $r(S^*) r(\bar{S}^*)/R_0$, so $V \le r(S^*) r(\bar{S}^*)/R_0$. The result follows. $\Box$
In the case that the rewards are integers, the problem of finding such a subset $S^*$ to minimize $|r(S) - r(\bar{S})|$ is the optimization version of the [*number partitioning problem*]{}, which is the problem of deciding whether a multiset of positive integers can be partitioned into two sets such that the sum of the integers in each set is equal. This problem is NP-hard, so that finding the value of the search game with $k=1$ is also NP-hard. There are, however, efficient algorithms to solve the problem in practice [@Korf].
Note that the value of the game for $k=1$ is $R_0/4$, if and only if the boxes can be partitioned into two subsets of equal total reward. It is tempting to conjecture that in general, the value of the game is $R_0/(k+1)^2$, if and only if the boxes can be partitioned into $k+1$ subsets of equal total rewards. This conjecture, however, is not true, as can be seen from the simple example with $n=6$ and $k=2$ when all the rewards are equal to $1$. By Theorem \[thm:eq\], the value of the game is $4/5$, but $R_0/(k+1)^2 = 6/3^2$. Nevertheless, the quantity $R_0/(k+1)^2$ is a lower bound for the value of the game, because the Searcher can choose one of the $k+1$ subsets uniformly at random, and receive an expected payoff of $R_0/(k+1)$ with probability at least $1/(k+1)$.
The case with $n=4$ boxes and $k=2$ booby traps
-----------------------------------------------
This section presents the solution to the game with $n=4$ boxes and $k=2$ booby traps. The Hider chooses two boxes to place the booby traps, so he has ${4 \choose 2} = 6$ pure strategies. The Searcher would want to open at most $n-k = 4-2=2$ boxes, so she has 10 viable pure strategies, including ${4 \choose 1} =4$ pure strategies that open just 1 box, and ${4 \choose 2} = 6$ pure strategies that open 2 boxes. While one can compute the value $V$ and optimal strategy of each player by a linear program, we will show that the optimal mixed strategy for the Searcher is one of the following three types:
1. Strategy A involves 4 active pure strategies: $\{1\}$, $\{2\}$, $\{3\}$, $\{4\}$. Specifically, the Searcher opens just 1 box, and chooses box $i$ with probability $$p_i = \frac{1/r_i}{1/r_1+1/r_2+1/r_3+1/r_4}, \qquad i=1,2,3,4.$$ Regardless of which two boxes contain booby traps, strategy A produces the same expected payoff $$V_A \equiv \frac{2}{\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} + \frac{1}{r_4}}.
\label{eq:V_A}$$ Intuitively, strategy A works well if $r_1, r_2, r_3, r_4$ are comparable.
2. Strategy B involves 3 active pure strategies: $\{1\}$, $\{2\}$, and $\{3,4\}$. Specifically, the Searcher opens box $i$, for $i=1,2$, with probability $$p_i = \frac{1/r_i}{1/r_1 + 1/r_2 + 1/(r_3+r_4)}, \qquad i=1,2,$$ or opens both boxes 3 and 4 with probability $$p_{34} = \frac{1/(r_3+r_4)}{1/r_1 + 1/r_2 + 1/(r_3+r_4)}.$$ Regardless of which two boxes contain booby traps, strategy B guarantees an expected payoff at least $$V_B \equiv \frac{1}{\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3 + r_4}}.
\label{eq:V_B}$$ Intuitively, strategy B works well if $r_3 + r_4$ is comparable to $r_1$ and $r_2$.
3. Strategy C involves 6 pure strategies: $\{1\}$, $\{2\}$, $\{3\}$, $\{1,4\}$, $\{2,4\}$, $\{3,4\}$. Specifically, the Searcher opens box $i$, for $i=1,2,3$, with probability $$p_i = \frac{1/r_i}{\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} + \frac{1}{r_1+r_4} + \frac{1}{r_2+r_4} + \frac{1}{r_3+r_4}}.$$ or opens both boxes $i$ and 4, for $i=1,2,3$, with probability $$p_{i4} = \frac{1/(r_i+r_4)}{\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} + \frac{1}{r_1+r_4} + \frac{1}{r_2+r_4} + \frac{1}{r_3+r_4}}.$$ Regardless of which two boxes contain booty traps, strategy C produces the same expected payoff $$V_C \equiv \frac{2}{\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} + \frac{1}{r_1+r_4} + \frac{1}{r_2+r_4} + \frac{1}{r_3+r_4}}.
\label{eq:V_C}$$ Intuitively, strategy C works well if $r_4$ is much smaller than the reward in each of the other three boxes.
The main result in this section is the following theorem.
\[th:n4k2\] One of the three strategies A, B, C is optimal for the Searcher. The value of the game is $$V = \max \{ V_A, V_B, V_C\},$$ where $V_A$, $V_B$, and $V_C$ are defined in , , and , respectively.
The proof of this theorem is lengthy, and we will present the three cases separately. Before doing so, we first offer some discussion to shed light on these three strategies. With some algebra, one can see that $V_A \geq V_B$ if and only if $$\frac{1}{r_1} + \frac{1}{r_2} \geq \frac{1}{r_3} + \frac{1}{r_4} - \frac{2}{r_3+r_4};
\label{eq:AB}$$ and $V_A \geq V_C$ if and only if $$\frac{1}{r_4} \leq \frac{1}{r_1+r_4} + \frac{1}{r_2+r_4} + \frac{1}{r_3+r_4};
\label{eq:AC}$$ and $V_B \geq V_C$ if and only if $$\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3+r_4} \leq \frac{1}{r_3} + \frac{1}{r_1+r_4} + \frac{1}{r_2+r_4}.
\label{eq:BC}$$
Strategy A treats each box equally. For Strategy A to work well, $r_1$ and $r_2$ cannot be too large compared with $r_3$ and $r_4$ (seen in ), and $r_4$ cannot be too small (seen in ). In other words, the four rewards need to be somewhat comparable. Strategy B combines the two boxes with smaller rewards together, and treats the problem as if there were only 3 boxes. For Strategy B to work well, $r_3$ and $r_4$ need to be substantially smaller than $r_1$ and $r_2$ (seen in ), and $r_3$ needs to be somewhat closer to $r_4$ rather than to $r_2$ (seen in ). Strategy C treats box 4—the one with the smallest reward—as a small add-on to one of the other three boxes. For Strategy C to work well, $r_4$ needs to be small enough (seen in ), and $r_1, r_2, r_3$ need to be somewhat close together (seen in ).
We next present the proof of Theorem \[th:n4k2\] in three sections, starting with the easiest case. The challenge in each of the three proofs is to show that the Hider has a mixed strategy to guarantee the payoff to be no more than the corresponding payoff guaranteed by the Searcher’s mixed strategy.
### Optimality of Strategy C
Strategy C is optimal for the Searcher and the value of the game is $V_C$ if and only if $$\frac{1}{r_1+r_4} + \frac{1}{r_2+r_4} + \frac{1}{r_3+r_4} \leq \frac{1}{r_4},
\label{eq:C>=A}$$ and $$\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3+r_4} \geq \frac{1}{r_3} + \frac{1}{r_1+r_4} + \frac{1}{r_2+r_4}.
\label{eq:C>=B}$$
*Proof.* If strategy C is optimal for the Searcher, then $V_C \geq V_A$, which is equivalent to , and $V_C \geq V_B$, which is equivalent to . Therefore, and are necessary conditions.
We next prove and are sufficient conditions. Since the Searcher can use strategy C to guarantee an expected payoff $V_C$, it remains to show that the Hider has a mixed strategy to guarantee an expected payoff no more than $V_C$. Let $q_{ij}$ denote the probability that the Hider hides the 2 booby traps in boxes $i$ and $j$, and let $$\begin{aligned}
q_{12} &= \frac{V_C}{r_3+r_4}, \qquad q_{13} = \frac{V_C}{r_2+r_4}, \qquad q_{23} = \frac{V_C}{r_1+r_4},
\\
q_{14} &= 1 - \frac{V_C}{r_2 + r_4} - \frac{V_C}{r_3 + r_4} - \frac{V_C}{r_1}, \\
q_{24} &= 1 - \frac{V_C}{r_1 + r_4} - \frac{V_C}{r_3 + r_4} - \frac{V_C}{r_2}, \\
q_{34} &= 1 - \frac{V_C}{r_1 + r_4} - \frac{V_C}{r_2 + r_4} - \frac{V_C}{r_3}.\end{aligned}$$
First, we show that the preceding is indeed a legitimate mixed strategy for the Hider. Using the definition in , one can verity that $\sum_{1 \leq i < j \leq 4} q_{ij} =1$. In addition, $0 \leq q_{23} \leq q_{13} \leq q_{12} \leq 1$ and $q_{34} \leq q_{24} \leq q_{14} \leq 1$, because $r_1 \geq r_2 \geq r_3 \geq r_4$. Finally, we see that $q_{34} \geq 0$, due to .
Next, we show that the Hider guarantees an expected payoff no more than $V_C$ regardless of what the Searcher does. Consider 4 cases.
1. If the Searcher opens $\{1,4\}$, then the expected payoff is $$(r_1+r_4) q_{23} = V_C.$$ A similar argument leads to the same conclusion if the Searcher opens $\{2,4\}$ or $\{3,4\}$.
2. If the Searcher opens {1}, then the expected payoff is $$r_1 (q_{23} + q_{24} + q_{34} ) = V_C.$$ A similar argument leads to the same conclusion if the Searcher opens {2} or {3}.
3. If the Searcher opens {4}, then the expected payoff is $$r_4 (q_{12}+q_{23}+q_{13}) = r_4 \left( \frac{1}{r_1+r_4} + \frac{1}{r_2+r_4} + \frac{1}{r_3+r_4} \right) V_C \leq V_C,$$ where the inequality follows from .
4. If the Searcher opens $\{1,2\}$, then the expected payoff is $$(r_1+r_2) q_{34} = (r_1+r_2) \left(
\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3+r_4} - \frac{1}{r_3} - \frac{1}{r_1+r_4} - \frac{1}{r_2+r_4} \right) \frac{V_C}{2}$$ To show that the preceding is no more than $V_C$, compute $$\begin{aligned}
& (r_1+r_2) \left(
\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3+r_4} - \frac{1}{r_3} - \frac{1}{r_1+r_4} - \frac{1}{r_2+r_4} \right) \\
& = (r_1+r_2) \left ( \frac{r_4}{r_1(r_1+r_4)} + \frac{r_4}{r_2(r_2+r_4)} - \frac{r_4}{r_3(r_3+r_4)} \right) \\
& = \frac{r_4}{r_1+r_4} + \frac{r_4}{r_2+r_4} + \left( \frac{r_1 r_4}{r_2(r_2+r_4)} - \frac{r_1 r_4}{r_3(r_3+r_4)} \right) + \left(\frac{r_2 r_4}{r_1(r_1+r_4)} - \frac{r_2 r_4}{r_3(r_3+r_4)}\right) \\
& \leq \frac{r_4}{r_1+r_4} + \frac{r_4}{r_2+r_4} + \frac{r_1 r_4}{r_1(r_1+r_4)} + \frac{r_2 r_4}{r_2(r_2+r_4)} \\
&= 2 \left(\frac{r_4}{r_1+r_4} + \frac{r_4}{r_2+r_4} \right) \\
& \leq 2 \left( 1 - \frac{r_4}{r_3+r_4} \right) \leq 2,\end{aligned}$$ where the first inequality follows from $q_{14} \geq 0$ and $q_{24} \geq 0$, and the second inequality follows from . A similar argument leads to the same conclusion if the Searcher opens $\{2,3\}$ or $\{1,3\}$.
The proof is complete. $\Box$
### Optimality of Strategy B
Strategy B is optimal for the Searcher and the value of the game is $V_B$ if and only if $$\frac{1}{r_1} + \frac{1}{r_2} \leq \frac{1}{r_3} + \frac{1}{r_4} - \frac{2}{r_3+r_4}.
\label{eq:B>=A}$$ and $$\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3+r_4} \leq \frac{1}{r_3} + \frac{1}{r_1+r_4} + \frac{1}{r_2+r_4}.
\label{eq:B>=C}$$
*Proof.* If strategy B is optimal for the Searcher, then $V_B \geq V_A$, which is equivalent to , and $V_B \geq V_C$, which is equivalent to . Therefore, and are necessary conditions.
We next prove and are sufficient conditions. Since the Searcher can use strategy B to guarantee an expected payoff at least $V_B$, it remains to show that the Hider has a mixed strategy to guarantee an expected payoff no more than $V_B$. Let $q_{ij}$ denote the probability that the Hider hides the 2 booby traps in boxes $i$ and $j$, and require $$\begin{aligned}
q_{12} &= \frac{V_B}{r_3+r_4}, \label{eq:12} \\
q_{34} &= 0, \label{eq:34} \\
q_{23} + q_{24} + q_{34} &= \frac{V_B}{r_1}, \label{eq:23+24} \\
q_{13} + q_{14} + q_{34} &= \frac{V_B}{r_2}. \label{eq:13+14}\end{aligned}$$
These constraints ensure that $\sum_{1 \leq i < j \leq 4} q_{ij} = 1$, and guarantee an expected payoff no more than $V_B$ if the Searcher uses pure strategies $\{3, 4\}$, $\{1, 2\}$, {1}, and {2}.
The Hider also needs to ensure an expected payoff no more than $V_B$ if the Searcher uses either $\{3\}$ or $\{4\}$, so we need to require $$\begin{aligned}
q_{12} + q_{14} + q_{24} &\leq \frac{V_B}{r_3}, \label{eq:12+14+24} \\
q_{12} + q_{13} + q_{23} &\leq \frac{V_B}{r_4}; \label{eq:12+13+23}\end{aligned}$$ and if the Searcher uses $\{1,3\}$, $\{2,3\}$, $\{2,4\}$, or $\{1,4\}$, so we also need to require $$\begin{aligned}
q_{24} &\leq \frac{V_B}{r_1+r_3}. \label{eq:24} \\
q_{14} &\leq \frac{V_B}{r_2+r_3}, \label{eq:14} \\
q_{13} &\leq \frac{V_B}{r_2+r_4}, \label{eq:13} \\
q_{23} &\leq \frac{V_B}{r_1+r_4}. \label{eq:23}\end{aligned}$$ To complete the proof, we need to show that there exists a feasible nonnegative solution to $q_{ij}$, $1 \leq i < j \leq 4$ subject to the constraints in through .
To proceed, write $$\frac{q_{13}}{V_B} = x, \qquad \frac{q_{23}}{V_B} = y,
\label{eq:13,23}$$ and use in and to obtain $$\frac{q_{24}}{V_B} = \frac{1}{r_1} - y, \qquad
\frac{q_{14}}{V_B} = \frac{1}{r_2} - x.
\label{eq:14,24}$$ To ensure $q_{13}, q_{23}, q_{14}, q_{24} \geq 0$, we need that $$0 \leq x \leq \frac{1}{r_2}, \qquad 0 \leq y \leq \frac{1}{r_1}.
\label{eq:x,y}$$
Next, subsitute and into – to rewrite the 6 inequalities constraints in terms of $x$ and $y$. Constraints and together become $$\frac{1}{r_1} + \frac{1}{r_2} - \frac{1}{r_3} + \frac{1}{r_3+r_4} \leq x + y \leq \frac{1}{r_4} - \frac{1}{r_3+r_4}.
\label{eq:x+y}$$ Constraints and together become $$\frac{1}{r_2} - \frac{1}{r_2 + r_3} \leq x \leq \frac{1}{r_2+r_4},
\label{eq:x}$$ and constraints and together become $$\frac{1}{r_1} - \frac{1}{r_1 + r_3} \leq y \leq \frac{1}{r_1+r_4}.
\label{eq:y}$$ Because constraints and make constraint redundant, it remains to show that there exists a feasible solution to $x$ and $y$ subject to constraints , , and .
First, note that in each of , , and , the unknown’s upper bound is greater than or equal to its lower bound. The feasibility of $x+y$ in follows directly from . The feasibility of $x$ in follows from $r_2 \geq r_3 \geq r_4$, and the feasibility of follows from $r_1 \geq r_3 \geq r_4$.
To complete the proof, we need to show that the sum between the upper bound (lower bound, respectively) of $x$ in and the upper bound (lower bound, respectively) of $y$ in is greater than or equal to the lower bound (upper bound, respectively) of $x+y$ in .
The first claim follows directly from . The second claim states that $$\frac{1}{r_2} - \frac{1}{r_2 + r_3} + \frac{1}{r_1} - \frac{1}{r_1 + r_3} \leq \frac{1}{r_4} - \frac{1}{r_3+r_4}.$$ To prove it, start with the left-hand side to obtain $$\begin{aligned}
r_3 \left( \frac{1}{r_2(r_2 + r_3)} + \frac{1}{r_1(r_1 + r_3)} \right) &\leq r_3 \left( \frac{1}{r_2(r_2 + r_4)} + \frac{1}{r_1(r_1 + r_4)} \right) \\ & \leq r_3 \left(\frac{1}{r_3(r_3+r_4)} \right) \\ & \leq r_3 \left(\frac{1}{r_4(r_3+r_4)} \right) \\
&= \frac{1}{r_4} - \frac{1}{r_3+r_4},\end{aligned}$$ where the first and third inequalities are due to $r_3 \geq r_4$, and the second inequality is due to . Consequently, we have proved that there exists a feasible solution to $x$ and $y$ that satisfy constraints , , and . In other words, we have proved that there exists a feasible solution to $q_{13}, q_{14}, q_{23}, q_{24}$ that satisfy the constraints in through . Therefore, we have shown that the Hider has a mixed strategy that guarantees the Searcher no more $V_B$, which completes the proof. $\Box$
### Optimality of Strategy A
We begin with two lemmas.
\[le:r1r2\] If $r_1 \geq r_2 \geq r_3 \geq r_4 \geq 0$, and holds, then $$\frac{1}{r_i} + \frac{1}{r_j} \geq \frac{1}{r_k} + \frac{1}{r_l} - \frac{2}{r_k+r_l},$$ where $i,j,k,l$ is any permutation of $\{1,2,3,4\}$.
*Proof.* Rewriting as $$\left(\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3+r_4} \right) \geq \left( \frac{1}{r_3} + \frac{1}{r_4} - \frac{1}{r_3+r_4} \right).$$ Because $r_1 \geq r_2 \geq r_3 \geq r_4 \geq 0$, with some algebra one can verify that any other permutation will make the left-hand side of the preceding larger and the right-hand side of the preceding smaller, so the inequality still holds. $\Box$
\[le:r4\] If $r_1 \geq r_2 \geq r_3 \geq r_4 \geq 0$, and holds, then $$\frac{1}{r_l} \leq \frac{1}{r_i+r_l} + \frac{1}{r_j+r_l} + \frac{1}{r_k+r_l},$$ where $i,j,k,l$ is any permutation of $\{1,2,3,4\}$.
*Proof.* Multiplying by $r_4 (r_1+r_4) (r_2+r_4) (r_3+r_4)$ on both sides of and canceling out common terms, we obtain $$r_1 r_2 r_3 \leq ((r_1 + r_2 + r_3 +r_4) + r_4) \, r_4^2.$$ Because $r_4$ is the smallest, it is clear that any other permutation will make the left-hand side of the preceding smaller and the right-hand side of the preceding larger, so the inequality still holds. $\Box$
We are now ready for the main result in this subsection.
\[th:A\] Strategy A is optimal for the Searcher and the value of the game is $V_A$ if and only if $$\frac{1}{r_1} + \frac{1}{r_2} \geq \frac{1}{r_3} + \frac{1}{r_4} - \frac{2}{r_3+r_4},
\label{eq:A>=B}$$ and $$\frac{1}{r_4} \leq \frac{1}{r_1+r_4} + \frac{1}{r_2+r_4} + \frac{1}{r_3+r_4},
\label{eq:A>=C}$$
*Proof.* If strategy A is optimal for the Searcher, then $V_A \geq V_B$, which is equivalent to , and $V_A \geq V_C$, which is equivalent to . Therefore, and are necessary conditions.
We next prove and are sufficient conditions. Since the Searcher can use strategy A to guarantee an expected payoff at least $V_A$, it remains to show that the Hider has a mixed strategy to guarantee an expected payoff no more than $V_A$. Let $q_{ij}$ denote the probability that the Hider hides the 2 booby traps in boxes $i$ and $j$, with probability $q_{ij} \geq 0$ and $\sum_{1 \leq i < j \leq 4} q_{ij} = 1$. In particular, we will show that the Hider has a feasible mixed strategy to achieve an expected payoff exactly $V_A$ if the Searcher opens any one box, and guarantees an expected payoff no more than $V_A$ if the Searcher opens any two boxes. In other words, we claim that there exists a feasible solution to $$\begin{aligned}
\sum_{1 \leq i < j \leq 4} q_{ij} &= 1, & \quad q_{ij} &\geq 0, \qquad \text{for } 1 \leq i < j \leq 4 \\
(q_{23} + q_{24} + q_{34} ) r_1 &= V_A, & \quad q_{12} (r_3 + r_4) &\leq V_A, \\
(q_{13} + q_{14} + q_{34} ) r_2 &= V_A, & \quad q_{13} (r_2 + r_4) &\leq V_A, \\
(q_{12} + q_{14} + q_{24} ) r_3 &= V_A, & \quad q_{14} (r_2 + r_3) &\leq V_A, \\
(q_{12} + q_{13} + q_{23} ) r_4 &= V_A, & \quad q_{23} (r_1 + r_4) &\leq V_A, \\
&& \quad q_{24} (r_1 + r_3) &\leq V_A, \\
&& \quad q_{34} (r_1 + r_2) &\leq V_A.\end{aligned}$$
To proceed, write $x = q_{34} / V_A$ and $y = q_{24} / V_A$, and use the first 5 equality constraints (in the left column) to solve $q_{ij}/V_A$ in terms of $x$ and $y$ for $1 \leq i < j \leq 4$. Use $q_{ij} \geq 0$ to obtain lower bounds for $q_{ij}/V_A$, for $1 \leq i < j \leq 4$, and the last 6 inequality constraints (in the right column) to obtain their upper bounds. The results are summarized below. $$\begin{aligned}
\frac{1}{(r_3 + r_4)} &\geq \frac{q_{12}}{V_A} = x + \frac{1}{2} \left( - \frac{1}{r_1} - \frac{1}{r_2} + \frac{1}{r_3} + \frac{1}{r_4} \right) \geq 0, \\
\frac{1}{(r_2 + r_4)} &\geq \frac{q_{13}}{V_A} = y + \frac{1}{2} \left( - \frac{1}{r_1} + \frac{1}{r_2} - \frac{1}{r_3} + \frac{1}{r_4} \right) \geq 0, \\
\frac{1}{(r_2 + r_3)} &\geq \frac{q_{14}}{V_A} = \frac{1}{2} \left( \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} -\frac{1}{r_4} \right) - x - y \geq 0, \\
\frac{1}{(r_1 + r_4)} &\geq \frac{q_{23}}{V_A} = \frac{1}{r_1} - x - y \geq 0, \\
\frac{1}{(r_1 + r_3)} &\geq \frac{q_{24}}{V_A} = y \geq 0, \\
\frac{1}{(r_1 + r_2)} &\geq \frac{q_{34}}{V_A} = x \geq 0.\end{aligned}$$ Rewrite the preceding in terms of $x$, $y$, and $x+y$, to get the following. $$\begin{aligned}
0 \leq x &\leq \frac{1}{r_1+r_2}, \label{eq:x1} \\
- \frac{1}{2} \left( -\frac{1}{r_1} - \frac{1}{r_2} + \frac{1}{r_3} + \frac{1}{r_4} \right) \leq x &\leq \frac{1}{r_3+r_4} - \frac{1}{2} \left( -\frac{1}{r_1} - \frac{1}{r_2} + \frac{1}{r_3} + \frac{1}{r_4} \right), \label{eq:x2} \\
0 \leq y &\leq \frac{1}{r_1+r_3}, \label{eq:y1} \\
- \frac{1}{2} \left( -\frac{1}{r_1} + \frac{1}{r_2} - \frac{1}{r_3} + \frac{1}{r_4} \right) \leq y &\leq \frac{1}{r_2+r_4} - \frac{1}{2} \left( -\frac{1}{r_1} + \frac{1}{r_2} - \frac{1}{r_3} + \frac{1}{r_4} \right) \label{eq:y2}, \\
\frac{1}{r_1} - \frac{1}{r_1+r_4} \leq x + y &\leq \frac{1}{r_1}, \label{eq:xy1} \\
\frac{1}{2} \left( \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} -\frac{1}{r_4} \right) - \frac{1}{r_2+r_3} \leq x + y &\leq \frac{1}{2} \left( \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} - \frac{1}{r_4} \right). \label{eq:xy2}\end{aligned}$$ It then remains to show that there exists a feasible solution to $x$ and $y$ that satisfies these six linear constraints.
First, we claim there exists a feasible solution to $x$ that satisfies the two constraints and . The larger lower bound for $x$ is clearly 0, since $r_1 \geq r_2 \geq r_3 \geq r_4$. While it is not clear which of the two upper bounds for $x$ is smaller, one can verify that both are nonnegative, due to . With a similar argument, there exists a feasible solution to $y$ that satisfies and , due to and Lemma \[le:r1r2\].
Second, there exists a feasible solution to $x+y$ that satisfies and , because each of the two upper bounds is greater than or equal to each of the two lower bounds, due to and Lemma \[le:r1r2\].
To complete the proof, we need to show that the sum between the upper bound (lower bound, respectively) of $x$ implied by and and the upper bound (lower bound, respectively) of $y$ implied by and is greater than or equal to the lower bound (upper bound, respectively) of $x+y$ implied by and .
From , , , and , the lower bound is 0 for $x$ and $y$, so we need to check the right-hand sides of and are both nonnegative. The part concerning is trivial, and the part concerning follows because $$\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} \geq \frac{1}{r_1+r_4} + \frac{1}{r_2+r_4} + \frac{1}{r_3+r_4} \geq \frac{1}{r_4},$$ where the second inequality follows from .
Finally, we need to show that the sum between the upper bound of $x$ implied by and and the upper bound of $y$ implied by and is greater than or equal to the lower bound of $x+y$ implied by and . We do so by showing that the sum of either upper bound of $x$ in or , and either upper bound of $y$ in or , is greater than or equal to either lower bound of $x+y$ in or . There are thus 8 inequalities to verify. For example, from , , , we need to show that $$\frac{1}{r_1+r_2} + \frac{1}{r_1+r_3} \geq \frac{1}{r_1} - \frac{1}{r_1+r_4},$$ which follows from and Lemma \[le:r4\]. Using and Lemma \[le:r4\], we can also verify the corresponding inequality involving , , , and that involving , , , and that involving , , .
We next verify the corresponding inequality involving , , , which requires $$\frac{1}{r_1+r_2} + \frac{1}{r_1+r_3} \geq \frac{1}{2} \left( \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} -\frac{1}{r_4} \right) - \frac{1}{r_2+r_3},$$ which is equivalent to $$\frac{2}{r_1+r_2} + \frac{2}{r_1+r_3} \geq \left( \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} -\frac{1}{r_4} \right) - \frac{2}{r_2+r_3}.
\label{eq:r2r3}$$ Starting from the right-hand side to get $$\begin{aligned}
\left( \frac{1}{r_2} + \frac{1}{r_3} - \frac{2}{r_2+r_3} \right) + \frac{1}{r_1} - \frac{1}{r_4} &\leq \left( \frac{1}{r_1} + \frac{1}{r_4} \right) + \frac{1}{r_1} - \frac{1}{r_4} \\
&= \frac{2}{r_1+r_1} + \frac{2}{r_1+r_1} \\
&\leq \frac{2}{r_1+r_2} + \frac{2}{r_1+r_3},\end{aligned}$$ where the first inequality follows from and Lemma \[le:r1r2\], and the last inequality follows from $r_1 \geq r_2$ and $r_1 \geq r_3$.
We can go through the same procedure to verify the corresponding inequality involving , , , and that involving , , , and that involving , , . Each of these three inequalities has the same form as in , with the bracket on the right-hand side having three positive terms and 1 negative term. The key to establish the inequality is to apply Lemma \[le:r1r2\] to the two positive terms with the largest indices among the three positive terms; for example, to prove we pick $r_2$ and $r_3$ to apply Lemma \[le:r1r2\].
Because there exists feasible solution to $x$ and $y$ that satisfies constraints –, we have shown that the Hider has a mixed strategy that guarantees an expected payoff no more than $V_A$, which completes the proof. $\Box$
General bounds
--------------
Here we give some general bounds on the value of the game, starting with an upper bound and a lower bound that are close to each other when $n$ is large and the rewards are small.
(Upper bound) The value $V$ of the game on the complete hypergraph satisfies $$\begin{aligned}
V \le \frac{R_0}{k+1}\left(1- \frac{1}{k+1} \right)^k, \label{eq:Vub}
\end{aligned}$$ where $R_0 = \sum_{i=1}^n r_i$.
*Proof.* First assume all the rewards are integers. If we define a new game by replacing a box of reward $r$ with two boxes of reward $r_1$ and $r_2$ with $r_1+r_2 = r$, then the value of the game can only increase, because any Searcher strategy in the original game can also be used in the new game. With a similar argument, we can replace each box $i$ with $r_i$ boxes each containing a reward of $1$, resulting in a new game with equal rewards of 1, whose value is at least as great as the original game. The value of the new game is equal to $U(R_0, k)$, as defined in Theorem \[thm:eq\]. Observe that by further replacing each box with $t$ new boxes each containing a reward of $1/t$, we obtain a game whose value $U(t R_0, k)/t$ is no smaller than that of the original game. Therefore, the value of the original game is bounded above by $$\lim_{t \rightarrow \infty} \frac{U(t R_0, k)}{t} = R_0 \lim_{t \rightarrow \infty} \frac{U(t R_0, k)}{t R_0} = \frac{R_0}{k+1}\left(1- \frac{1}{k+1} \right)^k,$$ where the last equality follows from .
If the rewards are all rational numbers, then we can obtain an equivalent game with integer rewards by multiplying them all by a common denominator $d$. All the payoffs in the resulting game will be larger by a factor of $d$, and therefore so will the value of the game and the parameter $R_0$. As a consequence, the left- and right-hand sides of (\[eq:Vub\]) will both be larger by a factor of $d$, so the inequality still holds. If the rewards are real numbers, then they can be approximated arbitrarily closely to rational numbers, so that the left- and right-hand sides of (\[eq:Vub\]) are also approximated arbitrarily closely, and the bound still holds. $\Box$
\[prop:indep\] (Lower bound) The value $V$ of the game on the complete hypergraph satisfies $$\begin{aligned}
V \ge \frac{R_0}{k+1}\left(1- \frac{1}{k+1} \right)^k \left( 1 - \frac{r([k])}{R_0} \right), \label{eq:Vlb}
\end{aligned}$$ where $R_0 = \sum_{i=1}^n r_i$ and $r([k]) = \sum_{i=1}^k r_i$.
*Proof.* Consider a Searcher strategy with which each box is independently opened with probability $1/(k+1)$. For a given Hider strategy $H \in [n]^{(k)}$, the probability that none of the boxes in $H$ is opened is $(1-1/(k+1))^k$. If the Searcher does not open any box in $H$, her expected payoff is $r(\bar{H})/(k+1)$; if she opens any boxes in $H$, her payoff is zero. Therefore, with such strategy the Searcher’s expected payoff is $$\left(1- \frac{1}{k+1} \right)^k \left( \frac{r(\bar H)}{k+1} \right).$$ The preceding in minimized when $r(H)$ is maximized; that is, for $H = [k]$. In this case, the expected payoff is the right-hand side of (\[eq:Vlb\]). $\Box$
It is worth pointing out that, among all the Searcher strategies that open each box independently at random with some given probability $p$, the one that guarantees the greatest expected payoff is given by $p=1/(k+1)$, namely the strategy of Proposition \[prop:indep\]. This claim can be verified via elementary calculus. The bounds in (\[eq:Vub\]) and (\[eq:Vlb\]) are close when $r([k])/R_0$ is close to zero. In particular, the bounds are asymptotically equal for constant $k$, as $n \rightarrow \infty$, if all the rewards are all $o(n)$. In this case, the Searcher strategy that opens each box independently with probability $1/(k+1)$ is asymptotically optimal.
Note that all the optimal Searcher strategies presented in this paper share the same form: the Searcher chooses each hyperedge $S$ with probability 0, or with probability proportional to $1/r(S)$. This observation gives rise to a set of lower bounds on the value, generalizing the Searcher strategy from Lemma \[lem:null\].
\[prop:partition\] Consider the search game played on an arbitrary hypergraph, and let $\mathcal{S} = \{S_1, \ldots, S_t \}$ be a set of hyperedges. Consider the Searcher strategy $p$ that chooses $S_j$ with probability $p(S_j) = \lambda/r(S_j)$, where $$\lambda \equiv \lambda(\mathcal S) \equiv \frac{1}{\sum_{j=1}^t 1/r(S_j)}.$$ This strategy guarantees an expected payoff of at least $m \lambda$, where $$m \equiv m(\mathcal S) \equiv \min_{H \in [n]^{(k)}} |\{S_j \in \mathcal{S}: S_j \cap H = \emptyset \} |$$ is the minimal—over all possible Hider strategies—number of hyperedges in $\mathcal S$ that contain no booby traps.
*Proof.* For a given Hider strategy $H$, let $ \mathcal{A} = \{S_j \in \mathcal{S}: S_j \cap H = \emptyset \} |$ be the set of hyperedges in $\mathcal{S}$ that contain no booby traps. By definition of $m$, we have $|\mathcal{A} | \ge m$. Hence, the expected payoff against $H$ is $$R(p,H) = \sum_{S \in \mathcal A} p(S) r(S) = \sum_{S \in \mathcal A} \lambda \ge m \lambda,$$ which completes the proof. $\Box$
If $\mathcal{S}$ is a partition of $[n]$, then the minimal number of hyepredges that contain no booby traps is $m(\mathcal{S})=t-k$, and Proposition \[prop:partition\] implies that the value is at least $(t-k) \lambda(\mathcal{S})$.
Based on the solutions to special cases presented in this paper, we make a conjecture on the Searcher’s optimal strategy.
Consider the booby trap search game played on a hypergraph. There exists an optimal Searcher strategy with which each hyperedge will not be chosen at all, or will be chosen with probability inversely proportional to the sum of the rewards on that hyperedge. In other words, the Searcher can achieve optimality by choosing the best subset of hyperedges and using the mixed strategy described in Proposition \[prop:partition\].
Conclusion {#sec:conclusion}
==========
This paper presents a new search game on a hypergraph between a Searcher and a Hider. The Searcher wants to collect maximum reward but has to avoid booby traps planted by the Hider. We present the solutions to a few special cases, based on which we make a conjecture about the form of the solution in general.
Two of the special cases presented in this paper involve the Searcher opening just one box, or opening any number of boxes. A relevant and practical situation may restrict the Searcher to opening a certain fixed number of boxes. If the booby trap only partially injures the Searcher but does not incapacitate her, then we can consider a model extension that allows the Searcher to keep going until she encounters a certain number of booby traps.
Acknowledgements {#acknowledgements .unnumbered}
================
This material is based upon work supported by the National Science Foundation under Grant No. IIS-1909446.
[^1]: Department of Management Science and Information Systems, Rutgers Business School, Newark, NJ 07102, tlidbetter@business.rutgers.edu
[^2]: Operations Research Department, Naval Postgraduate School, Monterey, CA 93943, kylin@nps.edu
|
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abstract: 'We show that any II$_1$ factor that has the same 4-quantifier theory as the hyperfinite II$_1$ factor $\R$ satisfies the conclusion of the Popa Factorial Commutant Embedding Problem (FCEP) and has the Brown property. These results improve recent results proving the same conclusions under the stronger assumption that the factor is actually elementarily equivalent to $\R$. In the same spirit, we improve a recent result of the first-named author, who showed that if (1) the amalgamated free product of embeddable factors over a property (T) base is once again embeddable, and (2) $\R$ is an infinitely generic embeddable factor, then the FCEP is true of all property (T) factors. In this paper, it is shown that item (2) can be weakened to assume that $\R$ has the same 3-quantifier theory as an infinitely generic embeddable factor.'
address:
- |
Department of Mathematics\
University of California, Irvine, 340 Rowland Hall (Bldg.\# 400), Irvine, CA 92697-3875
- 'Department of Mathematics and Statistics, McMaster University, 1280 Main St., Hamilton ON, Canada L8S 4K1'
author:
- Isaac Goldbring and Bradd Hart
title: 'Properties expressible in small fragments of the theory of the hyperfinite II$_1$ factor'
---
Introduction
============
The following problem of Popa is the main motivation for the work in this paper:
Suppose that $M$ is a separable embeddable factor. Does there exist an embedding $i:M\hookrightarrow \R^\u$ with factorial commutant, that is, such that $i(M)'\cap \R^\u$ is a factor?
Until recently, very little progress on the FCEP had been made. In [@jung], the following theorem was proven:
\[eeRFCEP\] If $M$ is elementarily equivalent to $\R$, then $M$ satisfies the FCEP.
Recall that II$_1$ factors $M$ and $N$ are elementarily equivalent, denoted $M\equiv N$, if, for any sentence $\sigma$ in the language of tracial von Neumann algebras, one has $\sigma^M=\sigma^N$. A logic-free definition can be given using the Keisler-Shelah Theorem: $M$ and $N$ are elementarily equivalent if and only if they have isomorphic ultrapowers.[^1] By [@MTOA3 Theorem 4.3], any separable II$_1$ factor $M$ has continuum many nonisomorphic separable II$_1$ factors elementarily equivalent to it, whence Theorem 1 gave continuum many new examples of separable II$_1$ factors satisfying the FCEP.
In this paper, we weaken the assumption of the previous theorem and arrive at the same conclusion. We say that II$_1$ factors $M$ and $N$ are $k$-elementarily equivalent, denoted $M\equiv_k N$, if they agree on all formulae of quantifier-complexity at most $k$. (This will be defined precisely in the last section.). The following is an imprecise version of our first main result:
\[4FCEP\] If $M\equiv_4 \R$, then $M$ satisfies the FCEP.
In another direction, one of the main results of [@Popa] was progress on the FCEP problem for embeddable[^2] property (T) factors:
\[propT\] Suppose that the following two statements are true:
1. Whenever $M_1$ and $M_2$ are embeddable II$_1$ factors with a common property (T) subfactor $N$, then the amalgamated free product $M_1*_N M_2$ is also embeddable.
2. $\R$ is an *infinitely generic* embeddable factor.
Then every embeddable property (T) factor satisfies the FCEP.
Infinitely generic factors form a large class of “rich” II$_1$ factors and more information about them can be found in [@ecfactor]. In [@ecfactor], it was claimed that $\R$ is an infinitely generic embeddable factor. However, the proof there is incredibly flawed and settling the question of whether or not $\R$ is actually an infinitely generic embeddable factor remains an important open question.
Ideally, one would like to remove the model-theoretic assumption (2) in the previous theorem, leaving only the operator-algebraic obstacle (1). Item (2) in the previous theorem is equivalent to the statement that $\R$ is elementarily equivalent to an infinitely generic embeddable factor. Consequently, the following theorem, a consequence of a more general result proven in Section 4, is a strengthening of the previous result:
\[newpropT\] Suppose that the following two statements are true:
1. Whenever $M_1$ and $M_2$ are embeddable II$_1$ factors with a common property (T) subfactor $N$, then the amalgamated free product $M_1*_N M_2$ is also embeddable.
2. There is an infinitely generic embeddable factor $M$ such that $M\equiv_3 \R$.
Then every embeddable property (T) factor satisfies the FCEP.
It is worth noting that any infinitely generic embeddable factor $M$ satisfies $M\equiv_2 \R$. In Section 4, we also note that the statement that there is an infinitely generic embeddable factor $M$ such that $M\equiv_3 \R$ is already known to be “halfway true.”
A crucial ingredient to the proof of Theorem 1 above is the following result of Nate Brown [@brown Theorem 6.9]:
If $N$ is a separable subfactor of $\R^\u$, then there is a separable subfactor $P$ of $\R^\u$ with $N\subseteq P$ such that $P'\cap \R^\u$ is a II$_1$ factor.
In [@jung], we said the II$_1$ factor $M$ had the **Brown property** if, for all separable subfactors $N$ of $M^\u$, then there is a separable subfactor $P$ of $M^\u$ with $N\subseteq P$ such that $P'\cap M^\u$ is a II$_1$ factor. It was shown in [@jung] that any $M\equiv \R$ has the Brown property. In the last section of this paper, we prove a strengthening of this result:
\[4Brown\] If $M\equiv_4 \R$, then $M$ has the Brown property.
An interesting question arises: are these results actually improvements of their predecessors? Indeed, perhaps it is the case that there is $k\in \mathbb N$ such that if $M\equiv_k \R$, then $M\equiv \R$. If this were to happen, then one would say that $\Th(\R)$ has *quantifier simplification*. Given recent results showing that the $\Th(\R)$ is very complicated from the model-theoretic perspective (see, e.g., [@ecfactor] and [@universaltheory]), we strongly believe in the following:
$\Th(\R)$ does not admit quantifier simplification.
For the rest of this paper, we work under the assumption that the previous Conjecture has a positive solution. In this case, Theorem A yields continuum many examples of factors satisfying the FCEP not covered by Theorem 1. Similarly, Theorem C yields continuum many new examples of factors with the Brown property.
Infinitely generic embeddable factors form a subclass of the more general class of **existentially closed embeddable factors**. An embeddable factor $M$ is existentially closed (e.c.) if: whenever $N$ is an embeddable factor with $M\subseteq N$, there is an embedding $N\hookrightarrow M^\u$ that restricts to the diagonal embedding $M\hookrightarrow M^\u$. It was noted in [@ecfactor] that $\R$ is an e.c. embeddable factor. Existentially closed embeddable factors have proven very important in applications of model-theoretic ideas to the study of II$_1$ factors. It is a major open question whether or not there are two non-elementarily equivalent e.c. embeddable factors. If $\R$ is not infinitely generic, then we would have an example of such a pair of e.c. embeddable factors. However, it could still be the case that all e.c. factors have the same 3-quantifier theory, in which case (2’) in Theorem B is actually satisfied.
In order to keep this note relatively self-contained, we do not include much model-theoretic or operator-algebraic background. A rather lengthy introduction to model-theoretic ideas as they pertain to problems around factorial commutants can be found in [@jung]. In Section 2, we prove the main model-theoretic tools needed in the proof of Theorem A. In Section 3 we prove Theorem A, in Section 4 we prove Theorem B, and in Section 5 we prove Theorem C.
Weak heirs and weak embeddings
==============================
In this section, we fix a continuous language $L$. We say that a formula $\varphi$ is in **prenex normal form** if it is of the form $$Q_1x_1\cdots Q_m x_m \psi(x_1,\ldots,x_m,\vec y),$$ with each $Q_i\in \{\sup,\inf\}$ and with $\psi$ quantifier-free. If the $Q_i$’s alternate type, then we say that $\varphi$ is $\forall_m$ (respectively $\exists_m$) if $Q_1=\sup$ (resp. $Q_1=\inf$).[^3] If a formula is equivalent to a $\forall_m$ or $\exists_m$ formula, we often abuse terminology and refer to the formula itself as $\forall_m$ or $\exists_m$.
By a **fragment** of $L$-formulae, we mean a set $\Delta$ consisting of all $\forall_m$-formulae or of all $\exists_m$-formulae for some $m$.
Fix an $L$-structure $M$, parameter sets $A\subseteq B\subseteq M$, and fragments $\Delta$ and $\Delta'$.
1. For $c\in M$, we set $\tp_\Delta^M(c/A)$ to be the set of all conditions $\varphi(x)=r$, where $\varphi\in \Delta$ has parameters from $A$ and $\varphi(c)^M=r$.
2. $S_\Delta^M(A)$ denotes the set of all $\tp_{\Delta}^M(c/A)$ for $c\in M$.
3. For $p\in S_\Delta^M(A)$ and $\varphi(x)$ a formula from $\Delta$ with parameters from $A$, we set $\varphi(x)^p$ to be the unique $r$ so that $\varphi(x)=r$ belongs to $p$.
4. For $c\in M$, we set $\tp_{\Delta,\Delta'}^M(c/A,B)$ to be the union of $\tp_\Delta^M(c/A)$ and $\tp_{\Delta'}^M(c/B)$.
5. We let $S_{\Delta,\Delta'}(A,B)$ denote the set of all $\tp_{\Delta,\Delta'}^M(c/A,B)$ for $c\in M$. We extend the notation $\varphi(x)^p$ to $S_{\Delta,\Delta'}(A,B)$ in the obvious way.
6. If $p\in S_\Delta(A)$, $q\in S_{\Delta,\Delta'}(A,B)$, and $\Delta'\subseteq \Delta$, we say that $q$ is an **heir** of $p$ if, for every $b\in B$, every $\varphi(x,y)\in \Delta'$, and every $\epsilon>0$, there is $a\in A$ such that $|\varphi(x,a)^p-\varphi(x,b)^q|<\epsilon$.
Suppose that $i:N\hookrightarrow M$ is an embedding between $L$-structures and $\Delta$ is a fragment. We say that $i$ is:
1. **downward $\Delta$** if, for any nonnegative formula $\varphi(x)\in \Delta$ and any $a\in N$, if $\varphi(i(a))^M=0$, then $\varphi(a)^N=0$;
2. **upward $\Delta$** if, for any nonnegative formula $\varphi(x)\in \Delta$ and any $a\in N$, if $\varphi(a)^N=0$, then $\varphi(i(a))^M=0$.
We note one obvious fact:
Given an embedding $i:N\hookrightarrow M$, we have that $i$ is downwards $\exists_m$ if and only if $i$ is upwards $\forall_m$.
Suppose that $i$ is not upwards $\forall_m$, so there is a nonnegative $\forall_m$ formula $\varphi(x)$ and $a\in N$ such that $\varphi(a)^N=0$ but $\varphi(i(a))^M=\epsilon>0$. Then $(\epsilon{
\buildrel\textstyle\ .\over{\hbox{
\vrule height3pt depth0pt width0pt}{\smash-}
}}\varphi(i(a)))^M=0$ and since this formula is equivalent to a $\exists_m$ formula, we have that $(\epsilon{
\buildrel\textstyle\ .\over{\hbox{
\vrule height3pt depth0pt width0pt}{\smash-}
}}\varphi(a))^N=0$, a contradiction. The other direction is similar.
The following is our main technical result concerning the existence of weak heirs. In the remainder of this paper, $\u$ denotes a countably incomplete ultrafilter on some index set (unless otherwise specified).
\[heir\] Suppose that $M$ is a separable $L$-structure. Fix a separable substructure $N$ of $M^\u$ such that the inclusion $N\subseteq M^\u$ is downward $\exists_{m+2}$. Fix also $p\in S_{\forall_m}(N)$. Then for any separable parameter set $A$ with $N\subseteq A\subseteq M^\u$ and any $n<m$, there is $q\in S_{\forall_m,\forall_n}(N,A)$ that is an heir of $p$.
We seek $a\in M^\u$ satisfying the following two kinds of conditions:
1. $\psi(a)=\psi(x)^p$ for any $\forall_m$-formula $\psi(x)$ with parameters from $N$;
2. $\varphi(a,c)^{M^\u}\geq \frac{\epsilon}{2}$ for any $\forall_{n+1}$-formula $\varphi(x,y)$ with parameters from $A$ and any $\epsilon>0$ such that $\varphi(x,b)^p\geq \epsilon$ for all $b\in N$.
Indeed, if $a$ is as above, we claim that $q:=\tp_{\forall_m,\forall_n}^{M^\u}(a/A)$ is an heir of $p$. By (1), $q$ is an extension of $p$. To see that $q$ is an heir, fix a $\forall_n$-formula $\varphi(x,c)$ with parameters from $A$ and set $s:=\varphi(x,c)^q=\varphi(a,c)^{M^\u}$. Suppose, towards a contradiction, that there is $\epsilon>0$ such that $|\varphi(x,b)^p-s|\geq \epsilon$ for all $b\in N$. It follows that $|\varphi(x,b)-s|^p\geq \epsilon$ for all $b\in N$. Since $|\varphi(x,b)-s|$ is logically equivalent to a $\forall_{n+1}$, whence, by (2), $|\varphi(a,c)^{M^\u}-s|\geq \frac{\epsilon}{2}$, leading to a contradiction.
Suppose now, towards a contradiction, that no such $a\in M^\u$ exists. By countable saturation, it follows that there are:
- a $\forall_m$-formula $\psi(x)$ with parameters from $N$ such that $\psi(x)^p=0$,
- a $\delta>0$, and
- formulae $\varphi_1(x,c_1),\ldots,\varphi_k(x,c_k)$ with parameters from $A$ as in (2)
such that, for any $a\in M^\u$, if $\psi(a)<\delta$, then $\varphi_i(a,c_i)<\frac{\epsilon}{2}$ for some $i=1,\ldots,k$.
In other words, $$\left(\sup_x\min\left(\delta{\mathbin{\mathpalette\dotminussym{}}}\psi(x),\min_{1\leq i\leq k}\left(\varphi_i(x,c_i){\mathbin{\mathpalette\dotminussym{}}}\frac{\epsilon}{2}\right)\right)\right)^{M^\u}=0.$$ Consequently, $$\left(\inf_{y_1}\cdots\inf_{y_k}\sup_x\min\left(\delta{\mathbin{\mathpalette\dotminussym{}}}\psi(x),\min_{1\leq i\leq k}\left(\varphi_i(x,y_i){\mathbin{\mathpalette\dotminussym{}}}\frac{\epsilon}{2}\right)\right)\right)^{M^\u}=0,$$ and thus, since the inclusion $N\subseteq M^\u$ is downward $\exists_{m+2}$, we have $$\left(\inf_{y_1}\cdots\inf_{y_m}\sup_x\min\left(\delta{\mathbin{\mathpalette\dotminussym{}}}\psi(x),\min_{1\leq i\leq m}\left(\varphi_i(x,y_i){\mathbin{\mathpalette\dotminussym{}}}\frac{\epsilon}{2}\right)\right)\right)^N=0.$$ Set $\eta:=\min(\delta,\frac{\epsilon}{2})$ and take $d_1,\ldots,d_k\in N$ such that $$\left(\sup_x\min\left(\delta{\mathbin{\mathpalette\dotminussym{}}}\psi(x),\min_{1\leq i\leq k}\left(\varphi_i(x,d_i){\mathbin{\mathpalette\dotminussym{}}}\frac{\epsilon}{2}\right)\right)\right)^N<\eta;$$ since the inclusion $N\subseteq M^\u$ is upward $\forall_{m+1}$, we have $$\left(\sup_x\min\left(\delta{\mathbin{\mathpalette\dotminussym{}}}\psi(x),\min_{1\leq i\leq k}\left(\varphi_i(x,d_i){\mathbin{\mathpalette\dotminussym{}}}\frac{\epsilon}{2}\right)\right)\right)^{M^\u}<\eta.$$ Take $a\in M^\u$ realizing $p$. Then $\psi(a)^{M^\u}=\psi(x)^p=0$, whence, since $\eta\leq \delta$, we have $\min_{1\leq i\leq k}(\varphi_i(x,d_i){\mathbin{\mathpalette\dotminussym{}}}\frac{\epsilon}{2})^{M^\u}<\eta\leq \frac{\epsilon}{2}$. Choosing $i$ such that $(\varphi_i(a,d_i){\mathbin{\mathpalette\dotminussym{}}}\frac{\epsilon}{2})^{M^\u}<\eta$, we get that $\varphi_i(x,d_i)^p=\varphi_i(a,d_i)^{M^\u}<\epsilon$, a contradiction.
We will be interested in the following special case of Theorem \[heir\]:
\[specialcase\] Suppose that $M$ is a separable $L$-structure. Fix a separable substructure $N$ of $M^\u$ such that the inclusion $N\subseteq M^\u$ is downward $\exists_{3}$. Fix also $p\in S_{\forall_1}(N)$. Then for any separable parameter set $A$ with $N\subseteq A\subseteq M^\u$, there is $q\in S_{\forall_1,\forall_0}(N,A)$ that is an heir of $p$.
Given a fragment $\Delta$ and an $L$-structure $M$, we set $$\Th_\Delta(M):=\{\sigma \ : \ \sigma \text{ is a nonnegative $L$-sentence from }\Delta \text{ and }\sigma^M=0\}.$$ If $N$ is another $L$-structure, we write $N\models \Th_\Delta(M)$ if $\sigma^N=0$ for all $\sigma\in \Th_\Delta(M)$.
We now prove a result connecting small quantifier-fragments of theories of structures with the existence of embeddings as in the previous theorem.
\[fragmentsandembeddings\] Suppose that $M$ and $N$ are separable $L$-structures and $m\in \mathbb N$. Then there is an embedding $i:N\hookrightarrow M^\u$ that is downwards $\exists_{m+2}$ if and only if $M\models \operatorname{Th}_{\exists_{m+3}}(N)$.
First suppose that a downwards $\exists_{m+2}$-embedding $i:N\hookrightarrow M^\u$ exists and $\sigma$ is a nonnegative $\exists_{m+3}$-sentence such that $\sigma^N=0$. Write $\sigma=\inf_x\varphi(x)$ with $\varphi$ a $\forall_{m+2}$-formula. Fix $\epsilon>0$ and take $a\in N$ such that $\varphi(a)<\epsilon$. Then $(\varphi(a){
\buildrel\textstyle\ .\over{\hbox{
\vrule height3pt depth0pt width0pt}{\smash-}
}}\epsilon)^N=0$, and since this formula is equivalent to a $\forall_{m+2}$-formula and $i$ is upwards $\forall_{m+2}$, we have that $(\varphi(i(a)){
\buildrel\textstyle\ .\over{\hbox{
\vrule height3pt depth0pt width0pt}{\smash-}
}}\epsilon)^{M^\u}=0$. Consequently, $(\inf_x(\varphi(x){
\buildrel\textstyle\ .\over{\hbox{
\vrule height3pt depth0pt width0pt}{\smash-}
}}\epsilon))^M=0$; since $M$ is arbitrary, we have that $\sigma^M=0$, as desired.
Conversely, suppose that $M\models \operatorname{Th}_{\exists_{m+3}}(N)$. Let $L_N$ be the language obtained by adding constants $c_a$ for $a\in N$. Set $\Gamma$ to be the following collection of $L_N$ sentences:
1. $\theta(c_{a_1},\ldots,c_{a_n})$, where $\theta$ is a nonnegative quantifier-free formula and $\theta(a_1,\ldots,a_n)^N=0$;
2. $\epsilon{
\buildrel\textstyle\ .\over{\hbox{
\vrule height3pt depth0pt width0pt}{\smash-}
}}\varphi(c_{a_1},\ldots,c_{a_n})$, where $\varphi$ is a $\exists_{m+2}$-formula with $\varphi(a_1,\ldots,a_n)^N\geq \epsilon$
If $\Gamma$ can be shown to be approximately finitely satisfiable in an expansion of $M$, then by countable saturation there is an expansion of $M^\u$ which is a model of $\Gamma$, and this yields the desired embedding. So suppose $\theta_1,\ldots,\theta_k$ are as in (1) and $\epsilon_j{
\buildrel\textstyle\ .\over{\hbox{
\vrule height3pt depth0pt width0pt}{\smash-}
}}\varphi_j$, $j=1,\ldots,l$, are as in (2). Then $$\inf_x\left(\max\left(\max_{i=1,\ldots,k}\theta_i(x),\max_{j=1,\ldots,l}(\epsilon_j{
\buildrel\textstyle\ .\over{\hbox{
\vrule height3pt depth0pt width0pt}{\smash-}
}}\varphi_j(x)\right)\right)$$ is equivalent to an $\exists_{m+3}$-sentence that evaluates to $0$ in $N$, whence, by assumption, also evaluates to $0$ in $M$. This completes the proof.
Combining Theorem \[heir\] and Proposition \[fragmentsandembeddings\], we arrive at:
\[whatwereallyuse\] Suppose that $M$ is a separable $L$-structure. Fix a separable substructure $N$ of $M^\u$ such that $M\models \operatorname{Th}_{\exists_{m+3}}(N)$. Fix also $p\in S_{\forall_m}^{M^\u}(N)$. Then for any separable parameter set $A$ with $N\subseteq A\subseteq M^\u$ and any $n<m$, there is $q\in S_{\forall_m,\forall_n}^{M^\u}(N,A)$ that is an heir of $p$. In particular, if $M\models \Th_{\exists_4}(N)$, then for any $p\in S_{\forall_1}^{M^\u}(N)$ and any separable parameter set $A$ with $N\subseteq A\subseteq M^\u$, there is $q\in S_{\forall_1,\forall_0}^{M^\u}(N,A)$ that is an heir of $p$.
Proof of Theorem A
==================
In this section, we apply the abstract results from the previous section to the setting of II$_1$ factors. Throughout this section, $L$ is the language of tracial von Neumann algebras and $T$ is the universal theory of embeddable tracial von Neumann algebras. All structures considered in this section will be models of $T$.
Suppose that $M$ and $N$ are separable with $N\subseteq M^\u$. Suppose also that $a,b\in M^\u$ are such that $a\in Z(N'\cap M^\u)$ and $\tp_{\forall_1}^{M^\u}(a/N)=\tp_{\forall_1}^{M^\u}(b/N)$. Then $b\in Z(N'\cap M^\u)$.
Since $\tp_{\forall_0}^{M^\u}(a/N)=\tp_{\forall_0}^{M^\u}(b/N)$, we have $b\in N'\cap M^\u$. Now fix $\epsilon>0$. By countable saturation, there are $e_1,\ldots,e_n\in N$ and $\delta>0$ such that, for all $c\in M^\u$, if $\|[c,e_i]\|_2<\delta$ for all $i=1,\ldots,n$, then $\|[c,a]\|_2<\epsilon$. Consequently, $$\sup_x\min\left(\delta{\mathbin{\mathpalette\dotminussym{}}}min_i\|[x,e_i]\|_2, \|[x,y]\|_2{\mathbin{\mathpalette\dotminussym{}}}\epsilon\right)$$ belongs to $\tp_{\forall_1}^{M^\u}(a/N)$, whence it also belongs to $\tp_{\forall_1}^{M^\u}(b/N)$. It follows that $b\in Z(N'\cap M^\u)$. So, if $c\in N'\cap M^\u$, then $\|[b,c\|_2\leq \epsilon$. Since $\epsilon$ was arbitrary, it follows that $[b,c]=0$, and thus $b\in Z(N'\cap M^\u)$, as desired.
\[commutantcorollary\] Suppose that $N\subseteq P\subseteq M^\u$, $P'\cap M^\u$ is a factor, and every element of $S_{\forall_1}^{M^\u}(N)$ admits an heir to $S_{\forall_1,\forall_0}^{M^\u}(N,P)$. Then $N'\cap M^\u$ is a factor.
Take $a\in Z(N'\cap M^\u)$ and let $p:=\tp_{\forall_1}(a/N)$. Let $q\in S_{\forall_1,\forall_0}(N,P)$ be an heir of $p$. Let $b\in M^\u$ satisfy $q$. By the heir property, $b\in P'\cap M^\u$. If $c\in P'\cap M^\u$, then $c\in N'\cap M^\u$, whence, by the previous lemma, $[b,c]=0$. It follows that $b\in Z(P'\cap M^\u)=\mathbb C$. So $b=\lambda\cdot 1$ for some $\lambda \in \mathbb C$, so $d(x,\lambda\cdot 1)=0$ belongs to $q$, whence it also belongs to $p$, and thus $a=\lambda\cdot 1$, as desired.
Recall the following fact of Nate Brown mentioned in the introduction:
For every separable $N\subseteq \R^\u$, there is a separable $P\subseteq \R^\u$ with $N\subseteq P$ such that $P'\cap \R^\u$ is a factor.
We are now able to prove the following more precise version of Theorem A:
Suppose that $N$ is an embeddable factor such that $\R\models \operatorname{Th}_{\exists_4}(N)$. Then $N$ satisfies the FCEP.
Fix $P$ as in the previous fact, so $N\subseteq P\subseteq \R^\u$ with $P'\cap \R^\u$ a factor. The proof then follows from Corollary \[whatwereallyuse\] and Corollary \[commutantcorollary\].
Proof of Theorem B
==================
Let (\*) denote the statement: the amalgamated free product of embeddable factors over a property (T) base is once again embeddable.
\[star\] Suppose that (\*) holds. Then whenever $N$ is a w-spectral gap subfactor of the e.c. embeddable factor $M$, then $(N'\cap M)'\cap M=N$.
In [@spectralgap], this was proven without a restriction to embeddable factors. The proof goes through in the embeddable case if one assumes (\*) holds.
Recall that if $N$ is a property (T) factor, then $N$ has a **Kazhdan set**, which is a finite subset $F$ of $N$ that satisfies the following property: there is a $K > 0$ such that for any II$_1$ factor $M$ containing $N$ as a subfactor, any $b\in M_1$, and any sufficiently small $\eta>0$, if $\|[a,b]\|_2<\eta$ for all $a\in F$, then there is $c\in N'\cap M$ such that $\|b-c\|_2<K\eta$. Since $\|b-E_{N'\cap M}(b)\|_2\leq \|b-c\|_2<K\eta$ and $E_{N'\cap M}$ is operator norm-contractive, it follows that we may assume that $c\in M_1$ as well. (See [@CJ Proposition 1] for a proof.)
\[T\] Suppose that (\*) holds. Suppose further that $N$ is an embeddable property (T) II$_1$ factor, $M$ is an e.c. embeddable factor containing $N$, and $j:M\hookrightarrow \R^\u$ is downward $\Sigma_2$. Then $j(N)'\cap \R^\u$ is a factor.
Suppose, towards a contradiction, that $a\in Z(j(N)'\cap \R^\u)$ but $d(a,\operatorname{tr}(a)\cdot 1)=\epsilon>0$. Without loss of generality, suppose $a$ is in the unit ball. Let $\{z_1,\ldots,z_n\}$ be a Kazhdan set for $N$ with Kazhdan constant $K$. Note that $$\R^\u\models \forall w\left(\max_{1\leq i\leq n}\|[w,j(z_i)]\|_2=0\rightarrow \|[w,a]\|_2=0\right),$$ whence, by [@mtfms Proposition 7.14], there is a continuous, nondecreasing function $\alpha:\mathbb{R}\to \mathbb R$ satisfying $\alpha(0)=0$ such that $$\R^\u\models \sup_w\left(\|[a,w]\|_2{\mathbin{\mathpalette\dotminussym{}}}\alpha\left(\max_{1\leq i\leq n}\|[w,j(z_i)]\|_2\right)\right)=0.$$ Set $\psi(x,\vec t):=\sup_w(\|[x,w]\|_2{\mathbin{\mathpalette\dotminussym{}}}\alpha(\max_{1\leq i\leq n}\|[w,t_i]\|_2))$, a universal formula such that $\R^\u\models \psi(a,j(\vec z))=0$ whence $$\R^\u\models \inf_x\max\left(\max_{1\leq i\leq n}\|[x,j(z_i)]\|_2, \psi(x,j(\vec z)),\epsilon{\mathbin{\mathpalette\dotminussym{}}}d(x,tr(x)\cdot 1)\right)=0.$$ Since the latter displayed formula is equivalent to a $\exists_2$-formula, by assumption we have $$M\models \inf_x\max\left(\max_{1\leq i\leq n}\|[x,z_i]\|_2, \psi(x,\vec z),\epsilon{\mathbin{\mathpalette\dotminussym{}}}d(x,tr(x)\cdot 1)\right)=0.$$ Fix $\eta>0$ sufficiently small and take $b\in M_1$ such that $$M\models \max\left(\max_{1\leq i\leq n}\|[b,z_i]\|_2, \psi(b,\vec z),\epsilon{\mathbin{\mathpalette\dotminussym{}}}d(b,tr(b)\cdot 1)\right)<\eta.$$ If $\eta$ is sufficiently small, there is $b'\in N'\cap M$ such that $d(b,b')<K\eta$. For simplicity, set $\beta:=K\eta$. Now suppose that $c\in N'\cap M$ is in the unit ball. Then $\|[b,c]\|_2<\eta$, whence $\|[b',c]\|_2<\eta+2\beta$. Since $c\in N'\cap M$ was arbitrary, we have $d(b',(N'\cap M)'\cap M)\leq \eta+2\beta$.[^4] By Lemma \[star\], since $M$ is e.c. and $N$ has w-spectral gap in $M$, we have that $(N'\cap M)'\cap M=N$, so $d(b',N)\leq \eta+2\beta$, that is, $d(b',E_N(b'))\leq \eta+2\beta$. However, $b'\in N'\cap M$ implies $E_N(b')\in Z(N)=\mathbb C$. It follows that $d(b',\operatorname{tr}(b')\cdot 1)=d(b,\mathbb C)\leq d(b,E_N(b'))\leq \eta+2\beta$. Since $\epsilon{\mathbin{\mathpalette\dotminussym{}}}d(b,\operatorname{tr}(b)\cdot 1)<\eta$, we have that $\epsilon{\mathbin{\mathpalette\dotminussym{}}}d(b',\operatorname{tr}(b')\cdot 1)<\eta+2d(b,b')<\eta+2\beta$, which is a contradiction as long as $2\eta+4\beta<\epsilon$. Recalling that $\beta=K\eta$, we have that $2\eta+4\beta=(2+4K)\eta$, whence choosing $\eta<\frac{\epsilon}{2+4K}$, we arrive at the desired contradiction.
The following is a more precise version of Theorem D; it follows immediately from Proposition \[fragmentsandembeddings\] and Theorem \[T\].
\[preciseD\] Suppose that (\*) holds and every embeddable factor $N$ embeds into an e.c. embeddable factor $M$ such that $M\models \Th_{\forall_3}(\R)$. Then every embeddable property (T) factor satisfies the FCEP.
The assumption in the previous corollary should be compared to:
If $M$ is an e.c. embeddable factor, then $M\models \operatorname{Th}_{\exists_3}(\R)$.
Since $M$ is a II$_1$ factor, we may assume that $\R\subseteq M$. Fix an $\exists_3$-sentence $\sigma=\inf_x\sup_y\inf_z \varphi(x,y,z)$ such that $\sigma^\R=0$. Fix $\epsilon>0$ and $a\in \R$ such that $(\inf_y\sup_z\varphi(a,y,z))^\R<\epsilon$. Fix $b\in M$ and an embedding $i:M\hookrightarrow \R^\u$. Then $(\inf_z\varphi(i(a),i(b),z)^{\R^\u}<\epsilon$, whence there is $c\in \R^\u$ such that $(\varphi(i(a),i(b),c)^{\R^\u}<\epsilon$. Since $M$ is e.c. there is $b'\in M$ such that $\varphi(a,b,c')<2\epsilon$. Since $\epsilon$ is arbitrary, we have that $\sigma^M=0$.
Thus, the assumption of Corollary \[preciseD\] comes tantalizingly close to removing any model-theoretic assumption at all, leaving only the operator-algebraic assumption (\*).
Proof of Theorem C
==================
We begin by explaining exactly what we mean for two structures to be $k$-elementarily equivalent.
If $\varphi$ is a formula and $k$ is a nonnegative integer, we recall what it means for $\varphi$ to have **quantifier depth at most $k$**, written $\depth(\varphi)\leq k$, by induction on the complexity of $\varphi$:
- If $\varphi$ is atomic, then $\depth(\varphi)\leq 0$.
- If $\varphi_1,\ldots,\varphi_n$ are formulae, $f:\mathbb R^n\to \mathbb R$ is a continuous function and $\varphi=f(\varphi_1,\ldots,\varphi_n)$, then $\depth(\varphi)\leq\max_{1\leq i\leq n}\depth(\varphi_i)$.
- If $\varphi=\sup_{\vec x} \psi$ or $\varphi=\inf_{\vec x} \psi$, then $\depth(\varphi)\leq \depth(\psi)+1$.
If $M$ and $N$ are $L$-structures, we write $M\equiv_k N$ if $\sigma^M=\sigma^N$ whenever $\operatorname{depth}(\sigma)\leq k$.
If $\sigma$ is an $\forall_m$-sentence or a $\exists_m$-sentence, then clearly $\depth(\sigma)=m$. Consequently, if $M\equiv_m N$, then $M\models \Th_{\forall_m}(N)$ and $N\models \Th_{\forall_m}(M)$.
We recall the following Ehrenfeucht-Fraisse game for continuous logic.
Let $M$ and $N$ be $L$-structures and let $k\in \mathbb N$. $\mathfrak G(M,N,k)$ denotes the following game played by two players. First, player I plays either a tuple[^5] $\vec{x_1}\in M$ or a tuple $\vec{y_1}\in N$. Player II then responds with a tuple $\vec{y_1}\in N$ or $\vec{x_1}\in M$. The play continues in this way for $k$ rounds. We say that *Player II wins $\mathfrak G(M,N,k)$* if there is an isomorphism between the substructures generated by $\{\vec{x_1},\ldots,\vec{x_k}\}$ and $\{\vec{y_1},\ldots,\vec{y_k}\}$ that maps $\vec{x_i}$ to $\vec{y_i}$.
If $M$ and $N$ are $L$-structures, we write $M\equiv_k^{EF} N$ if II has a winning strategy for $\mathfrak G(M,N,k)$.
It is a routine induction to show that $M\equiv_k^{EF} N$ implies $M\equiv_k N$. Conversely, one has the following result (see [@gamespaper Lemma 2.4]):
Suppose that $M$ and $N$ are countably saturated $L$-structures. Then $M\equiv_k N$ if and only if $M\equiv_k^{EF} N$.
We are now ready to prove Theorem C. Recall from the introduction that a II$_1$ factor $M$ has the Brown property if: for every separable subfactor $N$ of $M^\u$, there is a separable subfactor $P$ of $M^\u$ with $N\subseteq P$ such that $P'\cap M^\u$ is a II$_1$ factor.
Suppose that $M\equiv_4 \R$. Then $M$ has the Brown property.
Suppose $N$ is a separable subfactor of $M^\u$. It suffices to find a separable subfactor $P$ of $M^\u$ containing $N$ such that $P'\cap M^\u$ is a factor. Indeed, since $M\equiv_2 \R$, $M$ is McDuff, whence $P'\cap M^\u$ will contain a copy of $\R^\u$ and will thus be a II$_1$ factor, as desired.
Since $M\equiv_4 \R$ and $M^\u$ and $\R^\u$ are $\aleph_1$-saturated, we know that player II has a winning strategy in $\mathfrak G(M^\u,\R^\u,4)$. We assume in the following run of the game that player II plays according to this strategy. Let player I begin with $\vec a_1$, which is a countable sequence from the unit ball of $N$ which generates $N$. Let player II respond with $\vec b_1$ and let $N^*$ denote the separable subfactor of $\R^\u$ generated by $\vec b_1$. Since $\R$ has the Brown property, there is a separable subfactor $P^*$ of $\R^\u$ containing $N^*$ such that $(N^*)'\cap \R^\u$ is a factor. Let $\vec b_2$ be a countable subset of the unit ball of $P^*$ which, together with $\vec b_1$, generates $P^*$. Let player II respond with $\vec a_2$ and let $P$ be the separable subfactor of $M^\u$ generated by $\vec a_1$ and $\vec a_2$. We claim that this $P$ is as desired.
To see this, suppose that $a_3\in Z(P'\cap M^\u)$. We wish to show that $a_3\in \mathbb C$. To see this, let player II respond with $b_3\in \R^\u$. We claim that $b_3\in Z((P^*)'\cap \R^\u)$, whence $b_3\in \mathbb C$. To see this, suppose that $b_4\in (P^*)'\cap \R^\u$. Let player II respond with $a_4\in M^\u$. Since the map $\vec a_1\vec a_2 a_3 a_4\mapsto \vec b_1\vec b_2 b_3 b_4$ extends to an isomorphism between the subalgebras they generate, we see that $a_4\in P'\cap M^\u$. It follows that $a_3$ and $a_4$ commute, whence so do $b_3$ and $b_4$.
Now that we have established that $b_3\in \mathbb C$, the fact that the strategy is winning also shows that $a_3\in \mathbb C$, as desired.
Recall that a McDuff II$_1$ factor is **super McDuff** if $M'\cap M^\u$ is a II$_1$ factor. In [@jung Proposition 4.2.4], it was proven that $M$ has the Brown property if and only if all $N$ elementarily equivalent to $M$ are super McDuff. Consequently, we arrive at:
If $M\equiv_4 \R$, then $M$ is super McDuff.
As mentioned in the introduction, if $\Th(\R)$ does not admit quantifier simplification, then these results yield continuum many new examples of separable factors that are super McDuff and have the Brown property.
[99]{} S. Atkinson, I. Goldbring, and S. Kunnawalkam Elayavalli, *Factorial commutants and the generalized Jung property for II$_1$ factors*, preprint. arXiv 2004.02293. I. Ben Yaacov, A. Berenstein, C. W. Henson, and A. Usvyatsov, *Model theory for metric structures*, in Model theory with applications to algebra and analysis. Vol. 2, volume 350 of London Math. Soc. Lecture Note Ser., 315–427. Cambridge Univ. Press, Cambridge, 2008. N. Brown, *Topological dynamical systems associated to II$_1$ factors*, Adv. Math., **227** (2011), 1665-1699. With an appendix by N. Ozawa. A. Connes and V. Jones, *Property T for von Neumann algebras*, Bulletin of the London Mathematical Society **17** (1985), 57-62. I. Farah, I. Goldbring, B. Hart, and D. Sherman, *Existentially closed II$_1$ factors*, Fundamenta Mathematicae **233** (2016), 173-196. I. Farah, B. Hart, and D. Sherman, *Model theory of operator algebras III: Elementary equivalence and II$_1$ factors*, Bull. London Math. Soc. **46** (2014), 1-20. I. Goldbring, *On Popa’s factorial commutant embedding problem*, to appear in the Proceedings of the AMS. I. Goldbring, *Spectral gap and definability*, to appear in the book Beyond First Order Model Theory Volume 2. I. Goldbring and B. Hart, *The universal theory of the hyperfinite II$_1$ factor is not computable*, preprint. arXiv 2004.02299. I. Goldbring and B. Hart, *On the theories of McDuff’s II$_1$ factors*, International Mathematics Research Notices **27** (2017), 5609-5628.
[^1]: If one is willing to assume the continuum hypothesis, this can even be improved by saying that $M$ and $N$ are elementarily equivalent if and only if $M^\u\cong N^\u$ for any nonprincipal ultrafilter on $\mathbb N$.
[^2]: In this paper, we use the term **embeddable** as an abbreviation for $\R^\u$-embeddable.
[^3]: Technically we really should be speaking of $m-1$ alternations of *blocks* of quantifiers of the same length, but we blur this distinction here.
[^4]: This follows from the general fact that, for a subfactor $P$ of a II$_1$ factor $Q$ and $a\in Q_1$, one has $d(a,P'\cap Q)\leq \sup_{b\in P_1}\|[a,b]\|_2$.
[^5]: Here, tuples can be either of finite or countably infinite length.
|
---
abstract: 'In many situations a BCS-type superconductor will develop an imbalance between the populations of the hole-like and electron-like spectral branches. This imbalance suppresses the gap. It has been noted by Gal’perin, Kozub and Spivak [@spivak] that at large imbalance, when the gap is substantially suppressed, an instability develops. The analytic treatment of the system beyond the instability point is complicated by the fact that the Boltzmann approach breaks down. We study the short time behavior following the instability, in the collisionless regime, using methods developed by Yuzbashyan et al. [@kuznetsov; @kuznetsov; @2].'
author:
- 'A. Nahum'
- 'E. Bettelheim'
title: Dissipationless BCS Dynamics with Large Branch Imbalance
---
Introduction
============
The excitation spectrum of a BCS superconductor consists of an electron-like and a hole-like branch. While the two are equally populated in equilibrium, non-equilibrium states may have a ‘branch imbalance’. For example, if a superconductor is placed in an NSN junction as in the experiment of Clarke [@clarke; @tinkhamclarke; @tinkham], the injected quasi-particles can be primarily electron-like, and the electron-like branch more heavily populated than the hole-like branch in the steady state. Superconducting wires, where branch imbalance arises at phase-slip centres [@skocpol], give another example.
This imbalance suppresses the spectral gap [@galaiko]. If large enough, it returns the system to the normal state, which is known to be unstable below $T_c$ – this is the Cooper instability. It was realized by Gal’perin, Kozub and Spivak [@spivak; @spivak2] that the Cooper instability is a limiting case of a more general instability which afflicts any state beyond a critical level of imbalance and results in oscillations of the order parameter. Traditional approaches to the dynamics fail when the instability occurs, and for this reason the supercritical behavior of the superconductor is as yet unknown. It is important to resolve this, since situations in which large imbalance occurs are quite natural, for example in NSN junctions at large enough injection rate, or long superconducting wires held at sufficiently high voltage.
This paper makes a step in this direction by treating the short-time dynamics in the limit where dissipative processes act slowly in comparison with the BCS dynamics (in particular the oscillation of the gap). We work in the regime $T_c-T\ll T_c$, when the slow relaxation of imbalance allows the system to reach a ‘quasiequilibrium’ whose deviation from equilibrium can be characterized only by the amount of imbalance. We also assume the initial conditions are close to the unstable stationary solution, in a sense defined below.
Since the BCS dynamics are characterized by a time scale of order $1/(T_c-T)$, the assumed separation of scales is $1/(T_c-T)\ll\tau_\epsilon$, where $\tau_\epsilon$ is the time scale associated with energy relaxation. In a metallic superconductor with Debye energy $\Theta\gg T_c$ and $\tau_\epsilon\sim\Theta^2/T_c^3$, this gives only the unrestrictive $(T_c-T)/T_c\gg T_c^2/\Theta^2$.
Often one can avoid the complexities of the microscopic BCS dynamics of a superconductor with a simpler effective description such as the Ginzburg-Landau or the Boltzmann kinetic equation. Two time scales are important in deciding whether either is appropriate: the inelastic quasiparticle relaxation time, $\tau_\epsilon$, and the time scale $\tau_\Delta$ over which the order parameter varies significantly.
When $\tau_\epsilon\ll\tau_\Delta$, the quasiparticle distribution rapidly reaches a local equilibrium characterized by the order parameter $\Delta(\vec{r},t) = |\Delta(\vec{r},t)| e^{i
\chi(\vec{r},t)}$, with dynamics described by the Ginzburg-Landau equations for $\Delta(\vec{r},t)$. We are interested in the opposite limit, $\tau_\epsilon\gg\tau_\Delta$, which is usually tackled with the Boltzmann kinetic equation [@aronov] for the quasiparticle distribution function $n(\vec{r},\vec{p},t)$: $$\begin{aligned}
\label{Boltzmann}
& \frac{\partial n}{\partial t} + \frac{\partial \epsilon}{\partial
\vec{p}}
\frac{\partial n}{\partial \vec{r}} - \frac{ \partial
\epsilon}{\partial \vec{r}} \frac{\partial n}{\partial \vec{p}}=
-I\{n\}.\end{aligned}$$ Here $\epsilon$ is the energy of a quasiparticle state and the functional $I\{n\}$ accounts for impurity scattering and collisions between electrons or between electrons and phonons. The kinetic equation must be supplemented with the self- consistency equation, which determines $|\Delta (\vec{r},t)|$ and thus the quasiparticle energies: $$\begin{aligned}
\label{selfconsist1} & 1 = \frac{\lambda}{2} \int
\frac{1-2n_{\vec{p}}}{\epsilon (\xi_{\vec{p}})} d \xi_{\vec{p}}.\end{aligned}$$ Here $\xi_{\vec{p}}=p^2/2m-\mu$, $\mu$ is the chemical potential, and $\lambda$ the BCS coupling constant. In addition there is a ‘neutrality condition’ involving the phase $\chi$ of the order parameter. This condition arises from the continuity equation and ensures the conservation of charge – it is independent of (\[Boltzmann\]) in the case of a superconductor. In the spatially homogeneous case, if we define $\Phi\equiv\frac{\hbar}{2}
\frac{\partial \chi}{\partial t} + e \varphi$ (where $\varphi$ is the electric potential) and $\tilde{\xi}_{\vec{p}} = \xi_{\vec{p}} +
\Phi$, it takes the form $$\label{Phidef}
\int \frac{ n_{\vec{p} \phantom{|} } \tilde{\xi}_{\vec{p}}}{\epsilon
(\xi_{\vec{p}})}
d\xi_{\vec{p}}
=\Phi.$$ This integral quantifies the branch imbalance. The electron- and hole-like branches are distinguished by the sign of $\tilde\xi$. In equilibrium the two branches are identically populated since the quasiparticle energy ($\epsilon^2={\tilde\xi}^2+|\Delta|^2$) is even in this quantity, and $\Phi$ vanishes.
It is important that while branch imbalance is absent in equilibrium, its relaxation rate $\tau_Q$ diverges as $T_c$ is approached [@tinkham; @schmidschon]: $\tau_Q\sim\tau_\epsilon T_c/\Delta$. (This is in the absence of oscillations of the gap, and assuming relaxation due to electron-phonon collisions [@tinkham; @aronov].) Thus when $\Phi\neq 0$, the quasiparticle distribution reaches a ‘quasiequilibrium’ on a time of order $\tau_\epsilon$, characterized by distinct chemical potentials for the separately equilibrated hole-like and electron-like branches [@galaiko; @tinkham; @schmidschon; @artemenko]. This distribution is given below (\[npwithimbalance\]). Inserting it into the self-consistency equation (\[selfconsist\]) reveals that imbalance suppresses the gap relative to $\Delta_0$, its value when $\Phi=0$: $$\label{gapsuppression1}
|\Delta|^2 =\Delta_0^2 - 2 \Phi^2.$$ At $\Phi=\Delta_0/\sqrt{2}$, the gap is completely suppressed, and the system is returned to the unstable normal state. The instability appears earlier, at an intermediate value $\Phi_c$.
The Boltzmann description cannot handle the system after the instability takes hold, as modes are excited in which Cooper pairs posses non-zero relative phases (the relative phases of the $s_-(\xi)\equiv s_x(\xi)-is_y(\xi)$, in the language of Anderson pseudospins [@anderson]). Only the overall phase of the condensate is retained in the Boltzmann approach, effectively restricting the system to a subclass of solutions where the Cooper pairs precess in phase.
In this situation we must return to the Gorkov equations describing the mean-field dynamics of the individual Cooper pairs [@volkovkogan]. We study here only the limit in which dissipative processes are neglected ($\tau_\epsilon\rightarrow\infty$) and the dynamics controlled purely by the BCS Hamiltonian. This was done for the Cooper instability by Barankov, Levitov and Spivak in [@bls], yielding a ‘soliton train’ of peaks in the gap. The problem was further discussed by Warner and Leggett in [@leggett], by Barankov and Levitov in [@bl; @bl2], and by Yuzbashyan and Dzero in [@dynamicalvanishing]. The integrability of the mean-field BCS system was established by Yuzbashyan, Altshuler, Kuznetsov and Enolskii in the papers [@kuznetsov; @kuznetsov; @2] and a general framework for addressing its dynamics was developed, which we use here to tackle the more involved case of imbalance. We confine ourselves to the integrable BCS Hamiltonian because it can be treated analytically; it was however shown by Barankov and Levitov in [@levitovhiggs] for the case of the Cooper instability that the gap oscillations survive the breaking of integrability.
Our results show the emergence of oscillatory behavior at the instability point. On much larger time scales ($\sim
\tau_\epsilon$) dissipation will modulate the form of these oscillations and determine the final fate of the system. Such an analysis is beyond the scope of this paper, but the solutions we present may be thought of as candidates to be found either at very large times after the onset of the instability or at intermediate times on the way to the asymptotic behavior, and are a first step in calculating the long-time behavior. We discuss this briefly in Section \[conclusion\].
The ability to ‘switch on’ the pairing interaction in ultracold trapped gases via tunable Feshbach resonances [@f1; @f2; @f3; @f4; @f5; @f6] means that one might hope to observe the oscillations of the order parameter directly in the case of the Cooper instability [@bls; @leggett; @dynamicalvanishing; @bl2; @bl]. On the other hand, it is not clear whether one could create imbalance in a sufficiently controlled fashion to observe the instability it creates[^1]. In contrast, imbalance can be created easily in metals, for example in tunnel junctions [@clarke; @tinkham; @tinkhamclarke]. Here the direct observation of the collisionless dynamics is unlikely due to short dissipation times, and its significance is instead in its effect on processes at longer timescales such as the relaxation of imbalance.
The structure of the paper is as follows. Section \[gorkoveqns\] presents the equations of motion for the system and the relevant initial conditions, and includes the final result of the analysis. Section \[elements\] describes the formal solution to BCS dynamics [@kuznetsov; @kuznetsov; @2; @relaxationandpersistent; @erratum], which is applied to the relevant case in Section \[solutionwhereimbalance\]. We conclude in Section \[conclusion\].
Gorkov’s nonlinear equations & solution with imbalance {#gorkoveqns}
======================================================
The system of equations describing the BCS superconductor in the non-dissipative regime were derived by Volkov and Kogan [@volkovkogan] in the Keldysh Green’s function formalism: $$\begin{aligned}
\label{VolkovKogan}
\dot{\vec{s}}(\xi) = {\vec{s}}(\xi)\times (2 \Delta_x , 2 \Delta_y
, - 2 \xi),\end{aligned}$$ where $\vec{s}$ is defined by the following Keldysh Green’s functions: $$\begin{aligned}
\label{keldysh}
s_z (\xi) = {\langle}[c_\uparrow(\xi), c^\dagger_\uparrow(\xi) ] {\rangle}\nonumber\\
s_- (\xi) = {\langle}[c_\uparrow(\xi), c_\downarrow(\xi) ] {\rangle}\end{aligned}$$ $s_- = s_x - i s_y$, and: $$\begin{aligned}
\label{selfconsist}
\Delta = \Delta_x - i \Delta_y = \frac{\lambda}{2} \int s_-(\xi)
d\xi.\end{aligned}$$ This system of equations can be derived from a classical Hamiltonian: $$\begin{aligned}
H = \int 2 \xi s_z(\xi) d\xi - \frac{2}{\lambda}|\Delta|^2; \qquad
\{s_i(\xi), s_j(\xi')\}= \epsilon_{ijk} s_k(\xi)\delta(\xi-\xi').\end{aligned}$$ This is the mean-field BCS Hamiltonian written in terms of Anderson pseudo-spins, with an up (resp. down) spin representing a full (empty) Cooper pair. Singly-occupied pairs decouple from the order parameter dynamics (as can be seen from the BCS Hamiltonian which involves only pair-to-pair scattering) and correspond to zero-length spins. One may derive the Gorkov equations heuristically as a mean-field approximation to the BCS Hamiltonian.
The current paper aims to present solutions of (\[VolkovKogan\]) for imbalanced superconductors, using the approach of [@kuznetsov]. Before doing so, let us recall the instability of the stationary solutions of (\[VolkovKogan\]) in the presence of imbalance which was pointed out in [@spivak]. Generally, stationary solutions of (\[VolkovKogan\]) have the form: $$\begin{aligned}
\label{stationaryfg}
s_z(\xi) &=& -\frac{\xi + \Phi}{\sqrt{(\xi+\Phi)^2 + |\Delta|^2}}
(1-2 n(\xi))\\
s_-(\xi) &=& \phantom- \frac{\Delta}{\sqrt{(\xi+\Phi)^2 + |\Delta|^2}}
(1 - 2
n(\xi))\nonumber\end{aligned}$$ where $n(\xi)$ is the quasi-particle distribution function, which in the presence of imbalance is[^2] [@tinkham; @spivak] $$\begin{aligned}
\label{npwithimbalance}
n(\xi) = \frac{1}{\exp\left(\frac{\sqrt{(\xi+\Phi)^2 + |\Delta|^2} -
\Phi\, \mbox{sign}(\xi + \Phi)}{T}\right) + 1}\end{aligned}$$ which is a Fermi-Dirac distribution for the quasi-particles, with different chemical potentials for the hole-like and electron-like spectral branches implemented by the term $\Phi\,\mbox{sign}
(\xi +
\Phi)$ in the exponent. $\Phi$ parametrizes the amount of imbalance in the system. The expression for $n(\xi)$ is valid when $|\xi+\Phi| \gg |\Delta|$. The self consistency condition (\[selfconsist\]) is satisfied if [@galaiko]: $$\label{gapsuppression}
|\Delta|^2 =\Delta_0^2 - 2 \Phi^2$$ where $\Delta_0$ is the order parameter at the temperature $T$ in the absence of imbalance (for the case $\Phi=0$). Thus imbalance between electron-like and hole-like excitations suppresses the order parameter[^3].
One can easily see that the distribution is unstable when $\Phi
= \Delta_0/\sqrt{2}$, and $\Delta=0$ by (\[gapsuppression\]). At this point it becomes $$\begin{aligned}
\label{normalindisguise}
n(\xi) = \frac{1}{\exp \left(\frac{\xi \,
\mbox{sign}(\xi+\Phi)}{T}\right) +1}\end{aligned}$$ which is nothing but the quasi-particle distribution of a normal Fermi gas in the excitation representation, where an artifical distinction between hole-like and electron-like excitations is made at $\xi=-\Phi$. So, the peculiar form of (\[normalindisguise\]) is an artifact of the excitation representation, and it just describes a normal metal placed at $T<T_c$. The Cooper instability of this metal presents itself as a instability of the stationary solution of equation (\[VolkovKogan\]).
At $\Phi=0$ however the solution is stable, and represents the equilibrium superconducting state. There must therefore be an onset of instability at some finite $\Phi$ intermediate between $0$ and $\Delta_0/\sqrt{2}$. In order to find this point and in order to give a quantitative characterization of the instability, a linear stability analysis was performed around this solution in [@spivak], looking for the presence of an unstable mode $e^{-i\omega t}$, $\mbox{Im}
(\omega)>0$. This leads to an integral equation for $\omega$: $$\begin{aligned}
G(\sqrt{\omega^2/4-|\Delta|^2}) \equiv \int \frac{d\xi}{\sqrt{
\tilde\xi^2+|\Delta|^2}} \frac{1-2 n(\xi)}{\tilde\xi-
\sqrt{\omega^2/4-|\Delta|^2}} =0.\end{aligned}$$ Here $\tilde\xi=\xi+\Phi$. The solutions of this equation were found to be: $$\label{Gu=0}
\mbox{Re} (\omega) = 2 \Phi,\qquad |\mbox{Im} (\omega)|=
\frac{2}{\pi}((T - T_c) -
|\Delta|).$$ The instability arises when the right hand side of this expression for $|\mbox{Im} (\omega)|$ becomes positive at $\Phi_c$. When the imbalance completely suppresses the gap (\[Gu=0\]) agrees well with the usual Cooper instability case. In fact all we shall need in the following are the orders of magnitude of the the following quantities: $$\begin{aligned}
\label{orderofmag}
\Delta\sim s^2 T_c ,\qquad \gamma\equiv\frac{1}{2} \mbox{Im}
(\omega) \sim s^2 T_c, \qquad \Phi \sim s T_c\end{aligned}$$ where $s\equiv \sqrt{(T_c-T)/T_c}$, and we have also defined the parameter $\gamma$, the rate of instability, an important parameter which will appear frequently below. The order of magnitude of the different quantities could just as well have been taken from the Cooper instability case, as they remain the same when the imbalance completely suppresses the gap. Namely these orders of magnitude are not sensitive to the exact form of the distribution function, but rather to gap suppression and may be viewed as consequences of Eq. (\[gapsuppression\]).
The main result of the following analysis is the time dependence of the order parameter following a perturbation from the unstable stationary solution. This is a train of ‘solitons’ each of the form: $$\begin{aligned}
\label{gap}
\Delta(t)= \Delta
\left(1+
\frac{2\gamma}{\Delta}e^{-2 i \Phi (t-\chi)}\text{sech}
{(2\gamma t)}
\right)\end{aligned}$$ where $\Delta$ (as opposed to $\Delta (t)$) denotes the value of the order parameter, which we can take to be real and positive, in the stationary but unstable initial state. The solitons are separated in time by a period that increases logarithmically with the smallness of the perturbation, $$t_0=\gamma^{-1}\log\left(\frac{4\gamma}{\delta}\right)$$ in the notation used below. $\chi$ is a real constant (see below) which does not affect the qualitative nature of the soliton.
The behaviour of $|\Delta(t)|$ consists in fast (at frequency $2\Phi$) ocillations within an envelope given by ${|1\pm(2\gamma/\Delta)\text{sech} {(2\gamma t)}|}$. When $2\gamma/\Delta\gg 1$ (recall that as $\Delta\rightarrow 0$ we return to the finite-temperature normal metal) this formula gives a soliton of the same shape as that found in [@bls] for the order parameter oscillations associated with the Cooper instability. In the opposite limit where the instability is small and $2\gamma/\Delta\ll 1$, we find instead pulses of oscillations of $|\Delta(t)|$ during which the value of $|\Delta(t)|$ averaged over over a period of the fast oscillation rises: $|\Delta(t)|_{av}\simeq\Delta
(1+\gamma^2/\Delta^2 \text{sech}^2 (2\gamma t))$.
Elements of the integrable structure {#elements}
====================================
The instability discovered by Gal’perin et al. shows that at large times dissipation may take the system to a new state which is very different to the initial stationary solution. A similar situation exists when a normal metal is placed at $T<T_c$. For comparison we briefly review here the known results on this instability, the current paper expounding on these results to include the case where imbalance is present.
A normal metal at $T<T_c$ is unstable against the development of a gap (this is the celebrated Cooper instability). The short term dynamics of the system, just after the instability takes hold, consist in an oscillatory behavior of $\Delta$. These oscillations damp out at large times because of dissipation (this process is not described by the pure BCS Hamiltonian) and the material is left in the superconducting state. The oscillatory behavior of $\Delta$ consists of a ‘soliton train’ which can be found by solving (\[VolkovKogan\]) for a system placed near the normal metal state – namely with initial conditions (\[stationaryfg\]), where $n_p$ is the Fermi distribution function plus a small perturbation. This is discussed in [@bls; @leggett; @bl; @bl2].
The soliton train describes the short-time ($\ll \tau_\epsilon$) behavior of the order parameter following the onset of the Cooper instability. A similar analysis will be presented here for the case when the initial state is given by (\[stationaryfg\]) and (\[npwithimbalance\]), i.e. a superconductor with branch imbalance near $T_c$. We study the behavior of the system after a small perturbation is added to $n(\xi)$. This describes the short time behavior after the instability has taken hold. The approach also yields solutions where the perturbation is rather larger. We believe that these solutions may provide further intuition about the possible routes the system may take once dissipation effects are taken into account.
In order to find these solutions we apply the formalism developed in [@kuznetsov; @kuznetsov; @2] for the dynamics of the mean field BCS system, which draws heavily on the theory of integrable systems. We shall establish the notation and present the main concepts of the derivation, referring the reader interested in further details to the original papers. In these papers the spectrum is treated as discrete – here we assume a continuous spectrum.
An important object in the integrable structure is the Lax matrix of the system, a $2\times 2$ matrix depending on a complex parameter $u$, given by: $$\begin{aligned}
\label{laxmat}
\mathcal{L}(u)=\left(
\begin{array}{cc}
A(u) & B(u) \\
B^*(u) & -A(u)
\end{array}
\right)\end{aligned}$$ where: $$\begin{aligned}
& A(u) = \frac{2}{\lambda} - \int \frac{s_z(\xi)}{u-\xi}
d\xi, \\
& B(u) = \int \frac{s_-(\xi) }{u-\xi} d\xi \nonumber.\end{aligned}$$ For any $u$, the matrix $\mathcal{L}(u)$ is time dependent via $\vec s$, but its eigenvalues are not – a key to integrability. These eigenvalues constitute an infinite set of constants of motion[^4]. They are labeled $v(u)$, and given by: $$\begin{aligned}
\label{specpol}
v(u) = \pm \sqrt{-\det\mathcal{L}(u)}.\end{aligned}$$ The analytic structure of $v(u)$ is as follows: in general $v(u)$ has branch cuts, with square root behavior around the branch points, parallel to the imaginary $u$ axis, as well as a jump discontinuity on the real axis, on the support of the spectrum. The branch points $E_i$ satisfy $v(E_i)=0$. Other important quantities are the zeros of $B(u)$, dubbed $u_i$: $B(u_i)=0$.
An important simplification takes place if one is interested only in the long time behavior of the system (but still at times $\ll \tau_\epsilon$). After an initial transient the system exhibits periodic or quasi-periodic behavior whose frequency is dictated by the branch points of $v(u)$. The jump discontinuity is only relevant for the initial transient. The oscillatory behavior following the transient is captured by a simpler system containing only a finite number of spins $\vec{s}^{(i)}$, $i=1,...,g+1$, where $g+1$ is the number of branch cuts in $v(u)$. The system with a finite number of spins is integrable, with a similar integrable structure to the infinite system, integrals being replaced by sums; namely if $$\begin{aligned}
\label{3spin}
& A(u) = \frac{2}{\lambda} - \sum_{i=1}^{g+1}
\frac{s_z^{(i)}}{u-\xi_i} \\
& B(u) = \sum_{i=1}^{g+1} \frac{s_-^{(i)} }
{u-\xi_i}\nonumber\end{aligned}$$ are substituted into (\[laxmat\]) then the eigenvalues of the matrix are constants of motion. It is convenient to define $P(u) \equiv
\prod_i (u - \xi_i)$, and also a degree $2g+2$ polynomial $Q(u)$ with zeros at $E_i$, through $v(u) = \frac{2}{g}\sqrt{Q(u)}/P(u)$. The appearance of $\sqrt{Q(u)}$ signals the relevance of the algebraic Riemann surface defined by the curve $y(u) =
\sqrt{Q(u)}$ to this problem. To find the configuration of the finite number of spins at any given time one must know $Q(u)$, which is independent of time, and the time-dependent quantities $u_i(t)$, defined by $B(u_i(t))=0$.
\[overView\]
To find the dependence of the $u_i$ on time, it is best to make use of the connection of integrable systems and Riemann surfaces. A central theme in the study of Riemann surfaces are the cycles, which are the non-trivial closed curves on the Riemann surface (those that cannot be smoothly shrunk to a point). They will be denoted by $b_k$ and $h_k$, $k=1,...,g$. These cycles are depicted in Fig. \[cyclesfig\]. Another mainstay of the theory of Riemann surfaces are the Abelian (meromorphic) differentials on the surface. The so-called differentials of the first kind, which are everywhere holomorphic, form a $g$ dimensional vector space for which a basis is: $$\begin{aligned}
\hat{u}_k = \frac{d u}{\sqrt{Q(u)}} u^{k-1}\quad k=1,\dots,g.\end{aligned}$$ The differentials may be integrated around the non-trivial cycles to assist in the following definitions: $$\begin{aligned}
\label{cyclesdefinitions}
\omega_{k,l} =\frac{1}{2} \oint_{b_k} \hat{u}_l,\quad \omega'_{k,l} =
\frac{1}{2}
\oint_{h_k} \hat{u}_l,\quad \tau = \omega^{-1} \omega'.\end{aligned}$$ A familiar construction for genus one Riemann surfaces, which have the topology of a torus, allows us to represent the surface as a rectangle with opposite edges identified. The rectangle is characterized by its aspect ratio, which is an invariant of the Riemann surface as well. In the case of genus one the aspect ratio of the rectangle turns out to be equal to $-i \tau$ from (\[cyclesdefinitions\]). When the genus is higher than one we encounter a matrix, $\tau$, which is a generalization of the number $\tau$ of the genus-1 case. The rectangle with opposite sides identified also has an analogue for higher genera: it is replaced by $2g$ (real) dimensional volume in $\mathbb{C}^g$ given by $\mathbb{C}^g /
(\mathbb{Z}^g \omega + \mathbb{Z}^g \omega')$. This $2g$ dimensional volume is an analog of the $2$ dimensional rectangle in the genus-1 case, in that there exists an invertible map taking sets of points on the Riemann surface into it. This is given by $\vec{J}(\{u\}): \Omega
\to \mathbb{C}^g / (\mathbb{Z}^g \omega +
\mathbb{Z}^g \omega')$, where $\Omega$ is the space of sets of $g$ points $\{ u_1,\dots,u_g\}$ (these are not ordered sets, so permutations are considered equivalent): $$\begin{aligned}
J_j\left(\{u_i\}_{i=1}^g\right) = \sum_{i=1}^g \int_{P_0}^{u_i}
d\hat{u}_j.\end{aligned}$$ The space $\mathbb{C}^g / (\mathbb{Z}^g \omega +
\mathbb{Z}^g
\omega')$ is called the Jacobian. The contour of integration from the arbitrary initial point $P_0$ to the point $u$ can wind around any of the cycles any number of times, so as a mapping to $\mathbb{C}^g$ it is only defined up to the addition of an element of the lattice $\mathbb{Z}^g \omega + \mathbb{Z}^g \omega'$. This is however enough to give a well-defined map to $\mathbb{C}^g / (\mathbb{Z}^g \omega +
\mathbb{Z}^g \omega')$. One can check that in the case of genus one the mapping takes a point on the Riemann surfaces and maps it onto a rectangle whose aspect ratio is $-i\tau$, the rectangle being represented by $\mathbb{C} / (\mathbb{Z} \omega +
\mathbb{Z}
\omega')$.
The concept of the Jacobian is particularly important in the solution of the problem because one can show that the zeros of $B(u)$, which are denoted by $u_i$, satisfy the equation: $$\begin{aligned}
\vec{J}(\{u_i(t)\}_{i=1}^g) = (c_1,c_2,\dots,c_g+2it)\end{aligned}$$ where we have now written the time dependence of $u_i$ explicitly, while the $c_i$ are defined by: $$\begin{aligned}
\label{c}
c_i = \sum_{j=1}^g \int_{E_{2j}}^{u_j(t=0)} d\hat{u}_i.\end{aligned}$$ The roots $E_i$ of $Q(u)$ are listed for our case in (\[Es\]). Considered as constants of integration for the dynamics of the $g$ roots $u_i(t)$ of $B(u)$, the $c_i$ are $g$ free complex variables determining the initial values $u_i(0)$. However, not all initial values are permissible, i.e. an arbitrary $\vec c$ will not correspond to a configuration of the spins; there are $g$ constraints on $\vec c$.
The $u_i$ together with the spectral curve $Q(u)$ contain all the information needed to find the configuration of the spins. The problem of finding the $u_i$ is thus the problem of inverting the map $\vec{J}$. This is a solved mathematical problem with a long history, which goes by the name ‘the Jacobi Inversion problem’ [@enolskii]. The solution can be obtained in terms of the Riemann $\theta$-function. We are interested here in the order parameter, whose logarithmic derivative in time is given by $\sum_i u_i$. The explicit solution of the inversion problem for BCS dynamics is presented in [@kuznetsov], and gives the time dependence of the order parameter as: $$\begin{aligned}
\label{thetafunctionexpression}
\Delta(t) =
\frac{\lambda}{2}\sum_{i=1}^{g+1} s^{(i)}_-=
C \exp{\left( 2 \vec{d} \eta (\omega^T)^{-1} \vec{x}-
i\beta t\right)}
\frac{\theta\left((2\omega^T)^{-1}\left(\vec{x}+\vec{d}
\right)\left|\tau\right.\right)}
{\theta\left((2\omega^T)^{-1}\left(\vec{x}-\vec{d}
\right)\left|\tau\right.\right)}\end{aligned}$$ provided that $\eta$ and $d$ are given by: $$\begin{aligned}
\label{d}
\eta_{k,l} = - \sum_{j=l+1}^{2g+2-l} \frac{j-l}{4 (j+l)!}
\frac{d^{j+l} Q(u)} {d u^{j+l}} \omega_{k,j},\qquad d_j
=
\int_{E_0}^\infty d\hat{u}_j.\end{aligned}$$ The frequency $\beta$ can be written in terms of the roots $E_i$ of $Q(u)$ as $\beta=2 \sum_i E_i$.
Solution where imbalance is present {#solutionwhereimbalance}
===================================
In the stationary (but unstable) imbalanced state, $s_z(\xi)$ and $s_-(\xi)$ are given by the expressions (\[stationaryfg\]) and (\[npwithimbalance\]). In this case, the eigenvalue $v(u)$ is given by: $$\begin{aligned}
v^2(u-\Phi) &= (u^2 + \Delta^2) G(u)^2.\end{aligned}$$ Throughout we use $\Delta$ (as opposed to $\Delta(t)$) to denote the value of the gap before the perturbation. $G(u)$, which was defined in (\[Gu=0\]) because it appeared in the linear stability analysis, has appeared again as a common factor in $\det \mathcal{L}(u)=-A^2(u)-B(u)B^*(u)$. This is not a coincidence – the connection between roots of $v(u)$ and modes present in the solution is explained further in [@relaxationandpersistent]. Since $v(u)^2$ has six roots we are led to consider the three spin problem, which according to the arguments of the previous section represents the dynamics of the order parameter after an initial transient. The polynomial $Q(u)$ for the three-spin problem has the same roots as $v(u)^2$, given thus by: $$\begin{aligned}
\label{shiftedQ}
Q(u) = (u^2 + \Delta^2) (u-\Phi - i \gamma)^2
(u-\Phi + i \gamma)^2.\end{aligned}$$ (Here we have shifted $u$ by $\Phi$ – from the equations of motion in [@kuznetsov] we see that this can be compensated by giving the order parameter an additional phase factor.) $Q(u)$ has a very particular structure, related to the fact that it is a stationary solution. It has single roots only at $\pm i \Delta$ and the rest of its roots are double roots. This insures that the Riemann surface given by $Q(u) = y^2(u)$ is of genus $0$.
We now wish to perturb the initial conditions. Unless we fine-tune the perturbation to avoid doing so, we will lift the degeneracy associated with the double roots: the polynomial $Q(u)$ will now have $6$ single roots (which must still occur in complex conjugate pairs, since $Q$ has real coefficients): $$\begin{aligned}
Q(u) = (u^2 + \Delta^2)\left( (u-\Phi - i \gamma)^2 + \delta^2 \right)
\left( (u -\Phi+ i \gamma)^2 + \delta^2 \right).\end{aligned}$$ For the integrals in (\[c\]), (\[d\]) we need the definitions (Fig. \[cyclesfig\]): $$\begin{aligned}
\label{Es}
(E_0,E_1,E_2,E_3,E_4,E_5)=(-i\Delta,i\Delta,\Phi-i \gamma+i\delta,
\Phi+i\gamma-i\delta,
\Phi+i\gamma+i\delta,\Phi-i\gamma-i\delta).\end{aligned}$$ We neglect here (for example) the small shift in the position of the pair of roots around the origin: whereas the splitting of the double roots has a qualitative effect, such shifts have negligible effect on the solution, involving only small shifts in the parameters $\Delta$, $\gamma$ and $\Phi$, and a small change in the overall rate at which the phase of the order parameter rotates. Also, while in general $\delta$ can be complex, in the regime of interest to us the solution is not sensitive to the phase of $\delta$ except through the value of $\vec c$. In the following we treat $\delta$ as real unless otherwise stated.
After the perturbation the Riemann surface is of genus two and $\Delta(t)$ is given by (\[thetafunctionexpression\]) with genus two hyperelliptic $\theta$-functions. The expression is quite formidable, yet certain features can be clarified without much analysis. Most notably, the solution is quasi-periodic, with quasi-periods which can be deduced straightforwardly from the general periodicity properties of $\theta$ -functions together with the particular form the matrices $\omega$ and $\tau$ take in this case.
Before continuing to the analysis of the small perturbation case, we note that if the perturbation is large enough it can lead to the appearance of new roots for $Q(u)$, and to higher spin solutions (with higher genera) – this case is too general for us to say much about.
We assume that the parameter $\delta$ is small, as discussed in Section \[conclusion\]. We then take the leading order in $\delta$ of the expressions for the matrices $\tau$ and $( 2\omega^T )^{-1}
(\vec{x} \pm \vec{d} )$, which figure in (\[thetafunctionexpression\]). We also expand in $s=\sqrt{(T_c-T)/T_c}$, taking the lowest order terms for each element. Then we have, to leading order (in practice we must make sure lower order terms do not contribute): $$\begin{aligned}
\label{omega}
\omega^{-1}= \left(\begin{array}{cc}
-\frac{i\Phi^2}{\pi} & \frac{i\gamma\Delta^2 }{\Phi} \log^{-1}\left(
\frac{4\gamma}{\delta} \right) \\
\frac{i\Phi}{\pi}&
i\gamma \log^{-1}\left(\frac{4\gamma}{\delta}\right)
\end{array}\right)
;\qquad
\eta= \left(\begin{array}{cc}
0&0 \\
-\frac{i\pi\Delta^2}{2\Phi}&
\frac{i\Phi^2}{2\gamma}\log\left(\frac{\delta}{4\gamma} \right)
\end{array}\right)\nonumber;\end{aligned}$$
$$\begin{aligned}
\tau &= \left(\begin{array}{cc}
\frac{i}{2\pi} \log \left( \frac{4 \Phi^4}{\gamma \delta \Delta ^2}
\right) & -\frac{1}{2}
+ \frac{i\gamma}{\Phi}\log^{-1}{\left( \frac{4\gamma}{\delta}
\right )}
\\ -\frac{1}{2} + \frac{i\gamma}{\Phi}\log^{-1}{\left( \frac{4
\gamma}
{\delta}\right )}& \frac{i\pi}{2} \log^{-1} \left(\frac{4
\gamma}{\delta} \right)
\end{array}\right)\nonumber;
\\ \nonumber\\
(2\omega^T)^{-1} (\vec{x} \pm \vec{d}) &= \left(
\begin{array}{cc}
- \frac{ \Phi t}{\pi} \pm \frac{i}{2\pi} \log\left( \frac{i \Delta}{2
\Phi}
\right)
,\left(
\gamma t \pm
\frac{i \gamma}{2\Phi}
\right) \log^{-1} \left( \frac{\delta}{4\gamma}\right)
\end{array}
\right)+(2\omega^T)^{-1} \vec{c}.\end{aligned}$$
We will return to the vector $\vec{c}$, which requires further analysis. The period matrix $\tau$ is seen to have very large and very small elements on the diagonal, diverging or vanishing with $\log^{\pm1}(\delta)$. Because of this the $\theta$-function is well approximated by trigonometric functions. First we use the modular invariance of $\theta$-functions to trade in our period matrix for one whose elements are all of order $\log(\delta)$. This is done via the identity: $$\begin{aligned}
\theta \left(\vec{y} \left|
\left(\begin{array}{cc} i A & -\kappa \\
-\kappa & i h \end{array}\right)
\right)\right.=
\frac{e^{-\frac{\pi}{h}y_2^2}}{\sqrt{h}}
\theta \left(
\left(\begin{array}{cc} y_1-\frac{i \kappa}{h}y_2 \\ -
\frac{i}{h} y_2
\end{array}\right)
\left|
\left(\begin{array}{cc} i (A+\kappa^2/h) & i\kappa/h \\
i\kappa/h & i/h \end{array}\right)
\right)\right.\end{aligned}$$ which results in a ‘transformed’ $\tau$-matrix, $$\begin{aligned}
\label{tautr}
\tau_{tr}=\frac{i}{\pi} \left(\begin{array}{cc}\log{\left(
\frac{4\Phi^2}{\delta\Delta} \right)} &\log{\left(
\frac{4\gamma}{\delta}\right)}\\
\log{\left(\frac{4\gamma}{\delta}\right)} & 2 \log{\left(
\frac{4\gamma}{\delta} \right)}\end{array}\right) +
\frac{2\gamma}{\pi\Phi}\left(\begin{array}{cc}1 &1\\
1 & 0\end{array}\right).\end{aligned}$$ Once the $\theta$-function has been written in terms of a $\tau$ matrix with only large elements, its degeneration into trigonometric functions is easily obtained from the definition of the $\theta$-function in terms of an infinite sum, $$\begin{aligned}
\label{thetafunctionexpansion}
\theta \left( \vec{u} | \tau \right) \equiv \sum_{\vec{m} \in
\mathbb{Z}^g} e^{ i\pi ( \vec{m} \cdot \tau \vec{m} + 2
\vec{m} \cdot \vec{u}) }.\end{aligned}$$ The dominant terms in the sum will come from the $\vec m$ close to the the stationary point of the real part of the exponent: $$\begin{aligned}
\label{m0}
\vec{m}_0 =- (\mbox{Im}\tau)^{-1} \mbox{Im}\vec{u}\end{aligned}$$ ($\vec{m}_0$ is not necessarily a vector of integers). When $\vec{m}=(m,n)$ deviates much from $\vec{m}_0$ the exponential becomes rapidly smaller because of the largeness of $\tau$, such that the sum is dominated by only a few exponential terms – a (hyper-) trigonometric polynomial. Because $\tau$ is logarithmic in $\delta$ and appears linearly in the exponentials, the sub-dominant terms in the sum will be suppressed by powers of $\delta$.
Recovering the Cooper Instability
---------------------------------
By setting $\Phi = \Delta_0/\sqrt{2}$ we completely suppress the gap, returning the system to the normal-metal state. In this limit we should see the absolute value of the order parameter execute the train of $\cosh^{-1}$ solitons found in previous work on the Cooper instability [@bls]. This behaviour corresponds to a simpler two-spin solution. This degeneration into a system described by fewer spins (or lower-genus $\theta$-functions) is expected whenever we close a branch cut on the Riemann surface for $Q(u)$ (here that joining the roots at $\pm i \Delta$), so long as the initial conditions are such that there is a $u_i$ pinned at the resulting double root (here $u_1=0$) [@kuznetsov]. This is the case when the perturbation is such that $\Delta\ll\delta$, as is appropriate if the perturbation is most significant near the Fermi surface[^5]. Note that here $\Delta\neq\Delta(0)$, since the former does not include the effect of the perturbation. With these initial conditions, an appropriate vector of constants $\vec c$ is given by $$\begin{aligned}
\label{ccooper}
\vec c = \left(\begin{array}{cc}
\frac{i}{2\gamma\Phi} \log{\left( \frac{4\gamma}{\delta}\right)}
-\frac{1}{2\Phi^2}\left(\log \left( \frac{4 \Phi^4}{\gamma\delta
\Delta^2}\right)-2\right) ,
& \frac{1}{\Phi} \end{array} \right).\end{aligned}$$ Once we have expressions for the theta functions in terms of the ‘large’ $\tau$-matrix, we extract the asymptotes in the manner described above. Consider one $\theta$-function, say that in the denominator of (\[thetafunctionexpression\]). To begin with we find the stationary point $\vec m_0=(m_0,n_0)$ via (\[m0\]), and see that while the value of $n_0$ changes with time, $m_0 \simeq -1$ for small $\Delta$. If we take only this $m$, we have a genus-one $\theta$-function as expected on general grounds. The denominator degenerates similarly, and the ratio has a quasiperiod $$\begin{aligned}
\label{period}
t_0=\gamma^{-1} \log\left(\frac{4\gamma}{ \delta}\right)\end{aligned}$$ corresponding to the quasiperiod of the genus two $\theta$- functions. Numerator and denominator each have a stationary value of $n$, which can differ from that given by (\[m0\]), $$\begin{aligned}
n_0 (m)=-(\mbox{Im}(\tau_{22}))^{-1}\mbox{Im}(u_2+m
\tau_{12} )\end{aligned}$$ and the argument of each sum is a Gaussian in $n$ whose width is fixed by the $(2,2)$ component of the transformed $\tau$ matrix.
For generic $t$ we can ignore all but one $n$ for the denominator, giving a single exponential, but when the stationary value of $n$ is close to halfway between two integers two values of $n$ are of comparable importance, yielding an expression for $\theta$ in terms of a trigonometric function. (The numerator behaves similarly half a period later, but at these points the ratio is insignificantly small.) The end result is a train of solitons separated by $t_0$, each of the form (to leading order): $$\begin{aligned}
\label{coopergap}
\Delta(t)=\frac
{ 2 \gamma }
{\cosh{(2\gamma t)}}.\end{aligned}$$ The vector $\vec c$ and the overall normalization are fixed in the following way. The values of the $u_i$ (including $u_1=0$) determine $B(u)$ up to $\Delta(0)$: $B(u)=\frac{2\Delta(0)}
{\lambda}u (u-u_2) $. Equation (\[shiftedQ\]) gives $Q(u)$ in the $\Delta\rightarrow 0$ limit. Writing $Q(u)$ in terms of $A(u)$ and $B(u)$, we see that $A(u)=\frac{2}{\lambda} u (u-w_1)
(u-w_2)$ for some $w_i$ that are either both real, or conjugate to each other. Matching the coefficients of $Q(u)$ written in terms of $A(u)$ and $B(u)$ with (\[shiftedQ\]) gives a family of acceptable solutions for $u_2 (0)$ and $\Delta(0)$, corresponding to different stages in the time evolution. Choosing a particular $u_2(0)$ allows us to integrate to get the $\vec c$ above, and also fixes $\Delta(0)$ and thus the normalization of our solution. In the above formula we have omitted an overall phase rotation which corresponds to a redefinition of the chemical potential.
The form of the above soliton conforms exactly with previous results [@bls; @bl2] for the oscillations of the order parameter following a sudden turn-on of the BCS interaction. Interestingly, it also conforms exactly with the result of the next section, where we assume $\delta\ll\Delta$, in the limit that $\Delta\ll\gamma$. This is despite the difference in the relative sizes of $\Delta$ and $\delta$ in the two cases, which implies very different initial conditions for the $u_i$.
Main Case
---------
We now consider the case $\delta\ll\gamma,\Delta$.
### Initial Conditions for Small Perturbations
An important difference between this case and the previous is the value of $\vec c$, which (\[c\]) gives in terms of the initial positions of our variables $u_i(t=0)$. These are the zeroes of $B(u)$ at time $t=0$, which in the unperturbed case coincide with the zeroes of $G(u)$ as one can show using the self-consistency equation. For a small perturbation, the $u_i$ remain close to the branch cuts at $\Phi \pm i \gamma$; let us call them $u_\pm$. It is useful to define $z_\pm$ as the distance of $u_\pm$ from the centers of the branch cuts in units of $\delta$, which may be complex: $$\begin{aligned}
u_{+}(t=0)=\Phi+ i\gamma+z_{+}\delta;
\qquad u_{-}(t=0)=\Phi- i\gamma+z_{-}\delta^*.\end{aligned}$$ The integrals (\[c\]) defining $\vec c$ can then be given through $z_\pm$ using the function $I_\pm$ defined as $$\begin{aligned}
\label{Is}
I_\pm= - \int_{i}^{z_\pm} \frac{\mathrm{d}w}{\sqrt{w^2+1}}=
\frac{i\pi}{2}-\mbox{arcsinh} (z_\pm).\end{aligned}$$ The expression for $c_i$ is then given by[^6]: $$\begin{aligned}
c_1=\frac{
I_+ \left(1-i\frac{\gamma}{\Phi} \right)-I_- \left(1+i\frac{\gamma}
{\Phi} \right)
}{2i\gamma\Phi}, \qquad
c_2= \frac{ I_+-I_-}{2i\gamma};\end{aligned}$$ up to corrections suppressed by $\gamma^2/\Phi^2$ and $z\delta/\gamma$.
The form of the solution is sensitive to the values of $c_1$ and $c_2$. But as noted above, these are not independent parameters. The necessary constraints can be found in the following way. We first find expressions for $z_{\pm}$ in terms of the perturbations to $A(u)$ and $B(u)$. Let $u_I$ be the position of a root of $B(u)$ before the perturbation is added, and $u_F$ its position afterward. Expanding $B=B_0+\delta B$ about the initial position of the root $u_I$ tells us that, due to the perturbation, $u$ travels a distance given by $$\begin{aligned}
(u_F-u_I)\simeq -{\delta B(u_I)}/{B_0 '(u_I)}.\end{aligned}$$ Similarly we can expand $Q\propto A(u)^2+B(u)B^*(u)$, taking into account the fact that it has a double root to begin with, and the fact that before the perturbation $A_0$ and $B_0$ are related by $B_0(u)=\frac{\Delta}{\Phi+u}A_0(u)$. This yields both the width of the branch cut (i.e. $2\delta$ or $2\delta^*$) that is opened up by the perturbation, and the location of its centre, in terms of $\delta B(u_I)$ and $\delta A(u_I)$. We omit these formulas. Then the $z_\pm$ are given by $$\begin{aligned}
z=\frac{(u_F-u_I)-(\textrm{displacement of centre of branch cut})}
{(\textrm{complex half-length of branch cut})}.\end{aligned}$$ Defining $\delta B(u)=\frac{\Delta}{u} \delta A(u)+\delta C(u)$, all the $\delta A$s and $\delta B$s disappear in favor of $\delta C$s, and we can expand without worrying about the relative size of $\delta A$ versus $\delta B$: $$\begin{aligned}
\label{z}
z_\pm= \frac{\Phi\pm i\gamma}{\Delta}
\sqrt{\frac{\delta C(\Phi\pm i\gamma)}{\delta C^*(\Phi\pm
i\gamma)}}.\end{aligned}$$ Since $\delta C^*(\Phi+i\gamma)=\delta C(\Phi-i\gamma)^*$, this tells us that $|z_+ z_-|=\Phi^2/\Delta^2$ to leading order, and that[^7] $\arg z_+/z_-$ is of order $\gamma/\Phi$. Eq. (\[z\]) yields the following constraints on $c_i$ or $z_\pm$: $$\begin{aligned}
p_1&\equiv \mbox{Re }(c_1 \Phi^2 - c_2 \Phi)= \frac{1}{2}
\log 4 |z_+ z_-| =
\log \frac{2\Phi}{\Delta}, \\
p_2&\equiv\mbox{Re } (\gamma c_2)=\frac{1}{2}\arg\frac{z_-}
{z_+}\simeq0.\end{aligned}$$ These combinations of $c_1$ and $c_2$ are precisely those necessary for the correct expansion of the $\theta$-functions – for example $p_1$ dictates which integers $(m,n)$ give the leading order contributions to the representation of the theta function as a sum (\[thetafunctionexpansion\]).
Having derived these results by expanding $B(u)$, $A(u)$ and $Q(u)$, we must ask when they are valid. Assuming that $ \sqrt{ \delta C(u_I) / \delta C^*(u_I)}$ is approximately of order one, we find that the roots of $B(u)$ move a distance of order $\delta\Phi /\Delta$. This quantity must be much smaller than $\gamma$, the scale on which our initial polynomials vary. So a necessary condition for the validity of these approximations is $$\begin{aligned}
\delta \ll \gamma \Delta / \Phi.\end{aligned}$$ This excludes of course the Cooper instability case, where one root of $B(u)$ is a distance of order $\Phi$ from the start-points of the integrals in (\[Is\]). Since $B(u)$ vanishes as $\Delta\rightarrow0$, in this case it is not legitimate to assume that $\delta B\ll B_0$.
### Form of the Solution
Again we use the ‘transformed’ $\theta$-functions, and extract the dominant exponentials from the sums defining them, (\[thetafunctionexpansion\]). The precise values of $\vec c$ depend on the nature of the perturbation, but the information obtained above is enough to establish the nature of the solution up to (a) an overall shift in time and (b) a shift of the fast oscillations within their envelope. Up to such a shift, each soliton has the form (we neglect $\gamma/\Phi$ corrections): $$\begin{aligned}
\label{gap2}
\Delta(t)=\Delta\left(
1+
\frac{2\gamma}{\Delta}e^{-2 i \Phi t}\text{sech} {(2\gamma t)}
\right)\end{aligned}$$ and solitons occur at intervals of $t_0=\gamma^{-1}\log
(4\gamma/\delta)$.
More explicitly, taking into account the expressions for the shifts in terms of $\vec c$, the first soliton has the form $$\begin{aligned}
\label{gap3}
\Delta(t)= \Delta
\left(
1-
\frac{2i\gamma}{\Delta}
\exp{\left(-2 i \Phi t - c_2 \Phi + c_1 \Phi^2+\log\frac{\Delta}
{2\Phi} + i \arg \delta \right)}
\text{sech} {\bigg(2\gamma (t-t_0/2)-i\gamma c_2\bigg)}
\right).\end{aligned}$$ From (\[Is\],\[z\]), $\mbox{Im }c_2$ is of order $\gamma^{-1}
\log(\Phi/\Delta)$, so that if $\delta$ is sufficiently small the first soliton takes about half a period to appear.
Conclusion
==========
We have found the short-time behavior of a BCS superconductor following a small perturbation to the imbalanced initial conditions given by Eq. \[npwithimbalance\]. These initial conditions show a suppression of the gap [@galaiko] with increasing imbalance $\Phi$, and an instability [@spivak] when $\Phi>\Phi_c$ which becomes the celebrated Cooper instability when the gap is fully suppressed. As $\Phi$ is increased beyond $\Phi_c$, the gap oscillations following upon the instability grow in magnitude. They take the form of a train of solitons, each of duration $\sim\gamma^{-1}$ and magnitude $\sim\gamma$, and containing oscillations on the shorter timescale $\Phi^{-1}$ (Eq. \[gap2\], and pictured in Fig. \[overView\]). These oscillations should be observable if appropriate initial conditions can be prepared in a controlled fashion.
A stronger motivation for the work is that the oscillatory behavior is relevant to evolution on longer time scales ($\sim\tau_\epsilon$) in experimental situations with large imbalance. In particular, the oscillations are relevant to the relaxation of the imbalance, which in the absence of the instability occurs at a rate which vanishes with the gap as $T\rightarrow T_c$ [@tinkham; @schmidschon]. Understanding the short-time dynamics of (\[VolkovKogan\]) is a first step; to determine quantitatively what happens on long time scales it is necessary to compute how collisions modulate them. The moduli of the solution, i.e. the variables used to parameterize the kinds of short time behavior, can be taken to be the roots of $Q(u)$. These roots or moduli vary slowly with time on account of collisions. The non-equilibrium state of the system at long times may correspond for example either to an unchanging set of moduli, or to a limit cycle in moduli space. Such an analysis is beyond the scope of the current paper, but will involve (\[gap2\]) and possibly generalizations.
Our explicit expressions for the behavior of the order parameter apply when $\delta\ll \gamma$. In this limit, where the solitons are widely spaced, the expressions simplify greatly; but the generalization to larger $\delta$ may be necessary to treat imbalance relaxation in realistic situations. As an idealized Gendankenexperiment, our limit can be realized by instantaneously injecting electrons at the Fermi level to a system at the instability point – it can be shown using the definition (\[specpol\]) that such a perturbation increases the instability rate $\gamma$ while hardly increasing $\delta$[^8]. (A similar analysis shows that in a system with tunable interaction strength, increasing the coupling of a system with $\Phi\lesssim\Phi_c$ does the same.) The subsequent evolution of such systems on time scales $\tau_\epsilon$ may increase $\delta$ further.
The present work neglects spatial inhomogeneities. Whether they change the picture qualitatively in systems larger than the coherence length remains to be investigated. Gap oscillations can also parametrically excite inhomogeneous modes, as shown in [@cooperturbulence] for the Cooper instability.
We have already mentioned that thermal processes act by slowly perturbing the dynamics considered. We have not mentioned thermal fluctuations in the initial conditions, which were analysed for the Cooper instability in [@bl2; @developingpairing]. These fluctuations disappear when the level spacing, or the effective level spacing in a coherence length, goes to zero, but will cause variations in the parameters of the solution (e.g. our $\delta$) when it is finite. While the evolution is of the same form in each realization, it was shown that variations in the parameters between different realizations are qualitatively important for averages (e.g. of the absolute value of the gap) over realizations. Such averages would be relevant to experiments involving direct observation of gap fluctuations.\
\
The authors would like to thank I.A. Gruzberg, L.S. Levitov and P.B. Wiegmann for discussions. In particular we are indebted to B. Spivak for indispensable advice and guidance.\
\
EB was supported by grant number 206/07 from the ISF.
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[^1]: Population imbalance, i.e. having more spin-up than spin-down electrons, may be more easily created. The dynamics for this distinct situation is addressed in [@populationimbalance].
[^2]: These initial conditions (which will later be perturbed) are thermal averages. Typical initial conditions, as opposed to averaged initial conditions, need not have an $n(\xi)$ that is smooth on the scale of the level spacing. Justification for the use of thermally averaged initial conditions is given in [@bl2]: we can average our spins over a small energy range containing many levels to give a smooth $n(\xi)$ without changing the equations of motion (\[VolkovKogan\], \[selfconsist\]).
[^3]: To obtain (\[gapsuppression\]) one solves the self-consistency equations with (\[npwithimbalance\]). The self-consistency condition is dominated by spins in the energy range $|\xi+\Phi|
\sim T_C$, where (\[npwithimbalance\]) is valid.
[^4]: In the case where the spectrum is discrete only a finite number of these constants of motion are actually independent.
[^5]: To see this, consider the initial condition as deriving from a one-spin solution in which $\delta=\Delta=0$ (here all the $s^{(i)}_-=0$, so that $B=0$) by a perturbation which opens the branch cuts of size $\delta$ and $\Delta$. After the perturbation, the (three-spin) $B(u)$ is $\sim
\delta (u-w_0)(u-w_1)$ for some $w_{0,1}$. Deducing the size of the branch cuts in terms of $w_0$ and $w_1$ from the expression for $Q(u)$ in terms of $A(u)$ and $B(u)$, and demanding that $\Delta\ll\delta$, we see that one of the $w$ is close to the $\Delta$-sized cut and one is close to the $\delta$-sized cuts; as $\Delta\rightarrow 0$, the former root is pinned to the origin.
[^6]: The minus sign in front of the integral in (\[Is\]) is due to the fact that the $u_i$ have to be chosen on the lower Riemann sheet in this case. This can be ascertained by examination of the final result once derived.
[^7]: For this last we must resolve a sign ambiguity coming from the square roots, which is most easily done by examining the special case where $\delta$ is real and the centers of the branch cuts do not move.
[^8]: What is important is the structure of the Riemann surface given by the spectral polynomial (\[specpol\]), rather than the details of the distribution of the spins.
|
---
abstract: 'We develop a novel theoretical framework describing polariton-enhanced spin-orbit interaction of light on the surface of two-dimensional media. Starting from the integral formulation of electromagnetic scattering, we exploit the reduced dimensionality of the system to introduce a quantum-like formalism particularly suitable to fully take advantage of rotational invariance. Our description is closely related to that of a fictitious spin one quantum particle living in the atomically thin medium, whose orbital, spin and total angular momenta play a key role in the scattering process. Conservation of total angular momentum upon scattering enables to physically unveil the interaction between radiation and the two-dimensional material along with the detailed exchange processes among orbital and spin components. In addition, we specialize our model to doped extended graphene, finding such spin-orbit interaction to be dramatically enhanced by the excitation of surface plasmon polaritons propagating radially along the graphene sheet. We provide several examples of the enormous possibilities offered by plasmon-enhanced spin-orbit interaction of light including vortex generation, mixing, and engineering of tunable deep subwavelength arrays of optical traps in the near field. Our results hold great potential for the development of nano-scaled quantum active elements and logic gates for the manipulation of hyper-entangled photon states as well as for the design of artificial media imprinted by engineered photonic lattices tweezing cold atoms into the desired patterns.'
author:
- 'A. Ciattoni$^1$'
- 'C. Rizza$^1$'
- 'H. W. H. Lee$^2$'
- 'C. Conti$^{3,4}$'
- 'A. Marini$^5$'
title: 'Plasmon-enhanced spin-orbit interaction of light in graphene'
---
Introduction
============
Photons are light quanta characterized by energy and linear/angular momentum observables. While energy and linear momentum of photons are inherently linked to the wavelength, their angular momentum arises from both the spin and the phase pattern of the light wave, whose wavefront can be decomposed into orbital angular momentum (OAM) states of well defined topological charge [@Allen1992]. Standard protocols in quantum information are based on [*qbits*]{}, which in photonic realizations are composed of two-dimensional quantum states of photons with opposite spins arising from left and right polarizations ($|L\rangle$, $|R\rangle$) of the light wave. While the spin angular momenta of such states can take only values of $\pm \hbar$, the additional OAM of photons is generally unbound and can take any discrete value $l\hbar$, where $-\infty < l < +\infty$ is an integer accounting for the topological charge associated to the photon phase pattern. The resulting photon states – composed by the coherent superposition of different polarizations and helical spatial patterns with definite topological charge – live in a higher dimensional Hilbert space [@Mair2001; @Molina2007; @Nagali2010; @Cardano2015] and enable the possibility to encode information into hyper-entangled states, also named [*qdits*]{}, which provide superior capacity for quantum cryptography [@Bechmann2000; @Cerf2002; @Mirho2015]. In addition to such promising quantum applications, the OAM of twisted light is currently employed for classical communications in the atmosphere [@Paterson2005; @MalikOE2012; @Willner2015; @FarasSR2015; @Goyal2016], which are being demonstrated over increasing record-breaking distances [@TamburiniNJP2012; @WangNP2012; @Krenn2014; @KrennPNAS2016]. Furthermore, the non-trivial interaction of twisted light with matter [@Pfeifer2007] plays a major role in laser cooling [@Chu1998; @Phillips1998; @Ashkin2000], optical tweezing and manipulation [@Grier2003; @Dienero2008; @Bowman2013], and many other applications where the spin-orbit interaction (SOI) of light [@MarrucciJOPT2011; @Zayats2015] enables the controlling of the beam spatial pattern through its polarization.
SOI is an inherent property of the electromagnetic field that, similarly to relativistic quantum particles and electrons in solids [@Mathur1991; @Xiao2010], becomes relevant at spatial scales comparable with the optical wavelength [@Zhao2007] and is responsible for several unusual phenomena such as, for example, the spin-Hall effect of light [@Kavokin2005; @Hosten2008; @Ling2017; @Bliokh2006], Imbert-Fedorov shift [@Bliokh2013; @Aiello2012] and vortex generation [@Ciattoni2003; @Brasselet2009; @Yavorsky2012; @Ciattoni2017]. Owing to such inherent enhancement of SOI in the non-paraxial regime, focused beams or evanescently localized waves like surface plasmon polaritons (SPPs) [@OConnor2014; @Pan2016] undergo several spin-dependent effects such as unidirectional propagation at interfaces [@Rodrig2013]. Graphene - an atomically-thin lattice of carbon atoms arranged in hexagonal pattern [@Geim2007] - offers unique possibilities for confining light down to the nanometer scale [@Bonaccorso2010], with the further appealing ability of electrical tunability through gated injection of charge carriers. In particular, doped graphene enables the excitation of tunable surface plasmons in nano-islands and polaritons propagating along infinitely extended sheets [@JavierACSPhot] that play a relevant role in plasmon-enhanced light-matter interaction [@Koppens2011] leading to several applications in sensing [@Li2014; @Rodrigo2015; @Marini2015], harmonic generation [@Mikhailov2011; @Cox2017], complete optical absorption [@Sukosin2012], terahertz modulation [@Berardi2012; @Berardi2013], and many others. The extraordinary confinement offered by SPPs in extended graphene is expected to enhance dramatically the SOI of non-paraxial impinging light, leading to the development of active devices for the manipulation and control of OAM at the nanoscale.
\[Fig1\] {width="95.00000%"}
Here we provide for the first time, to the best of our knowledge, a first-principle theoretical approach able to describe the scattering of arbitrary tightly confined fields impinging on a generic two-dimensional medium. Our framework, which does not resort to any approximation on the vectorial electromagnetic field, is very similar in its kinematical traits to the description of a spin one quantum particle living in the atomically thin medium, whose orbital, spin and total angular momenta are the basic quantities allowing us to elucidate the physics of radiation matter interaction in the presence of two-dimensional materials. Rotational invariance is the key ingredient of our analysis and accordingly we find that the ensuing total angular momentum conservation rules radiation scattering. Indeed, the rotationally invariant electromagnetic coupling provided by the two-dimensional material, triggers photon transitions accompanied by an exchange between the orbital and spin angular momenta which is quantified by suitable selection rules. We further specialize our calculations to doped extended graphene, demonstrating that thanks to plasmon-enhanced SOI such an atomically-thin medium can generate efficiently optical vortices of order $m = \pm 2$ and actively manipulate the mixing of different OAM states by means of the external gate voltage, thus enabling fast electrical processing of information stored in OAM states. In addition, we demonstrate that such a tunable mixing of OAM vortices of different order can be exploited to devise arbitrary subwavelength arrays of optical traps in the near field able to pin cold atoms into desired patterns, thus enabling the engineering of artificial materials at will. Such extraordinary functionalities of graphene ensue from the very large momentum of SPPs which, conversely to state-of-the-art optical tweezers, enable the generation of an interference pattern at deep subwavelength scales.
Radiation scattering by two-dimensional media
=============================================
Lippman-Schwinger equation
--------------------------
Consider radiation scattering by a two-dimensional (2D) material lying on the plane $z=0$ embedded in vacuum and illuminated by a monochromatic radiation field (with time dependence $e^{-i \omega t}$, where $\omega$ is the angular frequency) impinging from the half-space $z<0$ (see Fig.1a). Owing to the reduced dimensionality of the system, the integral formalism is particularly suitable for describing radiation scattering. Exploiting the Green’s function method, in the presence of a time harmonic current density of complex amplitude ${\bf J}({\bf r})$, the overall radiation electric field can be written as [@Schwinger] $$\begin{aligned}
\label{rad_field}
&& {\bf{E}}\left( {\bf{r}} \right) = {\bf{E}}^{\left( i \right)} \left( {\bf{r}} \right) + \nonumber \\ &+& \frac{{i\omega \mu _0 }}{{4\pi }}\left( {1 + \frac{1}{{k_0^2 }}\nabla \nabla \cdot } \right) \int {d^3 {\bf{r}}'} h\left( {{\bf{r}} - {\bf{r}}'} \right){\bf{J}}\left( {{\bf{r}}'} \right),\end{aligned}$$ where $k_0 = \omega /c$, $c$ is the speed of light in vacuum and $\mu_0$ is the permeability of free space and $h\left( {\bf{r}} \right) = {e^{ik_0 r} } / r$ is the outgoing spherical wave; the field is the superposition of the incident field ${\bf{E}}^{\left( i \right)}$ (satisfying Maxwell equations in vacuum) and the field radiated by the current. In the limit of low impinging intensity, nonlinear effects are negligible and the current density induced by the external field on the atomically-thin medium is provided by $$\label{gr_cur_den}
{\bf{J}}\left( {\bf{r}} \right) = \sigma {\bf{E}}_ \bot \left( {{\bf{r}}_ \bot} \right)\delta \left( z \right),$$ where hereafter we set ${\bf{A}}_ \bot = A_x{\bf{ e}}_x + A_y{\bf{ e}}_y$ for the transverse part of a vector ${\bf{A}}$. Here $\sigma$ is the surface conductivity of the 2D medium (see Sec. II.D for the particular case of graphene), which generally depends on the radiation frequency $\omega$ and the in-plane radiation wavevector ${\bf k}_{\bot}$. Inserting Eq.(\[gr\_cur\_den\]) into Eq.(\[rad\_field\]) and separating transverse and longitudinal components we get $$\begin{aligned}
\label{scat}
&& {\bf{E}}_ \bot \left( {\bf{r}} \right) = {\bf{E}}_ \bot ^{\left( i \right)} \left( {\bf{r}} \right) + \nonumber \\
&+& \xi \left( {1 + \frac{1}{{k_0^2 }}\nabla _ \bot \nabla _ \bot \cdot } \right)\int {d^2 {\bf{r}}'_ \bot } \frac{{ik_0 }}{{2\pi }} h \left( {{\bf{r}} - {\bf{r}}'_ \bot } \right){\bf{E}}_ \bot \left( {{\bf{r}}'_ \bot } \right), \nonumber \\
&& E_z \left( {\bf{r}} \right) = E_z^{\left( i \right)} \left( {\bf{r}} \right) + \nonumber \\
&+& \xi \left( {\frac{1}{{k_0^2 }}\frac{\partial }{{\partial z}}\nabla _ \bot \cdot } \right)\int {d^2 {\bf{r}}'_ \bot } \frac{{ik_0 }}{{2\pi }} h \left( {{\bf{r}} - {\bf{r}}'_ \bot } \right){\bf{E}}_ \bot \left( {{\bf{r}}'_ \bot } \right),\end{aligned}$$ where $\xi = \left( \mu_0 c/2 \right) \sigma$, and these equations are fully equivalent to differential Maxwell’s equations supplemented with the boundary conditions. Evaluating the first of Eqs.(\[scat\]) at $z=0$ (i.e. setting ${\bf r} = {\bf r}_\bot$) we obtain the Lippman-Schwinger (LS) equation for ${\bf{E}}_ \bot \left( {{\bf{r}}_ \bot } \right)$ describing radiation scattering from the 2D medium
$$\label{lip-sch}
{\bf{E}}_ \bot \left( {{\bf{r}}_ \bot } \right) = {\bf{E}}_ \bot ^{\left( i \right)} \left( {{\bf{r}}_ \bot } \right) + \xi V {\bf{E}}_ \bot \left( {{\bf{r}}_ \bot } \right),$$
where the interaction operator is $V=MK$ with the operators $M$ and $K$ acting on transverse vectors ${\bf{A}}_ \bot$ according to $$\begin{aligned}
M{\bf{A}}_ \bot & =& \left( {1 + \frac{1}{{k_0^2 }}\nabla _ \bot \nabla _ \bot \cdot } \right){\bf{A}}_ \bot, \nonumber \\
K{\bf{A}}_ \bot & =& \int {d^2 {\bf{r}}'_ \bot } \frac{{ik_0 }}{{2\pi }}h\left( {{\bf{r}}_ \bot - {\bf{r}}'_ \bot } \right){\bf{A}}_ \bot \left( {{\bf{r}}'_ \bot } \right).\end{aligned}$$ Once the LS equation is solved for an incoming field ${\bf{E}}_ \bot ^{\left( i \right)} \left( {{\bf{r}}_ \bot } \right)$, Eqs.(\[scat\]) provides the field ${\bf{E}} \left( {{\bf{r}} } \right)$ in the whole space. It is worth noting that, conversely to standard bulk photonic media, it is possible to obtain an integral equation for the [*single*]{} transverse field ${\bf{E}}_ \bot \left( {{\bf{r}}_ \bot } \right)$ owing to the in-plane surface current $\sigma {\bf{E}}_\bot$ flowing through the 2D medium. In particular, this enables us to perform the electromagnetic scattering analysis in the Hilbert space of square-integrable complex transverse vectors.
In order to shed light on the radiation SOI mechanism accompanying the scattering, it is convenient to exploit its rotational invariance around the $z$ axis: if $R_\vartheta$ is the rotation operator around the z-axis of an angle $\vartheta$ and if ${\bf{E}}_\bot$ is the field produced by the incoming field ${\bf{E}}_\bot^{(i)}$ than $R_\vartheta {\bf{E}}_\bot$ is the field produced by the incoming field $R_\vartheta {\bf{E}}_\bot^{(i)}$. From the LS equation it is straightforward to observe that the rotation and interaction operators commute, i.e. $[ R_\vartheta ,V ] = 0$. Since we are dealing only with transverse vectors, the rotation operator is $R_\vartheta = e^{ - iJ \vartheta }$ where $J =L +S$ and $$\begin{aligned}
\label{ang-mom-ope}
L &=& \frac{1}{i} \left( {x\frac{\partial }{{\partial y}} - y\frac{\partial }{{\partial x}}} \right), \nonumber \\
S &=& \frac{1}{i} \left( {\begin{array}{*{20}c}
0 & 1 \\
{ - 1} & 0 \\
\end{array}} \right).\end{aligned}$$ The operator $J$ is the infinitesimal generator of rotations around the $z$-axis so that, since other rotations are not involved here, by definition it is the total angular momentum operator and it is the sum of the orbital $L$ and spin $S$ angular momentum operators. For clarity, we note that the radiation angular momentum and its orbital and spin parts (which are vectors) should not be confused with $J$, $L$ and $S$ since the latter are purely geometrical operators suitably describing rotations around the z axis of transverse fields. Rotational invariance implies that total angular momentum and interaction operators commute, i.e. $[ J,V ] = 0$. This implies the conservation of angular momentum, i.e. the fact that if ${\bf{E}}_\bot^{(i)}$ is an eigenvector of $J$ then ${\bf{E}}_\bot$ is an eigenvector of $J$ as well with the same eigenvalue. Nevertheless, the separate orbital and spin angular momenta are not conserved, i.e. $[ L,V ] \neq 0$ and $[ S,V ] \neq 0$. This implies an effective [*exchange*]{} between them amounting to a [*fundamental*]{} SOI produced by the 2D medium, which fundamental physics is the main subject of this paper.
Dirac’s formalism and Vortex Representation
-------------------------------------------
The analysis of the physical mechanism supporting radiation scattering by a generic 2D medium is greatly simplified by using the Dirac’s abstract vector space formalism. The Hilbert space which is suitable for our purposes is $\mathcal{H}=\mathcal{H}_{\rm orb} \otimes \mathcal{H}_{\rm spi}$, the tensor product of the orbital and spin (polarization) state spaces ${\cal H}_{\rm orb} = \mathop {\rm span}\limits_{{\bf{r}}_ \bot } \left\{ {\left| {{\bf{r}}_ \bot } \right\rangle } \right\}$ and $\mathcal{H}_{\rm spi} = {\rm span} \left\{ {\left| {{\bf{e}}_x } \right\rangle ,\left| {{\bf{e}}_y } \right\rangle } \right\}$, respectively. Here ${\left| {{\bf{r}}_ \bot } \right\rangle } $ are the eigenvectors of the transverse position operator and ${\left| {{\bf{e}}_x } \right\rangle ,\left| {{\bf{e}}_y } \right\rangle }$ are the abstract counterparts of the cartesian unit vectors. These two sets of vectors are orthonormal bases, i.e. $\int {d^2 {\bf{r}}_ \bot } \left| {{\bf{r}}_ \bot } \right\rangle \left\langle {{\bf{r}}_ \bot } \right| = \hat{I}_{\rm orb}$, $\left\langle {{{\bf{r}}_ \bot }} \mathrel{\left | {\vphantom {{{\bf{r}}_ \bot } {{\bf{r}}_ \bot '}}} \right. \kern-\nulldelimiterspace} {{{\bf{r}}_ \bot '}} \right\rangle = \delta \left( {{\bf{r}}_ \bot - {\bf{r}}_ \bot '} \right)$ and ${\sum\nolimits_j {\left| {{\bf{e}}_j } \right\rangle \left\langle {{\bf{e}}_j } \right|} = \hat I_{\rm spi} }$, $\left\langle {{{\bf{e}}_j }} \mathrel{\left | {\vphantom {{{\bf{e}}_j } {{\bf{e}}_{j'} }}} \right. \kern-\nulldelimiterspace} {{{\bf{e}}_{j'} }} \right\rangle = \delta _{jj'}$, where $\hat{I}_{\rm orb}$ and $\hat{I}_{\rm spi}$ are the identity operators of $\mathcal{H}_{\rm orb}$ and $\mathcal{H}_{\rm spi}$, respectively (we hereafter use the caret $\wedge$ to label operators on abstract vector spaces). Accordingly the vectors $\left| {{\bf{r}}_ \bot ,{\bf{e}}_j } \right\rangle = \left| {{\bf{r}}_ \bot } \right\rangle \otimes \left| {{\bf{e}}_j } \right\rangle$ form an orthonormal basis of $\mathcal{H}$ and each transverse field ${\bf{A}}_ \bot = A_x {\bf{e}}_x + A_y {\bf{e}}_y$ is associated in $\mathcal{H}$ with the ket $\left| \Phi \right\rangle = \int {d^2 {\bf{r}}_ \bot } \sum\nolimits_j {A_j \left( {{\bf{r}}_ \bot } \right)\left| {{\bf{r}}_ \bot ,{\bf{e}}_j } \right\rangle }$ with $A_j \left( {{\bf{r}}_ \bot } \right) = \left\langle {{{\bf{r}}_ \bot ,{\bf{e}}_j }} \mathrel{\left | {\vphantom {{{\bf{r}}_ \bot ,{\bf{e}}_j } \Phi }} \right. } {\Phi } \right\rangle$.
The total angular momentum operator acting on $\mathcal{H}$ is $\hat J = \hat L + \hat S$ where, from Eqs.(\[ang-mom-ope\]), the orbital and spin angular momentum operators are $$\begin{aligned}
\label{ang-mom-ope-abs}
\hat L &=& \frac{1}{i}\left( {\hat X\hat D_y - \hat Y\hat D_x } \right) \otimes \hat I_{\rm spi}, \nonumber \\
\hat S &=& {\hat I}_{\rm orb} \otimes \frac{1}{i}\left( {\left| {{\bf{e}}_x } \right\rangle \left\langle {{\bf{e}}_y } \right| - \left| {{\bf{e}}_y } \right\rangle \left\langle {{\bf{e}}_x } \right|} \right),\end{aligned}$$ where $\hat X$, $\hat Y$ are the position operators (defined by $\left\langle {{\bf{r}}_ \bot } \right|\hat X_i \left| {{\bf{r}}'_ \bot } \right\rangle = x_i \delta \left( {{\bf{r}}_ \bot - {\bf{r}}'_ \bot } \right)$) and $\hat D_x$, $\hat D_y$ are the derivatives operators (defined by $\left\langle {{\bf{r}}_ \bot } \right|\hat D_i \left| {{\bf{r}}'_ \bot } \right\rangle = \partial _i \delta \left( {{\bf{r}}_ \bot - {\bf{r}}'_ \bot } \right)$). From the expression above it is straightforward to conclude that $\left[ {\hat L,\hat S} \right] = 0$.
In the space $\mathcal{H}$, the LS equation (Eq. \[lip-sch\]) reads $$\label{lip-sch-abs}
\left| \Psi \right\rangle = \left| {\Psi ^{\left( i \right)} } \right\rangle + \xi \hat V\left| \Psi \right\rangle$$ where $\left| {\Psi } \right\rangle$ and $\left| {\Psi ^{\left( i \right)} } \right\rangle$ are the ket associated with the field ${\bf{E}}_\bot \left( {\bf r}_\bot \right)$ and the incoming field ${\bf{E}}_\bot^{(i)} \left( {\bf r}_\bot \right)$, respectively (see Fig.1a) and the interaction operator is $\hat V = \hat M \hat K$ with $$\begin{aligned}
\label{MK}
\hat M &=& \hat{I} + \frac{1}{{k_0^2 }}\sum\limits_{j,j'} {\hat D_j \hat D_{j'} \otimes \left| {{\bf{e}}_j } \right\rangle \left\langle {{\bf{e}}_{j'} } \right|}, \nonumber \\
\hat K &=& \int {d^2 {\bf{r}}_ \bot } \int {d^2 {\bf{r}}'_ \bot } \frac{{ik_0 }}{{2\pi }}h\left( {{\bf{r}}_ \bot - {\bf{r}}'_ \bot } \right)\left| {{\bf{r}}_ \bot } \right\rangle \left\langle {{\bf{r}}'_ \bot } \right| \otimes \hat{I}_{\rm spi}, \nonumber \\\end{aligned}$$ where $\hat{I}$ is the identity operator in $\mathcal{H}$.
The interaction operator $\hat V$ has a rather complicated structure which hampers the solution of the LS equation of Eq.(\[lip-sch-abs\]). A convenient simplification is gained by resorting to a different representation whose basis vectors are more suitable to deal with the rotational invariance of the system. The basic observation is that the operator $\hat K$ commutes with both the orbital and spin angular momentum operators (mainly since it produces the convolution with the spherical wave and it does not affect the polarization of the field) so that $\hat K, \hat L , \hat S$ are a complete set of commuting operators in $\mathcal{H}$ and accordingly their common eigenvectors turn out to form an orthonormal basis of $\mathcal{H}$. In Appendix \[APP-eigenvectors\] we show that these eigenvectors are $$\begin{aligned}
\label{vortex}
\left| {k_ \bot ,m,s} \right\rangle &=& \int {d^2 {\bf{r}}_ \bot } \sqrt {\frac{{k_ \bot }}{{2\pi }}} J_m \left( {k_ \bot r_ \bot } \right)e^{im\varphi } \left| {{\bf{r}}_ \bot } \right\rangle \otimes \nonumber \\
&\otimes& \frac{1}{{\sqrt 2 }}\left( {\left| {{\bf{e}}_x } \right\rangle + is\left| {{\bf{e}}_y } \right\rangle } \right)\end{aligned}$$ where $k_ \bot$ is any real and positive number, $m$ is an arbitrary integer and $s=\pm 1$. We also prove they satisfy the relations $$\begin{aligned}
\left\langle {k_ \bot ,m,s} \right.|\left. {k'_ \bot ,m',s'} \right\rangle &=& \delta \left( {k_ \bot - k'_ \bot } \right)\delta _{mm'} \delta _{ss'}, \nonumber \\
\sum\limits_{k_ \bot ,m,s} {|\left. {k_ \bot ,m,s} \right\rangle \left\langle {k_ \bot ,m,s} \right.|\;} &=& \hat I,\end{aligned}$$ where, according to Eq.(\[shorthand\]) of Appendix \[APP-eigenvectors\], $\sum\limits_{k_ \bot ,m,s}$ is a shorthand for the integration over $k_ \bot$ and the sum over $m$ and $s$. By construction, we have $$\begin{aligned}
\label{eigen}
\hat K\left| {k_ \bot ,m,s} \right\rangle &=& u \left( k_\bot \right) \left| {k_ \bot ,m,s} \right\rangle, \nonumber \\
\hat L\left| {k_ \bot ,m,s} \right\rangle &=& m \left| {k_ \bot ,m,s} \right\rangle, \nonumber \\
\hat S\left| {k_ \bot ,m,s} \right\rangle &=& s \left| {k_ \bot ,m,s} \right\rangle,\end{aligned}$$ where $u \left( {k_ \bot } \right) = - \left( {1 - k_ \bot ^2 /k_0^2 } \right)^{ - 1/2}$. From the electromagnetic point of view the field associated with $\left| {k_ \bot ,m,s} \right\rangle$ has the spatial profile $\sim J_m \left( {k_ \bot r_ \bot } \right)e^{im\varphi }$, i.e. it is has a Bessel profile of order $m$ endowed with a vortex of topological charge $m$ and it is left and right handed circularly polarized for $s=+1$ and $s=-1$, respectively, since these two values are associated to the unit vectors ${\bf{e}}_L = \frac{1}{\sqrt 2} \left( {{\bf{e}}_x + i{\bf{e}}_y } \right)$, ${\bf{e}}_R = \frac{1}{\sqrt 2} \left( {{\bf{e}}_x - i{\bf{e}}_y } \right)$, respectively. Hence $k_\bot$ is the radial transverse momentum, i.e. it is the radius of the Bessel ring in the momentum space. The second and the third of Eqs.(\[eigen\]) imply that $m$ and $s$ are the orbital and spin angular momenta of the state $\left| {k_ \bot ,m,s} \right\rangle$. This allows to unambiguously identify the orbital angular momentum with the vortex topological charge and the spin with the vortex polarization. Again, we emphasize that such angular momenta [*are not*]{} the standard electromagnetic ones but are the eigenvalues of the infinitesimal generators of rotations in the orbital and spin state spaces. We hereafter use the basis $\left| {k_ \bot ,m,s} \right\rangle$, which we name the [*vortex basis*]{}, since it provides the most suitable representation for our purposes (the [*vortex representation*]{}).
Owing to the rotational symmetry, the vortex $\left| {k_ \bot ,m,s} \right\rangle$ is an eigenvector of the total angular momentum $\hat J = \hat L + \hat S$ $$\hat J|\left. {k_\bot,m,s} \right\rangle = \left( {m + s} \right)|\left. {k_\bot,m,s} \right\rangle$$ with eigenvalue $j=m+s$. Therefore each eigenvalue of $\hat J$ is two-fold degenerate and its two-dimensional eigenspace is $$\label{E(kj)}
\mathcal{E} \left( {k_ \bot ,j} \right) = {\rm span} \left\{ {|\left. {k_\bot,j - 1, + 1} \right\rangle ,\;|\left. {k_\bot ,j + 1, - 1} \right\rangle } \right\}.$$ Since the vortex basis is orthonormal, it follows that the Hilbert space $\mathcal{H}$ is the direct sum of the eigenspaces $\mathcal{E} \left( {k_ \bot ,j} \right)$ \[i.e. $\mathcal{H} = \bigoplus _{k_ \bot ,j} \mathcal{E}\left( {k_ \bot ,j} \right)$\].
A geometrical characterization of the vortex representation is gained by exploiting the fact that here only rotations in the plane are allowed and hence the symmetry group is $SO(2)$ which, being Abelian, has only one-dimensional irreducible representation. In Appendix \[APP-eigenvectors\], we prove that each vector $|\left. {k_ \bot ,m,s} \right\rangle$ is the basis of an irreducible representation of $SO(2)$.
Fundamental radiation SOI processes {#SOI}
-----------------------------------
The vortex representation introduced in the above section, besides providing a framework for associating the topological charge with the orbital angular momentum and the circular polarizations with the spin states, is also very useful for handling the interaction of light with 2D media. To achieve this goal, let us consider the operators $$\begin{array}{*{20}c}
{\hat L_ + = \left( {\hat D_x + i\hat D_y } \right) \otimes \hat I_{\rm spi} ,} & \quad {\hat S_ + = \hat I_{\rm orb} \otimes \left| { + 1} \right\rangle \left\langle { - 1} \right|,} \\
{\hat L_ - = \left( {\hat D_x - i\hat D_y } \right) \otimes \hat I_{\rm spi} ,} & \quad {\hat S_ - = \hat I_{\rm orb} \otimes \left| { - 1} \right\rangle \left\langle { + 1} \right|,} \\
\end{array}$$ which, as shown in Appendix \[APP-ladder\], are such that $$\begin{aligned}
\label{ladder-ops}
\hat L_ \pm \left| {k_ \bot ,m,s} \right\rangle &=& \mp k_ \bot \left| {k_ \bot ,m \pm 1,s} \right\rangle, \nonumber \\
\hat S_ \pm |\left. {k_ \bot ,m,s} \right\rangle &=& \delta _{s, \mp 1} |\left. {k_ \bot ,m,s \pm 2} \right\rangle,\end{aligned}$$ so that $\hat L_\pm$ and $\hat S_\pm$ are orbital and spin ladder operators (the former raising and lowering $m$ by one unit and the latter raising and lowering $s$ by two units). The crucial point, that fully justifies our choice to use the vortex representation, is that the operators $\hat M$ and $\hat K$ of Eqs.(\[MK\]) can be expressed only in terms of the ladder operators as $$\begin{aligned}
\hat M &=& \hat I + \frac{1}{{2k_0^2 }} \left( \hat L_ + \hat L_ - +\hat L_ + ^2 \hat S_ - + \hat L_ - ^2 \hat S_ + \right), \nonumber \\
\hat K &=& - \left( {\hat I + \frac{1}{{k_0^2 }}\hat L_ + \hat L_ - } \right)^{ - 1/2},\end{aligned}$$ and these expressions are very compact and sufficiently simple to unveil the physical mechanisms supporting radiation scattering from the 2D medium (see Appendix \[APP-inter\]). Note that in Appendix \[APP-inter\] we also prove (as anticipated above) that both $\hat M$ and $\hat K$ are rotationally invariant operators (i.e. $[\hat M, \hat J] = [\hat K, \hat J] = 0$) so that the interaction operator $\hat V = \hat M \hat K$ is also a rotationally invariant operator ($[\hat V, \hat J]=0$). In particular, this implies that the eigenspaces $\mathcal{E} \left( {k_ \bot ,j} \right)$ are invariant for the interaction operator $\hat V$ so that the total angular momentum $\hat J$ is conserved in the scattering process. However, as opposed to $\hat K$, the operator $\hat M$ conserves neither the orbital nor the spin angular momentum so that the interaction operator $\hat V$ will in general trigger transitions among states of different $m$ and $s$ while conserving their sum $j=m+s$.
Before discussing the general solution of the LS equation \[see Eq.(\[lip-sch-abs\])\], we first examine its solution in the Born approximation (i.e. up to the first order in $\xi$), namely $$\label{LS-Sol-app}
\left| \Psi \right\rangle = \left( {\hat I + \xi \hat M\hat K} \right)\left| {\Psi ^{\left( i \right)} } \right\rangle.$$ If the incoming field is a vector of the vortex basis, $\left| {\Psi ^{\left( i \right)} } \right\rangle = \left| {k_ \bot ,m,s} \right\rangle$, by using Eqs.(\[ladder-ops\]), we obtain $$\begin{aligned}
\left| \Psi \right\rangle &=& \left[ {1 + \xi \left( {1 - \frac{{k_ \bot ^2 }}{{2k_0^2 }}} \right)u} \right] \left| {k_ \bot ,m,s} \right\rangle + \nonumber \\
&-& \left( \xi \frac{{k_ \bot ^2 }}{{2k_0^2 }}u \right) \delta _{s, + 1} \left| {k_ \bot ,m + 2,s - 2} \right\rangle + \nonumber \\
&-& \left( \xi \frac{{k_ \bot ^2 }}{{2k_0^2 }}u \right) \delta _{s, - 1} \left| {k_ \bot ,m - 2,s + 2} \right\rangle.\end{aligned}$$ Therefore the 2D medium triggers transitions from a vortex of orbital momentum $m$ and spin $s=+1$ to one of angular momentum $m+2$ and spin $s-2=-1$ (due to the interaction operator $\hat L_ + ^2 \hat S_ -$) and from a vortex of orbital momentum $m$ and spin $s=-1$ to one of angular momentum $m-2$ and spin $s+2=+1$ (due to the interaction operator $\hat L_ - ^2 \hat S_ +$). These two scattering processes (see Fig.1b) are very fundamental (see below) and they provide the physical background for understanding all the photonic spin-orbit interaction produced by 2D media. Note that the total angular momentum $j=m+s$ is always conserved in the two processes, as expected due to the above discussed rotation invariance of the system. Total angular momentum conservation also explains the selection rule $\Delta m = \pm 2$ since in each process the orbital angular momentum change has to compensate the change $\Delta s = \mp 2$ accompanying the flip of the spin (which has only the eigenvalues $+1$ and $-1$, see Fig.1b). Translated into the electromagnetic language, if we illuminate the 2D material with a left (right) handed circularly polarized vortex of topological charge $m$ the scattered field will also contain a right (left) handed circularly polarized vortex of topological charge $m+2$ ($m-2$). Note that the scattering amplitude $\xi \frac{{k_ \bot ^2 }}{{2k_0^2 }}u\left( {k_ \bot } \right)$ reveals that by increasing the ratio $k_\bot / k_0$ the transitions’ efficiency increases as well: the more transversely confined the incoming field is, the more strong are the scattered vortices.
The general solution of the LS equation (Eq.(\[lip-sch-abs\])) is (see Appendix \[APP-LS\]) $$\begin{aligned}
\label{LS-Sol}
\left| \Psi \right\rangle &=& \sum\limits_{k_ \bot ,m,s} {\psi _{k_ \bot ,m,s}^{\left( i \right)} Q\left( {k_ \bot } \right)|\left. {k_ \bot ,m,s} \right\rangle } + \nonumber \\
&+& \sum\limits_{k_ \bot ,m,s} {\psi _{k_ \bot ,m,s}^{\left( i \right)} P\left( {k_ \bot } \right)|\left. {k_ \bot ,m + 2s, - s} \right\rangle }\end{aligned}$$ where $\psi _{k_ \bot ,m,s}^{\left( i \right)} = \left\langle {{k_ \bot ,m,s}} \mathrel{\left | {\vphantom {{k_ \bot ,m,s} {\Psi ^{\left( i \right)} }}} \right. \kern-\nulldelimiterspace} {{\Psi ^{\left( i \right)} }} \right\rangle$ are the Fourier coefficients of the expansion of the incoming state in the vortex basis and $$\begin{aligned}
\label{QP}
Q &=& \frac{1}{2}\left( {\frac{1}{{1 - \xi u}} + \frac{1}{{1 - \xi u^{ - 1} }}} \right), \nonumber \\
P &=& \frac{1}{2}\left( {\frac{1}{{1 - \xi u}} - \frac{1}{{1 - \xi u^{ - 1} }}} \right). \label{GRESEQ}\end{aligned}$$ Note that the first term in the RHS of Eq.(\[LS-Sol\]) has the same vortex structure of the incoming state whereas the second term describes the very same scattering processes $\left( {k_ \bot ,m, + 1} \right) \to \left( {k_ \bot ,m + 2, - 1} \right)$ and $\left( {k_ \bot ,m, - 1} \right) \to \left( {k_ \bot ,m - 2, + 1} \right)$ described above in the Born approximation. Conversely to Eq.(\[LS-Sol-app\]), the solution of Eq.(\[LS-Sol\]) is non-perturbative and accounts for transverse magnetic (TM) and transverse electric (TE) resonances provided by the conditions ${\rm Re}\left(1 - \xi u\right)=0$ and ${\rm Re}\left(1 - \xi u^{-1}\right)=0$, respectively.
Graphene Plasmonic Resonance
----------------------------
The novel theoretical treatment of SOI of light discussed above is general for any kind of 2D material, which physical properties are fully incorporated within the parameter $\xi$ depending on the surface conductivity $\sigma$. In the particular case of graphene, at optical and infrared frequencies the response is dominated by the conical band structure ${\cal E} = \pm v_{\rm F} p$ around the two Dirac points of the first Brillouin zone, where $v_{\rm F} \approx 10^6$ m$/$s is the Fermi velocity and ${\cal E},{\bf p}$ are the electron energy and momentum, respectively [@CastroNeto2009]. While in undoped graphene the Fermi energy lies at the Dirac points, injection of charge carriers through electrical gating [@Chen2011] or chemical doping [@Liu2011] shifts efficiently the Fermi level up to $E_{\rm F} \approx 1-2$ eV owing to the conical dispersion and the 2D electron confinement. Thus, within the photon energy range $\hbar \omega < 2 E_{\rm F}$, where $\hbar$ is the reduced Planck constant, interband transitions are inhibited by the Pauli exclusion principle and graphene acquires a metal-like behavior [@Bonaccorso2010]. In turn, doping affects enormously the optical response of graphene from an efficient dispersionless absorber of $\approx 2.3 \%$ of impinging radiation (and universal conductivity $\sigma_0 = e^2/4\hbar$, where $-e$ is the electron charge) to a 2D metal with long relaxation time $\tau = \mu E_{\rm F}/ev_{\rm F}^2$, where $\mu$ is the electron mobility, which conversely to noble metals can reach the picosecond time scale at moderate doping and purity (affecting electron mobility) [@JavierACSPhot]. In principle the graphene surface conductivity $\sigma({\bf k}_{\bot},\omega)$ is nonlocal and depends on the in-plane radiation wavevector ${\bf k}_{\bot}$. However, when $k_{\bot}<k_{\rm F} = E_{\rm F}/\hbar v_{\rm F}$ electron dynamics is local and the graphene conductivity can be evaluated within the local random phase approximation (RPA) providing the integral expression $$\begin{aligned}
\sigma(\omega) & = & \frac{-ie^2}{\pi\hbar^2(\omega+i/\tau)}\int_{-\infty}^{+\infty}d{\cal E} \left\{|{\cal E}| \frac{ \partial f_{\cal E} }{\partial {\cal E}} + \right. \nonumber \\
& & \left. + \frac{ f_{\cal E} {\cal E}/|{\cal E}| }{1 - 4 {\cal E}^2/[\hbar(\omega+i/\tau)]^2} \right\},\end{aligned}$$ where $f_{\cal E} = \{ 1 + {\rm exp}[-({\cal E} - E_{\rm F})/k_{\rm B} T]\}$ is the Fermi-Dirac occupation density of states up to the Fermi energy $E_{\rm F}$, $k_{\rm B}$ is the Boltzmann constant and $T$ is the electron temperature.
Such metal-like behavior of doped graphene enables the tight coupling of photons with in-plane plasma oscillations (pictorially depicted in Fig. 1a as $\sigma_{\rm pl}$, leading to the existence of exponentially localized TM electromagnetic modes propagating along the infinitely extended sheet with wavevector ${\bf k}_{\bot}$ [@JavierACSPhot]. The dispersion relation $\omega(k_{\bot})$ of such modes is identified exactly by the quasi-pole ${\rm Re}\left(1 - \xi K\right)=0$ derived in Eq. (\[GRESEQ\]) and is depicted in Fig.1c, from which one can observe that graphene can confine SPPs down to approximately $10$ nm before non-local effects come into play counteracting further localization. Such an extraordinary confinement at the deep subwavelength nanoscale provided by graphene SPPs enables the enhancement of SOI of light by several orders of magnitude, as we discuss below.
For monochromatic fields, the graphene surface plasmon resonance occurs at a specific transverse wavevector $k_\bot$ which strongly depends on the Fermi energy of the sheet. In Fig.1d, we set $\lambda = 1.55 \: \mu$m and we depict the logarithmic plot of $|Q|$ introduced in the first of Eqs.(\[QP\]) as a function of $k_\bot / k_0$ for different Fermi energies from $0.7$ to $1.3$ eV. The peaks identify the spectral position of the graphene SPPs, whose transverse wavevectors $k_\bot$ shift from $50 \: k_0$ to $120 \: k_0$ for different Fermi energies.
Vortex Management
=================
Electromagnetic vortex scattering
---------------------------------
The above discussed theory on radiation scattering by graphene can be profitably exploited, as we are going to prove, in a number of relevant applications. As depicted in Fig.1a, the general scheme deals with illuminating the graphene sheet with an optical radiation whose circularly polarized components contain suitable vortices which are scattered into novel vortices setting up a desired profile of the transmitted field. As discussed in subsection \[SOI\], the vortex scattering efficiency is very small if the impinging field has a transverse confinement much greater than the wavelength. Besides, in order to make efficient the radiation-plasmon coupling and hence to take advantage of the associated resonance, the evanescent content of the incoming field necessarily has to be large. Both these conditions (which are essentially equivalent) can be achieved by using a near-field optical probe (the tip in Fig.1a) to generate the incoming field.
{width="95.00000%"}
In order to illustrate the phenomenology of the radiation SOI and to discuss its relevant applications, we evaluate the transmitted field ${\bf{E}} = E_L {\bf{e}}_L + E_R {\bf{e}}_L + E_z {\bf{e}}_z$ at $z=0^+$ in turn produced by the incoming field ${\bf E}^{(i)}_\bot \left( \bf{r}_\bot \right)$. We hereafter use circularly polarized components to describe the transverse part of the field which, in turn, satisfies the LS equation (due to the continuity of tangential component of the electric field across the graphene plane) so that $E_L \left( {{\bf{r}}_ \bot } \right) = \left\langle {{{\bf{r}}_ \bot , + 1}} \mathrel{\left | {\vphantom {{{\bf{r}}_ \bot , + 1} \Psi }} \right. \kern-\nulldelimiterspace} {\Psi } \right\rangle$ and $E_R \left( {{\bf{r}}_ \bot } \right) = \left\langle {{{\bf{r}}_ \bot , - 1}} \mathrel{\left | {\vphantom {{{\bf{r}}_ \bot , - 1} \Psi }} \right. \kern-\nulldelimiterspace} {\Psi } \right\rangle$, where $\left| {{\bf{r}}_ \bot ,s} \right\rangle = \left| {{\bf{r}}_ \bot } \right\rangle \otimes \left| s \right\rangle$. The longitudinal component $E_z$ can be evaluated both from the second of Eqs.(\[scat\]) or from the divergence-free property of the electric field in vacuum (see Appendix \[APP-field\]). Here such longitudinal component can not be neglected since, as above anticipated, we are going to discuss near-field optical applications where the fields are highly confined in the transverse plane and plasmonic resonance generally plays a key role. In addition, in order to discuss the spatial energy redistribution accompanying graphene SOI of light, we also consider the time-averaged Poynting vector pertaining the transmitted field at $z=0^+$, i.e. ${\bf{S}} = \frac{1}{2}{\mathop{\rm Re}\nolimits} \left( {{\bf{E}} \times {\bf{H}}^* } \right)$ (see Appendix \[APP-field\] for the evaluation of the magnetic field ${\bf{H}}$).
Vortex Generation {#VG}
-----------------
As a first application of the SOI of light produced by the graphene sheet, we discuss the generation of optical vortices out of an incoming field with vanishing topological charge. Consider an impinging monochromatic field of wavelength $\lambda = 1.55 \: \mu$m whose profile at the graphene sample (with Fermi energy $E_{\rm F} = 1.14$ eV) is assumed
$$\begin{aligned}
{\bf{E}}_ \bot ^{\left( i \right)} \left( {{\bf{r}}_ \bot } \right) &=& E_L^{\left( i \right)} \left( {r_ \bot } \right){\bf{e}}_L = \nonumber \\
&=& E_0 \frac{{e^2 }}{4}\left( {\frac{{r_ \bot }}{w}} \right)^4 e^{ - \left( {\frac{{r_ \bot }}{w}} \right)^2 } {\bf{e}}_L.\end{aligned}$$
This is a left hand circularly polarized field with vanishing topological charge (i.e. it has no vortex singularity at $\bf{r}_ \bot = \bf{0}$.) and the profile of $E_L^{\left( i \right)}$ is depicted in the left inset of Fig.2a. Apart from the amplitude $E_0$ (which we hereafter set equal to one), its only feature which is relevant for our discussion is the width $w$. Such impinging field is the superposition of only the basis vortices $(m,s)=(0,+1)$ so that its total angular momentum is $j=+1$ and its spectral content is provided by the Hankel transform of order zero (see Eq.(\[psi\])) $$\tilde E_L^{\left( i \right)} \left( {k_ \bot } \right) = \int\limits_0^{ + \infty } {dr_ \bot r_ \bot J_0 \left( {k_ \bot r_ \bot } \right)} E_L^{\left( i \right)} \left( {r_ \bot } \right),$$ which is depicted in the right inset of Fig.2a.
Due to SOI of light, the field scattered by the graphene sheet has the structure $$\begin{aligned}
\label{scatVort}
E_L \left( {{\bf{r}}_ \bot } \right) &=& E_{LL} \left( {r_ \bot } \right), \nonumber \\
E_R \left( {{\bf{r}}_ \bot } \right) &=& E_{RL} \left( {r_ \bot } \right)e^{i2\varphi }, \nonumber \\
E_z \left( {{\bf{r}}_ \bot } \right) &=& E_{zL} \left( {r_ \bot } \right)e^{i\varphi },\end{aligned}$$ whose amplitudes have close form integral expressions (see Appendix \[APP-field\]) which can be numerically evaluated. This scattered field has both left and right hand circularly polarized components, the former with circular symmetry and the latter with topological charge two (both with the total angular momentum $j=+1$). In other words, a vortex of topological charge two has been generated by the graphene sheet. The longitudinal component has topological charge $1$ (as a consequence of the divergence-free property of the electric field in vacuum) and, if we associate the spin value $s=0$ to $E_z$, it also has total angular momentum $j=+1$.
{width="45.00000%"}
In Figs.2a and 2b we plot the profiles of $|E_L|$ and $|E_R|$, respectively, for six different values of the incoming field width, i.e. $w = 200, 20, 18, 16, 14, 9 \cdot 10^{-3} \lambda$. Note from Fig.2b that the amplitude of the generated vortex changes from $\simeq 0.01$ to $\simeq 2.5$ when $w$ is reduced from $200 \cdot 10^{-3} \lambda$ to $9 \cdot 10^{-3} \lambda$. The fact that the efficiency of the vortex generation is increasingly higher for smaller widths $w$ is a general trait of SOI of light produced by graphene, as discussed in Sec. \[SOI\]. A second remarkable feature which is evident from Figs.2a and 2b is that, as the width $w$ is decreased, an increasingly stronger ripple appears on the radial profiles of both the circularly polarized components $E_L$ and $E_R$. Such a ripple is the signature of the coupling between the impinging field and the surface plasma oscillations of the graphene sheet, coupling resulting into the excitation of a radial SPP accompanied by a strong plasmon resonance. Since the chosen Fermi energy is $1.14$ eV, such plasmon resonance occurs at $k_\bot \simeq 54 \: k_0$ (see Fig.1d) and in the right inset of Fig.2a we have denoted the spectral position of the resonances $k_\bot w$ by vertical segments whose colors correspond to the values of $w$ considered in the simulations. Evidently, the smaller $w$, the closer the plasmonic resonance $k_\bot w$ to the region where the field spectrum has it central lobe. In other words, the more the field is tightly confined, the more strong is its evanescent spectral tail overlapping the plasmon resonance. Note also that, in addition to the radial ripple, the plasmonic resonance also increases the efficiency of the SOI of light produced by graphene. This is particularly evident from Fig.2a since the amplitude of the left hand component $E_L$ is practically equal to $1$ when the width $w$ is greater than $18 \cdot 10^{-3} \: \lambda$ (yellow curve) and it raises to $\sim 5$ for $w = 9 \cdot 10^{-3} \: \lambda$.
In Fig.2c we focus on the considered most tight confined field with $w=9 \cdot 10^{-3} \: \lambda = 14$ nm and we discuss some of its spatial features on the transverse plane. For comparison purposes, all the amplitudes are plotted on a disk centered at ${\bf r}_\bot = {\bf 0}$ and diameter of $200$ nm with the same color scale reported on right side. The amplitudes $|E_L^{(i)}|$, $|E_L|$ and $|E_R|$ are the two-dimensional counterparts of those considered in Figs.2a and 2b. Note that the scattered field has a multiple ring structure with a spatial extension much wider that the incoming field and this is a consequence of the large radial SPP decay length. The comparison between the longitudinal components of the incoming and scattered field, $|E_z^{(i)}|$ and $|E_z|$, respectively, reveals a strong enhancement of the latter (whose maximum is about $4.5$ as opposed to the maximum of the former which is about $0.5$) and this is an essential further signature of the plasmonic resonance. For completeness we have also plotted the amplitude $|1+E_R|$ which corresponds to the superposition of the generated vortex with an additional right hand circularly polarized plane wave. The vortical nature of $E_R$ (i.e. its topological charge is two) is strikingly evident from the spiral shape of the resulting interference which, due to the plasmon resonance, is also radially modulated thus providing an highly nontrivial pattern.
{width="95.00000%"}
In Fig.3 we again focus on the above considered field with $w=9 \cdot 10^{-3} \: \lambda = 14$ nm and we provide a comparison between the energy flows of the incoming and scattered field. From the expression of the magnetic field derived in Appendix \[APP-field\], it turns out that in these conditions both the incoming and scattered magnetic fields have the same vortical structure of the electric field in Eq.(\[scatVort\]) and this implies that the time-averaged Poynting vector has circularly symmetric cylindrical components, i.e ${\bf{S}}\left({\bf r}_\bot\right) = S_r \left(r_\bot\right) {\bf{e}}_r + S_\varphi \left(r_\bot\right) {\bf{e}}_\varphi + S_z \left(r_\bot\right){\bf{e}}_z$. In Figs.3a we plot the cylindrical components of the incoming Poynting vector and we note that $S^{(i)}_\varphi$ is much greater than $S^{(i)}_r$ and $S^{(i)}_z$ so that ${\bf S}^{(i)}$ is essentially an azimuthal field. In Fig.3b we consider a disk centered at ${\bf r} = {\bf 0}$ and diameter of $56$ nm, depicting the transverse part ${\bf{S}}_\bot^{(i)} = S_r^{(i)} {\bf{e}}_r + S_\varphi^{(i)} {\bf{e}}_\varphi$ as a vector field and the longitudinal component $S_z^{(i)}$ as a color plot. This pictorially shows that the energy flow of the incoming field mainly circulates around the $z$ axis, a property which is consistent with the circular symmetry and the left hand circular polarization of such field. In Fig.3c we plot the cylindrical components of the scattered Poynting vector and a comparison with Fig.3a reveals a particularly evident redistribution of the energy flow accompanying the vortex generation process. The most striking features are that all the three components are much higher than those of the incoming field, their profile shows a ripple and their amplitudes are mutually comparable. Such features are entailed by the interplay between graphene SOI of light and plasmon resonance. It is well known that the excitation of SPPs is accompanied by a non-trivial energy flow occurring in the plane containing the plasmon wavevector and the normal unit vector. In our case the SPP has a radial distribution and accordingly it affects the $S_r$ and $S_z$ components of the scattered field. The fact that there is a considerable enhancement of such components of the Poynting vector does not evidently violate power conservation of the scattering process since the SPP dramatically affects only the evanescent portion of the field which does not contribute to the overall power carried by the field. In Fig.3d, in analogy with Fig.3b, we depict ${\bf{S}} = S_r {\bf{e}}_r + S_\varphi {\bf{e}}_\varphi$ and $S_z$ pertaining the scattered field to pictorially show the energy flow redistribution accompanying vortex generation. The occurrence of a prominent longitudinal component $S_z$ in the scattered field entails a remarkable feature of the transverse part ${\bf{S}}_\bot$: its stream lines connecting the rings where $S_z$ is maximum to the ones where $S_z$ is minimum owing to the divergence-free property of the Poynting vector in vacuum, i.e. $\nabla \cdot {\bf S} = 0$.
Vortex Mixing
-------------
As a second application of graphene SOI of light, we discuss the mixing of optical vortices with ensuing generation of complex deep subwavelength optical lattices. Consider an impinging monochromatic field of wavelength $\lambda = 1.55 \: \mu$m with spatial profile at the graphene sample (with Fermi energy $E_{\rm F} = 1.14$ eV) $$\begin{aligned}
\label{incVort}
{\bf{E}}_ \bot^{(i)} \left( {{\bf{r}}_ \bot ,0} \right) &=& E_L^{(i)} \left( {{\bf{r}}_ \bot } \right){\bf{e}}_L + E_R^{(i)} \left( {{\bf{r}}_ \bot } \right){\bf{e}}_R = \nonumber \\
&=& E_0 \frac{{e^2 }}{4}\left( {\frac{{r_ \bot }}{w}} \right)^4 e^{ - \left( {\frac{{r_ \bot }}{w}} \right)^2 } \left( {e^{i\varphi } {\bf e}_L + 2e^{ - i\varphi } {\bf e}_R } \right). \nonumber \\\end{aligned}$$ This field is very much different from the one considered in the vortex generation process of Sec.\[VG\] in that it has both left and right hand circularly polarized components and, besides, such components carry the topological charges $+1$ and $-1$, respectively. The fields $E_L^{(i)} {\bf{e}}_L $ and $E_R^{(i)}{\bf{e}}_R$ are genuinely vortex fields with well defined values of orbital and spin angular momenta since they are the superposition of only the basis vortices $(m,s)=(+1,+1)$ (with total angular momentum $j=+2$) and $(m,s)=(-1,-1)$ (with total angular momentum $j=-2$), respectively. As a consequence the incoming field ${\bf{E}}_ \bot^{(i)}$ has not well defined values of the angular momenta but rather it is the superposition of two vortex fields. The spatial profile of $E_L^{\left( i \right)}$ is the same as the one considered in Sec.\[VG\] whereas $E_R^{\left( i \right)}$ has an amplitude which is twice the other (again $E_0$ is irrelevant and we hereafter set it equal to one).
Following the procedure described in Appendix \[APP-field\], the components of the field scattered by the graphene sheet is $$\begin{aligned}
\label{mixVort}
E_L \left( {{\bf{r}}_ \bot } \right) &=& E_{LL} \left( r_\bot \right) e^{i\varphi } + E_{LR} \left( r_\bot \right)e^{ - i3\varphi }, \nonumber \\
E_R \left( {{\bf{r}}_ \bot } \right) &=& E_{RL} \left( r_\bot \right)e^{i3\varphi } + E_{RR} \left( r_\bot \right)e^{ - i\varphi }, \nonumber \\
E_z \left( {{\bf{r}}_ \bot } \right) &=& E_{zL} \left( r_\bot \right)e^{i2\varphi } + E_{zR} \left( r_\bot \right)e^{ - i2\varphi },\end{aligned}$$ whose amplitudes have close form integral expressions which can be numerically evaluated. Note that the left and right hand circularly polarized components display additional vortex contributions of topological charges $-3$ and $+3$, respectively. These novel contributions are produced by the vortex transitions $$\begin{aligned}
(m,s) &=& (+1,+1) \rightarrow (+3,-1), \nonumber \\
(m,s) &=& (-1,-1) \rightarrow (-3,+1)\end{aligned}$$ triggered by graphene SOI of light (each separately conserving the total angular momenta $j=+2$ and $j=-2$). Therefore the scattering produces a mixing of the incoming vortices, each circular component being the superposition of different vortices. Note that the longitudinal component is the superposition of two terms with topological charges $+2$ and $-2$ so that, again, if we associate the spin value $s=0$ to $E_z$, these two contributions have total angular momenta $j=+2$ and $j=-2$, respectively.
In Fig.4 we consider the mixing of the vortices of Eq.(\[incVort\]) produced by graphene SOI of light for $w=9 \cdot 10^{-3} \: \lambda = 14$ nm. In Fig.4a we plot all the relevant amplitudes on a disk centered at ${\bf r}_\bot = {\bf 0}$ and a diameter of $100$ nm using for all of them the same color scale reported on the bottom for comparison purposes. The first and second rows of Fig.4a are related to the incoming and scattered field, respectively. The amplitudes $\left| {E_L^{\left( i \right)} } \right|$ and $\left| {E_R^{\left( i \right)} } \right|$ have circularly symmetric profiles, the latter being twice the former, whereas $\left| {E_z^{\left( i \right)} } \right|$ has a four lobe structured profile whose magnitude is comparable with the transverse components due to the tight transverse confinement of the field. The resulting electric field magnitude $|{\bf{E}}^{(i)}| = \sqrt{|E_L^{(i)}|^2 + |E_R^{(i)}|^2 + |E_z^{(i)}|^2}$ has an overall ring structure which is not circularly symmetric. The above discussed vortex mixing dramatically affects the amplitudes $\left| {E_L } \right|$ and $\left| {E_R } \right|$ since their patterns are very much different from the corresponding incoming ones and are equipped with very deep sub-wavelength spatial features. Even more complex profiles are observed in the amplitudes $|E_z|$ and $|{\bf E}| = \sqrt{|E_L|^2 + |E_R|^2 + |E_z|^2 }$. The richness of spatial details of such patterns ensues from two basic ingredients. First, each component is the superposition of two vortices with different topological charges so that their interference entails an involved angular structuring. Note that the two vortices in each component have topological charges differing by $\pm 4$ so that the interference pattern is a periodic function of $4\varphi$ thus explaining the fact that all the reported patterns are left invariant by a rotation of $\pi/2$. Second, due to the tight transverse confinement of the field, in analogy to the vortex generation discussed in Sec.\[VG\], vortex mixing is here accompanied by the excitation of SPPs with distinct radial modulation patterns illustrated in the lower row of Fig.4a. In addition plasmon resonance also yields a remarkable enhancement of the longitudinal component $|E_z|$ whose lobes reach a maximum five times larger than the maximum of $\left| {{\bf{E}}^{\left( i \right)} } \right|$. All these features merge to yield a very structured profile of $\left| {\bf{E}} \right|$ whose square has lobes which are about 40 times stronger than the maximum of $\left| {\bf{E}}^{(i)} \right|$.
{width="95.00000%"}
Such a remarkable qualitative field transformation operated by the atomically thin graphene sheet is accompanied by an equally strong spatial redistribution of the energy flow. In Fig.4b we compare the Poynting vectors of the incoming and transmitted fields by illustrating their transverse parts ${\bf{S}}_\bot^{(i)}$ and ${\bf{S}}_\bot$ as vector fields and their longitudinal components $S_z^{(i)}$ and $S_z$ as color plots on a disk centered at ${\bf r} = {\bf 0}$ and diameter of $70$ nm. The energy flow of the incoming field mainly circulates around the $z$ axis since $S_z^{(i)}$ is negligible, a feature which ensues from the vortical nature of the circular components of such a field. On the contrary, in the scattered field both ${\bf{S}}_\bot$ and $S_z$ exhibit structured profiles very much different from the incoming ones. Note that $S_z$ is very much stronger than $S_z^{(i)}$ and it exhibits, in addition to an overall ring structure, four lobes surrounding the central dark spot. Again, as in the above considered vortex generation process, the strong longitudinal component $S_z$ of the scattered field arise from the divergence-free nature of the Poynting vector in vacuum (i.e. $\nabla \cdot {\bf S} = 0$), implying that the stream lines of ${\bf S}_\bot$ connect regions where $S_z$ is maximum to regions where $S_z$ is minimum.
Discussion
==========
The SOI of light discussed above, including vortex generation and mixing, is the main physical process enabling the simultaneous entanglement of spin and orbital angular momentum components of photons. Currently, state-of-the-art macroscopic [*q -*]{}plates [@MarrucciJOPT2011] are being extensively used to encode information into hyper-entangled [*qdits*]{} of high dimension for quantum cryptography [@Bechmann2000; @Cerf2002; @Mirho2015]. While such game-changing devices can even be miniaturized to some extent, their size is inherently limited by the optical wavelength hampering functionality at the nanoscale, which would be desirable for the development of quantum computers. Recently, gold-based metasurfaces of subwavelength thickness have been shown to overcome this limitation thanks to the plasmonic confinement of the metal and are considered as a viable platform for quantum computation with [*qdits*]{} [@Karimi2014]. However, such devices require advanced fabrication techniques for the realization of involved and precise nanostructured arrays and inherently lack active tunability.
In view of such current limitations and our findings, graphene has all the features for aspiring to become the best material for quantum computation at the nanoscale. Indeed, as discussed in Sec. IIIB and illustrated in Fig. 2, such an atomically thin medium is able to generate efficiently vortices thanks to plasmon-enhanced SOI of light. A striking feature of such a generation process is that, in addition to the vortex angular modulation, the radial oscillations arising from graphene SPP excitation occur at the deep subwavelength nanometer scale, which is unachievable with [*q -*]{}plates and metasurfaces.
Such a feature becomes even more relevant in the plasmon-assisted vortex mixing discussed in Sec. IIIC and illustrated in Fig. 4, which enables the generation of deep subwavelength optical lattices. Remarkably, by impinging through the nanotip vortices with distinct topological charge and polarizations, plasmon-assisted SOI of light in graphene enables the engineering of arbitrary lattices with angular and radial features of few nanometers, which is inherently unachievable with standard optical tweezers. In turn, thanks also to the extraordinary field enhancement provided by graphene SPPs, such lattices are promising for devising artificial media at will by pinning cold atoms in the desired pattern. Indeed, the time-averaged optical force operated by the electromagnetic field on a generic atom or molecule with polarizability $\alpha>0$ is given by ${\bf F} = (1/2)\alpha\nabla|{\bf E}|^2$ and thus, e.g., the optical lattice depicted in Fig. 4a would arrange atoms/molecules into a square array resulting from the peaks of $|{\bf E}|^2$.
In this context, a further feature that is unique to graphene as compared to other photonic materials lies in the external tunability through the injection of charge carriers, which enables the active manipulation of vortex generation efficiency and subwavelength optical lattices through an external gate voltage. Tunability mainly arises from the fact that the spectral position and width of the plasmonic resonance are highly dependent on the Fermi energy of the graphene sample (see Fig. 1d), and in Fig. 5 we illustrate the potential offered by external control. In Fig. 5a we consider the same vortex generation process examined in Fig. 2c and we plot the maximum of $|{\bf E}|^2$ for Fermi energies $E_{\rm F}$ in the range $0.3 \div 1.8$ eV. For small values of $E_{\rm F}$ ($< 0.8$ eV) vortex generation is not very efficient with ${\rm max} \: |{\bf E}|^2$ close to $1$ since the plasmonic resonance is located at $k_\bot > 100 \: k_0$, far outside the main spectral content of the incoming field (see the right inset of Fig. 2a). For higher Fermi energies, the plasmonic resonance efficiently enhances vortex generation since its spectral profile fully overlaps with the incoming field spectrum. From an application point of view, such behavior could be extremely useful for achieving ultrafast amplitude modulation of deep-subwavelength confined vortices. Note that the decrease of ${\rm max} \: |{\bf E}|^2$ for $E_{\rm F} > 1.1$ eV is not due to some kind of plasmonic resonance quenching, but it is rather associated to the small lateral spectral lobe of ${\tilde E}_L^{(i)}$ (see again the right inset of Fig. 2a). In Fig.5b we examine the spatial profile of $|{\bf E}|^2$ of the same vortex mixing process considered in Fig.4a for different values of the Fermi energy $E_{\rm F}$. In this process plasmonic tunability exhibits an even more dramatic phenomenology since $|{\bf E}|^2$ displays both different amplidude and different shapes at different Fermi energies. In turn, the atomically-thin graphene sheet may constitute the core of a device able to generate extreme deep-subwavelength optical lattices with shape tunable in real time by means of the external bias voltage.
{width="45.00000%"}
Although our formalism and predictions are fully classical, they can be straightforwardly generalized to the quantum regime since all the operators introduced to describe light-matter interaction are either diagonal in the eigenstates of the total angular momentum or are expressed in terms of ladder operators (see Sec. IIIA and Appendices). In turn, in view of the results discussed above, the extended doped graphene sheet considered in our calculations can perform tunable logic operations with single photons carrying information in [*qdits*]{}. For example, in Fig.6 we pictorially illustrate a possible way a exploiting the SOI of light to achieve nontrivial information manipulation. The information can be encoded in the radiation state through its vortex $(m,s)$ distribution. The change of the vortex distribution through scattering effectively amounts, using the Shannon terminology, to the conversion of latent information (stored in $\left| {\Psi ^{\left( i \right)} } \right\rangle$) into manifest information (stored in $\left| \Psi \right\rangle$), i.e. information processing occurs. Such manipulation functionality encompasses various different channels since different bits can be activated by vortex generation or switched off by suitable interference entailed by vortex mixing. In addition, the range of possible channels is also increased by tunability, thus enabling to control efficiency and features of the overall information processing functionality.
Conclusions
===========
In summary, without taking any approximation, we develop a new theoretical framework accounting for the spin orbit interaction of arbitrary light fields, showing that their tight confinement along with the plasmonic resonance supported by doped extended graphene leads to the generation of optical vortices at the nanoscale with unprecedentedly high efficiency and deep-subwavelength spatial features. Remarkably, in spite of the atomic thickness of such a two-dimensional material, we demonstrate that it outperforms [*q -*]{}plates and metasurfaces with the further advantage of nano-operation and electrical tunability. While the novel theoretical model derived captures the spin orbit interaction of light mediated by an arbitrary 2D material, we show that extended graphene is ideal since it hosts surface plasmon polaritons with long lifetime and high quality factor, increasing the vortex generation efficiency. Furthermore, thanks to the mixing and interference of distinct vortices, we demonstrate the ability of graphene to generate deep subwavelength optical lattices of arbitrary shape and pace of few nanometers, enabling to devise artificial media at will. Although future work is required to extend our results to the quantum regime, we envisage that they will constitute a solid theoretical ground for the development of nano-scaled active elements and logic gates for enhanced quantum computation based on hyper-entangled photon states.
Acknowledgments
===============
A.C. and C.R. acknowledge support from U.S. Army International Technology Center Atlantic for financial support (Grant No. W911NF-14-1-0315). A.M. acknowledges support from the Rita Levi Montalcini Fellowship (Project No. PGR15PCCQ5) funded by the Italian Ministry of Education, Universities and Research (MIUR).
Eigenvectors of $\hat K$, $\hat L$ and $\hat S$: the Vortex Basis {#APP-eigenvectors}
=================================================================
Let us consider the simultaneous eigenvalue problem of the operators $\hat K$, $\hat L$ and $\hat S$ $$\begin{aligned}
\label{sim-eig}
\begin{array}{l}
\hat K\left| {k_ \bot ,m,s} \right\rangle = u\left( {k_ \bot } \right)\left| {k_ \bot ,m,s} \right\rangle, \\
\hat L\left| {k_ \bot ,m,s} \right\rangle = m\left| {k_ \bot ,m,s} \right\rangle, \\
\hat S\left| {k_ \bot ,m,s} \right\rangle = s\left| {k_ \bot ,m,s} \right\rangle. \\
\end{array}\end{aligned}$$ Since the operators $\hat K$ and $\hat L$ have no effect on the spin degrees of freedoms and the spin operator $ \hat S$ has no effect on the orbital state, it is possible to set $$\label{eigenvect}
\left| {k_ \bot ,m,s} \right\rangle = \left| {k_ \bot,m} \right\rangle \otimes \left| s \right\rangle.$$ The diagonalization of the spin operator $\hat S_{\rm spi} = \frac{1}{i}\left( {\left| {{\bf{e}}_x } \right\rangle \left\langle {{\bf{e}}_y } \right| - \left| {{\bf{e}}_y } \right\rangle \left\langle {{\bf{e}}_x } \right|} \right)$ is straightforward and its eigenvalues are $s=+1$ and $s=-1$ with corresponding eigenvectors $$\label{spin-states}
\left| s \right\rangle = \frac{1}{{\sqrt 2 }}\left( {\left| {{\bf{e}}_x } \right\rangle + is\left| {{\bf{e}}_y } \right\rangle } \right),$$ which are an orthonormal basis of $\mathcal{H}_{\rm spi}$, i.e. $$\begin{aligned}
\label{bas-spi}
\left\langle {s}
\mathrel{\left | {\vphantom {s {s'}}}
\right. \kern-\nulldelimiterspace}
{{s'}} \right\rangle &=& \delta _{ss'}, \nonumber \\
\sum\limits_{s = -1,1} {\left| s \right\rangle \left\langle s \right|} &=& \hat I_{\rm spi}.\end{aligned}$$ In order to find the orbital eigenvectors $\left| {k_\bot,m} \right\rangle$ it is convenient to cast the operator $\hat K$ in a different form. Setting $\hat K = \hat K_{\rm orb} \otimes \hat I_{\rm spi}$ and using the Weyl representation of the spherical wave [@App_Mandle] $$h\left( {\bf r} \right)= \frac{i}{{2\pi }}\int {d^2 {\bf{k}}_ \bot } e^{i{\bf{k}}_ \bot \cdot {\bf{r}}_ \bot } \frac{{e^{i\sqrt {k_0^2 - k_ \bot ^2 } \left| z \right|} }}{{\sqrt {k_0^2 - k_ \bot ^2 } }},$$ we have $$\begin{aligned}
\label{K-orb}
\hat K_{\rm orb} &=& - \frac{{k_0 }}{{4\pi ^2 }}\int {d^2 {\bf{r}}_ \bot } \int {d^2 {\bf{r}}'_ \bot } \times \nonumber \\
&\times& \int {d^2 {\bf{k}}_ \bot } \frac{{e^{i{\bf{k}}_ \bot \cdot \left( {{\bf{r}}_ \bot - {\bf{r}}'_ \bot } \right)} }}{{\sqrt {k_0^2 - k_ \bot ^2 } }}\left| {{\bf{r}}_ \bot } \right\rangle \left\langle {{\bf{r}}'_ \bot } \right| = \nonumber \\
&=& \int {d^2 {\bf{k}}_ \bot } \frac{ - k_0}{{\sqrt {k_0^2 - k_ \bot ^2 } }}\left| {{\bf{k}}_ \bot } \right\rangle \left\langle {{\bf{k}}_ \bot } \right| = \nonumber \\
&=& - \left( {1 + \frac{1}{{k_0^2 }}\hat D_ \bot ^2 } \right)^{ - 1/2},\end{aligned}$$ where $\left| {{\bf{k}}_ \bot } \right\rangle = \frac{1}{{2\pi }}\int {d^2 {\bf{r}}_ \bot } e^{i{\bf{k}}_ \bot \cdot {\bf{r}}_ \bot } \left| {{\bf{r}}_ \bot } \right\rangle$ are the plane waves states (which form an orthonormal basis of $\mathcal{H}_{\rm orb}$, i.e. $\left\langle {{{\bf{k}}_ \bot }} \mathrel{\left | {\vphantom {{{\bf{k}}_ \bot } {{\bf{k}}'_ \bot }}} \right. } {{{\bf{k}}'_ \bot }} \right\rangle = \delta \left( {{\bf{k}}_ \bot - {\bf{k}}'_ \bot } \right)$, $\int {d^2 {\bf{k}}_ \bot } \left| {{\bf{k}}_ \bot } \right\rangle \left\langle {{\bf{k}}_ \bot } \right| = I_{\rm orb}$)), $$\hat D_ \bot ^2 = \hat D_x^2 + \hat D_y^2$$ is the transverse Laplacian operator whose eigenvectors are the plane wave states, i.e. $$\hat D_ \bot ^2 \left| {{\bf{k}}_ \bot } \right\rangle = - k_ \bot ^2 \left| {{\bf{k}}_ \bot } \right\rangle.$$ Equation (\[K-orb\]) reveals that $\hat K_{\rm orb}$ is strictly a function of the transverse Laplacian operator and therefore $\left| {k_\bot,m} \right\rangle$ are the simultaneous eigenvectors of $\hat D_ \bot ^2$ and $\hat L_{\rm orb} = \frac{1}{i}\left( {\hat X\hat D_y - \hat Y\hat D_x } \right)$, i.e. $$\begin{aligned}
\hat D_ \bot ^2 \left| {k_ \bot ,m} \right\rangle &=& - k_ \bot ^2 \left| {k_ \bot ,m} \right\rangle, \nonumber \\
\hat L_{\rm orb} \left| {k_ \bot ,m} \right\rangle &=& m\left| {k_ \bot ,m} \right\rangle.\end{aligned}$$ Expanding the eigenvector on the basis $\left| {{\bf{r}}_ \bot } \right\rangle$ by setting $\left| {k_ \bot ,m} \right\rangle = \int {d^2 {\bf{r}}_ \bot } f_{k_ \bot ,m} \left( {{\bf{r}}_ \bot } \right)\left| {{\bf{r}}_ \bot } \right\rangle$, we get $$\begin{aligned}
\nabla _ \bot ^2 f_{k_ \bot ,m} &=& - k_ \bot ^2 f_{k_ \bot ,m}, \nonumber \\
\frac{\partial f_{k_ \bot ,m}}{{\partial \varphi }} &=& imf_{k_ \bot ,m}.\end{aligned}$$ where ${\bf{r}}_ \bot = r_ \bot \left( {\cos \varphi {\bf{e}}_x + \sin \varphi {\bf{e}}_y } \right)$ so that the eigenfunctions $f_{k_ \bot ,m}$ are the well-known cylindrical harmonics [@Schwinger] where $k_\bot$ is any positive real number and $m$ is any integer number. Accordingly the eigenvectors are $$\left| {k_ \bot ,m} \right\rangle = \int {d^2 {\bf{r}}_ \bot } \sqrt {\frac{{k_ \bot }}{{2\pi }}} J_m \left( {k_ \bot r_ \bot } \right)e^{im\varphi } \left| {{\bf{r}}_ \bot } \right\rangle,$$ where $J_m (\zeta)$ is the Bessel function of the first kind of order $m$ and they form an orthonormal basis of $\mathcal{H}_{\rm orb}$, i.e. $$\begin{aligned}
\label{bas-orb}
\left\langle {{k_ \bot ,m}}
\mathrel{\left | {\vphantom {{k_ \bot ,m} {k'_ \bot ,m'}}}
\right. \kern-\nulldelimiterspace}
{{k'_ \bot ,m'}} \right\rangle = \delta \left( {k_ \bot - k'_ \bot } \right) \delta _{mm'}, \nonumber \\
\int\limits_0^\infty {dk_ \bot } \sum\limits_{m = - \infty }^\infty {\left| {k_ \bot ,m} \right\rangle } \left\langle {k_ \bot ,m} \right| = I_{\rm orb}.\end{aligned}$$ We conclude that the eigenvectors of Eq.(\[eigenvect\]) are $$\begin{aligned}
\left| {k_ \bot ,m,s} \right\rangle &=& \int {d^2 {\bf{r}}_ \bot } \sqrt {\frac{{k_ \bot }}{{2\pi }}} J_m \left( {k_ \bot r_ \bot } \right)e^{im\varphi } \left| {{\bf{r}}_ \bot } \right\rangle \otimes \nonumber \\
&\otimes& \frac{1}{{\sqrt 2 }}\left( {\left| {{\bf{e}}_x } \right\rangle + is\left| {{\bf{e}}_y } \right\rangle } \right)\end{aligned}$$ and, due to Eqs.(\[bas-spi\]) and (\[bas-orb\]), they form an orthonormal basis of $\mathcal{H}$, i.e. $$\begin{aligned}
\left\langle {k_ \bot ,m,s} \right.|\left. {k'_ \bot ,m',s'} \right\rangle &=& \delta \left( {k_ \bot - k'_ \bot } \right)\delta _{mm'} \delta _{ss'}, \nonumber \\
\sum\limits_{k_ \bot ,m,s} {|\left. {k_ \bot ,m,s} \right\rangle \left\langle {k_ \bot ,m,s} \right.|\;} &=& \hat I,\end{aligned}$$ where, for notation convenience, we hereafter set $$\label{shorthand}
\sum\limits_{k_ \bot ,m,s} {} \equiv \int\limits_0^{ + \infty } {dk_\bot} \sum\limits_{m = - \infty }^\infty {\sum\limits_{s = - 1,1} }.$$ In the representation induced by the eigenvectors $\left| {k_ \bot ,m,s} \right\rangle$, the operators $\hat K$, $\hat L$ and $\hat S$ are $$\begin{aligned}
\label{KLS}
\hat K &=& \sum\limits_{k_ \bot ,m,s} {} u \left(k_\bot\right)
|\left. {k_ \bot ,m,s} \right\rangle \left\langle {k_ \bot ,m,s} \right.|, \nonumber \\
\hat L &=& \sum\limits_{k_ \bot ,m,s} {m|\left. {k_ \bot ,m,s} \right\rangle \left\langle {k_ \bot ,m,s} \right.|}, \nonumber \\
\hat S &=& \sum\limits_{k_ \bot ,m,s} {s|\left. {k_ \bot ,m,s} \right\rangle \left\langle {k_ \bot ,m,s} \right.|},\end{aligned}$$ where $$u\left( {k_ \bot } \right) = - \left( {1 - \frac{k_ \bot ^2}{ k_0^2} } \right)^{ - 1/2}.$$
A deeper understanding of the vortex representation can be gained from some geometrical consideration. In the presence of graphene, only rotations around the $z$ axis are allowed and hence the symmetry group is $SO(2)$. This group has infinitely many irreducible representations which are all one-dimensional since it is Abelian. First, the unitary operators $e^{ - i \hat L \vartheta }$ provide a representation of $SO(2)$ on $\mathcal{H}$ and, since $e^{ - i\hat L\vartheta } \left| {k_ \bot ,m,s} \right\rangle = e^{ - im\vartheta } \left| {k_ \bot ,m,s} \right\rangle$, each vector $\left| {k_ \bot ,m,s} \right \rangle$ spans a one-dimensional invariant subspace, i.e. it is the basis of the one-dimensional irreducible representation whose character is $e^{ - im\vartheta }$. This observation explains why the orbital angular momentum $m$ is not bounded as opposed to the constraint $-l<m<l$ pertaining the general $(2l+1)-$dimensional irreducible representation of the rotation group $SO(3)$. Second, the operators $e^{ - i\hat S\vartheta }$ provide a different representation of $SO(2)$ on $\mathcal{H}$ and, since $e^{ - i\hat S\vartheta } \left| {k_ \bot ,m,s} \right\rangle = e^{ - is\vartheta } \left| {k_ \bot ,m,s} \right\rangle$, the vectors $\left| {k_ \bot ,m, + 1} \right\rangle$ and $\left| {k_ \bot ,m, - 1} \right\rangle$ are bases of the two irreducible representations whose characters are $e^{ - i\vartheta }$ and $e^{i\vartheta }$, respectively. This situation should be compared with that of a spin-1 particle whose spin state is three-dimensional and whose spin operator $\hat S_z$ has three eigenvectors providing the three irreducible representations of $SO(2)$ whose characters are $e^{ - i\vartheta }$, $e^{i\vartheta }$ and $1$. As a matter of fact such three eigenvectors correspond in the 3D cartesian space to the unit vectors ${\bf{e}}_L$, ${\bf{e}}_R$ and ${\bf{e}}_z$. In the presence of graphene, the transverse electric field is not coupled with its longitudinal component $E_z$ (see Eqs.(\[scat\])) so that it has been possible to deal with a two dimensional polarization (spin) state space and with the spin operator $\hat{S}$ which is the restriction of its three-dimensional counterpart $\hat S_z$ to the transverse space. Third, the unitary rotation operators $e^{ - i \hat J \vartheta }$ on $\mathcal{H}$ provide a further representation of $SO(2)$. Since each subspace $\mathcal{E} \left( {k_ \bot ,j} \right)$ of Eq.(\[E(kj)\]) is an eigenspace of $\hat{J}$, it is invariant invariant for the rotation operators, and hence it provides the carrier space of a two-dimensional reducible representation of $SO(2)$. Now $e^{ - i\hat J\vartheta } \left| {k_ \bot ,j - 1,s + 1} \right\rangle = e^{ - ij\vartheta } \left| {k_ \bot ,j - 1,s + 1} \right\rangle$ and $e^{ - i\hat J\vartheta } \left| {k_ \bot ,j + 1,s - 1} \right\rangle = e^{ - ij\vartheta } \left| {k_ \bot ,j + 1,s - 1} \right\rangle$ so that each one of these two-dimensional representation is the direct sum of two one-dimensional irreducible representation having the same character $e^{ - ij\vartheta }$.
Orbital and spin ladder opertors {#APP-ladder}
================================
Consider the two operators $$\begin{aligned}
\label{L+L-}
\hat L_ + &=& \left( {\hat D_x + i\hat D_y } \right) \otimes \hat I_{\rm spi}, \nonumber \\
\hat L_ - &=& \left( {\hat D_x - i\hat D_y } \right) \otimes \hat I_{\rm spi}.\end{aligned}$$ which are straightforwardly seen to satisfy the commutation relations $$\begin{aligned}
\label{alge}
[ \hat L_ + ,\hat L_ - ] &=& 0, \nonumber \\
{ [ \hat L, \hat L_\pm ]} &=& \pm \hat L_\pm.\end{aligned}$$ The second of these equations shows that $\hat L_+$ and $\hat L_-$ are orbital ladder operators, raising and lowering the orbital angular momentum $m$, respectively, by one unit. By letting the operators $\hat L_ \pm$ to act on the basis vectors $\left| {k_ \bot ,m,s} \right\rangle$, and using polar coordinates ${\bf{r}}_ \bot = r_ \bot \left( {\cos \varphi {\bf{e}}_x + \sin \varphi {\bf{e}}_y } \right)$ in the transverse plane, we have
$$\begin{aligned}
\label{orb-ladder}
\hat L_ \pm \left| {k_ \bot ,m,s} \right\rangle &=& \int\limits_0^{ + \infty } {dr_ \bot } r_ \bot \int\limits_0^{2\pi } {d\varphi } \;e^{ \pm i\varphi } \left( {\frac{\partial }{{\partial r_ \bot }} \pm \frac{i}{{r_ \bot }}\frac{\partial }{{\partial \varphi }}} \right)\sqrt {\frac{{k_ \bot }}{{2\pi }}} J_m \left( {k_ \bot r_ \bot } \right)e^{im\varphi } \left| {{\bf{r}}_ \bot } \right\rangle \otimes \left| s \right\rangle = \nonumber \\
&=& \sqrt {\frac{{k_ \bot }}{{2\pi }}} \int\limits_0^{ + \infty } {dr_ \bot } r_ \bot \int\limits_0^{2\pi } {d\varphi } \;\left[ {\frac{{\partial J_m \left( {k_ \bot r_ \bot } \right)}}{{\partial r_ \bot }} \mp \frac{m}{{r_ \bot }}J_m \left( {k_ \bot r_ \bot } \right)} \right]e^{i\left( {m \pm 1} \right)\varphi } \left| {{\bf{r}}_ \bot } \right\rangle \otimes \left| s \right\rangle = \nonumber \\
&=& \mp k_ \bot \int\limits_0^{ + \infty } {dr_ \bot } r_ \bot \int\limits_0^{2\pi } {d\varphi } \;\sqrt {\frac{{k_ \bot }}{{2\pi }}} J_{m \pm 1} \left( {k_ \bot r_ \bot } \right)e^{i\left( {m \pm 1} \right)\varphi } \left| {{\bf{r}}_ \bot } \right\rangle \otimes \left| s \right\rangle = \nonumber \\
&=& \mp k_ \bot \left| {k_ \bot ,m \pm 1,s} \right\rangle,\end{aligned}$$
where the Bessel function identity $J_m '\left( \zeta \right) \mp mJ_m \left( \zeta \right)/\zeta = \mp J_{m \pm 1} \left( \zeta \right)$ has been exploited in the third step.
Note that such ladder operators are different from those commonly used when dealing with the full three-dimensional rotation group (i.e. $\hat L_x \pm \hat L_y$). As discussed in Appendix \[APP-eigenvectors\], each vector $\left| {k_ \bot ,m,s} \right\rangle$ spans the one-dimensional carrier space of an irreducible representation of the Abelian group $SO(2)$ (rotations in the plane). Therefore the operators $\hat L_+$ and $\hat L_-$ connect the carrier spaces of different irreducible representation of $SO(2)$ as opposed to the standard ladder operators whose action is restricted to the carrier space of a fixed $(2l+1)$-dimensional irreducible representation of $SO(3)$.
Consider now the two operators $$\begin{aligned}
\label{S+S-}
\hat S_ + &=& \hat I_{\rm orb} \otimes \left| { + 1} \right\rangle \left\langle { - 1} \right|, \nonumber \\
\hat S_ - &=& \hat I_{\rm orb} \otimes \left| { - 1} \right\rangle \left\langle { + 1} \right|,\end{aligned}$$ where the $\left| { \pm 1} \right\rangle$ are the eigenvectors of the spin operator $\hat{S}_{\rm spi}$. They are straightforwardly seen to satisfy the commutation relations $$\begin{aligned}
\label{alge2}
[ \hat S_ + ,\hat S_ - ] & =& \hat S, \nonumber \\
{ [ \hat S,\hat S_ \pm ] } &=& \pm 2\hat S_ \pm,\end{aligned}$$ the second of which shows that $\hat S_ + $ and $\hat S_ - $ are spin ladder operators, raising and lowering the spin $s$, respectively, by two units. The action of these operators on the basis vectors $|\left. {k_ \bot ,m, s} \right\rangle$ is given by $$\begin{aligned}
\hat S_ + |\left. {k_ \bot ,m, + 1} \right\rangle &=& 0, \nonumber \\
\hat S_ + |\left. {k_ \bot ,m, - 1} \right\rangle &=& |\left. {k_ \bot ,m, + 1} \right\rangle, \nonumber \\
\hat S_ - |\left. {k_ \bot ,m, + 1} \right\rangle &=& |\left. {k_ \bot ,m, - 1} \right\rangle, \nonumber \\
\hat S_ - |\left. {k_ \bot ,m, - 1} \right\rangle &=& 0,\end{aligned}$$ which can be summarized as $$\hat S_ \pm |\left. {k_ \bot ,m,s} \right\rangle = \delta _{s, \mp 1} |\left. {k_ \bot ,m,s \pm 2} \right\rangle,$$ From this relation it is particularly evident that the spin raising and lowering are accompanied by the changes $\Delta s = + 2$ and $\Delta s = -2$, respectively.
The interaction operators {#APP-inter}
=========================
The orbital ladder operators provide a natural factorization of the transverse Laplacian operator since $$\label{transv-lap}
\hat D_\bot^2 \otimes \hat{I}_{\rm spi} = \hat L_ + \hat L_ - = \hat L_ - \hat L_ +$$ and accordingly, from Eq.(\[orb-ladder\]) we have $$\hat L_ + \hat L_ - \left| {k_ \bot ,m,s} \right\rangle = - k_ \bot ^2 \left| {k_ \bot ,m,s} \right\rangle.$$ From Eqs.(\[alge\]) it easily seen that $[ \hat L_ + \hat L_ - ,\hat J ] = 0$, which states the well-known rotational invariance of the transverse Laplacian operator. From the first of Eqs.(\[KLS\]), we obtain $$\hat K = - \left( {\hat I + \frac{1}{{k_0^2 }}\hat L_ + \hat L_ - } \right)^{ - 1/2},$$ from which it is evident that $$[ {\hat K,\hat L} ] = [ {\hat K,\hat S} ] = [ {\hat K,\hat J}] =0,$$ so that the operator $\hat K$ is rotationally invariant (it conserves the total angular momentum) and it also conserves both the orbital and spin angular momenta.
By inverting Eq.(\[spin-states\]) we get $$\begin{aligned}
\left| {{\bf{e}}_x } \right\rangle &=& \frac{1}{{\sqrt 2 }}\left( {\left| { + 1} \right\rangle + \left| { - 1} \right\rangle } \right), \nonumber \\
\left| {{\bf{e}}_y } \right\rangle &=& \frac{1}{{\sqrt 2 i}}\left( {\left| { + 1} \right\rangle - \left| { - 1} \right\rangle } \right)\end{aligned}$$ which inserted into the first of Eq.(\[MK\]), after some algebra, yield $$\begin{aligned}
\hat M &=& 1 + \frac{1}{{2k_0^2 }}\left[ \left( {\hat D_x^2 + \hat D_y^2 } \right) \otimes I_{\rm spi} \right. + \nonumber \\
&+& \left. \left( {\hat D_x + i\hat D_y } \right)^2 \left| { - 1} \right\rangle \left\langle { + 1} \right| + \left( {\hat D_x - i\hat D_y } \right)^2 \left| { + 1} \right\rangle \left\langle { - 1} \right| \right]. \nonumber \\\end{aligned}$$ Using Eqs.(\[L+L-\]), (\[transv-lap\]) and (\[S+S-\]), we obtain $$\hat M = \hat I + \frac{1}{{2k_0^2 }}\left( {\hat L_ + \hat L_ - + \hat L_ + ^2 \hat S_ - + \hat L_ - ^2 \hat S_ + } \right).$$ By using Eqs.(\[alge\]) and (\[alge2\]) we obtain the commutation relations $$\begin{aligned}
[ {\hat M,\hat L} ] &=& - [ {\hat M,\hat S} ] = \frac{1}{{k_0^2 }}\left( {- \hat L_ + ^2 \hat S_ - + \hat L_ - ^2 \hat S_ + } \right), \nonumber \\
{[ {\hat M,\hat J} ] }&=& 0.\end{aligned}$$ which shows that the operator $\hat M$ is rotationally invariant (it conserves the total angular momentum) but it does not conserve both the orbital and spin angular momenta.
Solution of the LS equation {#APP-LS}
===========================
In order to solve the LS equation (Eq.(\[lip-sch-abs\])), we expand both the incoming and unknown states in the vortex basis, i.e. $$\begin{aligned}
\label{expa}
\left| {\Psi ^{\left( i \right)} } \right\rangle &=& \sum\limits_{k_ \bot ,m,s} {\psi _{k_ \bot ,m,s}^{\left( i \right)} |\left. {k_ \bot ,m,s} \right\rangle \;}, \nonumber \\
\left| \Psi \right\rangle &=& \sum\limits_{k_ \bot ,m,s} {\psi _{k_ \bot ,m,s} |\left. {k_ \bot ,m,s} \right\rangle },\end{aligned}$$ where $\psi _{k_ \bot ,m,s}^{\left( i \right)} = \left\langle {{k_ \bot ,m,s}} \mathrel{\left | {\vphantom {{k_ \bot ,m,s} {\Psi ^{\left( i \right)} }}} \right. \kern-\nulldelimiterspace} {{\Psi ^{\left( i \right)} }} \right\rangle$ and ${\psi _{k_ \bot ,m,s} }$ are unknown. Substituting Eqs.(\[expa\]) into Eq.(\[lip-sch-abs\]) and exploiting the ladder operators properties, after some algebra we get $$\begin{aligned}
\label{recu}
&&\left[ {\left( {1 - \xi u} \right) + \left( {1 - \xi u^{ - 1} } \right)} \right]\psi _{k_ \bot ,m,s} +\nonumber \\
&+& \left[ {\left( {1 - \xi u} \right) - \left( {1 - \xi u^{ - 1} } \right)} \right]\psi _{k_ \bot ,m + 2s, - s} = 2\psi _{k_ \bot ,m,s}^{\left( i \right)}, \nonumber \\\end{aligned}$$ where $u(k_\bot)$ is the eigenvalue of the operator $\hat K$, as above. Note that each term in Eq.(\[recu\]) has the same index $k_\bot$ and the same total angular momentum $j=m+s$, so that we are effectively solving the LS equation in each subspace $\mathcal{E}\left( {k_ \bot ,j} \right)$ in agreement with the rotationally invariance of the system. Substituting the trial solution $$\psi _{k_ \bot ,m,s} = Q\psi _{k_ \bot ,m,s}^{\left( i \right)} + P\psi _{k_ \bot ,m + 2s, - s}^{\left( i \right)}$$ into Eq.(\[recu\]) we obtain a system of two linear equations for $Q$ and $P$ whose solution is $$\begin{aligned}
Q &=& \frac{1}{2}\left( {\frac{1}{{1 - \xi u}} + \frac{1}{{1 - \xi u^{ - 1} }}} \right), \nonumber \\
P &=& \frac{1}{2}\left( {\frac{1}{{1 - \xi u}} - \frac{1}{{1 - \xi u^{ - 1} }}} \right).\end{aligned}$$ The solution of the LS equation accordingly is $$\begin{aligned}
\left| \Psi \right\rangle &=& \sum\limits_{k_ \bot ,m,s} {\psi _{k_ \bot ,m,s}^{\left( i \right)} Q\left( {k_ \bot } \right)|\left. {k_ \bot ,m,s} \right\rangle } + \nonumber \\
&+& \sum\limits_{k_ \bot ,m,s} {\psi _{k_ \bot ,m,s}^{\left( i \right)} P\left( {k_ \bot } \right)|\left. {k_ \bot ,m + 2s, - s} \right\rangle }\end{aligned}$$ where a suitable index relabelling has been performed in the second term of the RHS.
The scattered electromagnetic field {#APP-field}
===================================
From the relation $$\psi _{k_ \bot ,m,s}^{\left( i \right)} = \left\langle {{k_ \bot ,m,s}} \mathrel{\left | {\vphantom {{k_ \bot ,m,s} {\Psi ^{\left( i \right)} }}} \right. \kern-\nulldelimiterspace} {{\Psi ^{\left( i \right)} }} \right\rangle,$$ by exploiting Eq.(\[vortex\]) and the identification of the $s=+1$ and $s=-1$ spin eigenvalues with the left hand and right hand circular polarizations $L$ and $R$, respectively, we get $$\begin{aligned}
\label{psi}
\begin{pmatrix}
{\psi _{k_ \bot ,m, + 1}^{\left( i \right)} } \\
{\psi _{k_ \bot ,m, - 1}^{\left( i \right)} } \\
\end{pmatrix} &=&
\sqrt {\frac{{k_ \bot }}{{2\pi }}} \int\limits_0^{ + \infty } {dr_ \bot \;} r_ \bot \;J_m \left( {k_ \bot r_ \bot } \right)\int\limits_0^{2\pi } {d\varphi } e^{ - im\varphi } \times \nonumber \\
&\times& \begin{pmatrix}
{E_L^{\left( i \right)} \left( {r_ \bot ,\varphi } \right)} \\
{E_R^{\left( i \right)} \left( {r_ \bot ,\varphi } \right)} \\
\end{pmatrix}.\end{aligned}$$ The circularly polarized components of the field ${\bf{E}}_ \bot \left( {{\bf{r}}_ \bot } \right)$ are straightforwardly obtained by projecting the state $\left| \Psi \right\rangle$ of Eq.(\[LS-Sol\]) onto the basis $\left| {{\bf{r}}_ \bot ,s} \right\rangle = \left| {{\bf{r}}_ \bot } \right\rangle \otimes \left| s \right\rangle$, i.e. $$\begin{aligned}
\label{circ}
E_L \left( {{\bf{r}}_ \bot } \right) &=& \left\langle {{{\bf{r}}_ \bot , + 1}} \mathrel{\left | {\vphantom {{{\bf{r}}_ \bot , + 1} \Psi }} \right. \kern-\nulldelimiterspace} {\Psi } \right\rangle, \nonumber \\
E_R \left( {{\bf{r}}_ \bot } \right) &=& \left\langle {{{\bf{r}}_ \bot , - 1}} \mathrel{\left | {\vphantom {{{\bf{r}}_ \bot , - 1} \Psi }} \right. \kern-\nulldelimiterspace} {\Psi } \right\rangle.\end{aligned}$$ The longitudinal component $E_z$ is discontinuous across the graphene plane (due to the surface charge oscillation hosted by graphene) and its value on the right-side at $z=0^+$ can be evaluated from the second of Eqs.(\[scat\]). However, it is more convenient resorting to the full vectorial angular spectrum representation of the forward propagating field since it also provides the related magnetic field $\bf H$ (which we need for evaluating the Poynting vector). Considering the 2D Fourier transform of the transverse part of the field at $z=0$ $$\label{2dfour}
{\bf{\tilde E}}_ \bot \left( {{\bf{k}}_ \bot } \right) = \frac{1}{{\left( {2\pi } \right)^2 }}\int {d^2 {\bf{r}}_ \bot } e^{ - i{\bf{k}}_ \bot \cdot {\bf{r}}_ \bot } {\bf{E}}_ \bot \left( {{\bf{r}}_ \bot } \right),$$ the forward propagating electric and magnetic fields are
$$\begin{aligned}
\label{angulspec}
\begin{pmatrix}
{E_x } \\
{E_y } \\
{E_z } \\
\end{pmatrix} &=& \int {d^2 {\bf{k}}_ \bot } e^{i{\bf{k}}_ \bot \cdot {\bf{r}}_ \bot } \frac{{e^{ik_z z} }}{{k_z }}
\begin{pmatrix}
{k_z } & 0 \\
0 & {k_z } \\
{ - k_x } & { - k_y } \\
\end{pmatrix}
\begin{pmatrix}
{\tilde E_x } \\
{\tilde E_y } \\
\end{pmatrix}, \nonumber \\
\begin{pmatrix}
{H_x } \\
{H_y } \\
{H_z } \\
\end{pmatrix} &=& \frac{1}{{\omega \mu _0 }}\int {d^2 {\bf{k}}_ \bot } e^{i{\bf{k}}_ \bot \cdot {\bf{r}}_ \bot } \frac{{e^{ik_z z} }}{{k_z }} \begin{pmatrix}
{ - k_x k_y } & { - \left( {k_y^2 + k_z^2 } \right)} \\
{\left( {k_x^2 + k_z^2 } \right)} & {k_x k_y } \\
{ - k_y k_z } & {k_x k_z } \\
\end{pmatrix} \begin{pmatrix}
{\tilde E_x } \\
{\tilde E_y } \\
\end{pmatrix},\end{aligned}$$
where $k_z = \sqrt{k_0^2-k_\bot^2}$. Inserting Eqs.(\[circ\]) into Eq.(\[2dfour\]), after using circularly polarized components ${\bf{\tilde E}}_ \bot = \tilde E_L {\bf{e}}_L + \tilde E_R {\bf{e}}_R$, introducing polar coordinates in both direct and reciprocal space by setting ${\bf{r}}_ \bot = r_ \bot \left( {\cos \varphi {\bf{e}}_x + \sin \varphi {\bf{e}}_y } \right)$, ${\bf{k}}_ \bot = k_ \bot \left( {\cos \theta {\bf{e}}_x + \sin \theta {\bf{e}}_y } \right)$ and resorting to the Anger-Jacobi formula $$e^{i\zeta \cos \Phi } = \sum\limits_{m = - \infty }^{ + \infty } {i^m } J_m \left( \zeta \right)e^{im\Phi },$$ we get $$\label{LR}
\begin{pmatrix}
{\tilde E_L } \\
{\tilde E_R } \\
\end{pmatrix} = \sum\limits_{m = - \infty }^\infty {\frac{{e^{im\theta } }}{{i^m \sqrt {8\pi ^3 k_ \bot } }}} \begin{pmatrix}
{Q\psi _{k_ \bot ,m, + 1}^{\left( i \right)} + P\psi _{k_ \bot ,m + 2, - 1}^{\left( i \right)} } \\
{P\psi _{k_ \bot ,m - 2, + 1}^{\left( i \right)} + Q\psi _{k_ \bot ,m, - 1}^{\left( i \right)} } \\
\end{pmatrix}.$$ Using circularly polarized components for the transverse parts of the fields in Eqs.(\[angulspec\]) at $z=0$ and using Eq.(\[LR\]), after some tedious but straightforward algebra, we obtain
$$\begin{aligned}
\label{FIELDS}
\begin{pmatrix}
{E_L } \\
{E_R } \\
{E_z } \\
\end{pmatrix} &=& \int\limits_0^{ + \infty } {dk_ \bot } \sum\limits_{m = - \infty }^\infty {\sqrt {\frac{{k_ \bot }}{{2\pi }}} \begin{pmatrix}
{J_m \left( {k_ \bot r_ \bot } \right)e^{im\varphi } Q} & {J_{m - 2} \left( {k_ \bot r_ \bot } \right)e^{i\left( {m - 2} \right)\varphi } P} \\
{J_{m + 2} \left( {k_ \bot r_ \bot } \right)e^{i\left( {m + 2} \right)\varphi } P} & {J_m \left( {k_ \bot r_ \bot } \right)e^{im\varphi } Q} \\
{J_{m + 1} \left( {k_ \bot r_ \bot } \right)e^{i\left( {m + 1} \right)\varphi } \frac{{i }}{{k_z }}W_ - } & { - J_{m - 1} \left( {k_ \bot r_ \bot } \right)e^{i\left( {m - 1} \right)\varphi } \frac{{i }}{{k_z }} W_ -} \\
\end{pmatrix}\begin{pmatrix}
{\psi _{k_ \bot ,m, + 1}^{\left( i \right)} } \\
{\psi _{k_ \bot ,m, - 1}^{\left( i \right)} } \\
\end{pmatrix}}, \nonumber \\
\begin{pmatrix}
{H_L } \\
{H_R } \\
{H_z } \\
\end{pmatrix} &=& \frac{1}{{\omega \mu _0 }}\int\limits_0^{ + \infty } {dk_ \bot } \sum\limits_{m = - \infty }^\infty {\sqrt {\frac{{k_ \bot }}{{2\pi }}} } \begin{pmatrix}
{ - J_m \left( {k_ \bot r_ \bot } \right)e^{im\varphi } \tilde Q} & { - J_{m - 2} \left( {k_ \bot r_ \bot } \right)e^{i\left( {m - 2} \right)\varphi } \tilde P} \\
{J_{m + 2} \left( {k_ \bot r_ \bot } \right)e^{i\left( {m + 2} \right)\varphi } \tilde P} & {J_m \left( {k_ \bot r_ \bot } \right)e^{im\varphi } \tilde Q} \\
{J_{m + 1} \left( {k_ \bot r_ \bot } \right)e^{i\left( {m + 1} \right)\varphi } W_ + } & {J_{m - 1} \left( {k_ \bot r_ \bot } \right)e^{i\left( {m - 1} \right)\varphi } W_ + } \\
\end{pmatrix}\begin{pmatrix}
{\psi _{k_ \bot ,m, + 1}^{\left( i \right)} } \\
{\psi _{k_ \bot ,m, - 1}^{\left( i \right)} } \\
\end{pmatrix},\end{aligned}$$
where $$\begin{aligned}
W_ \pm &=& - \frac{{k_ \bot }}{{\sqrt 2 }}\left[ {Q \pm P} \right], \nonumber \\
\tilde Q &=& \frac{i}{{2k_z }}\left[ {\left( {k_z^2 + k_0^2 } \right)Q - k_ \bot ^2 P} \right], \nonumber \\
\tilde P &=& \frac{i}{{2k_z }}\left[ { - k_ \bot ^2 Q + \left( {k_z^2 + k_0^2 } \right)P} \right].\end{aligned}$$ Equations (\[FIELDS\]) together with Eq.(\[psi\]) allows to predict the full electromagnetic field on the right side of the graphene sheet (i.e. at $z=0^+$) once the transverse part of the incoming field is known.
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|
---
abstract: 'An equilateral triangle cannot be dissected into finitely many mutually incongruent equilateral triangles [@tutte1948]. Therefore Tuza [@tuza1991] asked for the largest number $s=s(n)$ such that there is a tiling of an equilateral triangle by $n$ equilateral triangles of $s(n)$ different sizes. We solve that problem completely and consider the analogous questions for dissections of convex $k$-gons into equilateral triangles, $k=4,5,6$. Moreover, we discuss all these questions for the subclass of tilings such that no two tiles are translates of each other.'
address: 'Institute of Mathematics, Friedrich Schiller University, D-07737 Jena, Germany'
author:
- Christian Richter
title: Tilings of convex polygons by equilateral triangles of many different sizes
---
Overview
========
Motivation
----------
A (finite) *tiling* of a subset $A$ of the Euclidean plane $\mathbb{R}^2$ is a family $\mathcal{T}=\{A_1,\ldots,A_n\}$ of subsets of $A$, called *tiles*, such that $A=A_1 \cup \ldots \cup A_n$ and the interiors of $A_1,\ldots,A_n$ are mutually disjoint. A tiling is called *perfect* if all tiles are images of each other under similarity transformations, but mutually incongruent under isometries. The concept of a perfect tiling has been introduced in [@brooks1940] for the construction of perfect dissections of rectangles and even of squares into squares. Tutte showed that there are no perfect tilings of triangles by (at least two) equilateral triangles [@tutte1948 Theorem 2$\cdot$12] (see also [@brooks1940 Section 10.3]). In fact, no convex polygon admits a perfect tiling by equilateral triangles, as has been shown by Buchman [@buchman1981 Section 2] and Tuza [@tuza1991 Theorem 1].
Therefore Tuza considered tilings of triangles by equilateral triangles that are nearly perfect in the sense that only few of the tiles are isometrically congruent: The *size* ${\operatorname{size}}(T)$ of an equilateral triangle $T$ is measured by the length of (one of) its sides. For a family $\mathcal{T}=\{T_1,\ldots,T_n\}$ of equilateral triangles, we define $s(\mathcal{T})=|\{{\operatorname{size}}(T_1),\ldots,{\operatorname{size}}(T_n)\}|$, where $|\cdot|$ denotes the cardinality of a set. Tuza introduced the numbers $${\operatorname{s}_\text{\scriptsize\rm tri}}(n)=\max\{s(\mathcal{T}): \mathcal{T} \mbox{ is a tiling of a triangle by } n \mbox{ equilateral triangles}\}.$$ He showed by an inflation argument that there exists a constant $c$, $\frac{5}{7} \le c \le 1$, such that ${\operatorname{s}_\text{\scriptsize\rm tri}}(n)=cn-o(n)$ as $n \to \infty$ [@tuza1991 Theorem 2] and he asked for the exact values of ${\operatorname{s}_\text{\scriptsize\rm tri}}(n)$ [@tuza1991 Problem 1] or at least for the exact value of $c$. We solve that problem completely in Theorem \[thm:general\](a).
If a convex polygon $P$ admits a tiling by equilateral triangles, its inner angles are of size $\frac{\pi}{3}$ or $\frac{2\pi}{3}$ and $P$ must be an equilateral triangle (all inner angles have size $\frac{\pi}{3})$, a trapezoid (sizes of angles are $\frac{\pi}{3}, \frac{\pi}{3}, \frac{2\pi}{3}, \frac{2\pi}{3}$ in cyclic order), a parallelogram ($\frac{\pi}{3}, \frac{2\pi}{3}, \frac{\pi}{3}, \frac{2\pi}{3}$), a pentagon ($\frac{\pi}{3}, \frac{2\pi}{3}, \frac{2\pi}{3}, \frac{2\pi}{3}, \frac{2\pi}{3}$) or a hexagon (six times $\frac{2\pi}{3}$). *(When speaking of trapezoids in the present paper we mean trapezoids that are no parallelograms.)* As we know that no convex polygon has a perfect tiling by equilateral triangles, we introduce relatives of Tuza’s numbers, namely $$\begin{aligned}
{\operatorname{s}_\text{\scriptsize\rm trap}}(n)=\max\{s(\mathcal{T}):\;& \mathcal{T} \mbox{ is a tiling of some convex}\\
&\mbox{trapezoid by } n \mbox{ equilateral triangles}\}\end{aligned}$$ and, similarly, ${\operatorname{s}_\text{\scriptsize\rm par}}(n)$, ${\operatorname{s}_\text{\scriptsize\rm pent}}(n)$ and ${\operatorname{s}_\text{\scriptsize\rm hex}}(n)$ with ‘trapezoid’ replaced by ‘parallelogram’, ‘pentagon’ and ‘hexagon’, respectively. Theorem \[thm:general\] gives our results on these numbers including the determination of the *domains* of the functions ${\operatorname{s}_\text{\scriptsize\rm tri}},\ldots,{\operatorname{s}_\text{\scriptsize\rm hex}}$. These are the sets of all integers $n$ such that there exists a tiling of a suitable convex polygon of the respective shape by $n$ equilateral triangles.
Another possibility of weakening the property of perfectness is based on the fact that, if two tiles of a tiling of a convex set by equilateral triangles are congruent under isometries, then they are either congruent under some translation or under some rotation by an angle of $\pi$. Some authors call the tiling already perfect if no two tiles are congruent under translations [@tutte1948; @drapal2010]. In order to avoid confusion with (isometric) perfectness, we shall speak of translational perfectness (or t-perfectness for short). That is, a tiling by equilateral triangles is called *t-perfect* if no two tiles are translates of each other. First examples of t-perfect tilings of parallelograms and triangles by equilateral triangles are given in [@tutte1948]. Drápal and Hämäläinen [@drapal2010] present a systematic computational approach to dissections of equilateral triangles into equilateral triangles with a particular emphasis on the t-perfect case. Illustrations of all t-perfect tilings of triangles by up to $19$ equilateral triangles are given in [@hamalainen].
We are interested in t-perfect tilings of convex polygons by equilateral triangles that are close to be (isometrically) perfect in so far as the number of tiles of equal size is as small as possible. More precisely, we study the numbers $$\begin{aligned}
{\operatorname{s}_\text{\scriptsize\rm tri}^\text{\scriptsize\rm t-perf}}(n)=\max\{s(\mathcal{T}):\,& \mathcal{T} \mbox{ is a t-perfect tiling of a}\\
&\mbox{triangle by } n \mbox{ equilateral triangles}\}\end{aligned}$$ as well as the analogous variants ${\operatorname{s}_\text{\scriptsize\rm trap}^\text{\scriptsize\rm t-perf}}(n)$, ${\operatorname{s}_\text{\scriptsize\rm par}^\text{\scriptsize\rm t-perf}}(n)$, ${\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}(n)$ and ${\operatorname{s}_\text{\scriptsize\rm hex}^\text{\scriptsize\rm t-perf}}(n)$ of ${\operatorname{s}_\text{\scriptsize\rm trap}}(n)$, ${\operatorname{s}_\text{\scriptsize\rm par}}(n)$, ${\operatorname{s}_\text{\scriptsize\rm pent}}(n)$ and ${\operatorname{s}_\text{\scriptsize\rm hex}}(n)$, respectively. Our corresponding results are summarized in Theorem \[thm:perfect\].
Finally, let us point out that the study of dissections into incongruent equilateral triangles is a fruitful field of ongoing research, see e.g. [@croft1991 Section C11], [@gruenbaum1987 Exercise 2.4.10], [@nandakumar Problem 4] and [@aduddell2017; @klaassen1995; @pach2018; @richter2012; @richter2018; @richter-wirth; @scherer1983].
Main results and open problems
------------------------------
\[thm:general\]
1. ${\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm tri}})=\{1,4\} \cup \{6,7,\ldots\}$ and $${\operatorname{s}_\text{\scriptsize\rm tri}}(n)=\left\{
\begin{array}{cl}
1, & n=1,4,\\
2, & n=6,\\
n-5, &n=7,8,\ldots
\end{array}
\right.$$
2. ${\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm trap}})=\{3\} \cup \{5,6,\ldots\}$ and $${\operatorname{s}_\text{\scriptsize\rm trap}}(n)=\left\{
\begin{array}{cl}
1, & n=3,\\
2, & n=5,\\
n-4, &n=6,7,\ldots
\end{array}
\right.$$
3. ${\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm par}})=\{2\} \cup \{4,5,\ldots\}$ and $${\operatorname{s}_\text{\scriptsize\rm par}}(n)=\left\{
\begin{array}{cl}
1, & n=2,4,\\
2, & n=5,\\
n-4, &n=6,7,\ldots
\end{array}
\right.$$
4. ${\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm pent}})=\{4,5,\ldots\}$ and $${\operatorname{s}_\text{\scriptsize\rm pent}}(n)=\left\{
\begin{array}{cl}
2, & n=4,\\
n-3, &n=5,6,\ldots
\end{array}
\right.$$
5. ${\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm hex}})=\{6,7,\ldots\}$ and $${\operatorname{s}_\text{\scriptsize\rm hex}}(n)\left\{
\begin{array}{ll}
=n-5, & n=6,7,8,\\
=n-4, & n=9,10,\ldots,19;\;21,22;\;24,25,\\
\in \{n-5,n-4\}, & n=20;\;23;\;26,27,\ldots
\end{array}
\right.$$
Here only the case of hexagons is not completely solved. All tilings we know to attain the upper bound ${\operatorname{s}_\text{\scriptsize\rm hex}}(n) \le n-4$ ($n \ge 9$) are displayed in the appendix (tilings (a)-(s)). Is this list complete?
The situation of t-perfect tilings appears more difficult. Only for pentagons we have a full solution. However, in the open cases the differences between upper and lower estimates are at most $2$. In the case of hexagons it remains open if there are t-perfect tilings by $12$ or $13$ triangles. We conjecture that $12,13 \notin{\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm hex}^\text{\scriptsize\rm t-perf}}\right)$.
\[thm:perfect\]
1. ${\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm tri}^\text{\scriptsize\rm t-perf}}\right)=\{1\}\cup\{15,16,\ldots\}$ and $${\operatorname{s}_\text{\scriptsize\rm tri}^\text{\scriptsize\rm t-perf}}(n)\left\{
\begin{array}{ll}
=1, & n=1,\\
=n-5, & n=15;\;17,18,\ldots,26;\;28,\\
=n-6, & n=16,\\
\in \{n-6,n-5\}, &n=27;\;29,30,\ldots
\end{array}
\right.$$
2. ${\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm trap}^\text{\scriptsize\rm t-perf}}\right)= \{13,14,\ldots\}$ and $${\operatorname{s}_\text{\scriptsize\rm trap}^\text{\scriptsize\rm t-perf}}(n)\left\{
\begin{array}{ll}
=n-4, & n=14;\;16,17,\ldots,25;\;27,\\
\in\{n-5,n-4\}, &n=13;\;15;\;26;\;28,29,\ldots
\end{array}
\right.$$
3. ${\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm par}^\text{\scriptsize\rm t-perf}}\right)= \{2\} \cup \{13,14,\ldots\}$ and $${\operatorname{s}_\text{\scriptsize\rm par}^\text{\scriptsize\rm t-perf}}(n)\left\{
\begin{array}{ll}
=1, & n=2,\\
=n-4, &n=15;\;18,19;\;21,22,23;\;26,\\
\in\{n-5,n-4\}, &n=13,14;\;16,17;\;20;\;24,25;\;27,28,\ldots
\end{array}
\right.$$
4. ${\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}\right)= \{12,13,\ldots\}$ and $${\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}(n)=n-4 \mbox{ for all } n=12,13,\ldots$$
5. $\{11\} \cup \{14,15,\ldots\} \subseteq {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm hex}^\text{\scriptsize\rm t-perf}}\right) \subseteq \{11,12,\ldots\}$ and $${\operatorname{s}_\text{\scriptsize\rm hex}^\text{\scriptsize\rm t-perf}}(n)\left\{
\begin{array}{ll}
=n-4, & n=11;\;14,15;\;17,18,19;\;22,\\
\in \{n-5,n-4\}, & n=16;\;20,21;\;23,\\
\in \{n-6,n-5,n-4\}, & n=24,25,\ldots
\end{array}
\right.$$
Proof of Theorem \[thm:general\]
================================
Spiral pentagons and related tilings
------------------------------------
The *Padovan spiral numbers* [@steward1996; @oeis] are defined recursively by $$\label{eq:Pad}
p(0)=p(1)=p(2)=1 \quad\mbox{and}\quad p(n)=p(n-3)+p(n-2) \mbox{ for } n=3,4,\ldots$$
\[lem:spiral\]
1. The Padovan spiral numbers satisfy
- $p(0)=p(1)=p(2)=1$, $p(3)=p(4)=2$ and $p(n)>p(n-1)$ for $n \ge 5$,
- $p(n-3) < \frac{1}{2}p(n) < p(n-2)$ for $n \notin \{3,4,6\}$.
2. For every $n \in \{4,5,\ldots\}$, there is a convex pentagon $P_n$ with sides of lengths $p(n-4)$, $p(n-3)$, $p(n-2)$, $p(n-1)$ and $p(n)$ that admits a tiling by $n$ equilateral triangles $T_i$ with ${\operatorname{size}}(T_i)=p(i-1)$, $i=1,\ldots,n$ (see [@steward1996]).
The first part of (a) follows from by induction. It implies the second one by $p(n)=p(n-3)+p(n-2)$. Claim (b) can be found in [@steward1996], see Figure \[fig:spiral\]: the pentagon $P_{n+1}$ is obtained from $P_n$ by adding a triangle of size $p(n)$ at the longest side of $P_n$.
(22,15)–(6,15)–(14,3)–cycle ; (6,15)–(0,6)–(12,6)–cycle ; (0,6)–(4,0)–(8,6)–cycle ; (4,0)–(12,0)–(8,6)–cycle ; (12,0)–(14,3)–(10,3)–cycle ; (14,3)–(10,3)–(12,6)–cycle ; (10,3)–(12,6)–(8,6)–cycle ;
(0,6)–(4,0)–(32,0)–(22,15)–(6,15)–(0,6)–(12,6)–(10,3)–(14,3)–(6,15) (4,0)–(8,6)–(12,0)–(22,15) (10,5) node [$1$]{} (12,4) node [$1$]{} (12,2) node [$1$]{} (8,2) node [$2$]{} (4,4) node [$2$]{} (6,9) node [$3$]{} (14,11) node [$4$]{} (22,5) node [$5$]{} ;
\[cor:lower\]
1. $\{6,7,\ldots\}\subseteq{\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm tri}})$ and $${\operatorname{s}_\text{\scriptsize\rm tri}}(n)\ge\left\{
\begin{array}{cl}
2, & n=6,\\
n-5, &n=7,8,\ldots
\end{array}
\right.$$
2. $\{5,6,\ldots\}\subseteq{\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm trap}})$ and $${\operatorname{s}_\text{\scriptsize\rm trap}}(n)\ge\left\{
\begin{array}{cl}
2, & n=5,\\
n-4, &n=6,7,\ldots
\end{array}
\right.$$
3. $\{5,6,\ldots\}\subseteq{\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm par}})$ and $${\operatorname{s}_\text{\scriptsize\rm par}}(n)\ge\left\{
\begin{array}{cl}
2, & n=5,\\
n-4, &n=6,7,\ldots
\end{array}
\right.$$
4. $\{4,5,\ldots\}\subseteq{\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm pent}})$ and $${\operatorname{s}_\text{\scriptsize\rm pent}}(n)\ge\left\{
\begin{array}{cl}
2, & n=4,\\
n-3, &n=5,6,\ldots
\end{array}
\right.$$
5. $\{7,8,\ldots\}\subseteq{\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm hex}})$ and ${\operatorname{s}_\text{\scriptsize\rm hex}}(n) \ge n-5$ for all $n=7,8,\ldots$
The pentagons $P_n$, $n= 4,5,\ldots$, from Lemma \[lem:spiral\](b) prove (d). For (a), we add two triangles of sizes $p(n-4)$ and $p(n-2)$ at the respective sides of $P_n$, cf. Figure \[fig:derived polygons\].
at (0,0) ; at (2.5,0) ; at (5,0) ; at (7.5,0) ; at (0,-1) [(a)]{}; at (2.5,-1) [(b)]{}; at (5,-1) [(c)]{}; at (7.5,-1) [(e)]{};
For (b), we add only one of these two triangles. For (c), we add a triangle of size $p(n-3)$. Every added triangle is of the same size as one of the original tiling of $P_n$ from Lemma \[lem:spiral\](b).
For (e), we add three triangles of size $\frac{1}{2}p(n)$ at the longest side of length $p(n)$ if $n \ne 6$, see Figure \[fig:derived polygons\]. By Lemma \[lem:spiral\](a), the size of the new triangles differs from all sizes of the given tiling of $P_n$ if $n=5$ or $n \ge 7$, whereas that size appears already in the tiling of $P_n$ if $n=4$. For $n=6$, we add three triangles of size $\frac{1}{2}p(5)=\frac{3}{2}\notin\{p(0),\ldots,p(5)\}$ at the side of size $p(5)=3$ of $P_6$.
Domains and lower estimates\[subsec:realizations\]
--------------------------------------------------
If a convex polygon $P$ has a tiling $\mathcal{T}$ by equilateral triangles, its inner angles are of sizes $\frac{\pi}{3}$ and $\frac{2\pi}{3}$. A side between two vertices of sizes $\alpha$ and $\beta$ is called an *$(\alpha,\beta)$-side of $P$*. All vertices of triangles from $\mathcal{T}$ are called *vertices of $\mathcal{T}$*. We speak of *$\frac{\pi}{3}$-vertices or $\frac{2\pi}{3}$-vertices of $\mathcal{T}$*, if they coincide with vertices of $P$ of the respective sizes, of *$\pi$-vertices of $\mathcal{T}$*, if they are no vertices of $P$ but on the boundary of $P$, and of *$2\pi$-vertices of $\mathcal{T}$* if they are in the interior of $P$. A triangle $T \in \mathcal{T}$ is called *exposed* if it contains a $\frac{\pi}{3}$-vertex of $\mathcal{T}$ or, equivalently, if $P \setminus T$ is still convex.
Let us show that $$\begin{aligned}
2,3;\; 5 & \notin & {\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm tri}}),\label{eq:dom3}\\
1,2;\; 4 & \notin & {\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm trap}}),\label{eq:dom4t}\\
1;\; 3 & \notin & {\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm par}}),\label{eq:dom4p}\\
1,2,3 & \notin & {\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm pent}}),\label{eq:dom5}\\
1,2,3,4,5 & \notin & {\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm hex}})\label{eq:dom6}.\end{aligned}$$
To see , let $n$ be the cardinality of a tiling $\mathcal{T}$ of some triangle $T$ and let $v_\pi$ be the number of $\pi$-vertices of $\mathcal{T}$. If $v_\pi=0$ then $n=1$. If $1 \le v_\pi \le 3$ then $v_\pi=3$ and $\mathcal{T}$ splits into three congruent exposed triangles and a tiling $\mathcal{T}'$ of the remaining triangle $T'$ of $T$. The cardinality $|\mathcal{T}'|=n-3$ cannot be two, whence $n=4$ (if $|\mathcal{T}'|=1$) or $n \ge 6$ (if $|\mathcal{T}'| \ge 3$). If $v_\pi \ge 4$ then one side of $T$ contains two consecutive $\pi$-vertices $v_1,v_2$. Hence $n \ge 6$, because $\mathcal{T}$ contains three exposed triangles, one triangle covering the segment $v_1v_2$ and two triangles for covering the yet uncovered remainders of neighbourhoods of $v_1$ and $v_2$.
For , note that $n \in {\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm trap}})$ implies $n+1 \in {\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm tri}})$, since a trapezoid can be transformed into a triangle by adding one triangle at its $\left(\frac{2\pi}{3},\frac{2\pi}{3}\right)$-side. Hence implies . Similarly, $n \in {\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm par}})$ implies $n+1 \in {\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm trap}})$ by adding a suitable triangle. Thus $\eqref{eq:dom4p}$ is a consequence of $\eqref{eq:dom4t}$.
A tiling of a convex pentagon contains at least four tiles, because it must contain one exposed triangle at the $\frac{\pi}{3}$-vertex and at least one tile for each of the three $\left(\frac{2\pi}{3},\frac{2\pi}{3}\right)$-sides. This gives . Similarly, we obtain , since a tiling of a convex hexagon contains at least one tile for each of the six sides of the hexagon.
Claims - together with Figure \[fig:small tilings\] and Corollary \[cor:lower\] show that the domains of ${\operatorname{s}_\text{\scriptsize\rm tri}}$, ${\operatorname{s}_\text{\scriptsize\rm trap}}$, ${\operatorname{s}_\text{\scriptsize\rm par}}$, ${\operatorname{s}_\text{\scriptsize\rm pent}}$ and ${\operatorname{s}_\text{\scriptsize\rm hex}}$ are as claimed in Theorem \[thm:general\].
(1.5,1)–(3.5,1)–(2.5,3)–(1.5,1) (4,0)–(8,0)–(6,4)–(4,0) (6,0)–(7,2)–(5,2)–(6,0)
(11,3)–(10,1)–(14,1)–(13,3)–(11,3)–(12,1)–(13,3)
(17,3)–(16,1)–(18,1)–(19,3)–(17,3)–(18,1) (20,3)–(19,1)–(23,1)–(24,3)–(20,3)–(21,1)–(22,3)–(23,1)
(26,2)–(27,0)–(29,0)–(30,2)–(29,4)–(27,4)–(26,2)–(30,2) (27,0)–(29,4) (29,0)–(27,4) ;
Moreover, Corollary \[cor:lower\] gives the lower estimates of these functions from Theorem \[thm:general\] apart from ${\operatorname{s}_\text{\scriptsize\rm hex}}(n) \ge n-4$ for $n \in \{9,\ldots,19\} \cup \{21,22,24,25\}$. These last estimates are justified by the tilings (a)-(s) presented in the appendix. We do not know other tilings of hexagons that realize ${\operatorname{s}_\text{\scriptsize\rm hex}}(n) \ge n-4$. Table \[tab:hexagons\] gives the parameters of the illustrated tilings.
A necessary condition
---------------------
\[lem:necessary\] Let $\mathcal{T}$ be a tiling of a convex $m$-gon $P$ by at least two equilateral triangles, and let $v_\pi$ be the number of $\pi$-vertices of $\mathcal{T}$. Then there are at least $m+v_\pi-3$ pairs of distinct triangles from $\mathcal{T}$ having a side in common.
We can assume that $$\label{eq:1}
v_\pi \ge 9-2m.$$ Indeed, this is trivial if $m \ge 5$. If $m=3$ then $v_\pi \ge 3$, since the trivial tiling $\mathcal{T}=\{P\}$ is excluded. If $m=4$ and $v_\pi=0$ then $\mathcal{T}$ is a tiling of a rhombus $P$ by two equilateral triangles, and the claim of the lemma is obvious.
The following graph-theoretic arguments generalize a similar approach from [@tutte1948; @buchman1981]. We associate a bipartite planar graph $\Gamma$ to $\mathcal{T}$: In the interior of every triangle from $\mathcal{T}$ we place a white node of $\Gamma$. All vertices of $\mathcal{T}$, except for the $\frac{\pi}{3}$-vertices, are the black nodes of $\Gamma$. A black and a white node of $\Gamma$ are joined by an edge if the black one is a vertex of the triangle represented by the white one, see Figure \[fig:graph\].
(2,3.464)–(0,0) (0,0)–(6,0) (6,0)–(8,3.464) (8,3.464)–(2,3.464)–(4,0)–(6,3.464)–(7,1.732)–(3,1.732)–(4,3.464)–(6,0) ;
(2,1.155) circle (.2) (2,3.464)–(2,1.355) (4,0)–(2.173,1.055) (4,1.155) circle (.2) (4,.955)–(4,0) (3.827,1.255)–(3,1.732) (4.173,1.255)–(5,1.732) (6,1.155) circle (.2) (6,.955)–(6,0) (5.827,1.255)–(5,1.732) (6.173,1.255)–(7,1.732) (5,.577) circle (.2) (5,.777)–(5,1.732) (4.827,.477)–(4,0) (5.173,.477)–(6,0) (3,2.887) circle (.2) (3,2.687)–(3,1.732) (2.827,2.987)–(2,3.464) (3.173,2.987)–(4,3.464) (5,2.887) circle (.2) (5,2.687)–(5,1.732) (4.827,2.987)–(4,3.464) (5.173,2.987)–(6,3.464) (7,2.887) circle (.2) (7,2.687)–(7,1.732) (6.827,2.987)–(6,3.464) (4,2.309) circle (.2) (4,2.509)–(4,3.464) (3.827,2.209)–(3,1.732) (4.173,2.209)–(5,1.732) (6,2.309) circle (.2) (6,2.509)–(6,3.464) (5.827,2.209)–(5,1.732) (6.173,2.209)–(7,1.732) ;
(2,3.464) circle (.2) (4,3.464) circle (.2) (6,3.464) circle (.2) (3,1.732) circle (.2) (5,1.732) circle (.2) (7,1.732) circle (.2) (4,0) circle (.2) (6,0) circle (.2) ;
Let $v$, $e$ and $f$ be the numbers of nodes, edges and faces of $\Gamma$, respectively.
All nodes of $\Gamma$ have degree $2$, $3$ or $6$. Denoting the respective numbers by $v_2$, $v_3$ and $v_6$, we have $$\label{eq:2}
v=v_2+v_3+v_6.$$ Since $P$ is a convex $m$-gon whose inner angles have sizes of $\frac{\pi}{3}$ or $\frac{2\pi}{3}$, it has $6-m$ inner angles of size $\frac{\pi}{3}$ and $2m-6$ inner angles of size $\frac{2\pi}{3}$. A white node of $\Gamma$ has degree $2$ if and only if it represents a triangle that covers an angle of $P$ of size $\frac{\pi}{3}$. A black node of $\Gamma$ has degree $2$ if and only if it corresponds to an angle of $P$ of size $\frac{2\pi}{3}$. Thus $$\label{eq:3}
v_2=(6-m)+(2m-6)=m.$$ Counting the edges of $\Gamma$ in terms of the nodes we obtain $$\label{eq:4}
2e=2v_2+3v_3+6v_6.$$
Since $\Gamma$ is bipartite, we have $$\label{eq:5}
f=\sum_{i=2}^\infty f_{2i}$$ where $f_j$ is the number of faces with $j$ edges. The boundary of $P$ contains exactly $(2m-6)+v_\pi$ black nodes of $\Gamma$: $2m-6$ of them represent inner angles of size $\frac{2\pi}{3}$ and $v_\pi$ of them are $\pi$-vertices. Hence the unbounded face of $\Gamma$ has $2((2m-6)+v_\pi)$ edges, and in turn $$\label{eq:6}
f_{2(2m-6+v_\pi)} \ge 1.$$ Counting the edges of $\Gamma$ in terms of the faces we obtain $$\label{eq:7}
2e=\sum_{i=2}^\infty 2if_{2i}.$$
Euler’s formula for $\Gamma$ gives $$\begin{aligned}
2 &=& f-e+v\\
&=& \left(f-\frac{1}{6}\,2e\right)+\left(v-\frac{1}{3}\,2e\right)\\
&\stackrel{\text{(\ref{eq:2},\ref{eq:4},\ref{eq:5},\ref{eq:7})}}{=}& \left(\sum_{i=2}^\infty f_{2i}-\frac{1}{6}\sum_{i=2}^\infty 2if_{2i}\right)+\left(v_2+v_3+v_6-\frac{1}{3}(2v_2+3v_3+6v_6)\right)\\
&\stackrel{\text{(\ref{eq:3})}}{=}& \frac{1}{3}\left(f_4-\sum_{i=3}^\infty (i-3)f_{2i}\right)+\frac{1}{3}\left(m-3v_6\right).\end{aligned}$$ Thus $$f_4=6+\sum_{i=3}^\infty (i-3)f_{2i}-m+3v_6.$$ Using $2m-6+v_\pi \ge 3$ (as a consequence of ) and $v_6 \ge 0$ we get $$f_4 \ge 6+ ((2m-6+v_\pi)-3)f_{2(2m-6+v_\pi)}-m$$ and, by , $$f_4 \ge 6+ ((2m-6+v_\pi)-3)-m=m+v_\pi-3.$$ This estimate completes the proof, because every face of $\Gamma$ with four edges represents two triangles of $\mathcal{T}$ having a side in common.
Upper estimates\[subsec:upgen\]
-------------------------------
We shall use the following observation.
\[lem:trapezoid\] If a tiling $\mathcal{T}$ of a trapezoid by equilateral triangles satisfies one of the conditions
- all $\pi$-vertices of $\mathcal{T}$ are contained in the $\left(\frac{\pi}{3},\frac{\pi}{3}\right)$-side of the trapezoid or
- $|\mathcal{T}|=3$,
then $\mathcal{T}$ is a similar image of the respective tiling from Figure \[fig:small tilings\] and $s(\mathcal{T})=1$.
We can assume ($\alpha$), since ($\beta$) implies ($\alpha$). Let $T_1,T_2 \in \mathcal{T}$ be the tiles covering the two $\left( \frac{\pi}{3},\frac{2\pi}{3} \right)$-sides of the trapezoid and let $T_3 \in \mathcal{T}$ be the triangle that covers the $\left( \frac{2\pi}{3},\frac{2\pi}{3} \right)$-side. Since the two $\left( \frac{\pi}{3},\frac{2\pi}{3} \right)$-sides have the same length, ${\operatorname{size}}(T_1)={\operatorname{size}}(T_2)$. We can exclude the situations ${\operatorname{size}}(T_3) < {\operatorname{size}}(T_1)$, because then the interiors of $T_1$ and $T_2$ would overlap, and ${\operatorname{size}}(T_3) > {\operatorname{size}}(T_1)$, since then the third vertex of $T_3$ would be outside the trapezoid. Hence ${\operatorname{size}}(T_1)={\operatorname{size}}(T_2)={\operatorname{size}}(T_3)$ and $T_3$ covers the remainder of the trapezoid between $T_1$ and $T_2$. The claim follows.
For completing the proof of Theorem \[thm:general\], it remains to show that $$\begin{aligned}
{\operatorname{s}_\text{\scriptsize\rm tri}}(n) &\le& \left\{
\begin{array}{cl}
1, & n=1,4,\\
2, & n=6,\\
n-5, &n=7,8,\ldots,
\end{array}
\right.
\label{eq:uptri}\\
{\operatorname{s}_\text{\scriptsize\rm trap}}(n) &\le& \left\{
\begin{array}{cl}
1, & n=3,\\
2, & n=5,\\
n-4, &n=6,7,\ldots,
\end{array}
\right.
\label{eq:uptra}\\
{\operatorname{s}_\text{\scriptsize\rm par}}(n) &\le& \left\{
\begin{array}{cl}
1, & n=2,4,\\
2, & n=5,\\
n-4, &n=6,7,\ldots,
\end{array}
\right.
\label{eq:uppar}\\
{\operatorname{s}_\text{\scriptsize\rm pent}}(n) &\le& \left\{
\begin{array}{cl}
2, & n=4,\\
n-3, &n=5,6,\ldots,
\end{array}
\right.
\label{eq:uppen}\\
{\operatorname{s}_\text{\scriptsize\rm hex}}(n) &\le& \left\{
\begin{array}{cl}
n-5, & n=6,7,8,\\
n-4, &n=9,10,\ldots
\end{array}
\right.
\label{eq:uphex}\end{aligned}$$ In all situations we consider a tiling $\mathcal{T}$ of a respective polygon $P$. The cardinality and the number of $\pi$-vertices of $\mathcal{T}$ are denoted by $n=|\mathcal{T}|$ and $v_\pi$, respectively. We have to show that $s(\mathcal{T})$ is bounded from above by the claimed upper estimate of ${\operatorname{s}_\text{\scriptsize\rm tri}}(n)$, ${\operatorname{s}_\text{\scriptsize\rm trap}}(n)$, ${\operatorname{s}_\text{\scriptsize\rm par}}(n)$, ${\operatorname{s}_\text{\scriptsize\rm pent}}(n)$ or ${\operatorname{s}_\text{\scriptsize\rm hex}}(n)$, respectively.
We proceed by induction over $n$. The claim is trivial if $n=1$. If $n=4$ then $\mathcal{T}$ contains three exposed triangles and the fourth tile covers the remainder of $P$. We obtain the respective tiling from Figure \[fig:small tilings\] by four congruent tiles. This yields $s(\mathcal{T})=1$. Now let $n \ge 6$.
*Case 1: $v_\pi \le 3$.* Then $v_\pi=3$ and $\mathcal{T}$ splits into three exposed triangles of the same size and a tiling $\mathcal{T}'$ of the remaining equilateral triangle $P'$ of $P$, $P'$ being of that size as well. Now $|\mathcal{T}'|=n-3\ge 3$, and the induction hypothesis gives $s(\mathcal{T}') \le |\mathcal{T}'|-3=n-6$. Since the tiles in $P'$ are smaller than the exposed tiles of $\mathcal{T}$, we obtain $s(\mathcal{T}) = s(\mathcal{T}')+1 \le (n-6)+1=n-5$.
*Case 2: $v_\pi=4$ and $n \ne 6$.* Now $\mathcal{T}$ contains three exposed triangles $T_1$, $T_2$ and $T_3$ such that $T_1$ shares a vertex with each of $T_2$ and $T_3$. Thus ${\operatorname{size}}(T_2)={\operatorname{size}}(T_3)$, but $T_2$ and $T_3$ are disjoint. By Lemma \[lem:necessary\], $\mathcal{T}$ contains at least $3+v_\pi-3=4$ pairs of triangles that share a side. We introduce a graph $\Gamma$ whose nodes are the triangles of $\mathcal{T}$. Two triangles form an edge if these are $T_2$ and $T_3$ or if they have a side in common. This graph has at least $1+4=5$ edges and does not contain a cycle of size less than six. (Cycles not involving the edge $\{T_2,T_3\}$ have size at least six. If a cycle contains $\{T_2,T_3\}$ then all triangles of $\mathcal{T} \setminus \{T_1\}$ are of the same size. By $v_\pi=4$, the side of $P$ that meets both $T_2$ and $T_3$ contains exactly two $\pi$-vertices. Hence there are exactly five triangles in $\mathcal{T} \setminus \{T_1\}$ and in turn $n=|\mathcal{T}|=6$, which is excluded.) We pick five edges of $\Gamma$. Since they do not contain a cycle and since each of them connects two congruent triangles, we have $s(\mathcal{T}) \le |\mathcal{T}|-5=n-5$.
*Case 3: $v_\pi=4$ and $n=6$.* Lemma \[lem:necessary\] shows that $\mathcal{T}$ contains at least $3+v_\pi-3=4$ pairs of triangles having a side in common. This yields $s(\mathcal{T}) \le |\mathcal{T}|-4=2$.
*Case 4: $v_\pi \ge 5$.* Now Lemma \[lem:necessary\] shows that $\mathcal{T}$ contains at least $3+v_\pi-3 \ge 5$ pairs of triangles having a side in common. This yields $s(\mathcal{T}) \le |\mathcal{T}|-5=n-5$.
We add an equilateral triangle at the $\left(\frac{2\pi}{3},\frac{2\pi}{3}\right)$-side of $P$ and obtain a triangle $P'$ and a corresponding tiling $\mathcal{T}'$ of $P'$ with $|\mathcal{T}'|=n+1$. Then $$s(\mathcal{T}) \le s(\mathcal{T}') \le {\operatorname{s}_\text{\scriptsize\rm tri}}(n+1) \stackrel{\text{\eqref{eq:uptri}}}{\le}
\left\{
\begin{array}{cl}
1, & n=3,\\
2, & n=5,\\
n-4, &n=6,7,\ldots
\end{array}
\right.$$
W.l.o.g. $n \ge 4$, since $n \in {\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm par}})=\{2\} \cup \{4,5,\ldots\}$ and the situation is clear for $n=2$.
*Case 1: One side length of $P$ is larger than $\max\{{\operatorname{size}}(T): T \in \mathcal{T}\}$.* We add two triangles at two sides of $P$ that meet at a $\frac{\pi}{3}$-vertex of $P$. This way we obtain a tiling $\mathcal{T}'$ of a triangle $P'$ with $|\mathcal{T}'|=n+2$ and $s(\mathcal{T}') \ge s(\mathcal{T})+1$, since one of the new triangles has a new size. Consequently, $$s(\mathcal{T}) \le s(\mathcal{T}')-1 \le {\operatorname{s}_\text{\scriptsize\rm tri}}(n+2)-1 \stackrel{\text{\eqref{eq:uptri}}}{\le}\left\{
\begin{array}{ll}
2-1=1, & n=4,\\
2-1 < 2, & n=5,\\
((n+2)-5)-1=n-4, &n=6,7,\ldots
\end{array}
\right.$$
*Case 2: No side length of $P$ is larger than $\max\{{\operatorname{size}}(T): T \in \mathcal{T}\}$.* Since no side of $P$ can be shorter than $\max\{{\operatorname{size}}(T): T \in \mathcal{T}\}$, $P$ is a rhombus and one tile $T_0 \in \mathcal{P}$ represents half of $P$. Then $\mathcal{T}'=\mathcal{T}\setminus\{T_0\}$ is a tiling of the other half $P'$ of $P$. In particular, $P'$ is a triangle, $s(\mathcal{T}')\ge s(\mathcal{T})-1$ and $$n \in \{5\} \cup \{7,8,\ldots\},$$ because $n \in {\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm par}}) \setminus \{2\}=\{4,5,\ldots\} $ and $n-1= |\mathcal{T}'| \in {\operatorname{dom}}({\operatorname{s}_\text{\scriptsize\rm tri}})=\{1,4\}\cup\{6,7,\ldots\}$. We obtain $$s(\mathcal{T}) \le s(\mathcal{T}')+1 \le {\operatorname{s}_\text{\scriptsize\rm tri}}(n-1)+1 \stackrel{\text{\eqref{eq:uptri}}}{\le}\left\{
\begin{array}{ll}
1+1=2, & n=5,\\
2+1=n-4, & n=7,\\
((n-1)-5)+1< n-4, &n=8,9,\ldots
\end{array}
\right.$$
*Case 1: $v_\pi=0$.* After removing the exposed triangle from $\mathcal{T}$ as well as from $P$, we obtain a tiling $\mathcal{T}'$ of a trapezoid $P'$ such that all $\pi$-vertices of $\mathcal{T}'$ are on the $\left(\frac{\pi}{3},\frac{\pi}{3}\right)$-side of $P'$. By Lemma \[lem:trapezoid\], $\mathcal{T}'$ is a similar image of the tiling of cardinality three from Figure \[fig:small tilings\]. Hence $\mathcal{T}$ consists of $n=4$ tiles, three of them being of the same size. So we have $s(\mathcal{T}) \le 2$, the required upper estimate for $n=4$.
*Case 2: $v_\pi\ge 1$.* By Lemma \[lem:necessary\], $\mathcal{T}$ contains at least $5+v_\pi-3 \ge 3$ pairs of triangles that have a side in common. This yields $s(\mathcal{T}) \le |\mathcal{T}|-3=n-3$.
*Case 1: $v_\pi=0$ or $n \le 6$.* We have $v_\pi=0$. Indeed, if $n \le 6$ then $n=6$ by . Since no tile from $\mathcal{T}$ covers segments of more than one side of the hexagon $P$, each side of $P$ is a side of some triangle from $\mathcal{T}$ and in turn $v_\pi=0$.
Let $T_1,\ldots,T_6 \in \mathcal{T}$ be the tiles that cover the sides of $P$ in successive order. It is enough to show that ${\operatorname{size}}(T_1)=\ldots={\operatorname{size}}(T_6)$, since then $P$ is a regular hexagon of side length ${\operatorname{size}}(T_1)$ that is completely tiled by $T_1,\ldots,T_6$, hence $n=6$ and $s(\mathcal{T})=1$, as claimed in .
To obtain a contradiction, suppose that, say, ${\operatorname{size}}(T_6) > {\operatorname{size}}(T_1)$. Then $$\label{eq:hex1.1}
{\operatorname{size}}(T_6) > {\operatorname{size}}(T_1) \ge {\operatorname{size}}(T_2) \ge {\operatorname{size}}(T_3),$$ because tiles do not overlap. The side lengths of $P$ satisfy $$\label{eq:hex1.2}
{\operatorname{size}}(T_2)+{\operatorname{size}}(T_3)={\operatorname{size}}(T_5)+{\operatorname{size}}(T_6).$$ Then shows that ${\operatorname{size}}(T_6)>{\operatorname{size}}(T_5)$, which implies $$\label{eq:hex1.3}
{\operatorname{size}}(T_6) > {\operatorname{size}}(T_5) \ge {\operatorname{size}}(T_4) \ge {\operatorname{size}}(T_3)$$ as above. Adding the inequalities ${\operatorname{size}}(T_2) < {\operatorname{size}}(T_6)$ from and ${\operatorname{size}}(T_3) \le {\operatorname{size}}(T_5)$ from gives a contradiction to .
*Case 2: $v_\pi=1$ and $n\in\{7,8\}$.* Let $v_0$ be the $\pi$-vertex, let $T_1,\ldots,T_7 \in \mathcal{T}$ be the tiles that form the boundary of $P$ ordered consecutively such that $v_0$ is a common vertex of $T_1$ and $T_7$, and let $T^\ast \in \mathcal{T}$ be the third tile having $v_0$ as a vertex, see Figure \[fig:case2\](a).
at (0,-.22)
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;
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(0,30)–(15,0)–(65,0)–(75,20)–(55,60)–(15,60)–(0,30)–(30,30)–(15,0) (15,60)–(45,0)–(55,20)–(65,0) (75,20)–(35,20)–(55,60) (0,55) node [(f)]{} ;
;
If $n=7$ then necessarily $T^\ast=T_4$, the hexagon $P$ splits into $T_4$ and two trapezoids tiled by $T_1,T_2,T_3$ and $T_5,T_6,T_7$, respectively, Lemma \[lem:trapezoid\] shows that $\mathcal{T}$ is similar to Figure \[fig:case2\](b), and we obtain our claim $s(\mathcal{T})\le 2$.
Now we consider $n=8$. If $v_0$ was a vertex of $T_4$ then one of the two trapezoids besides $T_4$ would be tiled by four triangles, contrary to . Thus $v_0$ is a vertex of $T_8 \in \mathcal{T}$. Consequently, the union of the lower sides of $T_3$ and $T_5$ is the union of the upper sides of $T_2$, $T_6$ and $T_8$.
If the lower sides of $T_3$ and $T_5$ together form a segment, then ${\operatorname{size}}(T_3)={\operatorname{size}}(T_5)$ and ${\operatorname{size}}(T_2)={\operatorname{size}}(T_6)={\operatorname{size}}(T_8)$, the tiling $\mathcal{T}$ is similar to Figure \[fig:case2\](c), and we obtain $s(\mathcal{T})=2<3$.
If the lower sides of $T_3$ and $T_5$ do not form a segment, then w.l.o.g. the lower side of $T_3$ is the union of the upper sides of $T_2$ and $T_8$ and the lower side of $T_5$ coincides with the upper one of $T_6$, see Figure \[fig:case2\](d). Then ${\operatorname{size}}(T_1)={\operatorname{size}}(T_2)={\operatorname{size}}(T_8)$, ${\operatorname{size}}(T_3)=2{\operatorname{size}}(T_1)$ and ${\operatorname{size}}(T_5)={\operatorname{size}}(T_6)=\frac{3}{2}{\operatorname{size}}(T_1)$. Now it follows easily that either ${\operatorname{size}}(T_4)=\frac{3}{2}{\operatorname{size}}(T_1)$ (and in turn ${\operatorname{size}}(T_7)={\operatorname{size}}(T_1)$, see Figure \[fig:case2\](e)) or ${\operatorname{size}}(T_4)=2{\operatorname{size}}(T_1)$ (and in turn ${\operatorname{size}}(T_7)=\frac{3}{2}{\operatorname{size}}(T_1)$, see Figure \[fig:case2\](f)). In both the situations we obtain our claim $s(\mathcal{T}) \le 3$.
*Case 3: $v_\pi=1$ and $n \ge 9$.* Lemma \[lem:necessary\] gives at least $6+v_\pi-3=4$ pairs of triangles in $\mathcal{T}$ that have a common side. This yields $s(\mathcal{T}) \le |\mathcal{T}|-4=n-4$.
*Case 4: $v_\pi \ge 2$ and $n \ge 7$.* Lemma \[lem:necessary\] shows that there are at least $6+v_\pi-3 \ge 5$ pairs of tiles in $\mathcal{T}$ having a side in common. This gives $$s(\mathcal{T}) \le |\mathcal{T}|-5=n-5\le\left\{
\begin{array}{ll}
n-5, & n=7,8,\\
n-4, & n=9,10,\ldots
\end{array}
\right.$$
Proof of Theorem \[thm:perfect\]
================================
Negative results on domains
---------------------------
For $$\label{eq:nDTRIp}
2,3,\ldots,14 \notin {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm tri}^\text{\scriptsize\rm t-perf}}\right)$$ we refer to [@drapal2010 Section 3.2]. It remains to show that $$\begin{aligned}
1,2,\ldots,12 & \notin & {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm trap}^\text{\scriptsize\rm t-perf}}\right), \label{eq:nDTRAp}\\
1;\; 3,4,\ldots,12 & \notin & {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm par}^\text{\scriptsize\rm t-perf}}\right), \label{eq:nDPARp}\\
1,2,\ldots,11 & \notin & {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}\right), \label{eq:nDPENTp}\\
1,2,\ldots,10 & \notin & {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm hex}^\text{\scriptsize\rm t-perf}}\right). \label{eq:nDHEXp}\end{aligned}$$ These claims are prepared by a lemma.
\[lem:perf\_dom\_neg\]
1. Let $\ast$ stand for $\rm tri$, $\rm trap$, $\rm par$, $\rm pent$ or $\rm hex$. If $n \in {\operatorname{dom}}\left({\operatorname{s}_\ast^\text{\scriptsize\rm t-perf}}\right)$ then $2{\operatorname{s}_\ast}(n) \ge n$.
2. If $n \in {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm trap}^\text{\scriptsize\rm t-perf}}\right)$ then $n-2 \in {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm trap}^\text{\scriptsize\rm t-perf}}\right) \cup {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}\right) \cup {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm hex}^\text{\scriptsize\rm t-perf}}\right)$.
3. If $n \in {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm par}^\text{\scriptsize\rm t-perf}}\right)$ then\
$n-2 \in \{0\} \cup {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm trap}^\text{\scriptsize\rm t-perf}}\right) \cup {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm par}^\text{\scriptsize\rm t-perf}}\right) \cup {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}\right) \cup {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm hex}^\text{\scriptsize\rm t-perf}}\right)$.
4. If $n \in {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}\right)$ then $n-1 \in {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm trap}^\text{\scriptsize\rm t-perf}}\right) \cup {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}\right) \cup {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm hex}^\text{\scriptsize\rm t-perf}}\right)$.
\(a) If $n \in {\operatorname{dom}}\left({\operatorname{s}_\ast^\text{\scriptsize\rm t-perf}}\right)$ then there exists a corresponding t-perfect tiling $\mathcal{T}$ of cardinality $n$. By t-perfectness, there are no three triangles of the same size in $\mathcal{T}$. Consequently, $n \le 2{\operatorname{s}_\ast^\text{\scriptsize\rm t-perf}}(n)$. Combining this with the trivial estimate ${\operatorname{s}_\ast^\text{\scriptsize\rm t-perf}}(n) \le {\operatorname{s}_\ast}(n)$ we obtain the claim $n \le 2{\operatorname{s}_\ast}(n)$.
\(b) Let $n \in {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm trap}^\text{\scriptsize\rm t-perf}}\right)$ be represented by a t-perfect tiling of some trapezoid $P$. Cutting off the two exposed triangles of the tiling from $P$ results in a smaller polygon $P'$ with a t-perfect tiling of cardinality $n-2$. The polygon $P'$ is neither a parallelogram nor a triangle, since in the latter case the two exposed triangles would have been congruent under a translation. So $P'$ is a trapezoid, a pentagon or a hexagon. Thus $n-2 \in {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm trap}^\text{\scriptsize\rm t-perf}}\right) \cup {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}\right) \cup {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm hex}^\text{\scriptsize\rm t-perf}}\right)$.
\(c) Now let $n \in {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm par}^\text{\scriptsize\rm t-perf}}\right)$ be represented by a t-perfect tiling of some parallelogram $P$. Cutting off the two exposed triangles from $P$, we get a polygon $P'$ that may be empty (if $n=2$) or a trapezoid, a parallelogram, a pentagon or a hexagon and has a t-perfect tiling of cardinality $n-2$. This yields (c).
\(d) Here we start with a t-perfect tiling of cardinality $n$ of some pentagon $P$. We cut off the one exposed triangle and get a t-perfect tiling of cardinality $n-1$ of some trapezoid, pentagon or hexagon $P'$.
\[cor:perf\_neg\] $$\begin{aligned}
1,2,\ldots,10 & \notin & {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm trap}^\text{\scriptsize\rm t-perf}}\right),\\
1;\;3,4,\ldots,10 & \notin & {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm par}^\text{\scriptsize\rm t-perf}}\right),\\
1,2,\ldots,9 & \notin & {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}\right),\\
1,2,\ldots,8 & \notin & {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm hex}^\text{\scriptsize\rm t-perf}}\right).\end{aligned}$$
The trivial inclusion ${\operatorname{dom}}\left({\operatorname{s}_\ast^\text{\scriptsize\rm t-perf}}\right)\subseteq{\operatorname{dom}}({\operatorname{s}_\ast})$, Theorem \[thm:general\] and Lemma \[lem:perf\_dom\_neg\](a) give $$\begin{aligned}
1,2,3,4,5,6,7 & \notin & {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm trap}^\text{\scriptsize\rm t-perf}}\right),\\
1;\; 3,4,5,6,7 & \notin & {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm par}^\text{\scriptsize\rm t-perf}}\right),\\
1,2,3;\; 5 & \notin & {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}\right),\\
1,2,3,4,5,6,7,8 & \notin & {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm hex}^\text{\scriptsize\rm t-perf}}\right).\end{aligned}$$ Now fourfold application of Lemma \[lem:perf\_dom\_neg\](d) yields $$4;\; 6,7,8 \notin {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}\right).$$ Then Lemma \[lem:perf\_dom\_neg\](b) and (c) show that $$8,9,10 \notin {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm trap}^\text{\scriptsize\rm t-perf}}\right) \quad\mbox{ and }\quad
8,9,10 \notin {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm par}^\text{\scriptsize\rm t-perf}}\right).$$ Finally, by Lemma \[lem:perf\_dom\_neg\](d), $$9 \notin {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}\right).$$
\[lem:perf\_hex\_dom\_neg\]
$9,10 \notin {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm hex}^\text{\scriptsize\rm t-perf}}\right)$.
To obtain a contradiction, suppose that there is a t-perfect tiling $\mathcal{T}=\{T_1,\ldots,T_n\}$ of some convex hexagon $P$ with $n \in \{9,10\}$. Let $v_\pi$ be the number of $\pi$-vertices of $\mathcal{T}$.
*Case 1: $v_\pi=0$.* As in Case 1 of the proof of in Subsection \[subsec:upgen\], we see that $n=6$, a contradiction.
*Case 2: $v_\pi \ge 3$.* Lemma \[lem:necessary\] says that $\mathcal{T}$ contains at least $6-v_\pi+3 \ge 6$ pairs of triangles that share a side. Since $n < 12$, there must be three triangles of the same size, which contradicts the t-perfectness of $\mathcal{T}$.
*Case 3: $v_\pi=2$.* Now Lemma \[lem:necessary\] gives $6+v_\pi-3=5$ pairs of triangles with a common side. Since $\mathcal{T}$ is t-perfect, this shows that $n=10$ and the five pairs constitute five rhombi $R_1,\ldots,R_5$ of mutually different side lengths that tile $P$.
For a vector $w \in \mathbb{R}^2 \setminus \{0\}$, we define the functional $\varphi_w(Q)$ of a polygon $Q$ by $$\varphi_w(Q)=\sum_{\genfrac{}{}{0pt}{2}{\mbox{\scriptsize $s$ is a side of $Q$ with}}{\mbox{\scriptsize outer normal vector $w$}}} {\operatorname{length}}(s)-
\sum_{\genfrac{}{}{0pt}{2}{\mbox{\scriptsize $s$ is a side of $Q$ with}}{\mbox{\scriptsize outer normal vector $-w$}}} {\operatorname{length}}(s).$$ Clearly $\varphi_w(R_i)=0$ and, since $\varphi_w$ is additive under tiling, $\varphi_w(P)=\varphi_w(R_1)+\ldots+\varphi_w(R_5)=0$ for every $w$. That is, opposite sides of $P$ have the same length. By the assumption $v_\pi=2$, there are two opposite sides $s_1$, $s_2$ of $P$ that do not contain $\pi$-vertices of $\mathcal{T}$. So $s_1$ and $s_2$ are sides of two triangles $T_1,T_2 \in \mathcal{T}$ whose union is one of the rhombi, say $R_1$, because ${\operatorname{length}}(s_1)={\operatorname{length}}(s_2)$. But then $R_1,\ldots,R_5$ cannot tile $P$, a contradiction.
*Case 4: $v_\pi=1$.* Now we are in the situation of the left-hand side of Figure \[fig:case4\], where $T_9$ (and possibly $T_{10}$) are not displayed.
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;
Clearly $T_4 \ne T_8$, because otherwise the trapezoids $v_0v_1v_2v_3$ and $v_0v_4v_5v_6$ would be tiled by a total number of eight or nine triangles, a contradiction to Corollary \[cor:perf\_neg\]. The $2\pi$-vertices of the triangles $T_1,\ldots,T_8$ are denoted by $v_1',\ldots,v_7',v_8^l,v_8^r$. We will discuss coincidences between these vertices.
First note that two of these nine points cannot agree if their lower indices differ by more than one modulo $8$: Indeed, if a $2\pi$-vertex of $T_i$ coincides with a $2\pi$-vertex of $T_j$ such that $T_i$ and $T_j$ are no neighbours in the cyclic order of $T_1,\ldots,T_8$, then there is a polygonal arc $a$ along sides of $T_i$ and $T_j$ that connects a vertex of $T_i$ with a vertex of $T_j$ and passes through their common $2\pi$-vertex such that $a$ dissects $P$ into two polygons $P'$ and $P''$ both being tiled by at least three (and in turn at most seven) triangles of $\mathcal{T}$ and such that one of $P'$ and $P''$ is convex. (E.g., if $v_1'=v_3'$ we can pick $a=v_0v_1'v_3$, if $v_2'=v_8^r$ we can pick $a=v_2v_2'v_8^lv_0$.) Thus some convex polygon admits a t-perfect tiling of a cardinality between $3$ and $7$. This is impossible by and Corollary \[cor:perf\_neg\]. Consequently, *the only possible coincidences among $v_1',\ldots,v_7',v_8^l,v_8^r$ are $$v_1'=v_2',\;v_2'=v_3',\;v_3'=v_4',\;v_4'=v_5',\;v_5'=v_6',\;v_6'=v_7',\;v_7'=v_8^l,\;v_8^r=v_1'.$$ When counted cyclically, no two consecutive of these identities are satisfied,* because they would give rise to three tiles of the same size, which contradicts t-perfectness.
The total size of all inner angles of $T_1,\ldots,T_9(,T_{10})$ is $9\pi$ or $10\pi$. The size of those who are placed on the boundary of $P$ is $5\pi$. Thus the sizes of inner angles in $2\pi$-vertices of $\mathcal{T}$ sum up to $4\pi$ or $5\pi$. Since every $2\pi$-vertex requires inner angles of total size $\pi$ or $2\pi$, $\mathcal{T}$ has at most five $2\pi$-vertices. All of $v_1',\ldots,v_7',v_8^l,v_8^r$ are $2\pi$-vertices. So there are at least four identities between them. Taking the above observation into account, we obtain $$v_1'=v_2',\quad v_3'=v_4',\quad v_5'=v_6',\quad v_7'=v_8^l$$ (the alternative situation $v_2'=v_3',v_4'=v_5',v_6'=v_7',v_8^r=v_1'$ would be equivalent).
So $T_1 \cup T_2$, $T_3 \cup T_4$, $T_5 \cup T_6$ and $T_7 \cup T_8$ are rhombi and the sizes of $T_1,T_3,T_5,T_7$ are mutually different. Since ${\operatorname{size}}(T_2)+{\operatorname{size}}(T_3)={\operatorname{size}}(T_5)+{\operatorname{size}}(T_6)=2{\operatorname{size}}(T_5)$, we have either ${\operatorname{size}}(T_2) < {\operatorname{size}}(T_5) < {\operatorname{size}}(T_3)$ or ${\operatorname{size}}(T_2) > {\operatorname{size}}(T_5) > {\operatorname{size}}(T_3)$. As the latter situation would cause an overlap of $T_2$ and $T_5$, the former inequality applies and we are in the situation of the right-hand side of Figure \[fig:case4\]. The remainder $P'$ of $P$ after cutting off $T_1,\ldots,T_8$ must be a pentagon with inner angles of sizes $\frac{2\pi}{3}$, $\frac{\pi}{3}$, $\frac{4\pi}{3}$, $\frac{\pi}{3}$ and $\frac{\pi}{3}$ in consecutive order. This excludes $n=9$, because $P'$ cannot be tiled by one single triangle $T_9$. If $n=10$, the only possibility of tiling $P'$ by $T_9$ and $T_{10}$ is dotted in the figure. But then the tiling of $P$ is not t-perfect. This contradiction completes the proof.
By Corollary \[cor:perf\_neg\] and Lemma \[lem:perf\_hex\_dom\_neg\], it remains to prove that $$\begin{aligned}
11,12 \notin {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm trap}^\text{\scriptsize\rm t-perf}}\right), \quad
11,12 \notin {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm par}^\text{\scriptsize\rm t-perf}}\right), \quad
10,11 \notin {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}\right).\end{aligned}$$ The third claim follows from Corollary \[cor:perf\_neg\] and Lemma \[lem:perf\_hex\_dom\_neg\] by twofold application of Lemma \[lem:perf\_dom\_neg\](d). Then the remainder is a consequence of Lemma \[lem:perf\_dom\_neg\](b) and (c).
New spiral pentagons and corresponding t-perfect tilings
--------------------------------------------------------
We define a sequence of integers $q(n)$, $n=8,9,\ldots$, by $$\label{eq:t-Pad}
q(8)=8,\; q(9)=11,\; q(10)=9 \quad\mbox{and}\quad q(n)=q(n-3)+q(n-2) \mbox{ for } n \ge 11.$$
\[lem:t-spiral\]
1. The above numbers satisfy
- $q(n)>q(n-1)$ and $q(n)>12$ for $n \ge 11$,
- $\frac{1}{3}(q(n-1)-q(n-2)) \notin \{2,3,5,7,8,9,11,12\} \cup \{q(m): m \ge 8\}$ for $n=12,13,14,15;\; 17,18,\ldots$
2. For every $n \in \{12,13,\ldots\}$, there is a convex pentagon $Q_n$ with sides of lengths $q(n-4)$, $q(n-3)$, $q(n-2)$, $q(n-1)$ and $q(n)$ (in consecutive order) with an inner angle of size $\frac{\pi}{3}$ between the sides of lengths $q(n-1)$ and $q(n)$. That pentagon admits a t-perfect tiling $\mathcal{T}_n$ by $n$ equilateral triangles $T_i$, $i=1,\ldots,n$, such that
- ${\operatorname{size}}(T_1)={\operatorname{size}}(T_2)=2$, ${\operatorname{size}}(T_3)=3$, ${\operatorname{size}}(T_4)={\operatorname{size}}(T_5)=5$, ${\operatorname{size}}(T_6)={\operatorname{size}}(T_7)=7$, ${\operatorname{size}}(T_8)={\operatorname{size}}(T_9)=8$, ${\operatorname{size}}(T_{10})=9$, ${\operatorname{size}}(T_{11})=11$, ${\operatorname{size}}(T_{12})=12$ and ${\operatorname{size}}(T_i)=q(i-1)$ for $i=13,\ldots,n$,
- if one side length of $Q_n$ is a side length of some triangle $T_i \in \mathcal{T}_n$ and different from $8$ (the latter is always the case if $n \ge 13$), then that side of $Q_n$ is a side of $T_i$ and there is only one triangle of that size in $\mathcal{T}_n$.
The first part of (a) follows from by induction. The second part of (a) is shown for $n \le 24$ by explicit computation. For $n \ge 25$, it is proved by inductive verification of $$q(n-10) < \frac{1}{3}(q(n-1)-q(n-2))=\frac{q(n-6)}{3} < q(n-9)$$ as follows: First note that $$q(n-1)-q(n-2)=q(n-4)+q(n-3)-q(n-5)-q(n-4)=q(n-3)-q(n-5)=q(n-6).$$ The base cases for $n=25,26,27$, obtained by computation, are $q(15)=48 < \frac{q(19)}{3}=\frac{154}{3} < q(16)=67 < \frac{q(20)}{3}=\frac{202}{3} < q(17)=87 < \frac{q(21)}{3}=\frac{269}{3} < q(18)=115$. Finally, the step case for $n \ge 28$ is $$\frac{q(n-6)}{3}=\frac{q(n-9)}{3}+\frac{q(n-8)}{3}
\left\{
\begin{array}{l}
>q(n-13)+q(n-12)=q(n-10),\\[.5ex]
< q(n-12)+q(n-11)=q(n-9).
\end{array}
\right.$$
Also claim (b) is shown by induction: The pentagon $Q_{12}$ and the tiling $\mathcal{T}_{12}$ is illustrated in Figure \[fig:t-spiral\] (see also tiling (c) in the appendix).
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;
Then $Q_{n+1}$ and $\mathcal{T}_{n+1}$ are obtained from $Q_n$ and $\mathcal{T}_n$ by adding a triangle $T_{n+1}$ of size $q(n)$ at the side of size $q(n)$ of $Q_n$, cf. Figure \[fig:t-spiral\].
Positive results on domains and lower estimates
-----------------------------------------------
We start with a direct consequence of Lemma \[lem:t-spiral\].
\[cor:t-lower\]
1. $\{15,16,\ldots\}\subseteq{\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm tri}^\text{\scriptsize\rm t-perf}}\right)$ and ${\operatorname{s}_\text{\scriptsize\rm tri}^\text{\scriptsize\rm t-perf}}(n) \ge n-6$ for all $n=15,16,\ldots$
2. $\{13,14,\ldots\}\subseteq{\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm trap}^\text{\scriptsize\rm t-perf}}\right)$ and ${\operatorname{s}_\text{\scriptsize\rm trap}^\text{\scriptsize\rm t-perf}}(n)\ge n-5$ for all $n=13,14,\ldots$
3. $\{13,14,\ldots\}\subseteq{\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm par}^\text{\scriptsize\rm t-perf}}\right)$ and ${\operatorname{s}_\text{\scriptsize\rm par}^\text{\scriptsize\rm t-perf}}(n)\ge n-5$ for all $n=13,14,\ldots$
4. $\{12,13,\ldots\}\subseteq{\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}\right)$ and ${\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}(n)\ge n-4$ for all $n=12,13,\ldots$
5. $\{17,18,\ldots\} \setminus \{21\} \subseteq{\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm hex}^\text{\scriptsize\rm t-perf}}\right)$ and ${\operatorname{s}_\text{\scriptsize\rm hex}^\text{\scriptsize\rm t-perf}}(n) \ge n-6$ for all $n=17,18,19,20;\; 22,23,\ldots$
The pentagons $Q_n$ and tilings $\mathcal{T}_n$ from Lemma \[lem:t-spiral\](b) prove (d).
For (a), we add two triangles of sizes $q(n-4)$ and $q(n-2)$ at the respective sides of $Q_n$, cf. Figure \[fig:derived polygons\](a). This gives a t-perfect tiling of a triangle by $n+2$ tiles of at least $n-4=(n+2)-6$ different sizes if $n \ge 13$. (For $n=12$, t-perfectness is violated, since the size $q(12-4)=8$ is not allowed, see the last claim of Lemma \[lem:t-spiral\](b).)
For (b) and (c), we add a triangle of size $q(n-2)$ or $q(n-3)$, respectively, to $Q_n$ and $\mathcal{T}_n$ for $n \ge 12$, cf. Figure \[fig:derived polygons\].
For (e), we add five triangles $T'_{n+1},\ldots,T'_{n+5}$ of sizes $${\operatorname{size}}(T'_{k})=\left\{
\begin{array}{ll}
\frac{1}{3}\big(q(n-1)-q(n-2)\big),& k=n+1, \\[1ex]
\frac{2}{3}q(n-2)+\frac{1}{3}q(n-1),& k=n+2,n+3,\\[1ex]
\frac{1}{3}q(n-2)+\frac{2}{3}q(n-1),& k=n+4,n+5,\\
\end{array}
\right.$$ over the sides of lengths $q(n-2)$ and $q(n-1)$ of $Q_n$, see Figure \[fig:thex\].
(52,0)–(68,8)–(52,16)–(24,16)–(0,4)–(8,0)–(52,0)–(34,9) (10,9)–(38,9)–(24,16) (52,16)–(36,8)–(68,8) (36,8.5)–(39.3,8.9) (12,3.5) node [$Q_n$]{} (45,8.9) node [$T'_{n+1}$]{} (24,11.33) node [$T'_{n+2}$]{} (38,13.67) node [$T'_{n+3}$]{} (52,10.67) node [$T'_{n+4}$]{} (52,5.33) node [$T'_{n+5}$]{} (22,8) node [$q(n-2)$]{} (32.5,4.5) node [$q(n-1)$]{} (30,1) node [$q(n)$]{} ;
By Lemma \[lem:t-spiral\](a), ${\operatorname{size}}(T'_{n+1}) \ne {\operatorname{size}}(T_i)$ for $i=1,\ldots,n$ if $n \in \{12,13,\ldots\} \setminus \{16\}$. Moreover, the monotonicity from Lemma \[lem:t-spiral\](a) gives $$q(n-2) < {\operatorname{size}}(T'_{n+2})={\operatorname{size}}(T'_{n+3}) < {\operatorname{size}}(T'_{n+4})={\operatorname{size}}(T'_{n+5}) < q(n-1),$$ so that the sizes of the new triangles are different from those of $\mathcal{T}_n$. Thus $\mathcal{T}_n$ together with $T'_{n+1},\ldots,T'_{n+5}$ constitutes a t-perfect tiling by $n+5$ triangles of $(n-4)+3=(n+5)-6$ different sizes. This proves (e).
Now the remaining (non-trivial) claims from Theorem \[thm:perfect\] on domains and lower estimates are $$\begin{aligned}
&11;\;14,15,16;\;21 \in {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm hex}^\text{\scriptsize\rm t-perf}}\right),\label{eq:dthex}\\
&{\operatorname{s}_\text{\scriptsize\rm tri}^\text{\scriptsize\rm t-perf}}(n) \ge n-5, \quad n=15;\; 17,18,\ldots,26;\; 28,\label{eq:sttri}\\
&{\operatorname{s}_\text{\scriptsize\rm trap}^\text{\scriptsize\rm t-perf}}(n) \ge n-4, \quad n= 14;\; 16,17,\ldots,25;\; 27,\label{eq:sttra}\\
&{\operatorname{s}_\text{\scriptsize\rm par}^\text{\scriptsize\rm t-perf}}(n) \ge n-4, \quad n= 15;\; 18,19;\; 21,22,23;\; 26,\label{eq:stpar}\\
&{\operatorname{s}_\text{\scriptsize\rm hex}^\text{\scriptsize\rm t-perf}}(n) \ge \left\{
\begin{array}{ll}
n-4, & n=11;\; 14,15;\; 17,18,19;\; 22,\\
n-5, & n=16;\; 20,21;\; 23.
\end{array}
\right.
\label{eq:sthex}\end{aligned}$$
It is enough to show that the lower bounds from are attained by t-perfect tilings of the respective cardinalities. We refer to the tilings (c), (j), (k), (m), (n), (o), (q) and (t), (u), (v), (w) from the appendix and Table \[tab:hexagons\].
These estimates are established by tilings constructed as follows: We start with a particular tiling from the appendix. Then we add successively triangles of prescribed sizes such that in each step the new triangle is placed over the side of the respective length of the previously tiled polygon. Table \[tab:parameters\] summarizes all parameters.
Several of the above tilings showing that ${\operatorname{s}_\text{\scriptsize\rm tri}^\text{\scriptsize\rm t-perf}}(n) \ge n-5$ can be found in the literature: for $n=15$ see [@tutte1948 Figure 1 and the text thereafter], [@tuza1991 Figure 4], [@drapal2010 Figure 8, second tiling] and [@hamalainen perfectdissectionsize15595r5c3], for $n=17$ see [@drapal2010 Figure 10, first tiling] and [@hamalainen perfectdissectionsize173091r3c0], for $n=18$ see [@hamalainen perfectdissectionsize1830413r6c3], for $n=19$ see [@hamalainen perfectdissectionsize19130975r6c4].
[|c|p[50mm]{}||c|p[50mm]{}|]{} &\
$n$ && $n$ &\
15 & (c); 12, 19, 20, 11 & 14 & (c); 12, 19, 20\
17 & (c); 12, 19, 28, 39, 20, 28 & 16 & (c); 12, 19, 28, 39, 20\
18 & (j); 27, 44, 47, 24 & 17 & (j); 27, 44, 47\
19 & (k); 20, 32, 33, 17 & 18 & (k); 20, 32, 33\
20 & (j); 27, 44, 67, 91, 47, 67 & 19 & (j); 27, 44, 67, 91, 47\
21 & (m); 64, 106, 111, 59 & 20 & (m); 64, 106, 111\
22 & (n); 102, 157, 162, 84 & 21 & (n); 102, 157, 162\
23 & (o); 138, 213, 220, 114 & 22 & (o); 138, 213, 220\
24 & (n); 102, 157, 235, 319, 162, 235 & 23 & (n); 102, 157, 235, 319, 162\
25 & (o); 138, 213, 319, 433, 220, 319 & 24 & (o); 138, 213, 319, 433, 220\
26 & (q); 325, 533, 534, 283 & 25 & (q); 325, 533, 534\
28 & (q); 325, 533, 784, 1067, 534, 784 & 27 & (q); 325, 533, 784, 1067, 534\
\
$n$ && $n$ &\
15 & (c); 12, 19, 28, 20 & 22 & (n); 102, 157, 235, 162\
18 & (j); 27, 44, 67, 47 & 23 & (o); 138, 213, 319, 220\
19 & (k); 20, 32, 48, 33 & 26 & (q); 325, 533, 784, 534\
21 & (m); 64, 106, 158, 111 &&\
Upper estimates
---------------
The property of t-perfectness gives a new upper estimate for pentagons.
\[lem:necessary\_t-perfect\] ${\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}(n) \le n-4$ for all $n \in {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}\right)$.
Assume that Lemma \[lem:necessary\_t-perfect\] fails. Let $n_0 \in {\operatorname{dom}}\left({\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}\right) \stackrel{\mbox{\eqref{eq:nDPENTp}}}{\subseteq}\{12,13,\ldots\}$ be minimal such that ${\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}(n_0) \ge n_0-3$, and let $\mathcal{T}=\{T_1,\ldots,T_{n_0}\}$ be a tiling of some convex pentagon $P$ realizing $s(\mathcal{T})={\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}(n_0)$. Cutting off the exposed triangle, we obtain a t-perfect tiling $\mathcal{T}'$ of some convex polygon $P'$ with $|\mathcal{T}'|=n_0-1$ and $$s(\mathcal{T}') \ge s(\mathcal{T})-1 = {\operatorname{s}_\text{\scriptsize\rm pent}^\text{\scriptsize\rm t-perf}}(n_0)-1 \ge (n_0-3)-1= |\mathcal{T}'|-3.$$ By $|\mathcal{T}'|=n_0-1 \ge 11$, Theorem \[thm:general\] tells us that $s(\mathcal{T}')=|\mathcal{T}'|-3$ and $P'$ is a pentagon. This contradicts the minimality of $n_0$.
The estimate ${\operatorname{s}_\text{\scriptsize\rm tri}^\text{\scriptsize\rm t-perf}}(16)\le 10$ from Theorem \[thm:perfect\](a) is shown in [@drapal2010 Section 3.2 and Figure 9]. The remaining upper estimates in Theorem \[thm:perfect\] follow from Theorem \[thm:general\] and the trivial relations ${\operatorname{s}_\ast^\text{\scriptsize\rm t-perf}}(n) \le {\operatorname{s}_\ast}(n)$ for $n \in {\operatorname{dom}}\left({\operatorname{s}_\ast^\text{\scriptsize\rm t-perf}}\right)$.
Appendix. Particular tilings of hexagons {#appendix.-particular-tilings-of-hexagons .unnumbered}
========================================
The following illustrations are scaled such that the smallest tiles appear congruent. Sizes of larger tiles are indicated in the figures. Sizes of all tiles are summarized in Table \[tab:hexagons\].\
(0,3)–(3,9)–(8,9)–(10.5,4)–(8.5,0)–(1.5,0)–(0,3)–(6,3)–(3,9) (8,9)–(5.5, 4)–(10.5,4) (1.5,0)–(3,3)–(4.5,0)–(6.5,4)–(8.5,0) ; at (6.5,1.33) [$4$]{}; at (8.5,2.67) [$4$]{}; at (5.5,7.33) [$5$]{}; at (8,5.67) [$5$]{}; at (3,5) [$6$]{};
(a)
(0,4)–(4,12)–(11,12)–(14,6)–(11,0)–(2,0)–(0,4)–(8,4)–(4,12) (2,0)–(4,4)–(6,0)–(8.5,5)–(7.5,5)–(11,12) (14,6)–(8,6)–(11,0) ; at (2,2.67) [$4$]{}; at (4,1.33) [$4$]{}; at (6,2.67) [$4$]{}; at (8.5,1.67) [$5$]{}; at (11,4) [$6$]{}; at (11,8) [$6$]{}; at (7.5,9.67) [$7$]{}; at (4,6.67) [$8$]{};
(b)
(0,8)–(4,16)–(15,16)–(19.5,7)–(16,0)–(4,0)–(0,8)–(8,8)–(4,0) (19.5,7)–(10.5,7)–(11.5,5)–(6.5,5)–(9,0)–(12.5,7)–(16,0) (15,16)–(9.5,5)–(4,16) ; at (9.5,12.67) [$11$]{}; at (15,10) [$9$]{}; at (4,10.67) [$8$]{}; at (4,5.33) [$8$]{}; at (16,4.67) [$7$]{}; at (12.5,2.33) [$7$]{};
(c)
(0,7)–(5,17)–(14,17)–(18.5,8)–(14.5,0)–(3.5,0)–(0,7)–(10,7)–(5,17) (3.5,0)–(7,7)–(10.5,0)–(12.5,4)–(8.5,4)–(10.5,8)–(14.5,0) (14,17)–(9.5,8)–(18.5,8) ; at (10.5,5.33) [$4$]{}; at (10.5,2.67) [$4$]{}; at (12.5,1.33) [$4$]{}; at (3.5,4.67) [$7$]{}; at (7,2.33) [$7$]{}; at (14.5,5.33) [$8$]{}; at (14,11) [$9$]{}; at (9.5,14) [$9$]{}; at (5,10.33) [$10$]{};
(d)
(0,10)–(6.5,23)–(18.5,23)–(24.5,11)–(19,0)–(5,0)–(0,10)–(13,10)–(6.5,23) (5,0)–(10,10)–(11.5,7)–(13.5,11)–(19,0) (12,0)–(8.5,7)–(15.5,7)–(12,0) (18.5,23)–(12.5,11)–(24.5,11) ; at (13.5,8.33) [$4$]{}; at (8.5,2.33) [$7$]{}; at (12,4.67) [$7$]{}; at (15.5,2.33) [$7$]{}; at (5,6.67) [$10$]{}; at (19,7.33) [$11$]{}; at (18.5,15) [$12$]{}; at (12.5,19) [$12$]{}; at (6.5,14.33) [$13$]{};
(e)\
(0,7)–(7,21)–(19,21)–(24.5,10)–(19.5,0)–(3.5,0)–(0,7)–(14,7)–(7,21) (3.5,0)–(7,7)–(10.5,0)–(15,9)–(13,9)–(13.5,10)–(14,9)–(14.5,10)–(19.5,0) (19,21)–(13.5,10)–(24.5,10) ; at (3.5,4.67) [$7$]{}; at (7,2.33) [$7$]{}; at (10.5,4.67) [$7$]{}; at (15,3) [$9$]{}; at (19.5,6.67) [$10$]{}; at (19,13.67) [$11$]{}; at (13,17) [$12$]{}; at (7,11.67) [$14$]{};
(f)
(0,14)–(7,28)–(26,28)–(33.5,13)–(27,0)–(7,0)–(0,14)–(14,14)–(7,0) (27,0)–(20.5,13)–(19.5,11)–(21.5,11)–(16,0)–(11.5,9)–(20.5,9)–(18.5,13)–(33.5,13) (7,28)–(16.5,9)–(26,28) ; at (11.5,3) [$9$]{}; at (16,6) [$9$]{}; at (21.5,4) [$11$]{}; at (27,8.67) [$13$]{}; at (7,9.33) [$14$]{}; at (7,18.67) [$14$]{}; at (26,18) [$15$]{}; at (16.5,21.67) [$19$]{};
(g)
(0,16)–(10.5,37)–(36.5,37)–(46.5,17)–(38,0)–(8,0)–(0,16)–(21,16)–(18.5,11)–(29.5,11)–(26.5,17)–(46.5,17) (8,0)–(16,16)–(24,0)–(31,14)–(28,14)–(29.5,17)–(38,0) (10.5,37)–(23.5,11)–(36.5,37) ; at (8,10.67) [$16$]{}; at (16,5.33) [$16$]{}; at (38,11.33) [$17$]{}; at (36.5,23.67) [$20$]{}; at (10.5,23) [$21$]{}; at (23.5,28.33) [$26$]{};
(h)
(0,17)–(11.5,40)–(32.5,40)–(43,19)–(33.5,0)–(8.5,0)–(0,17)–(23,17)–(11.5,40) (8.5,0)–(17,17)–(20,11)–(24,19)–(33.5,0) (22.5,0)–(15.5,14)–(18.5,14)–(17,11)–(28,11)–(22.5,0) (32.5,40)–(22,19)–(43,19) ; at (24,13.67) [$8$]{}; at (28,3.67) [$11$]{}; at (22.5,7.33) [$11$]{}; at (15.5,4.67) [$14$]{}; at (8.5,11.33) [$17$]{}; at (33.5,12.67) [$19$]{}; at (32.5,26) [$21$]{}; at (22,33) [$21$]{}; at (11.5,24.67) [$23$]{};
(i)\
(0,17)–(11.5,40)–(35.5,40)–(45.5,20)–(35.5,0)–(8.5,0)–(0,17)–(23,17)–(20,11)–(17,17)–(8.5,0) (19.5,0)–(14,11)–(25,11)–(22.5,16)–(27.5,16)–(19.5,0) (11.5,40)–(23.5,16)–(35.5,40) (45.5,20)–(25.5,20)–(35.5,0) ; at (25.5,17.33) [$4$]{}; at (25,14.33) [$5$]{}; at (22.5,12.67) [$5$]{}; at (20,15) [$6$]{}; at (17,13) [$6$]{}; at (14,3.67) [$11$]{}; at (19.5,7.33) [$11$]{}; at (27.5,5.33) [$16$]{}; at (8.5,10.67) [$17$]{}; at (35.5,13.33) [$20$]{}; at (35.5,26.67) [$20$]{}; at (11.5,24.33) [$23$]{}; at (23.5,32) [$24$]{};
(j)
(0,12)–(8,28)–(25,28)–(32.5,13)–(26,0)–(6,0)–(0,12)–(16,12)–(14,8)–(22,8)–(18,0)–(12,12)–(6,0) (26,0)–(19.5,13)–(17,8)–(15.5,11)–(18.5,11)–(17.5,13)–(32.5,13) (8,28)–(16.5,11)–(25,28) ; at (14,10.67) [$4$]{}; at (19.5,9.67) [$5$]{}; at (18,5.33) [$8$]{}; at (22,2.67) [$8$]{}; at (6,8) [$12$]{}; at (12,4) [$12$]{}; at (26,8.67) [$13$]{}; at (25,18) [$15$]{}; at (8,17.33) [$16$]{}; at (16.5,22.33) [$17$]{};
(k)
(0,30)–(19,68)–(62,68)–(79,34)–(62,0)–(15,0)–(0,30)–(38,30)–(34,22)–(30,30)–(15,0) (37,0)–(26,22)–(37,22)–(31.5,11)–(42.5,11)–(35.5,25)–(49.5,25)–(37,0) (19,68)–(40.5,25)–(62,68) (79,34)–(45,34)–(62,0) ; at (26,7.33) [$22$]{}; at (49.5,8.33) [$25$]{}; at (15,20) [$30$]{}; at (62,22.66) [$34$]{}; at (62,45.33) [$34$]{}; at (19,42.67) [$38$]{}; at (40,54) [$43$]{};
(l)\
(0,42)–(26,94)–(85,94)–(108.5,47)–(85,0)–(21,0)–(0,42)–(52,42)–(47,32)–(50,32)–(42,16)–(58,16)–(48.5,35)–(67.5,35)–(50,0)–(35.5,29)–(48.5,29)–(42,42)–(21,0) (26,94)–(55.5,35)–(85,94) (108.5,47)–(61.5,47)–(85,0) ; at (50,21.33) [$16$]{}; at (50,10.67) [$16$]{}; at (58,28.67) [$19$]{}; at (35.5,9.67) [$29$]{}; at (67.5,11.67) [$35$]{}; at (21,28) [$42$]{}; at (85,31.33) [$47$]{}; at (85,62.67) [$47$]{}; at (26,59.33) [$52$]{}; at (55.5,74.33) [$59$]{};
(m)
(0,60)–(36.5,133)–(120.5,133)–(159.5,55)–(132,0)–(30,0)–(0,60)–(73,60)–(66.5,47)–(60,60)–(30,0) (132,0)–(104.5,55)–(77,0)–(53.5,47)–(68.5,47)–(61,32)–(93,32)–(81.5,55)–(159.5,55) (76,32)–(67.5,49)–(84.5,49)–(76,32) (36.5,133)–(78.5,49)–(120.5,133) ; at (73,52.67) [$11$]{}; at (66.5,55.67) [$13$]{}; at (60,51.33) [$13$]{}; at (61,42) [$15$]{}; at (68.5,37) [$15$]{}; at (76,43.33) [$17$]{}; at (84.5,37.67) [$17$]{}; at (93,47.33) [$23$]{}; at (77,21.67) [$32$]{}; at (53.5,15.67) [$47$]{}; at (104.5,18.33) [$55$]{}; at (132,36.67) [$55$]{}; at (30,40) [$60$]{}; at (36.5,84.33) [$73$]{}; at (120.5,81) [$78$]{}; at (78.5,105) [$84$]{};
(n)\
(0,82)–(49.5,181)–(163.5,181)–(216.5,75)–(179,0)–(41,0)–(0,82)–(99,82)–(90.5,65)–(92.5,65)–(82,44)–(126,44)–(110.5,75)–(216.5,75) (41,0)–(82,82)–(91.5,63)–(72.5,63)–(104,0)–(141.5,75)–(179,0) (103,44)–(91.5,67)–(114.5,67)–(103,44) (49.5,181)–(106.5,67)–(163.5,181) ; at (110.5,69.67) [$8$]{}; at (99,72) [$15$]{}; at (90.5,76.33) [$17$]{}; at (82,69.33) [$19$]{}; at (82,56.67) [$19$]{}; at (92.5,51) [$21$]{}; at (103,59.33) [$23$]{}; at (114.5,51.67) [$23$]{}; at (126,64.67) [$31$]{}; at (105,29.33) [$44$]{}; at (72.5,21) [$63$]{}; at (141.5,25) [$75$]{}; at (179,50) [$75$]{}; at (41,54.67) [$82$]{}; at (49.5,115) [$99$]{}; at (163.5,110.33) [$106$]{}; at (106.5,143) [$114$]{};
(o)\
(53,71)–(86,71)–(78.5,56)–(80.5,56)–(71.5,38)–(109.5,38)–(96,65)–(133,65) (53,35)–(71,71)–(79,55)–(79.5,56)–(80,55)–(63,55)–(76.5,28) (104.5,28)–(123,65)–(133,45) (89.5,38)–(99.5,58)–(79.5,58)–(89.5,38) (81,81)–(92.5,58)–(104,81) ; (53,28)–(133,28)–(133,81)–(53,81)–cycle ; at (96,60.33) [$7$]{}; at (86,62.33) [$13$]{}; at (78.5,66) [$15$]{}; at (71,60.33) [$16$]{}; at (71.5,49.33) [$17$]{}; at (80.5,44) [$18$]{}; at (89.5,51.33) [$20$]{}; at (99.5,44.67) [$20$]{}; at (109.5,56) [$27$]{}; at (90.5,33) [$38$]{}; at (64,36) [$55$]{}; at (129.67,58.33) [$65$]{}; at (122,39) [$65$]{}; at (59,59) [$71$]{}; at (68,76) [$86$]{}; at (117,73) [$92$]{}; at (92.5,73.33) [$99$]{};
(p)\
(0,208)–(125.5,459)–(408.5,459)–(533.5,209)–(429,0)–(104,0)–(0,208)–(251,208)–(229.5,165)–(208,208)–(104,0) (429,0)–(324.5,209)–(300,160)–(349,160)–(269,0)–(186.5,165)–(240.5,165)–(213.5,111)–(324.5,111)–(296,168)–(304,168)–(283.5,209)–(533.5,209) (267.5,111)–(235,176)–(300,176)–(267.5,111) (125.5,459)–(267,176)–(408.5,459) ; at (304,195.33) [$41$]{}; at (229.5,193.67) [$43$]{}; at (208,179.33) [$43$]{}; at (324.5,176.33) [$49$]{}; at (324.5,143.67) [$49$]{}; at (240.5,129) [$54$]{}; at (213.5,147) [$54$]{}; at (296,130) [$57$]{}; at (267.5,154.33) [$65$]{}; at (269,74) [$111$]{}; at (349,53.33) [$160$]{}; at (186.5,55) [$165$]{}; at (104,138.66) [$208$]{}; at (429,139.33) [$209$]{}; at (408.5,292.33) [$250$]{}; at (125.5,291.67) [$251$]{}; at (267,364.67) [$283$]{};
(q)\
(0,36.1)–(21.7,79.5)–(70.8,79.5)–(92.55,36.0)–(74.55,0)–(18.05,0)–(0,36.1)–(43.4,36.1)–(39.75,28.8)–(41.35,28.8)–(36.5,19.1)–(55.6,19.1)–(50.9,28.5)–(60.3,28.5)–(46.05,0)–(32.05,28.0)–(40.95,28.0)–(40.55,28.8)–(40.15,28.0)–(36.1,36.1)–(18.05,0) (74.55,0)–(56.55,36.0)–(52.8,28.5)–(49.05,36.0)–(92.55,36.0) (46.2,19.1)–(40.55,30.4)–(51.85,30.4)–(46.2,19.1) (21.7,79.5)–(46.25,30.4)–(70.8,79.5) ; at (49.05,32.267) [$56$]{}; at (43.4,32.3) [$57$]{}; at (39.75,33.667) [$73$]{}; at (52.8,33.5) [$75$]{}; at (56.55,31) [$75$]{}; at (36.1,30.7) [$81$]{}; at (36.9,25.033) [$89$]{}; at (55.6,25.367) [$94$]{}; at (50.9,22.233) [$94$]{}; at (41.35,22.333) [$97$]{}; at (46.2,26.633) [$113$]{}; at (46.05,12.733) [$191$]{}; at (32.05,9.333) [$280$]{}; at (60.3,9.833) [$285$]{}; at (74.55,24) [$360$]{}; at (18.05,24.067) [$361$]{}; at (21.7,50.567) [$434$]{}; at (70.8,50.5) [$435$]{}; at (46.25,63.133) [$491$]{};
(r)\
(0,49.4)–(29.6,108.6)–(96.4,108.6)–(125.95,49.5)–(101.2,0)–(24.7,0)–(0,49.4)–(59.2,49.4)–(54.3,39.6)–(56.5,39.6)–(49.95,26.5)–(76.45,26.5)–(69.75,39.9)–(71.65,39.9)–(66.85,49.5)–(125.95,49.5) (24.7,0)–(49.4,49.4)–(54.85,38.5)–(55.4,39.6)–(55.95,38.5)–(43.95,38.5)–(63.2,0)–(82.2,38.0)–(70.7,38.0)–(76.45,49.5)–(101.2,0) (63.05,26.5)–(70.7,41.8)–(55.4,41.8)–(63.05,26.5) (29.6,108.6)–(63.0,41.8)–(96.4,108.6) ; at (59.2,44.333) [$76$]{}; at (66.85,44.367) [$77$]{}; at (71.65,46.3) [$96$]{}; at (54.3,46.133) [$98$]{}; at (49.4,42.133) [$109$]{}; at (76.45,34.167) [$115$]{}; at (76.45,41.833) [$115$]{}; at (49.95,34.5) [$120$]{}; at (56.5,30.867) [$131$]{}; at (69.75,30.967) [$134$]{}; at (63.05,36.7) [$153$]{}; at (63.2,17.667) [$265$]{}; at (82.2,12.667) [$380$]{}; at (43.95,12.833) [$385$]{}; at (24.7,32.933) [$494$]{}; at (101.2,33) [$495$]{}; at (96.35,69.133) [$591$]{}; at (29.55,69.2) [$592$]{}; at (62.95,86.333) [$668$]{};
(s)\
(0,33)–(19,71)–(62,71)–(82.5,30)–(67.5,0)–(16.5,0)–(0,33)–(38,33)–(35.5,28)–(42.5,28)–(41.5,30)–(82.5,30) (19,71)–(40.5,28)–(62,71) (16.5,0)–(33,33)–(39,21)–(43.5,30)–(48,21)–(27,21)–(37.5,0)–(52.5,30)–(67.5,0) ; at (40.5,56.67) [$43$]{}; at (62,43.67) [$41$]{}; at (18,45.67) [$38$]{}; at (16.5,22) [$33$]{}; at (67.5,20) [$30$]{}; at (52.5,10) [$30$]{}; at (27,7) [$21$]{}; at (37.5,14) [$21$]{}; at (33,25) [$12$]{}; at (43.5,24) [$9$]{}; at (48,27) [$9$]{}; at (39,25.67) [$7$]{};
(t)\
(89.5,115)–(115,115)–(110,125) (89.5,64)–(99.5,84) (120,43)–(115,53) (163,43)–(168,53) (190.5,58)–(180.5,78) (168,103)–(190.5,103) (155,125)–(144,103) (115,115)–(99.5,84) (115,115)–(132.5,80)–(144,103) (115,53)–(99.5,84)–(130.5,84)–(115,53)–(168,53)–(155,79)–(156,79)–(144,103)–(168,103) (168,53)–(180.5,78)–(155.5,78)–(168,103)–(180.5,78) (142,53)–(155.5,80)–(128.5,80)–(142,53) ; (89.5,43)–(190.5,43)–(190.5,125)–(89.5,125)–cycle ; at (130.5,81.33) [$4$]{}; at (115,94.33) [$31$]{}; at (115,73.67) [$31$]{}; at (128.5,62) [$27$]{}; at (142,71) [$27$]{}; at (155,61.67) [$26$]{}; at (168,69.67) [$25$]{}; at (168,86.33) [$25$]{}; at (156,95) [$24$]{}; at (144,87.33) [$23$]{}; at (141.5,48) [$53$]{}; at (180,53) [$78$]{}; at (101,55) [$84$]{}; at (183,88) [$103$]{}; at (100,120) [$115$]{}; at (99,98) [$115$]{}; at (170,114) [$127$]{}; at (132.5,110) [$150$]{};
(u)\
(0,39)–(19.5,0)–(92.5,0)–(109,33)–(92.5,66)–(60.5,66)–(45,35)–(21.5,82)–(0,39)–(43,39)–(41,35)–(76,35)–(58.5,0)–(39,39)–(19.5,0) (92.5,0)–(75.5,34)–(76.5,34)–(60.5,66) (109,33)–(76,33)–(92.5,66) (-10,82)–(21.5,82)–(16.5,92) (0,39)–(-10,19) (19.5,0)–(24.5,-10) (92.5,0)–(87.5,-10) (109,33)–(119,13) (92.5,66)–(119,66) (60.5,66)–(73.5,92) ; (-10,-10)–(119,-10)–(119,92)–(-10,92)–cycle ; at (45,73) [$202$]{}; at (93,79) [$171$]{}; at (5,87) [$155$]{}; at (-.5,61) [$155$]{}; at (110.17,48.33) [$139$]{}; at (2,6) [$112$]{}; at (107,3) [$106$]{}; at (56,-5) [$73$]{}; at (21.5,53.33) [$43$]{}; at (19.5,26) [$39$]{}; at (39,13) [$39$]{}; at (58.5,23.33) [$35$]{}; at (75.5,11.33) [$34$]{}; at (92.5,22) [$33$]{}; at (92.5,44) [$33$]{}; at (76.5,55.33) [$32$]{}; at (60.5,45.33) [$31$]{}; at (41,37.67) [$4$]{}; at (43,36.33) [$4$]{};
(v)\
(0,26.7)–(13.35,0)–(54.45,0)–(67.65,26.4)–(51.8,58.1)–(15.7,58.1)–(0,26.7)–(31.4,26.7)–(29.05,22.0)–(38.15,22.0)–(34.15,14.0)–(30.15,22.0)–(29.6,20.9)–(26.7,26.7)–(13.35,0) (54.45,0)–(41.25,26.4)–(38.15,20.2)–(44.35,20.2)–(34.25,0)–(23.8,20.9)–(30.7,20.9)–(27.25,14.0)–(41.25,14.0)–(37.7,21.1)–(38.6,21.1)–(35.95,26.4)–(67.65,26.4) (51.8,58.1)–(33.75,22.0)–(15.7,58.1) ; at (33.75,46.07) [$361$]{}; at (15.7,37.17) [$314$]{}; at (13.35,17.8) [$267$]{}; at (23.8,6.97) [$209$]{}; at (34.25,9.33) [$140$]{}; at (44.35,6.73) [$202$]{}; at (54.45,17.6) [$264$]{}; at (51.8,36.97) [$317$]{}; at (34.15,19.33) [$80$]{}; at (37.7,16.37) [$71$]{}; at (30.7,16.3) [$69$]{}; at (27.25,18.6) [$69$]{}; at (41.25,18.13) [$62$]{}; at (41.25,22.27) [$62$]{}; at (26.7,22.83) [$58$]{}; at (38.6,24.63) [$53$]{};
at (29.05,25.13) [$47$]{}; at (31.5,23.57) [$47$]{};
at (35.95,23.47) [$44$]{};
(w)
[|c|c|p[83mm]{}|c|]{}\
& $n$ & sizes of tiles (references of related tilings) & t-perfect\
(a) & 9 & 1, 3, 3, 3, 4, 4, 5, 5, 6 & no\
(b) & 10 & 1, 1, 4, 4, 4, 5, 6, 6, 7, 8 (cf. [@drapal2010 p. 1488, third tiling]) & no\
(c) & 11 & 2, 2, 3, 5, 5, 7, 7, 8, 8, 9, 11 (cf. [@tutte1948 Figure 1], [@tuza1991 Figure 4], [@drapal2010 Figure 8], [@hamalainen perfectdissectionsize15595r5c3]) & yes\
(d) & 11 & 1, 3, 4, 4, 4, 7, 7, 8, 9, 9, 10 & no\
(e) & 12 & 1, 3, 3, 4, 7, 7, 7, 10, 11, 12, 12, 13 & no\
(f) & 12 & 1, 1, 1, 2, 7, 7, 7, 9, 10, 11, 12, 14 & no\
(g) & 13 & 2, 2, 2, 4, 5, 9, 9, 11, 13, 14, 14, 15, 19 & no\
(h) & 14 & 3, 3, 3, 5, 5, 6, 11, 14, 16, 16, 17, 20, 21, 26 & no\
(i) & 14 & 2, 3, 3, 3, 6, 8, 11, 11, 14, 17, 19, 21, 21, 23 & no\
(j) & 14 & 1, 4, 5, 5, 6, 6, 11, 11, 16, 17, 20, 20, 23, 24 (cf. [@drapal2010 p. 1488, last tiling], [@hamalainen perfectdissectionsize1830413r6c3]) & yes\
(k) & 15 & 1, 2, 2, 3, 3, 4, 5, 8, 8, 12, 12, 13, 15, 16, 17 (cf. [@hamalainen perfectdissectionsize1830411r3c6])& yes\
(l) & 16 & 3, 5, 8, 8, 9, 11, 11, 11, 14, 22, 25, 30, 34, 34, 38, 43 & no\
(m) & 17 & 3, 3, 7, 10, 12, 13, 13, 16, 16, 19, 29, 35, 42, 47, 47, 52, 59 & yes\
(n) & 18 & 2, 6, 11, 13, 13, 15, 15, 17, 17, 23, 32, 47, 55, 55, 60, 73, 78, 84 & yes\
(o) & 19 & 2, 2, 8, 15, 17, 19, 19, 21, 23, 23, 31, 44, 63, 75, 75, 82, 99, 106, 114 & yes\
(p) & 21 & 1, 1, 1, 2, 7, 13, 15, 16, 17, 18, 20, 20, 27, 38, 55, 65, 65, 71, 86, 92, 99 & no\
(q) & 22 & 8, 8, 11, 32, 33, 41, 43, 43, 49, 49, 54, 54, 57, 65, 111, 160, 165, 208, 209, 250, 251, 283 & yes\
(r) & 24 & 8, 8, 8, 16, 19, 56, 57, 73, 75, 75, 81, 89, 94, 94, 97, 113, 191, 280, 285, 360, 361, 434, 435, 491 & no\
(s) & 25 & 11, 11, 11, 19, 19, 22, 76, 77, 96, 98, 109, 115, 115, 120, 131, 134, 153, 265, 380, 385, 494, 495, 591, 592, 668 & no\
\
& $n$ &\
(t) & 16 &\
(u) & 20 &\
(v) & 21 &\
(w) & 23 &\
[00]{}
R. Aduddell, M. Ascanio, A. Deaton, C. Mann: [*Unilateral and equitransitive tilings by equilateral triangles*]{}, Discrete Math. [**340**]{} (2017), 1669–1680.
R.L. Brooks, C.A.B. Smith, A.H. Stone, W.T. Tutte: [*The dissection of rectangles into squares*]{}, Duke Math. J. [**7**]{} (1940), 312–340.
E. Buchman: [*The impossibility of tiling a convex region with unequal equilateral triangles*]{}, Amer. Math. Monthly [**88**]{} (1981), 748–753.
H.T. Croft, K.J. Falconer, R.K. Guy: [*Unsolved problems in geometry*]{}, Problem Books in Mathematics. Unsolved Problems in Intuitive Mathematics, II. Springer-Verlag, New York, 1991.
A. Drápal, C. Hämäläinen: *An enumeration of equilateral triangle dissections*, Discrete Appl. Math. [**158**]{} (2010), 1479–1495.
B. Grünbaum, G.C. Shephard: [*Tilings and patterns*]{}, W.H. Freeman and Company, New York, 1987.
C. Hämäläinen: *Triangle dissections code*, http://bitbucket.org/carlohamalainen/dissections (choose “Downloads”).
B. Klaa[ß]{}en: *Infinite perfekte Dreieckszerlegungen auch für gleichseitige Dreiecke*, Elem. Math. [**50**]{} (1995), 116–121.
R. Nandakumar: [http://nandacumar.blogspot.com]{}, blog entry of June 14, 2016.
OEIS Foundation Inc. (2018), The On-Line Encyclopedia of Integer Sequences, http://oeis.org/A134816
J. Pach, G. Tardos: [*Tiling the plane with equilateral triangles*]{}, arXiv:1805.08840, May 22, 2018.
C. Richter: *Tiling by incongruent equilateral triangles without requiring local finiteness*, Elem. Math. [**67**]{} (2012), 157–163.
C. Richter: *Tiling by incongruent equilateral triangles without requiring local finiteness, Part II*, Elem. Math. [**73**]{} (2018), 15–22.
C. Richter, M. Wirth: *Tilings of convex sets by mutually incongruent equilateral triangles contain arbitrarily small tiles*, Discrete Comput. Geom. (to appear).
K. Scherer: [*The impossibility of a tessellation of the plane into equilateral triangles whose sidelengths are mutually different, one of them being minimal*]{}, Elem. Math. [**38**]{} (1983), 1–4. I. Steward: *Tales of a neglected number*, Sci. Amer. [**274**]{} (1996), 102–103.
W.T. Tutte: [*The dissection of equilateral triangles into equilateral triangles*]{}, Proc. Cambridge Philos. Soc. [**44**]{} (1948), 463–482.
Z. Tuza: [*Dissections into equilateral triangles*]{}, Elem. Math. [**46**]{} (1991), 153–158.
|
---
abstract: 'Kant and Laplace suggested the Solar System formed from a rotating gaseous disk in the 18th century, but convincing evidence that young stars are indeed surrounded by such disks was not presented for another 200 years. As we move into the 21st century the emphasis is now on disk formation, the role of disks in star formation, and on how planets form in those disks. Radio wavelengths play a key role in these studies, currently providing some of the highest spatial resolution images of disks, along with evidence of the growth of dust grains into planetesimals. The future capabilities of EVLA and ALMA provide extremely exciting prospects for resolving disk structure and kinematics, studying disk chemistry, directly detecting proto-planets, and imaging disks in formation.'
author:
- 'Claire J. Chandler and Debra S. Shepherd'
title: 'Disks, young stars, and radio waves: the quest for forming planetary systems'
---
Introduction
============
The search for forming planetary systems is ultimately a search for our origins. How do the gas and dust in molecular clouds evolve into rocky and gas giant planets, and how common are planetary systems like the Solar System? Radial velocity searches for extra solar planets indicate that the Solar System might actually be unusual, but such searches are only just achieving the sensitivities needed to detect the Solar System-like planetary systems. For several reasons it is much easier to study the disks from which planetary systems are expected to form, and this may shed light on the environment and processes taking place in the early Solar System.
The formation of disks at some stage during the star formation process is made inevitable by one of several “angular momentum problems” in star formation. Specific angular momentum is imparted to molecular clouds by differential Galactic rotation, and many orders of magnitude must be lost before a planetary system can form. The outcome of the star formation process for solar-type stars is therefore a T Tauri star, descending the Hayashi track towards the main sequence in the HR diagram, surrounded by a disk. Massive stars reach the main sequence while still deeply embedded and accreting.
A combination of radiative heating by the central star, accretion, and heating by the interstellar radiation field results in temperatures for disks on Solar System size scales of 10 to 20 K for Solar-type stars. The bulk of the disk mass is therefore best traced by millimeter and submillimeter wavelength emission. Indeed, the first convincing evidence for the ubiquity of disks surrounding T Tauri stars arose from a single-dish survey for 1.3 mm continuum emission by @chandler:Beckwith1990. Although those measurements were not able to resolve the emission the flux densities detected, and corresponding dust column densities, implied that the dust had to be distributed in flattened structures in order to explain the low optical extinctions to the central stars. The rest of this review focuses on the role of radio interferometry in spatially resolving the emission from circumstellar disks, and in understanding the disk properties.
Dust emission from disks
========================
Dust emission at centimeter through to submillimeter wavelengths has played a vital role in establishing the presence of circumstellar disks and for providing evidence of grain growth. The radiative transfer of the dust emission is relatively simple compared with that of molecular lines (with caveats regarding the chemical composition and shape of the dust grains), and may be the only tracer of mass in regions where molecules are depleted from the gas phase due to low temperatures. The dust emissivity is typically parameterized as a power-law function of frequency, $\kappa_\nu = \kappa_0 (\nu/\nu_0)^\beta$. The value of $\beta$ is potentially a good measure of the size of the dust grains, with $\beta = 2$ corresponding to grain sizes $a \ll \lambda$ where $\lambda$ is the observing wavelength. This is typical for interstellar dust. Lower values of $\beta$ are interpreted as evidence for larger dust particles, and in the extreme case of $a \gg \lambda$ the dust opacity is grey, with $\beta = 0$. For distributions of particle sizes the wavelength at which the slope of the dust opacity steepens indicates the size of the largest particles present.
If the dust emission is optically thin and in the Rayleigh-Jeans (RJ) part of the spectrum it is straightforward to derive $\beta$ from the spectral index. In this case, the flux density $F_\nu \propto \nu^2
(1-e^{-\tau}) \propto \nu^\alpha$, where $\alpha = 2+\beta$. Corrections for not being in the RJ are typically needed, and for the high column densities expected in T Tauri disks the optically-thin approximation may also not hold. This is obviously a problem if $\alpha$ is to be used to derive $\beta$, since high optical depth can mimic a low value of $\beta$. Key to interpreting the spectral index of dust emission at millimeter and submillimeter wavelengths is therefore the ability to resolve the emission, in order to separate optical depth from $\beta$. Furthermore, in order to demonstrate particle growth to cm-sized planetesimals or larger observations at the longest possible wavelength are needed.
Several recent studies have been made using the VLA at 7 mm to investigate the growth to cm-sized particles by resolving the emission from T Tauri disks. @chandler:Rodmann2006 observed 14 low-mass, pre-main sequence stars in the Taurus-Auriga star forming region, with a spatial resolution of $1.5''$, corresponding to approximately 200 AU. Ten were detected at $5\sigma$ or better, and all were resolved in at least one dimension. After applying RJ corrections and for possible contributions from free-free emission at this wavelength the distribution of $\beta$ (Figure \[chandler:beta\_dist\]) indicates a peak at $\beta \sim 1$, and shows evidence for cm-sized particles in these disks. Similar results have been found by @chandler:Natta2004 for intermediate-mass, pre-main sequence Herbig Ae stars, where 4 out of 9 disks observed were resolved.
Recent observations of the 10 Myr old T Tauri star TW Hya provide evidence of both planetesimal growth and disk clearing, the latter possibly due to the presence of a planet. The 7 mm continuum emission from this disk was first resolved on size scales of 10s of AU by the VLA in its most compact, D, configuration [@chandler:Wilner2000 Figure \[chandler:tw\_hya\_fig\], bottom left]. Subsequent modelling showed that a dip in its spectral energy distribution at $\sim 10\mu$m, and enhanced emission at $\sim 20\mu$m, was best accounted for by an inner hole extending to 4 AU, and direct heating of the inner edge of the disk at this radius by the central star [@chandler:Calvet2002 Figure \[chandler:tw\_hya\_fig\], top]. This hole has now been imaged by the VLA in its most extended, A, configuration [@chandler:Hughes2007 Figure \[chandler:tw\_hya\_fig\], bottom right].
Evidence for cm-sized particles in the TW Hya disk comes from 3.5 cm emission imaged with the VLA in its A configuration [@chandler:Wilner2005]. The emission is resolved with a brightness temperature $\sim 10$ K, and is consistent with dust emission. Further, the 3.5 cm emission is enhanced relative to that expected for a “standard,” continuous power-law distribution of grain sizes, but can be modelled as a bimodal grain size size distribution with some fraction of interstellar-type dust and an extra component of “pebbles.” Considerable amounts of mass could be hidden in large particles in disks in this way, the only evidence for which comes from observations at cm-wavelengths.
Disk kinematics from molecular line emission
============================================
The primary tracer of circumstellar disk kinematics is emission from the most abundant isotopomer of CO, $^{12}$CO. Interferometric observations of the emission in the $J$=1–0, 2–1, and 3–2 rotational transitions at 115, 230, and 345 GHz respectively have been used to show that in many cases the velocity structure in T Tauri disks is consistent with Keplerian rotation [@chandler:Simon2000; @chandler:Dutrey2003; @chandler:Qi2004; @chandler:Isella2007]. Indeed, modelling of cubes of spectral line emission can give the disk inclination, radial velocity profile, and provide independent and direct determinations of the central stellar mass to better than 10% [see, e.g., @chandler:Simon2000]. These methods are able to constrain theoretical pre-main sequence tracks, and are now limited by the uncertain distance determinations for the pre-main sequence stars. However, parallax measurements from radio VLBI observations are beginning to provide some of the most accurate distances known for young stars exhibiting non-thermal emission, and the distance to T Tau S has recently been pinned down to 0.4% using the VLBA [@chandler:Loinard2005; @chandler:Loinard2007].
While most disks around pre-main sequence stars studied to date do show Keplerian velocity profiles there is some evidence of deviations from Keplerian rotation for the young star AB Aur [@chandler:Pietu2005; @chandler:Lin2006]. This source shows molecular gas traced by $^{13}$CO(1–0) emission associated with a spiral feature at optical wavelengths [@chandler:Corder2005], on scales of $\sim 4''$ ($\sim 600$ AU). Both $^{12}$CO(3–2) and dust continuum emission show a possible spiral feature on even smaller scales [@chandler:Pietu2005; @chandler:Lin2006 Figure \[chandler:ab\_aur\_fig\]]. @chandler:Pietu2005 also show that if a power-law radial velocity profile is assumed the best fit is $V \propto
r^{-0.41\pm0.01}$, significantly shallower than Keplerian. As a result, estimates of the stellar mass derived [*assuming*]{} Keplerian rotation vary dramatically depending on the size scale and resolution of the observations used. Adding to the complexity, there is an inner hole of radius $\sim 70$ AU [@chandler:Pietu2005], and some indications of outward radial motion along the spiral features [@chandler:Pietu2005; @chandler:Lin2006].
Various possible explanations have been proposed for the observed structure of AB Aur [@chandler:Pietu2005; @chandler:Lin2006]. Perhaps most attractive is the idea that there may be a low mass companion in an inclined orbit that can produce both the inner hole and excite the spiral features. An encounter with a nearby star is also a possibility. Alternatively, AB Aur remains surrounded by an extended circumstellar envelope and also exhibits some evidence that its dust is not as evolved as has been observed in other proto-planetary disks, suggesting that the disk may just be sufficiently young that is has not yet relaxed to a Keplerian configuration.
Accretion and outflow
=====================
If accretion is to continue throughout the protostellar and T Tauri phases of young, low-mass stars, at the rates implied by the observed luminosities of those systems, some means of removing the angular momentum from the accreting material is needed. As is the case for accretion disks in almost all astrophysical environments, jets and winds are believed to be key to carrying away the angular momentum allowing accretion to continue. Measurements of rotation in protostellar jets have been very difficult to make, however. Perhaps the best example to date is for DG Tauri (Figure \[chandler:dg\_tau\_fig\]). The direction of rotation in its disk is traced by $^{13}$CO(2–1) emission [@chandler:Testi2002]. HST observations of the \[SII\] and \[OI\] emission from the jet, in slits oriented at offsets $\pm0.14''$ from the jet axis, show a velocity gradient in the same sense as the disk rotation [@chandler:Bacciotti2002]. This may be the first direct evidence of a disk/jet connection and the transport of angular momentum using jets.
In spite of the sparse direct observational evidence, linked accretion and outflow are fundamental ingredients in theories of star and planet formation. Outflow removes excess angular momentum from accreting gas, the disk and wind regulate the accretion, and of course the remnant disk ultimately becomes a planetary system. Models for how this takes place in detail fall into two broad categories: disk winds and X-winds. Disk winds are winds originating at a range of radii in the disk, and most models invoke rotating magnetic fields as the means by which material is transferred from the disk to a collimated wind or jet moving perpendicular to the disk surface [see @chandler:Konigl2000 for a review]. In X-winds the wind originates at the “X-point,” the radius at which magnetic field lines tied to the central star meet the inner radius of the disk, which is therefore also the disk radius co-rotating with the star [e.g., @chandler:Shu2000]. Assuming the jet co-rotates with a Keplerian disk, measured velocities imply the jet launching point is spread over a range of radii from about 0.1 AU to up to a few AU [e.g., @chandler:Coffey2007; @chandler:Tatulli2007]. These results tend to favour disk winds over X-winds, but are not yet conclusive.
Disks and massive star formation
================================
A reasonably well-established and understood picture of low-mass star formation has emerged over the last two decades, including the vital role of disks in mediating accretion and providing the building blocks for planetary systems. However, a simple scaling-up of this picture to higher mass stars faces a problem for stars more massive than $\sim
10 M_\odot$. Radiation pressure on infalling dust grains will reverse the infall, unless the dust absorption cross-section per unit mass can be significantly reduced (for example, by making the accreting material very optically thick, perhaps by having a very high accretion rate), or by reducing the effective luminosity by making the radiation field anisotropic [e.g., @chandler:Yorke2002]. Thus the properties of massive protostars forming via disk-mediated accretion may be expected to be somewhat different from their low-mass counterparts.
Direct observations of solar system-sized disks ($\sim 100$ AU) toward early B stars are not as well established as those around T Tauri stars. @chandler:Shepherd2001 inferred the existence of a 130 AU disk around a B2 young stellar object based on 7 mm continuum emission. However, the disk was not resolved, and a model was needed to isolate the disk emission (presumably dust) from ionized gas emission in the outflow and surrounding HII region. Further, evidence of rotation from molecular line emission was not available on these size scales. Several examples of 1000 to 2000 AU flattened, rotating structures have been reported around several early B (proto)stars which probably either trace the outer accretion disk/dense torus that may be feeding the inner disk, or are the result of confusion of multiple objects within the available resolution of the observations: e.g., IRAS 20126+4104 [@chandler:Cesaroni1999; @chandler:Cesaroni2005], Cep A HW2 [@chandler:Brogan2007; @chandler:Comito2007], IRAS 23151+5912 [@chandler:Beuther2007]. Given that these (proto)stars are still actively powering outflows, it is likely that an inner, as yet undetectable, accretion disk is present that launches the outflow. Above a mass of about $15 M_\odot$, observations of disks and disk winds/outflows are needed to verify whether they may also form via disk-mediated accretion.
One of the complications with attempts at direct imaging of disks around massive stars is that the nearest massive star-forming region, the Orion Molecular Cloud at $D \sim 450$ pc, is a factor of ten more distant than young, low-mass stars such as TW Hya, so that achieving the necessary spatial resolution can be a challenge. Masers provide a unique opportunity in this respect, since their high non-thermal brightness temperatures enable VLBI techniques to provide sub-milliarcsec resolution in some cases. Radio source I in Orion is associated with OH, H$_2$O, and SiO masers, and the SiO masers in particular trace hot, dense gas close to the protostar (the vibrational ground state traces densities $n \sim 10^6$ cm$^{-3}$, temperatures $T \sim 1000$ K, while the first vibrationally excited state traces $n \sim 10^{10\pm1}$ cm$^{-3}$, $T \sim
1000$–2000 K). Multi-epoch imaging of these masers using the VLBA have recently measured the 3-dimensional dynamics of the disk/wind interaction in the protostar [@chandler:GreenhillXX; @chandler:MatthewsXX].
The vibrational ground state SiO emission traces a bow tie structure, while the 7 mm continuum emission appears to trace a disk (Figure \[chandler:bowtie\_fig\]). The vibrationally excited masers predominantly outline a cross centered on the continuum, but there are also masers bridging the gap between the southern and western arms of the cross. The best model consistent with the remarkably systematic velocity patterns in the maser emission (Figure \[chandler:sourceI\_masers\_fig\], left) is that of an almost edge-on disk, with the SiO masers tracing the interaction of a wind with the disk surface, and subsequent collimation of the wind by magnetic fields and the surrounding molecular cloud. Proper motions show that both SiO and H$_2$O masers are moving approximately perpendicular to the disk surface (Figure \[chandler:sourceI\_masers\_fig\], right). The diameter of the disk is $\sim 150$ mas ($\sim 70$ AU), and the resolution with which the masers have been observed is $\sim 0.4$ mas ($\sim 0.2$ AU).
The future of disk observations: ALMA and EVLA
==============================================
The resolution and sensitivity of two new instruments that will become available within the next five years, the Atacama Large Millimeter Array (ALMA) and the Expanded Very Large Array (EVLA), will completely revolutionize the study of circumstellar disks. ALMA will operate at wavelengths between 9 mm and 320 $\mu$m, with up to 64 12-m antennas and resolutions as small as 0.01$''$ at the shortest wavelengths. It will be superb for resolving disk structure and kinematics for T Tauri stars and debris disks. Its 8 GHz of bandwidth and excellent continuum sensitivity makes it ideal for studying disk chemistry and the direct detection of proto-planets.
The EVLA will have 8 GHz of bandwidth at 1.3 cm and 7 mm, providing a factor of ten improvement in continuum sensitivity over the VLA and a factor of 2 to 4 better angular resolution than ALMA will be able to achieve at the equivalent wavelength. Long wavelength observations are vital for demonstrating planetesimal formation, and for imaging disk formation in the youngest protostars where the dust emission will be optically thick at the wavelengths where ALMA will have the highest spatial resolution.
Conclusions
===========
Over the last few years, submm, mm, and cm-wave observations of circumstellar disks around low-mass, pre-main sequence stars have demonstrated evidence for grain growth to cm-sized particles, a necessary precursor to planet formation. Keplerian velocity fields in those disks have enabled independent measurements of central stellar masses, and are able to constrain pre-main sequence evolutionary tracks. Deviations from Keplerian rotation have been observed in some cases, and raise the possibility that low-mass companions may be inferred from such observations. Evidence for angular momentum transport in disk winds has been demonstrated by the first observations of rotation in stellar jets, in a sense consistent with that of the accompanying disk. Masers are now able to trace disk and wind interactions for a massive protostar, suggesting that disk-mediated accretion may be a mechanism for forming high-mass stars. The future of disk studies is very bright; ALMA and EVLA will transform this field by providing images of gaps in disks, detecting dust heated by proto-planets, revealing the early phases of disk formation, and through detailed studies of the gas chemistry.
The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
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---
abstract: 'Several aspects of the recently proposed DMC-CIPSI approach consisting in using selected Configuration Interaction (SCI) approaches such as CIPSI (Configuration Interaction using a Perturbative Selection done Iteratively) to build accurate nodes for diffusion Monte Carlo (DMC) calculations are presented and discussed. The main ideas are illustrated with a number of calculations for diatomics molecules and for the benchmark G1 set.'
author:
- Michel Caffarel
- Thomas Applencourt
- Emmanuel Giner
- Anthony Scemama
title: Using CIPSI nodes in diffusion Monte Carlo
---
Introduction {#intro}
============
In recent years the present authors have reported a number of fixed-node DMC studies using trial wavefunctions whose determinantal part is built with the CIPSI approach. [@giner_2013; @giner_phd_2014; @scemama_jcp_2014; @giner_jcp_2015; @caffarel_jcp_2016] The purpose of this paper is to review the present situation, to clarify some important aspects of DMC-CIPSI, and to present some new illustrative results.
In section \[sci\] we briefly recall what Configuration Interaction (CI) methods are about and present the basic ideas of perturbatively selected CI approaches. We emphasize on the very high efficiency of SCI in approaching the exact Full CI limit using only a [*tiny*]{} fraction of the full Hilbert space of determinants. Selecting important determinants being a natural idea, it is no surprise that it has been introduced a long time ago and has been rediscovered many times under various forms since then. To the best of our knowledge selected CI appeared for the first time in 1969 in two independent works by Bender and Davidson[@bender_pr_1969] and Whitten and Hackmeyer.[@Whitten_1969] In practice, the flavor of SCI we employ is the CIPSI approach introduced by Malrieu and collaborators in 1973.[@huron_jcp_1973] CIPSI being our working algorithm for generating CI expansions, a brief description is given here. It is noted that the recent FCI-QMC method of Alavi [*et al.*]{}[@booth_jcp_2009; @cleland_jcp_2010] is essentially a SCI approach, except that selection of determinants in FCI-QMC is done stochastically instead of deterministically. In section \[results\] the performance of CIPSI is illustrated for the case of the water molecule at equilibrium geometry using the cc-pCV$n$Z family of basis sets, with $n=2$ to 5 and for the whole set of 55 molecules and 9 atoms of the G1 standard set.[@pople_jcp_1989; @curtiss_jcp_1990] It is shown that in all cases the FCI limit is closely approached.
In section \[dmc\] the use of CIPSI nodes in DMC is discussed. We first present our motivations and then comment on the key result observed, namely that in all applications realized so far the fixed-node error associated with the approximate nodes of the CIPSI expansion is found to systematically decrease both as a function of the number of selected determinants and as the size of the basis set. This remarkable property provides a convenient way of controlling the fixed-node error. Let us emphasize that in contrast with common practice in QMC the molecular orbitals are not stochastically re-optimized here. An illustrative application to the water molecule is presented.[@caffarel_jcp_2016] Of course, the main price to pay is the need of using much larger CI expansions than usual. The main ideas of our recently proposed approach[@preprint_multidets] to handle very large number of determinants in QMC are presented. In practice, converged DMC calculations using trial wavefunctions including up to a few millions of determinants are feasible. The computational increase with respect to single-determinant calculations is roughly proportional to $\sim \sqrt{N_{dets}}$ with a small prefactor.
In section \[pseudo\] the implementation of effective core potentials (ECP) in DMC using CIPSI trial wavefunctions is presented. As already proposed some time ago,[@Hurley_1987; @Hammond_1987] CI expansions allow to calculate analytically the action of the nonlinear pseudo-potential operator on the trial wavefunction. In this way, the use of quadrature points to integrate the wavefunction over the sphere as usually done[@mitas_jcp_1991] is avoided and a gain in computational effort essentially proportional to the number of grid points is achieved. The effectiveness of the approach is illustrated in the case of the atomization energy of the C$_2$ molecule.
Finally, Sec. \[conclu\] presents a detailed summary of the main features of the DMC-CIPSI approach and some lines of research presently under investigation are mentioned.
Selected Configuration Interaction {#sci}
==================================
Configuration Interaction methods
---------------------------------
In Configuration Interaction the wavefunction is written as a sum of Slater determinants $$|\Psi\rangle = \sum_i c_i |D_i\rangle
\label{ciexp}$$ where determinants are built over spin-orbitals. Let $\{ \phi_k \}$ be the set of $N_{\rm MO}$ orthonormal molecular orbitals used, the size of the full Hilbert space is given by the number of ways of distributing the $N_{\uparrow}$ electrons among the orbitals times the corresponding number for $N_{\downarrow}$ electrons. The total size of the full CI space is then (no symmetries are considered) $$N_{FCI}= \binom{N_{\rm MO}}{N_{\uparrow}} \binom{N_{\rm MO}}{N_{\downarrow}}$$ The CI eigenspectrum is obtained by diagonalizing the Hamiltonian matrix, $H_{ij}= {\langle D_i|H|D_j\rangle}$ within the orthonormal basis of determinants. In practice, the exponential increase of the FCI space restricts the use of FCI to small systems including a small number of electrons and molecular orbitals ($N_{FCI}$ not greater than about $10^9$). To go beyond, the FCI expansion has to be truncated. The most popular strategy consists in defining a subspace of determinants chosen [*a priori*]{}. Typically, the Hartree-Fock determinant (or a few determinants) is chosen as reference and all possible determinants built by promoting a given number of electrons from the HF occupied orbitals to the virtual ones are considered. In the CIS approach only single excitations are considered, in CISD all single and double excitations, etc.
![N$_2$ in the cc-pVTZ basis set (R$_{N-N}$=1.0977 Å). Variation of the number of determinants with $n$-excitations with respect to the Hartree-Fock determinant in the CIPSI expansion as a function of the number of selected determinants up to $5 \times 10^6$. []{data-label="n2"}](exc.pdf){width="1.\columnwidth"}
Now, numerical experience shows that among all possible determinants corresponding to a given number of excitations, only a [*tiny*]{} fraction plays a significant role in constructing the properties of the low-lying eigenstates. Furthermore, the weight of a determinant in the CI expansion is not directly related to its degree of excitation. For example, quadruply-excited determinants may play a more important role than some doubly- or singly-excited determinants. However, in practice, limiting the maximum number of excitations to about six is usually sufficient to get chemical accuracy. To give some quantitative illustration of these statements, Figure \[n2\] presents the number of determinants per class of excitations $n$ as a function of the number of determinants in the CIPSI wavefunction for the $N_2$ molecule at equilibrium geometry (cc-pVTZ basis set). Without entering now into the details of CIPSI presented below, let us just note that for 5$ \times 10^6$ determinants the CIPSI expansion has almost converged to the FCI solution. Accordingly, results presented in the figure for the distribution of excitations is essentially that of the FCI wavefunction.
As a consequence of the preceding remarks, it is clear that it is desirable to find a way of selecting only the most important determinants of the FCI expansion without considering all those of negligible weight (the vast majority). This is the purpose of selected configuration interaction approaches.
Selected CI and CIPSI algorithm
-------------------------------
To the best of our knowledge Bender and Davidson[@bender_pr_1969] and Whitten and Hackmeyer[@Whitten_1969] were the first in 1969 to introduce and exploit the idea of selecting determinants in CI approaches. In their work Bender and Davison proposed to select space configuration using an energy contribution criterion. Denoting $|\phi_0\rangle$ the restricted HF CSF-configuration, $|\phi_i^l \rangle$ all possible spin configurations issued from the space configuration, and $$\epsilon_i^{(2)} = \frac{1}{k} \sum_{l=1}^k \frac{ {|\langle \phi_i^l |H|\phi_0 \rangle |}^2 }
{ \langle \phi_0 |H|\phi_0 \rangle - \langle \phi_i^l |H|\phi_i^l \rangle }$$ the “average” perturbative energy contribution, the space configurations were ordered according to this contribution and those determinants contributing the most selected. The CI wavefunction was then constructed by using the selected configurations, $|\phi_0\rangle$, and all single excitations. A few months later, a similar idea using the very same perturbative criterion was introduced independently by Whitten and Hackmeyer.[@Whitten_1969] In addition, they proposed to improve step-by-step the CI expansion by iterating the selection step to reach the most important determinants beyond double-excitations.
In 1973 Malrieu and collaborators[@huron_jcp_1973] presented the CIPSI method (and later on an improved version of it[@cipsi_1983]). In CIPSI the construction of the multirefence variational space is essentially identical to that of Whiten and Hackmeyer. However, in order to better describe the dynamical correlation effects poorly reproduced by the multireference space, a perturbational calculation of the remaining correlation contributions was proposed. In applications the perturbational part is usually important from both a qualitative and quantitative point of view.\
The CIPSI algorithm being our practical scheme for generating selected CI expansions, let us now present its main steps.\
$\bullet$ Step 0: Start from a given determinant ([*e.g.*]{} the Hartree-Fock determinant) or set of determinants, thus defining an initial reference subspace: $S_0=\{|D_0\rangle,...\}$. Diagonalize $H$ within $S_0$ and get the ground-state energy $ E_0^{(0)}$ and eigenvector: $$|\Psi_0^{(0)}\rangle= \sum_{i \in S_0} c_i^{(0)} |D_i \rangle$$ Here and in what follows, a superscript on various quantities is used to indicate the iteration number.\
Then, do iteratively ($n=0,...$):\
[$\bullet$ Step 1]{}: Collect all [*different*]{} determinants $|D_{k}\rangle$ connected by $H$ to $|\Psi_0^{(n)}\rangle$, that is $$\langle \Psi_0^{(n)}|H|D_{k}\rangle \ne 0$$ and not belonging to the reference space $S_{n}$.\
[$\bullet$ Step 2]{}: Compute the small energy change of the total energy due to each connected determinant as evaluated at second-order perturbation theory $$\delta e(|D_{k}\rangle)=-\frac{{|\langle \Psi_0^{(n)}|H|D_{k}\rangle|}^2}{H_{kk}-E_0^{(n)}}
\label{e2pert}$$
[$\bullet$ Step 3]{}: Add the determinant $|D_{k^*}\rangle$ associated with the largest $|\delta e|$ to the reference subspace: $$S_{n} \rightarrow S_{n+1}= S_{n} \cup \{|D_{k*} \rangle\}$$\
Of course, instead of adding only one determinant a group of determinants can be selected using a threshold. This is what is actually done in practice.\
[$\bullet$ Step 4]{}: Diagonalize $H$ within $S_{n+1}$ to get: $$|\Psi_0^{(n+1)}\rangle= \sum_{i \in S_{n+1}} c_i^{(n+1)} |D_i\rangle \;\;\; {\rm with}\;\;\; E_0^{(n+1)}
\label{psi_cip}$$
[$\bullet$]{} Go to step 1 or stop if the target size for the reference subspace has been reached.\
Denoting $N_{\rm dets}$ the final number of determinants, the resulting ground-state $|\Psi_0(N_{\rm dets})\rangle$ is the variational CIPSI solution. It is the expansion used in DMC to contruct the determinantal part of the trial wavefunction.
A second step in CIPSI is the calculation of a perturbational estimate of the correlation energy left between the variational CIPSI energy and the exact FCI one. At second order, this contribution writes $$E_{PT2}= - \sum_{k \in \mathcal{M}} \frac{ { {|\langle \Psi_0(N_{\rm dets})|H|D_k} \rangle|}^2}{H_{kk}- E_0(N_{\rm dets})}
\label{pt2}$$ where $\mathcal{M}$ denotes the set of all determinants not belonging to the reference space and connected to the CIPSI expansion $|\Psi_0(N_{\rm dets})\rangle$ by $H$ (single and double excitations only) and $E_0(N_{\rm dets})$ the variational CIPSI energy. In practice, this contribution allows to recover a major part of the remaining correlation energy.\
At this point a number of remarks are in order:\
i.) Although the selection scheme is presented here for computing the ground-state eigenvector only, no special difficulties arise when generalizing the scheme to a finite number of states (see, [*e.g.*]{}[@cipsi_1983])\
ii.) The decomposition of the Hamiltonian $H$ underlying the perturbative second-order expression introduced in step 2 is known as the Epstein-Nesbet partition.[@en1; @en2] This decomposition is not unique, other possible choices are the M[ø]{}ller-Plesset partition[@mp] or the barycentric one,[@huron_jcp_1973] see discussion in [@cipsi_1983].\
iii.) Instead of calculating the energetic change perturbatively, expression (\[e2pert\]), it can be preferable to employ the non-perturbative expression resulting from the diagonalization of $H$ into the two-dimensional basis consisting of the vectors $|\Psi_0^{(n)}\rangle$ and $|D_{k}\rangle$. Simple algebra shows that the energetic change is given by $$\delta e(|D_k\rangle)=
\frac{1}{2} \left[H_{kk} - E_0(N_{\rm dets})\right]
\left[1-\sqrt{1 + \frac{4 {|\langle \Psi_0^{(n)}|H|D_k \rangle|}^2}{{[H_{kk}-E_0(N_{\rm dets})]}^2}}\right]
\label{e2ex}$$ In the limit of small transition matrix elements, $\langle \Psi_0^{(n)}|H|D_k \rangle$, both expressions (\[e2pert\]) and (\[e2ex\]) coincide. The non-perturbative formula is used in our applications.\
iv.) The implementation of this algorithm can be performed using limited amount of central memory. On the other hand, the CPU time required is essentially proportional to $N_{\rm dets} N_{\rm occ}^2 N_{\rm virt}^2$ where $N_{\rm occ}$ is the number of occupied molecular orbitals and $N_{\rm virt}$ the number of virtual orbitals.
Selected CI variants
--------------------
As already pointed out selecting the most important determinants of the FCI expansion is a so natural idea that, since the pioneering work of Bender and Davidson[@bender_pr_1969] and Whitten and Hackmeyer,[@Whitten_1969] several variants of SCI approaches have been proposed. In practice, the actual differences between approaches are usually rather minor and most ideas and technical aspects seem to have been re-discovered several times by independent groups. To give a fair account of the subject and an exhaustive list of references is thus difficult. Here, we limit ourselves to the references we are aware of, namely [@bender_pr_1969; @Whitten_1969; @Hackmeyer_1971; @langhoff_ijqc_1973; @huron_jcp_1973; @buenker_1974; @buenker_1975; @buenker_1978; @bruna_1980; @buenker_1981; @cipsi_1983; @cimiraglia_jcp_1985; @cimiraglia_jcc_1987; @harrison_1991; @cimiraglia_ijqc_1996; @Angeli_1997_I; @Angeli_1997_II; @Angeli_2001; @Bunge_2006; @Roth_2007; @Roth_2009; @mcci; @evangelista_jcp_2014; @tubman_2016; @tubman_cyrus_2016]. Regarding more specifically CIPSI, there has been a sustained research activity conducted during the 80’s and 90’s by research groups in Toulouse (Malrieu and coll.), Pisa (Angeli, Persico, Cimiraglia and coll.), and then Ferrara (Angeli, Cimiraglia) including the development at Pisa of a very efficient CIPSI code using diagrammatic techniques[@cipsi_code; @cimiraglia_jcp_1985; @cimiraglia_ijqc_1996]. Thanks to all this, CIPSI has been extensively applied for years by several groups to a variety of accurate studies of ground and excited states and potential energy surfaces (see, for example [@Povill_1992; @Illas_1991a; @Illas_1991; @Millie_1986; @Persico_1991; @Illas_1988; @Cabrol_1996; @Cabrol_1996; @Angeli_1996; @Milli__2000; @M_dl_1997; @Cattaneo_1999; @Li_2011; @Mennucci_2001; @Novoa_1988; @Aymar_2006; @Aymar_2005]) Finally, note that in the last years our group has developped its own CIPSI code, Quantum Package. This code has been designed to be particularly easy to install, run and modify; it can be freely downloaded at [@quantum_package].
FCI-QMC as a stochastic selected CI approach
--------------------------------------------
Full Configuration Interaction Quantum Monte Carlo (FCI-QMC) is a method for solving stochastically the FCI equations.[@booth_jcp_2009; @cleland_jcp_2010] Introducing as in DMC an imaginary time $t$ the coefficients $c_i$ of the CI expansion, Eq.(\[ciexp\]), are evolved in time using the operator $[1-\tau (H-E)]$ as small-time propagator $${\bf c}(t+\tau) =[1-\tau(H-E)] {\bf c}(t)
\label{ele}$$ ${\bf c}$ being the vector of coefficients, $E$ some reference energy, and $\tau$ the time step. After $n$ steps the coefficients are given by $${\bf c}(t) = [1-\tau(H-E)]^n {\bf c}(t=0).
\label{eqFCI}$$ In the long-time limit ($t=n\tau$ large) the vector ${\bf c}$ converges to the exact CI vector (independently on initial conditions ${\bf c}$(t=0) provided that $\langle {\bf c}$(t=0)$|{\bf c}\rangle \ne 0$ and for a sufficiently small time step). As in all QMC methods, a set of walkers is introduced for sampling coefficients and a few simple stochastic rules realizing [*in average*]{} the action of $H$ according to Eq.(\[ele\]) are introduced (spawning, death/cloning and annihilation). Note that equations of evolution (\[eqFCI\]) are similar to those of continuous DMC (electrons moving in ordinary space) where a small-time expression of operator $e^{-\tau (H-E)}$ is used, and are essentially identical to the equations of lattice DMC (see [*e.g.*]{},[@van_Bemmel_1994]) The two main differences of FCIQMC with other QMC approaches are the fact that no trial vector is introduced (thus, avoiding the fixed-node error) and that the stochastic rules used are particularly efficient in attenuating the sign instability inherent to all stochastic simulations of fermionic systems (annihilation at each MC step of walkers of opposite sign on occupied determinants and use of the initiator approximation).
At a given time $t$ the CI expansion is stochastically realized by the distribution of walkers as $$|\Psi\rangle = \sum_i n_i |D_i\rangle$$ where $n_i$ is the sum of the signed weight of walkers on Slater determinant $|D_i\rangle$ ($M=\sum_i |n_i|$= total number of walkers). This wavefunction is the counterpart of the CIPSI expansion at iteration $n$, Eq.(\[psi\_cip\]). As in CIPSI at the next step $t+\tau$ (next iteration $n+1$) new determinants will appear. In FCI-QMC it is realized through spawning. Some determinants may also disappear through the action of the diagonal part of the Hamiltonian $[1-\tau (H_{ii}-E)]$ (death/cloning step). These two steps are designed to reproduce in average the action of the propagator on determinant $D_i$ $$[1-\tau (H-E)]|D_i\rangle = [1-\tau (H_{ii}-E)]|D_i\rangle -\tau \sum_{k \ne i}
H_{ik}|D_k\rangle.$$ In CIPSI a given determinant $|D_i\rangle$ is selected only once during iterations via Eq.(\[e2pert\]). In latter iterations it is included in the reference space and does not participate anymore to the selection. Starting from some initial determinant (usually the HF determinant) the probability of selecting $|D_i\rangle$ at some given iteration $n$ is related to the existence of a series of $(n-1)$ intermediate determinants $(|D_{i_1}\rangle,|D_{i_2}\rangle,
...,|D_{i_k}\rangle,...)$ different from $|D_i\rangle$ and connecting it to the initial determinant so that the product $$\prod_k \frac{|H_{i_{k+1}i_k}|^2}{H_{i_{k+1}i_k}-E_0}$$ is large compared to products corresponding to other series of intermediate determinants.
In FCI-QMC a determinant $|D_i\rangle$ is spawned (selected) from $|D_j\rangle$ according to the magnitude of $H_{ii}$ and -in contrast with CIPSI- with no direct dependence on the inverse of $(H_{ii}-E_0)$. However, during MC iterations the number of walkers on a given determinant evolves in time according to the death/cloning step and leads to a weighted contribution of determinants to spawning. After integration in time the weight of the determinants $|D_i\rangle$ can be estimated to be about $\int dt e^{-t(H_{ii}-E_0)}$ that is, $\sim \frac{1}{H_{ii}-E_0}$ for large enough time. As seen, FCI-QMC and CIPSI are in close connection.
Applications of CIPSI {#results}
=====================
The water molecule {#cipsi_water}
------------------
To exemplify CIPSI all-electron calculations for the water molecule using basis sets of various sizes are presented. In our first example we propose to reproduce the density matrix renormalization group (DMRG) calculation of Chan and Head-Gordon[@Chan_2003] at geometry ($R_{OH}=1 \AA,\theta_{OH}=104.5^{\circ}$) and using the “Roos Augmented Double Zeta ANO” basis set consisting of 41 orbitals[@Schuchardt_2007; @JCC:JCC9]. The full CI Hilbert space contains about $5.6\;10^{11}$ determinants (no spin or space symmetries taken into account). Calculations have been carried out using our perturbatively selected CI program Quantum Package.[@quantum_package] The energy convergence as a function of the number of selected determinants in different situations is presented in Figure \[chan\].
Four different curves are shown together with the DMRG energy value of -76.31471(1) of Chan and Head-Gordon[@Chan_2003] (solid horizontal line). The two upper curves represent the CIPSI variational energy as a function of the number of selected determinants up to 750 000 using either canonical or natural molecular orbitals. Natural orbitals have been obtained by diagonalizing the first-order density matrix built with the largest expansion obtained using canonical orbitals. As seen the convergence of both variational energies is very rapid. Using canonical orbitals an energy of -76.31239 a.u. is obtained with 750 000 determinants, a value differing from the FCI one by only 2.3 millihartree (about 1.4 kcal/mol). As known the accuracy of CI calculations is significantly enhanced when using natural orbitals.[@Davidson1972235] Here, it is clearly the case and the lowest energy reached is now -76.31436 a.u. with an error of 0.35 millihartree (about 0.2 kcal/mol). When adding the second-order energy correction $E_{PT2}$, Eq.(\[pt2\]), the energy convergence is much improved (two lower curves of Figure \[chan\]). The kcal/mol (chemical) accuracy is reached with only 1000 and 4000 determinants using canonical and natural orbitals, respectively. The best CIPSI energy including second-order correction and obtained with canonical orbitals is -76.31452 a.u. When using natural orbitals the energy is found to converge with five decimal places to the value of -76.31471 a.u., in perfect agreement with the DMRG result of Chan and Head-Gordon, -76.31471(1) a.u.
Let us emphasize that approaching the FCI limit with such a level of accuracy and so few determinants (compared to the total number of $5.6\;10^{11}$) is particularly striking and is one of the most remarkable features of SCI approaches.
To illustrate the possibility of making calculations with much larger basis sets, results obtained with the correlation-consistent polarized core-valence basis sets, cc-pCV$n$Z, with $n$ going from 2 to 5 are presented. The geometry chosen is now the experimental equilibrium geometry, $R_{OH}=0.9572$ Å and $\theta_{OH}=104.52^{\circ}$. The number of basis set functions are 28, 71, 174 and 255 for cc-pCVDZ, cc-pCVTZ,cc-pCVQZ, and cc-pCV5Z, respectively. The total number of determinants of the FCI Hilbert space with such basis sets are about $1\,10^{10}$,$1.7\,10^{14}$,$1.6\,10^{18}$, and $7.5\,10^{19}$, respectively.
![Convergence of the energy with the number of selected determinants (logarithmic scale). The graph on the left displays the variational energy, and the graph on the right shows the energy with the perturbative correction, Eq.(\[pt2\]).[]{data-label="ecipsi"}](e_h2o_cipsi.pdf){width="1.\columnwidth"}
On the left part of Figure \[ecipsi\] the convergence of the ground-state variational energy obtained for each basis set is shown. As seen, the convergence is still possible with such larger basis sets. On the right part, the full CIPSI energy curves $(E_{var} + E_{PT2})$ are presented; each curve is found to converge with a good accuracy to the full CI limit.
Generalization: The G1 set
--------------------------
In contrast with the exact Full-CI approach which takes into account the entire set of determinants and is thus rapidly unfeasible, CIPSI can be used for much larger systems. The exact limits depend of course on the size of the basis set used, the number of electrons, and also on the level of convergence asked for when approaching the full CI limit. To illustrate the feasibility of CIPSI for larger systems we present systematic all-electron calculations for the G1 benchmark set of Pople and collaborators.[@g2] The set is composed of 55 molecules and 9 different atoms. The cc-pVDZ and cc-pVTZ basis sets have been used. For all systems and both basis sets a quasi-FCI convergence has been reached. In Figure \[g1\_1\] the number of selected determinants needed to recover 99% of the correlation energy at CIPSI variational level (cc-pVDZ basis set) is plotted for each molecule or atom. For each system results are given either for canonical or natural orbitals. Depending on the importance of the multiconfigurational character of the system, this number may vary considerably (from a few tens to about 10$^7$). As expected, the number of determinants needed using natural orbitals is most of the times smaller and sometimes comparable. Figure \[g1\_2\] is similar to the preceding figure, except that numbers are given now for a full CIPSI calculation including the second-order energy correction and that a much greater accuracy corresponding to 99.9% of the correlation energy is targeted. As seen, it is remarkable that such a high precision can be reached for all systems with a number of determinants not exceeding $\sim 10^7$. In contrast with variational calculations, it should be noted that the use of natural orbitals does not systematically improve the convergence. Finally, some comparison with accurate CCSD(T) calculations performed using the same basis sets and geometries are presented. In Figure \[ccsdt\] the distribution of errors in atomization energies calculated with both CCSD(T) and CIPSI methods are plotted. For the cc-pVDZ basis set, CCSD(T) and CIPSI curves are very similar, indicating that CCSD(T) calculations have also reached the quasi full CI limit. For the larger cc-pVTZ basis set, the two curves remain similar but some significant differences show up with CIPSI results more distributed toward small errors due to a better description of multireference systems.
![Number of selected determinants required to recover 99% of the total correlation energy at CIPSI/cc-pVDZ variational level. Results for canonical and natural orbitals are given.[]{data-label="g1_1"}](g2_Variational.pdf){width="1.1\columnwidth"}
![Number of selected determinants required to recover 99.9% of the total correlation energy at full CIPSI/cc-pVDZ level $(E_{var} + E_{PT2})$. Results for canonical and natural orbitals are given.[]{data-label="g1_2"}](g2_Perturbative.pdf){width="1.1\columnwidth"}
![Distribution of errors in atomization energies for the whole G1 set of atomic and molecular systems calculated with CIPSI and CCSD(T). Results shown for cc-pVDZ and cc-pVTZ basis sets.[]{data-label="ccsdt"}](g2.pdf){width="1.\columnwidth"}
Using CIPSI nodes in DMC {#dmc}
========================
Motivations
-----------
In DMC the standard practice is to introduce compact trial wavefunctions reproducing as much as possible the mathematical and physical properties of the exact wave function. Next, the “best” nodes are determined through optimization of the parameters of the trial wavefunction in a preliminary variational Monte Carlo (VMC) run. The objective function to minimize is either the variational energy associated with the trial wavefunction or the variance of the Hamiltonian (or a combination of both). A number of algorithms have been elaborated to perform this important practical step as efficiently as possible.[@Filippi_2000; @Schautz_2002; @Umrigar_2005; @Scemama_2006; @Toulouse_2007; @Toulouse_2008] No limitations existing in QMC for the choice of the functional form of the trial wavefunction, many different expressions have been introduced (see, [*e.g.*]{} [@mosko; @sorella; @mitas; @rios; @goddard; @fili_vb; @braida; @bouabca]). However, the most popular one is certainly the Jastrow-Slater trial wavefunction expressed as a short expansion over a set of Slater determinants multiplied by a global Jastrow factor describing explicitly the electron-electron and electron-electron-nucleus interactions and, in particular, imposing the electron-electron cusp conditions associated with the zero-interelectronic distance limit of the true wavefunction.
In the DMC-CIPSI approach the determinantal part of the trial wavefunction is built using systematic CIPSI expansions. The main motivation is that CI approaches provide a simple, deterministic, and systematic way of constructing wavefunctions of controllable quality. In a given one-particle basis set, the wavefunction is improved by increasing the number of determinants, up to the Full CI (FCI) limit. Then, by increasing the basis set, the wavefunction can be further improved, up to the complete basis set (CBS) limit where the exact solution of the continuous electronic Schr[ö]{}dinger equation is reached. The CI nodes, which are defined as the zeroes of the expansion, are also expected to follow such a systematic improvement, thus facilitating the control of the fixed-node error. A second important motivation is that the stochastic optimization step can be avoided since a systematic way of improving the wavefunction is now at our disposal. The optimal CI coefficients are obtained by the (deterministic) diagonalization of the Hamiltonian matrix in the basis set of Slater determinants. It is a simple and robust step which leads to a unique set of coefficients. Furthermore, it can be made automatic, an important feature in the perspective of designing a fully black-box QMC code. Finally, using [*deterministically*]{} constructed nodal structures greatly facilitates the use of nodes evolving [*smoothly*]{} as a function of any parameter of the Hamiltonian. It is important when calculating potential energy surfaces (see, our application to the F$_2$ molecule,[@giner_jcp_2015]) or response properties under external fields.
The main price to pay for such advantages is of course the need of considering much larger multideterminant expansions (from tens of thousands up to a few millions) than in standard DMC implementations where compactness of the trial wavefunction is searched for. However, efficient algorithms have been proposed to perform such calculations[@Nukala_2009; @Clark_2011; @Weerasinghe_2014]. Very recently, we have also presented an efficient algorithm for computing very large CI expansions. Its main ideas are briefly summarized in section \[largedets\] below.
Toward a better control of the fixed-node approximation
-------------------------------------------------------
A remarkable property systematically observed so far in our DMC applications using large CIPSI expansions[@giner_2013; @scemama_jcp_2014; @giner_jcp_2015] is that, except for a possible transient regime at small number of determinants,[@note2] the fixed-node error resulting from the use of CIPSI nodes is found to decrease monotonically, both as a function of the number of selected determinants, $N_{dets}$, and of the basis set size, $M$. This result is illustrated here in the case of the water molecule at equilibrium geometry. Results shown here complement our recent benchmark study on water.[@caffarel_jcp_2016]. In Figure \[fig:h2o\_dmc\] all-electron fixed-node energies obtained with DMC-CIPSI as a function of the number of selected determinants for the first four cc-pCVnZ basis set (n=2-5) are reported. Calculations have been performed using the variational CIPSI expansions of the preceding subsection. In practice, DMC simulations have been realized using our general-purpose QMC program QMC=Chem (downloadable at [@qmcchem]). A minimal Jastrow prefactor taking care of the electron-electron cusp condition is employed and molecular orbitals are slightly modified at very short electron-nucleus distances to impose exact electron-nucleus cusp conditions. The time step used, $\tau= 2 \times 10^{-4}$ a.u., has been chosen small enough to make the finite time step error not observable with statistical fluctuations. As seen on the figure the convergence of DMC energies both as a function of the number of determinants and of the basis set are almost reached. The value of $-76.43744(18)$ a.u. obtained with the largest basis set and 1 423 377 determinants is, to the best of our knowledge, the lowest upper bound reported so far, the experimentally derived estimate of the exact nonrelativistic energy being -76.4389(1) a.u.[@klopper_mp_2001] Thanks to our recent algorithm for calculating very large number of determinants in DMC[@preprint_multidets] (see, section \[largedets\] below), the increase of CPU time for the largest calculation including more than 1.4 million of determinants compared to the same calculation limited to the Hartree-Fock determinant is only $\sim 235$.
In practice, the possibility of calculating fixed-node energies displaying such a regular behavior as a function of the number of determinants and molecular orbitals is clearly attractive in terms of control of the fixed-node error. For example, in our benchmark study of the water molecule[@caffarel_jcp_2016] it was possible to extrapolate the DMC energies obtained with each cc-pCVnZ basis set as a function of the cardinal number $n$, as routinely done in deterministic CI calculations. Using a standard $1/n^3$ law a very accurate DMC-CIPSI energy value of -76.43894(12) a.u. was obtained, in full agreement with the estimate exact value of -76.4389(1) a.u.[@caffarel_jcp_2016]
At this point, we emphasize that the observed property of systematic decrease of the energy as a function of the number of determinants is known not to be systematically true for a general CI expansion (see, [*e.g.*]{} [@flad_book_1997]). Here, its validity may probably be attributed to the fact that determinants are selected in a hierarchical way (the most important ones first), so that the wavefunctions quality increases step by step, and so the quality of nodes. However, from a mathematical point of view, such a property is far from being trivial. There is no simple argument why the FCI nodes obtained from minimization of the [*variational*]{} energy with respect to the multideterminant coefficients would lead to the best nodal structure (minimum of the [*fixed-node*]{} energy with respect to such coefficients). In a general space (not necessarily a Hilbert space of determinants) it is easy to construct a wavefunction of poor quality having a high variational energy but exact nodes and, then, to exhibit a wavefunction with a much lower energy but wrong nodes. To demonstrate the validity or not of the observed property in a finite space of determinants built with molecular orbitals expanded in a finite basis set remains to be done.
![DMC energy of the water molecule as a function of the number of determinants in the trial wave function (logarithmic scale). The horizontal solid line indicates the experimentally derived estimate of the exact nonrelativistic energy.[@klopper_mp_2001][]{data-label="fig:h2o_dmc"}](e_h2o_dmc.pdf){width="90.00000%"}
Evaluating very large number of determinants in QMC {#largedets}
---------------------------------------------------
The algorithm we use to run DMC calculations with a very large number of determinants (presently up to a few millions) has been presented in detail in [@preprint_multidets]. Its efficiency is sufficiently high to perform converged DMC calculations with a number of determinants up to a few millions of determinants. In the case of the chlorine atom discussed in [@preprint_multidets] a trial wavefunction including about 750 000 determinants has been used with a computational increase of about 400 compared to a single-determinant calculation. As already mentioned above, in the benchmark calculation of the water molecule[@caffarel_jcp_2016] up to 1 423 377 determinants have been used for a computational increase of only $\sim$ 235.\
The main ideas of the algorithm are as follows.\
$\bullet$ [*O($\sqrt{N_{dets}}$)-scaling*]{}. A first observation is that the determinantal part of trial wavefunctions built with $N_{dets}$ determinants can be rewritten as a function of a set of [*different*]{} [*spin*]{}-specific determinants $D^\sigma_i({\bf R}_\sigma)$ ($\sigma=\uparrow,\downarrow$) as follows $$\label{eq:main}
\Psi_{Det}({\bf R}) = \sum_{i=1}^{N_{\rm dets}^\uparrow} \sum_{j=1}^{N_{\rm dets}^\downarrow} C_{ij}
D^\uparrow_i({\bf R}_\uparrow) D^\downarrow_j({\bf R}_\downarrow)$$ where ${\bf C}$ is a matrix of coefficients of size $N_{\rm dets}^\uparrow \times N_{\rm dets}^\downarrow$, ${\bf R}=({\bf r}_1,...,{\bf r}_N)$ denotes the full set of electron space coordinates, and ${\bf R}_\uparrow$ and ${\bf R}_\downarrow$ the two subsets of coordinates associated with $\uparrow$ and $\downarrow$ electrons.
In standard CI expansions the number of unique spin-specific determinants is much smaller than $N_{dets}$ and typically scales as $\sqrt{N_{dets}}$. It is true for FCI expansions where all possible determinants are considered. Indeed, $N_{\rm dets}^\sigma$ attains its maximal value of $\binom {N_{\rm MO}}{N_\sigma}$ and since $N_{\rm dets}$ is given as $N_{\rm dets}^\uparrow \times N_{\rm dets}^\downarrow$, the number of unique spin-specific determinants $D^\sigma({\bf R})$ is of order $\sqrt{N_{\rm dets}}$. However, it is in general also true for the usual truncated expansions (CASSCF, CISD, etc.) essentially because the numerous excitations implying multiple excitations of spin-like electrons plays a marginal role and have a vanishing weight.\
$\bullet$ [*Optimized Sherman-Morrison updates*]{}. As proposed in a number of works,[@Nukala_2009; @Clark_2011; @Weerasinghe_2014] we calculate the determinants and their derivatives using the Sherman-Morrison (SM) formula for updating the inverse Slater matrices. However, in contrast with other implementations, we have found more efficient not to compare the Slater matrix to a common reference (typically, the Hartree-Fock determinant) but instead to perform the Sherman-Morrison updates with respect to the previously computed determinant $D_{i-1}^\sigma$. To reduce the prefactor associated with this step the list of determinants is sorted with a suitably chosen order so that with high probability successive determinants in the list differ only by one- or two-column substitution, thus decreasing the average number of substitution performed.\
$\bullet$ [*Exploiting high-performance capabilities of present-day processors*]{}. This very practical aspect – which is in general too much underestimated – is far from being anecdotal since it allows us to gain important computational savings. A number of important features include the use of vector fused-multiply add (FMA) instructions (that is, the calculation of `a=a+b*c` in one CPU cycle) for the innermost loops. It is extremely efficient and should be systematically searched for. Using such instructions (present in general-purpose processors), up to eight FMA per CPU cycle can be performed in double precision. While computing loops, overheads are also very costly and should be reduced/eliminated. By taking care separately of the various parts of the loop (peeling loop, scalar loop, vector loop, and tail loop) through size-specific and/or hard-coded subroutines, a level of 100% vectorized loops is reached in our code. Another crucial point is to properly manage the data flow arriving to the processing unit. As known, to be able to move data from the memory to the CPU with a sufficiently high data transfer to keep the CPU busy is a major concern of modern calculations. Then, it is not only important to make maximum use of the low-latency cache memories to store intermediate data but also to maximize prefetching allowing the processor to anticipate the use of the right data and instructions in advance. To enhance prefetching the algorithm should allow the predictability of the data arrival in the CPU (that is, avoid random access as much as possible). It is this important aspect that has motivated us to use Sherman-Morrison updates, despite the fact that a method like the Table method[@Clark_2011] has formally a better scaling. Indeed, massive calculations of scalar products at the heart of repeated uses of SM updates are so ideally adapted to present-day processors that very high performances can be obtained.\
$\bullet$ [*Improved truncation scheme.*]{} Instead of truncating the CI expansion according to the magnitude of the multideterminant coefficients as usual done, we propose instead to remove spin-specific determinants according to their total contribution to the norm of the expansion. In this way, more determinants can be handled for a price corresponding to shorter expansions. To be more precise, we first observe that truncating the wavefunction according to the magnitude of coefficients has the effect of removing elements of the sparse matrix ${\bf C}$ of Eq.(\[eq:main\]). A reduction of the computational cost occurs only when a full line ($\uparrow$) or a full column ($\downarrow$) of ${\bf C}$ contains only zeroes, in that case the determinant ${\bf D}_\sigma$ can be removed from the calculation. Now, by expressing the norm of the wave function as $${\cal N} = \sum_{i=1}^{N_{\rm dets}^\uparrow} \sum_{j=1}^{N_{\rm dets}^\downarrow} C_{ij}^2 =
\sum_{i=1}^{N_{\rm dets}^\uparrow} {\cal N}_i^\uparrow = \sum_{j=1}^{N_{\rm dets}^\downarrow} {\cal N}_j^\downarrow.
\label{eq:norm}$$ it is possible to assign a contribution to the norm to each determinant. Then, all determinants whose contribution to the norm is below some threshold will be removed from the expansion. This truncation scheme allows to eliminate the smallest number of coefficients needed to obtain some computational gain. Moreover, the size-consistence property of the wave function is expected to be approximately preserved by such a truncation : when a $\sigma$-determinant is removed, it is equivalent to removing the product of ${\bf D}_\sigma$ with all the ${\bar \sigma}$-determinants of the wave function.
Pseudopotentials for DMC using CIPSI {#pseudo}
====================================
When using pseudopotentials a valence Hamiltonian is defined $$H_{val} = H_{\rm loc} + V_{\rm ECP}$$ where $H_{\rm loc}$ is the local part describing the kinetic energy, the Coulombic repulsion and the local part of the effective core potentials (ECP). $$H_{\rm loc} = -\frac{1}{2} \sum_i \nabla_i^2 + \sum_{i,\alpha} v_{\rm loc}(r_{i\alpha}) + \sum_{i<j}
\frac{1}{r_{ij}}$$ and $V_{\rm ECP}$ the non-local part written as $$V_{\rm ECP}= \sum_{i,\alpha} \sum_l v_l (r_{i \alpha}) \sum_{m=-l}^l Y_{lm}(\Omega_{i \alpha}) \int
d{\Omega^\prime_{i \alpha}} Y^*_{lm} ( \Omega^\prime_{i \alpha})$$ where $v_l$ is a radial pseudopotential, $Y_{lm}$ is the spherical harmonic, $\alpha$ labels pseudo-ions.
The action of a non-local operator being difficult to sample in DMC, $V_{\rm ECP}$ is “localized” by projecting it on the trial wavefunction. The localized form of the pseudo-potential is thus defined as $$V^{\rm loc}_{\rm ECP}= \frac{V_{\rm ECP}\Psi_T}{\Psi_T}$$ and we are led back to standard DMC simulations using only local operators at the price of introducing a new “localization approximation”. This error is usually minimized by optimization of the trial wavefunction, see ref.[@casula]. In practice, the necessity of numerically evaluating the localized potential is the main difference with standard DMC calculations.
For each nucleus $\alpha$ and electron $i$, the two-dimensional angular integrals of the product of each $Y_{lm}$ and the trial wavefunction (all electrons fixed except the $i$th-electron moved over the sphere centered on nucleus $\alpha$ and of radius $r_{i \alpha}$) must be performed. By choosing the axes oriented such that the $i$th electron is on the $z$ axis, the contribution coming from the pair $(i,\alpha)$ is given by[@mitas_jcp_1991] $$\sum_{i,\alpha} \sum_l \frac{2l+1}{4\pi} v_l (r_{i \alpha})
\int d{\Omega^\prime_{i \alpha}} P_l( cos \theta^\prime) \frac{ \psi_T ({\bf r}_1, ...{\bf r}^\prime_i,...,{\bf r}_N)}
{\psi_T({\bf r}_1,..{\bf r}_i,.. {\bf r}_N)}$$ where $P_l$ denotes a Legendre polynomial. Because of the Jastrow factor, the integrals involved cannot be computed analytically. The standard solution is to evaluate them numerically using some quadrature for the sphere. Here, the CI form allows to perform the integration exactly, as already proposed some time ago.[@Hurley_1987; @Hammond_1987] Note that although no Jastrow prefactor is used here when localizing the pseudo-potential operator, such a prefactor can still be used for the DMC simulation itself. A first advantage is that the calculation is significantly faster: in practice, the computational cost is the same as evaluating the Laplacian of the wave function and a gain proportional to the number of quadrature points is obtained. A second advantage is the possibility of a better control of the localization error by increasing the number of determinants.
To illustrate these statements, we have chosen to calculate the atomization energy of the C$_2$ molecule at the Hartree-Fock, CIPSI, DMC-HF and DMC-CIPSI levels with and without pseudopotentials. All-electron HF or CIPSI calculations have been performed with the cc-pVTZ basis set. To allow meaningful comparisons, $1s$ molecular orbitals have been kept frozen in all-electron CIPSI calculations. Pseudopotential calculations were done using the pseudopotentials of Burkatzki *et al.*[@burkatzki] with the corresponding VTZ basis set. The electron-nucleus cusps of all the wave functions were imposed,[@maCusp; @Kussmann_cusp; @per:cusp] and no Jastrow factor was used. For the sake of comparison, the same time step ($5 \times 10^{-4}$ au) was used for all-electron and pseudopotential calculations, although a much larger time step could have been taken with pseudopotentials.
[lccccc]{} & &\
& C (a.u.) & C$_2$ (a.u.) & AE (kcal/mol) & C & C$_2$\
\
all-$e$ & -37.6867 & -75.4015 & 17.6 & 1 & 1\
pseudo- & -5.3290 & -10.6880 & 18.8 & 1 & 1\
\
all-$e$ & -37.7810 & -75.7852 & 140.1 & 3796 & $10^6$\
pseudo- & -5.4280 & -11.0800 & 140.6 & 3882 & $10^6$\
\
all-$e$ & -37.8293(1) & -75.8597(3) & 126.3(2) & 1 & 1\
pseudo- & -5.4167(1) & -11.0362(3) & 127.2(2) & 1 & 1\
\
all-$e$ & -37.8431(2) & -75.9166(2) & 144.6(2) & 3497 & 173553\
pseudo- & -5.4334(1) & -11.0969(3) & 144.3(2) & 3532 & 231991\
& 147$\pm$2 & &\
The results presented in Table \[tab:pseudo\] show that all the atomization energies obtained using pseudopotentials are in very good agreement with those obtained with all-electron calculations at the same level of theory. The DMC energies obtained with CIPSI trial wave functions are always below those obtained with Hartree-Fock trial wave functions, and the error in the atomization energy is reduced from 20 kcal/mol with HF nodes down to 3 kcal/mol with CIPSI nodes.
Calculations were performed on Intel Xeon E5-2680v3 processors. Timings are given in Table \[tab:timing\]. For the carbon atom the computational time needed for one walker to perform one complete Monte Carlo step (all electrons moved) is the same with or without pseudopotentials. For the C$_2$ molecule, the calculation is even faster with pseudopotentials: A factor of about $1.5$ is gained with respect to the all-electron calculation. This can be explained by the computational effort saved due to the reduced size of Slater matrices in the pseudopotential case (from $6\times6$ to $4\times4$) but, more importantly, by the fact that the additional cost related to the calculation of the contributions due to the pseudopotential is not enough important to reverse the situation. In all-electron calculations, the variance is only slightly reduced when going from the Hartree-Fock trial wave function to the CIPSI wave function (with frozen core). Indeed, the largest part of the fluctuations comes from the lack of correlation of the core electrons. In the calculations involving pseudopotentials, the decrease of the variance is significant: a reduction by a factor of 2.4 and 3.2 is observed.
From a more general perspective, comparisons between all-electron and pseudopotential calculations must take into account both the computational effort required in each case and the level of fluctuations resulting from the quality of the trial wavefunction. To quantify this, we have reported in the table the number of CPU hours required to obtain an error bar of 1 kcal/mol. Using pseudopotentials for the C$_2$ molecule, it is found that the reduction of the variance due to the improvement of the wave function with the multideterminant expansion almost compensates the cost of the computation due to the additional 230 000 determinants : the CPU time needed to obtain a desired accuracy is only $1.2\times$ more than the single determinant calculation.
----------- --------- --------- --------- --------- ----------- -----------
all-$e$ pseudo- all-$e$ pseudo- all-$e$ pseudo-
DMC-HF
C 0.0076 0.0078 1.54 1.18 7.858(3) 0.3471(2)
C$_2$ 0.0286 0.0186 14.95 10.35 16.208(7) 1.1372(6)
DMC-CIPSI
C 0.193 0.201 5.61 0.70 7.620(8) 0.1084(4)
C$_2$ 10.1 8.12 91.05 12.72 15.61(3) 0.460(1)
----------- --------- --------- --------- --------- ----------- -----------
: CPU time for one complete Monte Carlo step (one walker, all electrons moved), CPU time needed to reach an error on 1 kcal/mol, and variances associated with the HF and CIPSI trial wave functions (electron-nucleus cusp corrected).[]{data-label="tab:timing"}
Summary and some perspectives {#conclu}
=============================
Let us first summarize the most important ideas and results presented in this work.\
\
i.) Selected Configuration Interaction approaches such as CIPSI are very efficient methods for approaching the full CI limit with a number of determinants representing only a tiny fraction of the full determinantal space. This is so because only the most important determinants of the FCI expansion are perturbatively selected at each step of the iterative process. We note that the recent FCI-QMC method of Alavi [*et al.*]{}[@booth_jcp_2009; @cleland_jcp_2010] uses essentially the same idea, except that in CIPSI the selection is done deterministically instead of stochastically.\
\
ii.) In constrast with exact FCI which becomes rapidly prohibitively expensive, CIPSI allows to treat larger systems, while maintaining results of near-Full CI quality. The exact practical limits depend of course on the size of the basis set used, the number of active electrons, and also on the level of convergence asked for when approaching the full CI limit. In this work, the CIPSI approach has been exemplified with near-FCI quality all-electron calculations for the water molecule using a series of basis sets of increasing size up to the cc-pCV5Z basis set and for the whole set of 55 molecules and 9 atoms of the benchmark G1 set (cc-pVDZ basis set). In each case, the huge size of the FCI space forbids exact FCI calculations. CIPSI has been applied to larger systems, for example for calculating accurate total energies for the atoms of the $3d$ series,[@scemama_jcp_2014] and for obtaining near-FCI quality results for the CuCl$_2$ molecule (calculations including 63 electrons and 25 active valence electrons).[@Caffarel_2014] Note that by using Effective Core Potentials as described in section \[pseudo\] even larger systems can be treated.\
\
iii.) We emphasize that the idea of selecting determinants is not limited to the entire space of determinants but can be used to make CI expansion to converge in a subset of determinants chosen [*a priori*]{}. For example, efficient and accurate selected CASCI, CISD, or even MRCC[@mrcc_jcp_2016] calculations can be performed. Note that going beyond CASCI and implementing a selected CASSCF approach (CASCI with optimization of molecular orbitals) is also possible; this is let for further work. However, note that a stochastic version of CASSCF within FCI-QMC framework has already been implemented by Alavi [*et al.*]{}[@Thomas_jctc_2015]\
\
iv.) CIPSI expansions can be used as determinantal part of the trial wavefunctions employed in DMC calculations. In others words, we propose to use selected CI nodes as approximation of the unknown exact nodes. The basic motivation is that CI approaches provide a simple, deterministic, and systematic way to build wavefunctions of controllable quality. In a given one-particle basis set, the wavefunction is improved by increasing the number of determinants, up to the FCI limit. Then, by increasing the basis set, the wavefunction can be further improved, up to the CBS limit where the exact solution of the continuous electronic Schrödinger equation is reached. CI nodes, defined as the zeroes of the CI expansions, are also expected to display such a systematic improvement.\
\
v.) The main result giving substance to the use of selected CIPSI nodes is that in all applications realized so far the fixed-node error is found to decrease both as a function of the number of selected determinants and of the size of the basis set. Mathematically speaking, such a result is far from being trivial. In practice, such a property is particularly useful in terms of control of the fixed-node error.\
\
vi.) From a practical point of view, the price to pay is the need of considering much larger multideterminant expansions (from tens of thousands up to a few millions) than in standard DMC where compactness of the trial wavefunction is usually searched for. Indeed, computing at each of Monte Carlo step the first and second derivatives of the trial wavefunction (drift vector and local energy) is the hot spot of DMC. However, efficient algorithms have been proposed to perform such calculations[@Nukala_2009; @Clark_2011; @Weerasinghe_2014]. Here, we have briefly summarized our recently introduced algorithm allowing to compute $N$-determinant expansions issued from selected CI calculations with a computational cost roughly proportional to $\sqrt{N}$ (with a small prefactor).\
\
vii.) One key advantage of using CIPSI nodes is that their construction can be made fully automatic. Coefficients of the CI expansion are obtained in a simple and deterministic way by diagonalizing the Hamiltonian matrix and the solution is unique. Furthermore, when approaching the FCI limit the resulting expansion becomes independent on the type of molecular orbitals used (canonical, natural, Kohn-Sham, see Figure 8 of ref.[@Caffarel_2014]). Another attractive feature is that the nodes built are reproducible and thus “DMC models” can be defined in the spirit of WFT or DFT [*ab initio*]{} approaches (HF/cc-pVnZ, MP2/6-31G, CCSD(T), DFT/B3LYP etc.) Indeed, once the basis set has been specified, the nodes are unambiguously defined at convergence of the DMC energy as a function of the number of selected determinants. Furthermore, in this limit the nodal surfaces vary continuously as a function of the parameters of the Hamiltonian. A particularly important example is the possibility of obtaining regular potential energy surface (PES). This idea has been illustrated in a previous work on the potential energy curve of the F$_2$ molecule.[@giner_jcp_2015] Furthermore, it is also possible to reduce the “non-parallelism” error resulting from the use of a trial wavefunction of non-uniform quality across the PES. This can be done for example by using a variable number of selected determinants depending on the geometry and chosen to lead to a constant second-order estimate of the remaining correlation energy (constant-PT2 approach,[@giner_jcp_2015] ).\
\
viii.) As in standard DMC approaches a Jastrow prefactor can be used to reduce statistical fluctuations. However, in contrast with what is usually done, we do not propose to re-optimize the determinantal CIPSI part in presence of this Jastrow term. The main reason for that is not to lose the advantages of using deterministically constructed nodal structures: Systematic improvement of nodes as a function of the number of determinants and of the size of the basis set, simplicity of construction of nodes and reproductibility, possibility of optimizing a very large number of small coefficients in the CI expansion (no noise limiting in practice the magnitude of optimizable coefficients), smooth evolution of nodes under variation of an external parameter (geometry, external field), etc.\
\
ix.) The price to pay for not re-optimizing the determinantal part in the presence of a Jastrow is that for small basis sets larger fixed-node errors are usually obtained. However, when increasing sufficiently the quality of basis set, it is no longer true as illustrated for example in the case of the oxygen atom,[@giner_2013] the water molecule,[@caffarel_jcp_2016] and the $3d$-transition metal atoms[@scemama_jcp_2014] for which benchmark total energies have been obtained.\
\
x.) CIPSI wavefunctions are particularly attractive when using non-local Effective Core Potentials (ECP). Indeed, as already proposed some time ago,[@Hurley_1987; @Hammond_1987] CI expansions allow the analytical calculation of the action of the non-linear part of the pseudo-potential operator on the trial wavefunction. In this way, the use of a numerical grid defined over the sphere is avoided and a gain in computational effort essentially proportional to the number of grid points is obtained. Here, this idea has been illustrated in the case of the C$_2$ molecule.\
\
Finally, let us briefly mention a number of topics presently under investigation.\
\
xi.) The slow part of the CI convergence is known to result from the absence of electron-electron cusp. In standard QMC approaches, the short distance electron-electron behavior is introduced into the Jastrow prefactor and its impact on nodes is taken into account by optimization of the full trial wavefunction. Under re-optimization, molecular orbitals are changed and the distribution of multideterminant coefficients is modified with a re-inforcement of coefficients associated with chemically meaningful determinants and a reduction of the numerous small coefficients associated with the absence of cusp. To keep the CIPSI as compact as possible and to eliminate this unphysical and uncoherent background of small coefficients a R12/F12 version of CIPSI is called for. We emphasize that such an analytical and deterministic construction of the R12/F12 expansion is necessary if we want to keep the advantages related to the deterministic construction of nodes.\
\
xii.) To treat even larger systems, the increase of the number of determinants in the CIPSI expansion must be kept under control. Instead of targeting the near full CI limit, simpler models can be used in the spirit of what is done in MRCC approaches[@mrcc_jcp_2016] or by defining effective Hamiltonians in the reference space modelling the effect of the external space (so-called internally decontracted approaches).\
\
xiii.) Finally, it is clear that systematic studies on difficult systems of various types are needed to explore the potential and limits of the DMC-CIPSI approach.\
\
[*Acknowledgments.*]{} We would like to thank C. Angeli and P-F. Loos for their useful comments on the manuscript. AS and MC thank the Agence Nationale pour la Recherche (ANR) for support through Grant No ANR 2011 BS08 004 01. This work was performed using HPC resources from CALMIP (Toulouse) under allocation 2016-0510 and from GENCI-TGCC (Grant 2016-08s015).
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abstract: 'Co single atom junctions on copper surfaces are studied by scanning tunneling microscopy and *ab-initio* calculations. The Kondo temperature of single cobalt atoms on the Cu(111) surface has been measured at various tip-sample distances ranging from tunneling to the point contact regime. The experiments show a constant Kondo temperature for a whole range of tip-substrate distances consistently with the predicted energy position of the spin-polarized d-levels of Co. This is in striking difference to experiments on Co/Cu(100) junctions, where a substantial increase of the Kondo temperature has been found. Our calculations reveal that the different behavior of the Co adatoms on the two Cu surfaces originates from the interplay between the structural relaxations and the electronic properties in the near-contact regime.'
author:
- Lucia Vitali
- Robin Ohmann
- Sebastian Stepanow
- Pietro Gambardella
- Kun Tao
- Renzhong Huang
- 'Valeri S. Stepanyuk'
- Patrick Bruno
- Klaus Kern
title: 'Kondo effect in single atom contacts: the importance of the atomic geometry'
---
The electron transport properties in a circuit, whose dimensions are reduced to single atom/molecule contact, is dominated by the quantum character of matter [@cuniberti05] and requires a deep understanding of its nanometer scale properties. Additionally, the electron transport through such a junction is strongly influenced by the coupling of the orbitals of the electrodes to the bridging molecule or atom [@tao08]. Accessing the correct information of the geometrical arrangement of the contact is fundamental to the interpretation of the experimental data. This is often challenging, despite the continuous progress in understanding the electron transport through nanometer scale junctions [@scheer98; @yanson98; @pascual95; @lortscher07; @burgi99; @neel07; @termirov08; @xiao04]. Among other techniques such as mechanical controlled break-junction [@scheer98; @yanson98; @lortscher07] or electro-migration [@park99], a junction achieved with the tip of a scanning tunneling microscope (STM) on a metal surface [@gimzewski95; @burgi99; @neel07; @termirov08] has been proven as a valuable tool to target and select the substrate configuration before and after the contact is formed. Nonetheless, a general picture on the influence of atomistic order on the electron conductance at nanometer scale junctions is, at present, still missing.
Co adatoms on copper surfaces constitute an ideal system for studying the electron transport through nanometer scale junctions. The interplay of the unpaired electrons of single Co atoms and the free electron states on copper surfaces leads to the formation of a narrow electronic resonance at the Fermi level known as the Kondo resonance. This has been extensively characterized in tunneling conditions [@madhavan98; @wahl04; @knorr02]. One main result of these studies is the evidence that on different supporting surfaces the width of this resonance coincides with a change of the occupation of the d electron levels of the Co adatom [@wahl04]. This inherently reflects a varied coupling of the magnetic impurity to the metal substrate. Indeed, a simple model has been suggested to relate the energy position of the electron d-levels of the impurity and the atomic arrangement in close proximity of the Co adatom [@wahl04; @knorr02]. Specifically, a narrower Kondo resonance is observed for Co adatoms on Cu(111) than on the Cu(100) surface in agreement with a shift in energy of the d-levels i.e with an increase in their occupation from the first to the second surface [@knorr02; @wahl04]. Due to this dependence, the width of the Kondo resonance is then a good reference parameter to characterize the influence of the tip at reduced distances. However, it is a priori not evident if the width of the Kondo resonance will follow a trend similar to the one observed in tunneling configuration on various surfaces also when the tip is approached to the point contact configuration.
In order to address this question we measured the current-voltage characteristics at different tip-sample distances ranging from tunneling to point contact on individual Co adatoms on a Cu(111) surface. As will be shown in the following, the width of the Kondo resonance is practically constant on this surface at all tip-substrate distances in an apparent contradiction with the previously reported results on Cu(100) [@neel07].
Based on *ab-initio* theoretical calculations aimed to determine the electronic and magnetic properties of these two systems, we will show that the opposing results observed on the two copper surfaces are not contradicting but demonstrate nicely the determining influence of the local atomic structure on the transport properties of a nanoscale junction.
The experiments were performed using a home built scanning tunneling microscope operated at 6K in ultra high vacuum (UHV) with a base pressure of 1\*10$^{-11}$mbar. The Cu(111) single crystal has been cleaned in UHV by cycles of Ar ion sputtering and annealing. Co single atoms have been deposited from a thoroughly degassed Co wire wound around a tungsten filament on the Cu surface at 20K. This resulted in a coverage of about 10$^{-3}$ML of isolated immobile Co adatoms. The STM tip, chemically etched from tungsten wire, was treated in vacuo by electron field emission and soft indentation into the copper surface. This assured a spectroscopically featureless tip near the Fermi energy. Given this preparation, the tip was most likely covered by copper atoms deriving from the substrate.
The inset in figure \[fig1\] shows the conductance of a single Co adatom at various tip-substrate displacements. This has been achieved by recording the current while approaching the tip towards the atom, in open feedback loop conditions. As the tip substrate distance is reduced the current increases smoothly from the tunneling to the point contact regime following the exponential dependence with the tip-substrate distance (z) characteristic of the electron tunneling process I(z)=I$_0$exp(-Az) (where A is proportional to the work function of tip and substrate). Fitting the experimental curve, we obtained a work function for the Co/Cu(111) system of 5 eV. As the point contact regime is reached the current is found to exhibit a characteristic quantization plateau with only a weak dependence on the distance. The plateau is observed to be 1G$_0$ where G$_0$ is the conductance quantum G$_0$=2e$^2$/h (h is Planck constant) in agreement with studies on Co/Cu(100)[@neel07]. Topographic images acquired before and after the tip was approached and retracted from the point contact configuration confirm that the contact region as well as the tip have not changed during the tip displacements.
![\[fig1\] (color on line) Conductance and dI/dV spectra for isolated Co atoms on a Cu(111) surface achieved at different tip-substrate distances $\Delta$Z from tunneling to point contact. In inset a representative current vs tip-displacement is shown. dI/dV spectra have been recorded at the position indicated by a circle in the inset. The curves are normalized to the tunneling current at the tip height location and vertical offset has been added for a better visualization. The Kondo temperature T$_K$ given on the right side of the image has been obtained fitting the curves with a Fano line shape (red line). []{data-label="fig1"}](fig1f.eps){width="0.9\linewidth"}
Information on the Kondo resonance have been obtained recording the dI/dV spectra on top of the Co adatom at various tip-substrate displacements. In figure \[fig1\], we report the spectra obtained at the tip-substrate separation indicated by the circles in the inset. The current versus voltage is measured with a lock-in amplifier applying a voltage modulation in the range of 1 to 0.1mV (rms). All the curves obtained in the range from the initial tunneling ($\Delta$Z=-2Å) to the point contact ($\Delta$Z=0.2Å) condition show a characteristic dip in the local density of states at an energy close to the Fermi level. This dip, which is due to the Kondo resonance can be characterized according to its width $\Delta$E, which is proportional to the Kondo temperature T$_K$, $\Delta$E=2k$_B$T$_K$, where k$_B$ is the Boltzmann constant [@madhavan98; @wahl04]. The Kondo Temperature can be extracted from these curves by fitting the experimental spectra with a Fano line function according to dI/dV$\propto$(q+$\epsilon$)$^2$/(1+$\epsilon^2$), with $\epsilon$=(eV-$\epsilon$$_K$)/k$_B$T$_K$ where q and $\epsilon_K$ define the asymmetry of the curve and the energy position of the resonance with respect to the Fermi energy [@fano61]. The fitted Kondo temperature T$_K$ is reported in figure \[fig1\] for each sampled tip position. As can be seen, the Kondo temperature for the Co on Cu(111) system is constant, within the experimental error, from the tunneling to the point contact regime.
The observed behavior of the Kondo temperature on the Cu(111) surface contrasts with the behavior previously reported for the Co/Cu(100) system, where a considerable increase of the Kondo temperature (from 70-90K in tunneling to 150K in point contact) was observed [@neel07]. As will be shown below this difference can be ascribed to the sensitivity of the Kondo effect to the local atomic geometry.
To obtain a physical understanding of the structural sensitivity, we have modelled at first the atomic relaxation in the single Co atom junction under the influence of the tip proximity and then considered its consequence on the electronic structure. Indeed, reducing the tip-substrate separation can induce a local perturbation in the atomic ordering at the junction which can affect the coupling between the orbitals of the electrodes and of the Co atom and consequently the electronic and the magnetic properties of the system. To simulate the nanoscale-junction on the atomic scale, we have performed molecule static (MS) calculations with many-body interatomic potentials [@levanov00]. In these simulations the tip has been represented as a pyramid consisting of 10 Cu atoms arranged in fcc(111) stacking. Figure \[fig2\] shows the variation of the tip-adatom and the adatom-substrate separations during the tip displacement (panel a and b, respectively). On a first glance one can see that beside an initial region, the tip- Co atom as well as the Co atom-substrate distances are not linearly proportional to the tip displacement. As the tip-substrate distance is reduced, the atomic order at the junction relaxes: the atoms of the tip, the Co impurity as well as the atoms of the substrate move to new equilibrium positions. The real tip-substrate distance is then a dynamic variable according to the specific location of the tip and to its attractive and repulsive interaction with the surface and the impurity. Specifically, up to the minimum distance of 5.3Å, the tip-Co atom distance is almost linear with the tip displacement. Approaching further, the distance between the opposite sides of the nanometer scale junction is reduced to a larger extend than the effectively applied tip displacement due to an attractive interaction (up to 4.7Å). Reducing the tip-substrate distance below 4.7Å, the interaction becomes repulsive. At this tip proximity, the adatom-substrate distance, defined as the vertical distance between the Co adatom and its first nearest neighbor, is strongly reduced while the distance between the tip and adatom is only slightly decreased. This implies that the Co adatom shifts towards the substrate. As a consequence when the point contact configuration is reached, the Co-Cu(111) surface distance compares to the equilibrium distance predicted for the tunneling condition (dotted line in panel b).
Figure \[fig2\] compares also the atomic relaxation process on the two copper surfaces. The general trend of attractive and repulsive interaction of tip-atom and surface can be observed in both cases. However, differences in the atom dynamics under the influence of the tip and in the Co-surface distance are obvious. Specifically, under the influence of the tip the Co impurity is pushed deeper into the Cu(100) surface in point contact configuration than it is in tunneling conditions (black dotted line in panel b). Therefore, it can be expected that the stronger interaction with the surface increases on the Cu(100) substrate the hybridization of the d-levels of the Co adatom with the sp states of the surface.
![\[fig2\] Atomic relaxation at the single Co atom junction as a function of the tip-substrate displacement. The tip-adatom (H) and the adatom-substrate (L) distances for Co adatom on Cu(111) and Cu(100) surface are shown in panel a and b, respectively. The dotted lines emphasize the relative Co-substrate distance in point contact and in tunneling conditions. A, B and C indicate the position where the LDOS shown in figure 3 have been calculated. []{data-label="fig2"}](fig2e.eps){width="0.95\linewidth"}
This and its consequence on the magnetic properties of the junction at various tip proximity, can be understood calculating the local density of states (LDOS) of the Co adatom. All the calculations were performed within the linear combination of atomic orbital (LCAO) formalism by means of density functional theory (DFT) implemented in SIESTA[@soler02]. The geometry was optimized by SIESTA until all residual forces on each atom are smaller than 0.01eV/Å [@MD-Sie; @siesta].
In figure \[fig3\] the d-levels of Co/Cu(111) is shown for different tip-substrate separations (denoted A, B and C in figure \[fig2\]) with the energies given with respect to the Fermi level. It can be seen that only the occupied density of states of the Co adatom on Cu(111) are slightly affected by the tip substrate distance. Moreover, the energy difference between the center of the occupied spin-up band (or majority states) and the center of the partially unoccupied spin-down band (or minority states) U for three tip-substrate separations are nearly the same. Consistently, also the magnetic moment of the Co adatom at these three tip-substrate separations (1,99$\mu_{B}$, 1.96$\mu_{B}$ and 1,78$\mu_{B}$, respectively) are only slightly affected by the tip proximity. On the contrary a large energy shift of the d-levels was reported for Co adsorbed on Cu(100) surface [@neel07; @huang06]. A comparison of the energy position of the occupied d-levels is shown in figure \[fig3\]b. On both surfaces the position of the occupied d-levels shifts towards higher energies under the influence of the tip proximity. On Cu(111) surface this shift is, however, much smaller. On Cu(100) the substantial change in the occupation of the d-levels is reflected in the increase of the Kondo temperature in point contact. Accurate calculations of the expected increase of the Kondo temperature on the these surfaces is, however, not straight forward. Nonetheless, the theoretical predictions and the experimentally observed Kondo temperature in point contact regime follows the trend described by the model proposed by Wahl *et al*. [@wahl04] for the tunneling regime. The increase of the occupation of the d-level effects sensibly the Kondo temperature on Cu(100) and almost negligibly on the Cu(111) surface.
![\[fig3\] Influence of the tip-proximity on the d-levels of Co atoms on Cu(111). a) Spin polarized LDOS for the d-levels. The curves are calculated for tip-substrate displacements as denoted in figure \[fig2\]. b) Energy position of the occupied d-levels on Cu(111) and Cu(100) at different tip-substrate positions. The lines connecting the points are a guide to the eye.[]{data-label="fig3"}](fig3c.eps){width="1\linewidth"}
In conclusion, the present experimental and theoretical work demonstrates that the local atomic geometry plays a major role in the electron transport properties of nanoscale junctions. The tip proximity in the point contact regime influences the atomic relaxation in the single atom junction and thereby determines the lattice equilibrium position. These structural relaxations induce a modification of the sp-d hybridization between the electrode surface and the bridging atom. While on the closed packed Cu surface the impurity d-level is less affected, it shifts substantially on the open (100) surface. This explains the striking difference observed in the behavior of the Kondo temperature of Co adatoms upon point contact formation on Cu(111) and Cu(100). We believe that these results have general validity and might clarify a few of the uncertainties in the electron transport through nanometer scale junctions characterized by break-junction experiments.
We acknowledge P.Wahl for fruitful discussions. The work was supported by Deutsche Forschungsgemeinschaft (DPG SPP 1165, SPP 1243 and SSP 1153)
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LDA approaches have been used for the exchange and correlation potential, and a 250 Ry energy cutoff is used to define real-space grid for numerical calculations involving the electron density. Core electrons of all elements are replaced by nonlocal norm-conserving Troullier-Martins pseudopotentials. Valence electrons of the Cu substrate are described using double-$\zeta$ plus polarization atomic orbital basis set, and a triple-$\zeta$ plus polarization atomic orbital basis sets for Co adatom.
We chose three different tip-substrate distances (denoted A, B and C in figure \[fig2\]) and perform a fully relaxation by Siesta code. The displacements of the Co adatom at three tip-substrate separations (denoted $a$, $b$ and $c$ in Fig. \[fig2\]) calculated by the MS method are 0.02Å, 0.21Å and 0.05Å, respectively. While these are 0.01Å, 0.26Åand 0.08Å obtained from Siesta code, coinciding with the MD results very well. Relaxation of the atoms in the tip is also taken into account in our calculations [@huang06].
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---
abstract: 'Event-based neuromorphic systems promise to reduce the energy consumption of deep learning tasks by replacing expensive floating point operations on dense matrices by low power sparse and asynchronous operations on spike events. While these systems can be trained increasingly well using approximations of the backpropagation algorithm, these implementations usually require high precision errors for training and are therefore incompatible with the typical communication infrastructure of neuromorphic circuits. In this work, we analyze how the gradient can be discretized into spike events when training a spiking neural network. To accelerate our simulation, we show that using a special implementation of the integrate-and-fire neuron allows us to describe the accumulated activations and errors of the spiking neural network in terms of an equivalent artificial neural network, allowing us to largely speed up training compared to an explicit simulation of all spike events. This way we are able to demonstrate that even for deep networks, the gradients can be discretized sufficiently well with spikes if the gradient is properly rescaled. This form of spike-based backpropagation enables us to achieve equivalent or better accuracies on the MNIST and CIFAR10 dataset than comparable state-of-the-art spiking neural networks trained with full precision gradients. The algorithm, which we call *SpikeGrad*, is based on accumulation and comparison operations and can naturally exploit sparsity in the gradient computation, which makes it an interesting choice for a spiking neuromorphic systems with on-chip learning capacities.'
author:
- |
Johannes C. Thiele\
CEA, LIST\
91191 Gif-sur-Yvette CEDEX, France\
`johannes.thiele@cea.fr`\
Olivier Bichler\
CEA, LIST\
91191 Gif-sur-Yvette CEDEX, France\
`olivier.bichler@cea.fr`\
Antoine Dupret\
CEA, LIST\
91191 Gif-sur-Yvette CEDEX, France\
`antoine.dupret@cea.fr`\
title: 'SpikeGrad: An ANN-equivalent Computation Model for Implementing Backpropagation with Spikes'
---
Introduction
============
Spiking neural networks (SNNs) are a new generation of artificial neural network models [@Maass:1997], which try to harness potentially useful properties of biological neurons for energy efficient neuromorphic systems. In traditional artificial neural networks (ANNs), processing is based on operations on dense, real valued tensors. In contrast to this, SNNs communicate with asynchronous spike events, which potentially allows them to process efficiently information with high temporal and spatial sparsity if implemented in custom event-based hardware (see for instance [@Merolla:2014] and [@Qiao:2015]).
#### Previous work on optimizing SNNs with backpropagation
The recent years have seen a large number of approaches devoted to optimization of spiking neural networks with the backpropagation algorithm, either by converting ANNs to SNNs [@Diehl:2015SpikeConversion][@OConnor:2016][@Esser:2016][@Rueckauer:2017] or by simulating spikes explicitly in the forward pass and optimizing these dynamics with full precision gradients [@Lee:2016][@Yin:2017][@Wu:2018SpikingBP][@Wu:2018SpikingBPDeep][@Severa:2019][@Jin:2018][@Belec:2018][@Zenke:2018SuperSpike]. These methods do usually not communicate gradients as spike signals (for a recent and more detailed review of training algorithms for SNNs, see [@Pfeiffer:2018] or [@Tavanaei:2019]). It would however be desirable to enable on-chip learning in neuromorphic chips using the power of the backpropagation algorithm, while maintaining the advantages of spike-based processing also in the backpropagation phase. Recent work of [@Binas:2016] and [@Wu:2019] has discussed how forward processing in an SNN could be mapped to an ANN. Our work extends this analysis to the backward propagation pass, to yield a fully spike-based implementation of the backpropagation algorithm.
#### Previous work on approximating backpropagation with spikes
In the works of [@Neftci:2017] and [@Samadi:2017] a spike-based version of the backpropagation algorithm is implemented, using direct feedback to neurons via spike propagation through fixed weights to each layer of the network. While good performance on the MNIST dataset is achieved, they do not demonstrate the capacity of their algorithm on large ANNs and more realistic benchmarks. The exact backpropagation algorithm, which backpropagates through symmetric weights might however be required to reach good performance on large-scale problems [@Baldi:2016][@Bartunov:2018]. [@Thiele:2019] uses an approximation of the backpropagation algorithm where the error is propagated via spike events to train a network for relational inference. However, no mathematical analysis of the approximate capacities of the algorithm is provided and no scalability to large scale classification problems is demonstrated.
#### Paper contributions
Our contributions are twofold: First, we demonstrate how backpropagation can be seamlessly integrated into the spiking neural network framework by using a second accumulation compartment for error propagation, which discretizes the error into spikes. This way we obtain a system that is able to perform learning and inference based on accumulations and comparisons alone. As for the forward pass, this allows us to exploit the dynamic precision and sparsity provided by the discretization of all operations into asynchronous spike events. Secondly, we show that the system obtained in this way can be mapped to an ANN with equivalent accumulated responses in all layers. This allows us to simulate training of large-scale SNNs efficiently on graphic processing units (GPUs), using their equivalent ANN. We demonstrate classification accuracies equivalent or superior to existing implementations of SNNs trained with full precision gradients, and comparable to the precision of standard ANNs. Based on our review of the literature, our work provides for the first time an analysis of how the sparsity of the gradient during backpropagation can be exploited within a large-scale SNN processing structure. This is the first time competitive classification performances are reported on a large-scale spiking network where training and inference are fully implemented with spikes.
The *SpikeGrad* algorithm {#sec:acc_based_model}
=========================
We begin with the description of *SpikeGrad*, the spike-based backpropagation algorithm. For each training example/mini-batch, integration is performed from $t=0$ to $t=T$ for the forward pass and from $t=T+\Delta t$ to $t=\mathcal{T}$ in the backward pass. Since no explicit time is used in the algorithm, $\Delta t$ represents symbolically the (very short) time between the arrival of an incoming spike and the response of the neuron, which is only used here to describe causality.
#### Integrate-and-fire neuron model
Our architecture consists of multiple layers (labeled by $l\in [0,L]$) of integrate-and-fire (IF) neurons with integration variable $V_i^l(t)$ and threshold ${\ensuremath{\Theta_\mathrm{ff}}}$: $$\label{update_potential}
V_i^l(t+ \Delta t) = V_i^l(t) - {\ensuremath{\Theta_\mathrm{ff}}}s_i^l(t) + \sum_j w^{l}_{ij}s^{l-1}_j(t),\quad V_i^l(0) = b^l_i.$$ The variable $w^{l}_{ij}$ is the weight and $b^l_i$ a bias value. The spike activation function $s^l_i(t) \in \{-1,0,1\}$ is a function which triggers a signed spike event depending on the internal variables of the neuron. It will be shown later that the specific choice of the activation function is fundamental for the mapping to an equivalent ANN. After a neuron has fired, its integration variable is decremented or incremented by the threshold value ${\ensuremath{\Theta_\mathrm{ff}}}$, which is represented by the second term on the r.h.s. of .
As a representation of the neuron activity, we use a trace $x^l_i(t)$ which accumulates spike information over a single example: $$\label{learning_rate_trace}
x_i^{l}(t+\Delta t) = x_i^{l}(t) + \eta s_i^{l}(t).$$ By weighting the activity with the learning rate $\eta$ we avoid performing a multiplication when weighting the input with the learning rate for the weight update .
#### Implementation of implicit ReLU and surrogate activation function derivative
It is possible to define an implicit activation function based on how the neuron variables affect the spike activation function $s^l_i(t)$. In our implementation, we use the following fully symmetric function to represent linear activation functions (used for instance in pooling layers): $$\label{spike_activation_function_linear}
s^{l,\mathrm{lin}}_i\left(V_i^l(t)\right) \coloneqq \begin{cases}
\; 1 & \mathrm{if}\;V^l_i(t)\geq{\ensuremath{\Theta_\mathrm{ff}}}\\
\; -1 & \mathrm{if}\;V^l_i(t)\leq-{\ensuremath{\Theta_\mathrm{ff}}}\\\
\; 0 & \mathrm{otherwise}
\end{cases}.$$ The following function corresponds to the rectified linear unit (ReLU) activation function: $$\label{spike_activation_function_ReLU}
s^{l,\mathrm{ReLU}}_i\left(V_i^l(t),x^l_i(t)\right) \coloneqq \begin{cases}
\; 1 & \mathrm{if}\;V^l_i(t)\geq{\ensuremath{\Theta_\mathrm{ff}}}\\
\; -1 & \mathrm{if}\;V^l_i(t)\leq-{\ensuremath{\Theta_\mathrm{ff}}}\;\mathrm{and}\; x^l_i(t)>0\\
\; 0 & \mathrm{otherwise}
\end{cases}.$$ The pseudo-derivative of the activation function is denoted symbolically by $S^{'l}_i$. We use $S_i^{'l,\mathrm{lin}}(T)=1$ for the linear case. For the ReLU, we use a surrogate of the form: $$\label{straight_through_estimator_x}
S_i^{'l,\mathrm{ReLU}}(T) \coloneqq \begin{cases}
\; 1 & \mathrm{if}\; V^l_i(T) > 0 \;\mathrm{or}\;x_i^l(T)>0 \\
\; 0 & \mathrm{otherwise}
\end{cases}.$$ These choices will be motivated in the following sections. Note that the derivatives depend only on the final states of the neurons at time $T$.
#### Discretization of gradient into spikes
For gradient backpropagation, we introduce a second compartment with threshold ${\ensuremath{\Theta_\mathrm{bp}}}$ in each neuron, which integrates error signals from higher layers. The process discretizes errors in the same fashion as the forward pass discretizes an input signal into a sequence of signed spike signals: $$\label{update_error}
U_i^l(t+\Delta t) = U_i^l(t) - {\ensuremath{\Theta_\mathrm{bp}}}z_i^l(t) + \sum_k w^{l+1}_{ki}\delta^{l+1}_k(t).$$ To this end, we introduce a ternary *error spike activation function* $z_i^{l}(t)\in \{-1,0,1\}$ which is defined in analogy to using the error integration variable $U_i^l(t)$ and the backpropagation threshold ${\ensuremath{\Theta_\mathrm{bp}}}$. The error is then obtained by gating this ternarized variable $z_i^{l}(t)$ with one of the surrogate activation function derivatives of the previous section (linear or ReLU): $$\label{ternarized error}
\delta_i^{l}(t) = z_i^{l}(t) S^{'l}_i(T).$$ This ternary spike signal is backpropagated through the weights to the lower layers and also applied in the update rule of the weight increment accumulator $\omega_{ij}^{l}$: $$\label{weight_update_final}
\omega_{ij}^{l}(t+\Delta t) = \omega_{ij}^{l}(t) -\delta_i^{l}(t)x_j^{l-1}(T),$$ which is triggered every time an error spike signal is backpropagated. The weight updates are accumulated during error propagation and are applied after propagation is finished to update each weight simultaneously. In this way, the backpropagation of errors and the weight update will, exactly as forward propagation, only involve additions and comparisons of floating point numbers.
The *SpikeGrad* algorithm can also be expressed in an event-based formulation, described in algorithms \[alg1\], \[alg2\] and \[alg3\]. This formulation is closer to how the algorithm would be implemented in an actual SNN hardware system.
#### Loss function and error scale
We use the cross entropy loss function in the final layer applied to the softmax of the total integrated signal $V^L_i(T)$ (no spikes are triggered in the top layer during inference). This requires more complex operations than accumulations, but is negligible if the number of classes is small. To make sure that sufficient error spikes are triggered in the top layer, and that error spikes arrive even in the lowest layer of the network, we apply a scaling factor $\alpha$ to the error values before transferring them to $U_i^L$. This scaling factor also implicitly sets the precision of the gradient, since a higher number of spikes means that a large range of values can be represented. To counteract the relative increase of the gradient scale, the learning rates have to be rescaled by a factor $\nicefrac{1}{\alpha}$.
#### Input encoding
As pointed out in [@Rueckauer:2017] and [@Wu:2018SpikingBPDeep], it is crucial to maintain the full precision of the input image to obtain good performances on complex standard benchmarks with SNNs. One possibility is to encode the input in a large number of spikes [@Sengupta:2019]. Another possibility, which has been shown to require a much lower number of spikes in the network, is to multiply the input values directly with the weights of the first layer (just like in a standard ANN). The drawback is that the first layer then requires multiplication operations. The additional cost of this procedure may however be negligible if all other layers can profit from spike-based computation. This problematic does not exist for stimuli which are natively encoded in spikes.
$V^l_i \gets V^l_i + s\cdot w^l_{ij}$ $s^l_i \gets s^l_i(V^l_i,x^l_i)$ $V^l_i \gets V^l_i - s^l_i\cdot {\ensuremath{\Theta_\mathrm{ff}}}$ $x^l_i \gets x^l_i + \eta s^l_i$
$U^l_i \gets U^l_i + \delta\cdot w^{l+1}_{ki}$ $z^l_i \gets z^l_i(U^l_i)$ $\delta^l_i \gets z^l_i \cdot S^{'l}_i$ $U^l_i \gets U^l_i - z^l_i\cdot {\ensuremath{\Theta_\mathrm{bp}}}$ $\omega^l_{ij} \gets \omega^l_{ij} - \delta^l_i\cdot x^{l-1}_j$
**init:** $\mathbf{V}\gets\mathbf{b}$, $\mathbf{U}\gets 0$, $\mathbf{x}\gets 0$, $\mathbf{\omega}\gets 0$ $\mathbf{S'} \gets \mathbf{S'}(\mathbf{V},\mathbf{x})$ $\mathbf{U^L} \gets \alpha \cdot \nicefrac{\partial\mathcal{L}}{\partial\,\mathrm{softmax}(\mathbf{V^L})}$ $\mathbf{w} \gets \mathbf{w} + \mathbf{\omega}$
Formulation of the equivalent ANN
=================================
The simulation of the temporal dynamics of spikes requires a large number of time steps or events if activations are large. It would therefore be extremely beneficial if we were able to map the SNN to an equivalent ANN that can be trained much faster on standard hardware. In this section, we demonstrate that it is possible to find such an ANN using the forward and backward propagation dynamics described in the previous section.
#### Spike discretization error
We start our analysis with equation . We reorder the terms and sum over the increments $\Delta V_i^l(t) = V_i^l(t+\Delta t) -V_i^l(t)$ every time the integration variable is changed either by a spike that arrives at time $t_j^s \in [0, T]$ via connection $j$, or by a spike that is triggered at time $t_i^s\in [0, T]$. With the initial conditions $V_i^l(0)=b^l_i$, $s_i^l(0)=0$, we obtain the final value $V_i^l(T)$: $$V_i^l(T) = \sum_{t_j^s,t_i^s}\Delta V_i^l = - {\ensuremath{\Theta_\mathrm{ff}}}\sum_{t_i^s} s_i^l(t_i^s) + \sum_j w^{l}_{ij}\sum_{t_j^s} s^{l-1}_j(t_j^s) + b^l_i$$ By defining the total transmitted output of a neuron as $S_i^{l} \coloneqq \sum_{t_i^s} s_i^l(t_i^s)$ we obtain: $$\label{total_activation}
\frac{1}{{\ensuremath{\Theta_\mathrm{ff}}}}V_i^l(T) = \mathbb{S}_i^{l} - S_i^{l},\quad \mathbb{S}_i^{l} \coloneqq \frac{1}{{\ensuremath{\Theta_\mathrm{ff}}}}\left(\sum_j w^{l}_{ij}S_j^{l-1} + b^l_i\right)$$ The same reasoning can be applied to backpropagation of the gradient. We define the summed responses over error spikes times $\tau_j^s \in [T+\Delta t,\mathcal{T}]$ as $Z_i^{l} \coloneqq \sum_{\tau_i^s} z_i^l(\tau_i^s)$ to obtain: $$\label{total_error}
\frac{1}{{\ensuremath{\Theta_\mathrm{bp}}}}U_i^l(\mathcal{T}) = \mathbb{Z}_i^{l} - Z_i^{l},\quad \mathbb{Z}_i^{l} \coloneqq \frac{1}{{\ensuremath{\Theta_\mathrm{bp}}}}\left(\sum_k w^{l+1}_{ki}E_k^{l+1}\right)$$ $$E_k^{l+1} = \sum_{\tau^s_k} \delta_k^{l+1}(\tau^s_k) = \sum_{\tau^s_k} S_k^{'l+1}(T) z_k^{l+1}(\tau^s_k) = S_k^{'l+1}(T) Z_k^{l+1}.$$ In both equation and , the terms $\mathbb{S}_i^{l}$ and $\mathbb{Z}_i^{l}$ are equivalent to the output of an ANN with signed integer inputs $S^{l-1}_j$ and $E^{l+1}_k$. The scaling factors $\nicefrac{1}{{\ensuremath{\Theta_\mathrm{ff}}}}$ and $\nicefrac{1}{{\ensuremath{\Theta_\mathrm{bp}}}}$ can be interpreted as a linear activation function in the case of the forward pass, and a gradient rescaling in the case of the backward pass. If gradients shall not be explicitly rescaled, backpropagation requires ${\ensuremath{\Theta_\mathrm{bp}}}={\ensuremath{\Theta_\mathrm{ff}}}$. The values of the residual integrations $\nicefrac{1}{{\ensuremath{\Theta_\mathrm{ff}}}}V_i^l(T)$ and $\nicefrac{1}{{\ensuremath{\Theta_\mathrm{bp}}}}U_i^l(\mathcal{T})$ therefore represent the *spike discretization error* $\mathrm{SDE}_\mathrm{ff} \coloneqq \mathbb{S}_i^{l} - S_i^{l}$ or $\mathrm{SDE}_\mathrm{ff} \coloneqq \mathbb{Z}_i^{l} - Z_i^{l}$ between the ANN outputs $\mathbb{S}_i^{l}$ and $\mathbb{Z}_i^{l}$ and the accumulated SNN outputs $S_i^{l}$ and $Z_i^{l}$. Since we know that $V_i^l(T)\in(-{\ensuremath{\Theta_\mathrm{ff}}},{\ensuremath{\Theta_\mathrm{ff}}})$ and $U_i^l(\mathcal{T})\in(-{\ensuremath{\Theta_\mathrm{bp}}},{\ensuremath{\Theta_\mathrm{bp}}})$, this gives bounds of $|\mathrm{SDE}_\mathrm{ff}| < 1$ and $|\mathrm{SDE}_\mathrm{bp}| < 1$.
So far we can only represent linear functions. We now consider an implementation where the ANN applies a ReLU activation function instead. The SDE in this case is: $$\label{total_activation_integer_ReLU}
\mathrm{SDE}^\mathrm{ReLU}_\mathrm{ff} \coloneqq \mathrm{ReLU}\left(\mathbb{S}^l_i\right) - S_i^{l}.$$ We can calculate the error by considering that forces the neuron in one of two regimes (note that $x_i^{l} > 0 \Leftrightarrow S_i^{l} > 0$): In one case, $S_i^{l} = 0,\,V_i^{l}(T) < {\ensuremath{\Theta_\mathrm{ff}}}$ (this includes $V_i^l(T) \leq -{\ensuremath{\Theta_\mathrm{ff}}}$). This implies $\mathbb{S}^l_i = \nicefrac{1}{{\ensuremath{\Theta_\mathrm{ff}}}}V_i^l(T)$ and therefore $|\mathrm{SDE}^\mathrm{ReLU}_\mathrm{ff}| < 1$ (or even $|\mathrm{SDE}^\mathrm{ReLU}_\mathrm{ff}| = 0$ if $V_i^{l}(T) \leq 0$). In the other case, $S_i^{l} > 0, \,V_i^{l}(t) \in (-{\ensuremath{\Theta_\mathrm{ff}}},{\ensuremath{\Theta_\mathrm{ff}}})$, where is equivalent to .
This equivalence motivates the choice of as a surrogate derivative for the SNN: the condition $(V^l_i(T) > 0 \;\mathrm{or}\;x_i^l(T)>0)$ can be seen to be equivalent to $\mathbb{S}^l_i(T) > 0$, which defines the derivative of a ReLU. Finally, for the total weight increment $\Delta w_{ij}^l$, it can be seen from and that: $$x_i^l(T) = \sum_{t_i^s}\Delta x_i^{l}(t_i^s) = \eta S_i^{l},\quad\Rightarrow \quad\Delta w_{ij}^l(\mathcal{T}) = \sum_{\tau^s_i} \Delta \omega_{ij}^{l}(\tau^s_i) = -\eta S_j^{l-1}E_i^l,$$ which is exactly the weight update formula of an ANN defined on the accumulated variables. We have therefore demonstrated that the SNN can be represented by an ANN by replacing recursively all $S$ and $Z$ by $\mathbb{S}$ and $\mathbb{Z}$ and applying the corresponding activation function directly on these variables. The error that will be caused by this substitution compared to using the accumulated variables $S$ and $Z$ of an SNN is described by the SDE. This ANN can now be used for training of the SNN on GPUs. The *SpikeGrad* algorithm formulated on the variables $s$, $z$, $\delta$ and $x$ represents the algorithm that would be implemented on a event-based *spiking* neural network hardware platform. We will now demonstrate how the SDE can be further reduced to obtain an ANN and SNN that are exactly equivalent.
#### Response equivalence
For a large number of spikes, the SDE may be negligible compared to the activation of the ANN. However, in a framework whose objective it is to minimize the number of spikes emitted by each neuron, this error can have a potentially large impact.
One option to reduce the error between the ANN and the SNN output is to constrain the ANN during training to integer values. One possibility is to round the ANN outputs: $$\label{total_activation_integer}
\mathbb{S}_i^{l,\mathrm{round}} \coloneqq \mathrm{round}[\mathbb{S}_i^{l}] = \mathrm{round}\left[{\frac{1}{{\ensuremath{\Theta_\mathrm{ff}}}}\left(\sum_j w^{l}_{ij}S_j^{l-1} + b^l_i\right)}\right],$$ The $\mathrm{round}$ function here rounds to the next integer value, with boundary cases rounded *away* from zero. This behavior can be implemented in the SNN by a modified spike activation function which is applied after the full stimulus has been propagated. To obtain the exact response as the ANN, we have to take into account the current value of $S_i^l$ and modify the threshold values: $$\label{spike_activation_residual}
s^{l,\mathrm{res}}_i\left(V_i^l(T), S_i^l\right) \coloneqq \begin{cases}
\; 1 & \mathrm{if}\;V^l_i(T) >\nicefrac{{\ensuremath{\Theta_\mathrm{ff}}}}{2}\;\mathrm{or}\;(S_i^l \geq 0,\;V^l_i(T) = \nicefrac{{\ensuremath{\Theta_\mathrm{ff}}}}{2})\\
\; -1 & \mathrm{if}\;V^l_i(T) <-\nicefrac{{\ensuremath{\Theta_\mathrm{ff}}}}{2}\;\mathrm{or}\;(S_i^l \leq 0,\;V^l_i(T) = -\nicefrac{{\ensuremath{\Theta_\mathrm{ff}}}}{2})\\
\; 0 & \mathrm{otherwise}
\end{cases}.$$ Because this spike activation function is applied only to the residual values, we call it the *residual spike activation function*. The function is applied to a layer after all spikes have been propagated with the standard spike activation function or . We start with the lowest layer and propagate all residual spikes to the higher layers, which use the standard activation function. We then proceed with setting the next layer to residual mode and propagate the residual spikes. This is continued until we arrive at the last layer of the network.
By considering all possible rounding scenarios, it can be seen that indeed implies: $$S_i^l + s^{l,\mathrm{res}}_i\left(V_i^l(T), S_i^l\right) = \mathrm{round}[S_i^l + \nicefrac{1}{{\ensuremath{\Theta_\mathrm{ff}}}} V_i^l(T)] = \mathrm{round}[\mathbb{S}_i^{l}].$$ The same principle can be applied to obtain integer-rounded error propagation: $$\label{total_error_integer}
\mathbb{Z}_i^{l,\mathrm{round}} \coloneqq \mathrm{round}\left[\mathbb{Z}_i^{l}\right] = \mathrm{round}\left[\frac{1}{{\ensuremath{\Theta_\mathrm{bp}}}}\left(\sum_k w^{l+1}_{ki}E_k^{l+1} \right)\right].$$ We have to apply the following modified spike activation function in the SNN after the full error has been propagated by the standard error spike activation function: $$\label{error_activation_residual}
z^{l,\mathrm{res}}_i\left(U_i^l(\mathcal{T},Z_i^l\right) \coloneqq \begin{cases}
\; 1 & \mathrm{if}\;U_i^l(\mathcal{T})) >\nicefrac{{\ensuremath{\Theta_\mathrm{bp}}}}{2}\;\mathrm{or}\;(Z_i^l \geq 0,\;U_i^l(\mathcal{T}) = \nicefrac{{\ensuremath{\Theta_\mathrm{bp}}}}{2})\\
\; -1 & \mathrm{if}\;U_i^l(\mathcal{T}) <-\nicefrac{{\ensuremath{\Theta_\mathrm{bp}}}}{2}\;\mathrm{or}\;(Z_i^l \leq 0,\;U_i^l(\mathcal{T}) = -\nicefrac{{\ensuremath{\Theta_\mathrm{bp}}}}{2})\\
\; 0 & \mathrm{otherwise}
\end{cases},$$ which implies: $$Z_i^l + z^{l,\mathrm{res}}_i\left(U_i^l(\mathcal{T}), Z_i^l\right) = \mathrm{round}[Z_i^l + \nicefrac{1}{{\ensuremath{\Theta_\mathrm{bp}}}}U_i^l(\mathcal{T})] = \mathrm{round}[\mathbb{Z}_i^{l}].$$ We have therefore shown that the SNN will after each propagation phase have exactly the same accumulated responses as the corresponding ANN. The same principle can be applied to obtain other forms of rounding (e.g. floor and ceil), if and are modified accordingly.
#### Computational complexity estimation
Note that we have only demonstrated the equivalence of the accumulated neurons responses. However, for each of the response values, there is a large number of possible combinations of $1$ and $-1$ values that lead to the same response. The computational complexity of the event-based algorithm depends therefore on the total number $n$ of these events. The best possible case is when the accumulated response value $S^l_i$ is represented by exactly $|S^l_i|$ spikes. In the worst case, a large number of additional redundant spikes is emitted which sum up to $0$. The maximal number of spikes in each layer is bounded by the largest possible integration value that can be obtained. This depends on the maximal weight value $w^l_\mathrm{max}$, the number of connections $N^l_\mathrm{in}$ and the number of spike events $n^{l-1}$ each connection receives, which is given by the maximal value of the previous layer (or the input in the first layer): $$n_\mathrm{min}^l = |S^l_i|, \quad n^l_\mathrm{max} = \left\lfloor\frac{1}{{\ensuremath{\Theta_\mathrm{ff}}}} N^l_\mathrm{in} w^l_\mathrm{max} n^{l-1}_\mathrm{max}\right\rfloor.$$ The same reasoning applies to backpropagation. Our experiments show that for input encodings where the input is provided in a continuous fashion, and weight values which are much smaller than the threshold value, the deviation from the best case scenario is rather small. This is because in this case the sub-threshold integration allows to average out the fluctuations in the signal. This way the firing rate stays rather close to its long term average and few redundant spikes are emitted. For the total number of spikes $n$ in the full network on the CIFAR10 test set, we obtain empirically $\nicefrac{n - n_\mathrm{min}}{n_\mathrm{min}} < 0.035$.
Experiments
===========
For all experiments, the means, errors and maximal values are calculated over 20 simulation runs.
#### Classification performance
Tables \[Compare\_performance\_MNIST\] and \[Compare\_performance\_cifar10\] compare the state-of-the-art results for SNNs on the MNIST and CIFAR10 datasets. It can be seen that in both cases our results are competitive with respect to the state-of-the-art results of other SNNs trained with high precision gradients. Compared to results using the same topology, our algorithm performs at least equivalently.
The final classification performance of the network as a function of the error scaling term $\alpha$ in the final layer can be seen in figure \[fig:bp-metrics-scaleVar\]. Previous work on low bitwidth gradients [@Zhou:2018DoReFa] found that gradients usually require a higher precision than both weights and activations. Our results also seem to indicate that a certain minimum number of error spikes is necessary to achieve convergence. This strongly depends on the depth of the network and if enough spikes are triggered to provide sufficient gradient signal in the bottom layers. For the CIFAR10 network, convergence becomes unstable for approximately $\alpha < 300$. If the number of operations is large enough for convergence, the required precision for the gradient does not seem to be extremely large. On the MNIST task, the difference in test performance between a gradient rescaled by a factor of 50 and a gradient rescaled by a factor of 100 becomes insignificant. In the CIFAR10 task, this is true for a rescaling by 400 or 500. Also the results obtained with the float precision gradients in tables \[Compare\_performance\_MNIST\] and \[Compare\_performance\_cifar10\] demonstrate the same performance, given the range of the error.
Architecture Method Rec. Rate (**max**\[mean$\pm$std\])
------------------------------------ -------------------------------- -------------------------------------
Wu et al. [@Wu:2018SpikingBP]\* Direct training float gradient $99.42\%$
Rueckauer et al. [@Rueckauer:2017] CNN converted to SNN $99.44\%$
Jin et al. [@Jin:2018]\* Direct Macro/Micro BP $99.49\%$
**This work**\* Direct float gradient $\mathbf{99.48}[99.36\pm 0.06]\%$
**This work**\* Direct spike gradient $\mathbf{99.52}[99.38\pm 0.06]\%$
: Comparison of different state-of-the-art spiking CNN architectures on MNIST. \* indicates that the same topology (28x28-15C5-P2-40C5-P2-300-10) was used. []{data-label="Compare_performance_MNIST"}
Architecture Method Rec. Rate (**max**\[mean$\pm$std\])
------------------------------------- ------------------------------------ -------------------------------------
Rueckauer et al. [@Rueckauer:2017] CNN converted SNN (with BatchNorm) $90.85\%$
Sengupta et al. [@Sengupta:2019] VGG-16 converted to SNN $91.55\%$
Wu et al. [@Wu:2018SpikingBPDeep]\* Float gradient (no NeuNorm) $89.32\%$
**This work**\* Direct float gradient $\mathbf{89.72}[89.38\pm 0.25]\%$
**This work**\* Direct spike gradient $\mathbf{89.99}[89.49\pm 0.28]\%$
: Comparison of different state-of-the-art spiking CNN architectures on CIFAR10. \* indicates that the same topology (32x32-128C3-256C3-P2-512C3-P2-1024C3-512C3-1024-512-10) was used. []{data-label="Compare_performance_cifar10"}
#### Sparsity in backpropagated gradient
To evaluate the potential efficiency of the spike coding scheme relative to an ANN, we use the metric of relative synaptic operations. A synaptic operation corresponds to a multiply-accumulate (MAC) in the case of an ANN, and a simple accumulation (ACC) in the case of an SNN. This metric allows us to compare networks based on their fundamental operation. The advantage of this metric is the fact that it does not depend on the exact implementation of the operations (for instance the number of bits used to represent each number). Since an ACC is however generally cheaper and easier to implement than a MAC, we can be sure that an SNN is more efficient in terms of its operations than the corresponding ANN if the number of ACCs is smaller than the number of MACs.
In figure \[fig:bp-metrics-scaleVar\] it can be seen that the number of operations (i.e. the number of spikes) decreases with increasing inference precision of the network. This is a result of the decrease of error in the classification layer, which leads to the emission of a smaller number of error spikes. Numbers were obtained with the integer activation of the equivalent ANN to keep simulation times tractable. As explained previously, the actual number of events and synaptic operations in an SNN may therefore slightly deviate from these numbers. Figure \[fig:operations\_layer\_bp\_forward\] demonstrates how the number of operations during the backpropagation phase is distributed in the layers of the network (float precision input layer and average pooling layers were omitted). While propagating deeper into the network, the relative number of operations decreases and the error becomes increasingly sparse. This tendency is consistent during the whole training process for different epochs.
[0.49]{} ![Number of relative synaptic operations during backpropagation for different error scaling factors $\alpha$ as a function of the epoch. Numbers are based on activation values of the equivalent ANN. Test performance with error is given for each $\alpha$.[]{data-label="fig:bp-metrics-scaleVar"}](bp-metrics-scaleVar-MNIST.pdf "fig:"){width="100.00000%"}
[0.49]{} ![Number of relative synaptic operations during backpropagation for different error scaling factors $\alpha$ as a function of the epoch. Numbers are based on activation values of the equivalent ANN. Test performance with error is given for each $\alpha$.[]{data-label="fig:bp-metrics-scaleVar"}](bp-metrics-scaleVar-CIFAR10.pdf "fig:"){width="100.00000%"}
[0.49]{} ![Number of relative synaptic operations during backpropagation in each layer (connections in direction of backpropagation) for different epochs. For MNIST $\alpha=100$, for CIFAR10 $\alpha=500$.[]{data-label="fig:operations_layer_bp_forward"}](operations_layer_bp_forward_MNIST.pdf "fig:"){width="100.00000%"}
[0.49]{} ![Number of relative synaptic operations during backpropagation in each layer (connections in direction of backpropagation) for different epochs. For MNIST $\alpha=100$, for CIFAR10 $\alpha=500$.[]{data-label="fig:operations_layer_bp_forward"}](operations_layer_bp_forward_CIFAR10.pdf "fig:"){width="100.00000%"}
Discussion and conclusion
=========================
Using spike-based propagation of the error gradient, we demonstrated that the paradigm of event-based information propagation can be translated to the backpropagation algorithm. We have not only shown that competitive inference performance can be achieved, but also that gradient propagation seems particularly suitable to leverage spike-based processing by exploiting high signal sparsity. For both forward and backward propagation, *SpikeGrad* requires a similar communication infrastructure between neurons, which simplifies a possible spiking hardware implementation. One restriction of our algorithm is the need for negative spikes, which could be problematic depending on the particular hardware implementation.
In particular the topology used for CIFAR10 classification is rather large for the given task. We decided to use the same topologies as the state-of-the-art to allow for better comparison. In an ANN implementation, it is generally undesirable to use a network with a large number of parameters, since it increases the need for memory and computation. The relatively large number of parameters may to a certain extent explain the very low number of relative synaptic operations we observed during backpropagation. In an SNN, a large number of parameters is however less problematic from a computational perspective, since only the neurons which are activated by input spikes will trigger computations. A large portion of the network will therefore remain inactive. It would still be interesting to investigate signal sparsity and performance of *SpikeGrad* in ANN topologies that were explicitly designed for minimal computation and memory requirements.
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---
abstract: 'We investigate the impact of a finite response time of Kerr nonlinearities over the onset of spontaneous oscillations (self-pulsing) occurring in a nanocavity. The complete characterization of the underlying Hopf bifurcation in the full parameter space allows us to show the existence of a critical value of the response time and to envisage different regimes of competition with bistability. The transition from a stable oscillatory state to chaos is found to occur only in cavities which are detuned far off-resonance, which turns out to be mutually exclusive with the region where the cavity can operate as a bistable switch.'
author:
- Andrea Armaroli
- Stefania Malaguti
- Gaetano Bellanca
- Alfredo de Rossi and Sylvain Combrié
- Stefano Trillo
title: 'Oscillatory dynamics in nano-cavities with non-instantaneous Kerr response'
---
INTRODUCTION
============
Self-pulsing (SP), the onset of spontaneous oscillations, is a universal feature of nonlinear structures with feedback. As long as passive systems are concerned SP has been investigated theoretically in settings ranging from isolated ring cavities [@Ikeda80; @Ikeda82; @Lugiato82] and parametric intracavity mixing [@SW83; @L88; @TH96] to Bragg gratings [@WC82; @P07] or grating-assisted backward frequency conversion schemes [@D95; @C05], and it is still a subject of active research [@Maes09; @GB11; @Malaguti11]. In particular, the dynamics of nonlinear passive cavities, whose study has been pioneered in the eighties [@Ikeda80; @Ikeda82; @Lugiato82], is extremely rich encompassing stable as well as chaotic SP which can compete with bistabilities and transverse effects [@LL87]. Historically, SP and chaos (the optical equivalent of strong turbulence) have been first analyzed by means of delay-differential models accounting for the round-trip delay at each passage, which can be large in ring cavities. An instability named after Ikeda occurs in this framework when the relaxation time of the nonlinear response is much shorter than the transit time [@Ikeda80] and has been tested experimentally [@expSP]. However SP and so-called weak turbulence occurs also in the opposite (say [*short cavity*]{}) limit, where the delay can be averaged out to end up with a differential mean-field model [@Ikeda82]. This regime becomes important nowadays where nano-cavities are employed for many modern photonic applications [@Vahala03], including bistability [@MIT02], demonstrated in photonic crystal (PhC) membranes which offer great flexibility of design as well as high nonlinear performances in semiconductors [@Barclay05; @Notomi05; @Uesugi06; @Weidner06; @Weidner07; @Combrie08; @deRossi09; @Yang07]. In these cavities, transverse effects are absent and their dimensions are so small that the response time of the medium can be much larger than the light transit time in the cavity, yet being comparable with the photon lifetime which is strongly enhanced on account of the large quality factor $Q$. SP in such nano-cavities has been recently predicted, owing to the free-carrier dispersion induced by two-photon absorption [@Malaguti11]. The main features of such mechanism is the existence of a critical value of the time relaxation constant $\tau$, as well as a wide region of the parameter space where stable (non-chaotic) SP can be potentially observed. In this paper we analyse the dynamics of a nano-resonator in the good-cavity limit when the underlying nonlinear mechanism is a Kerr-like nonlinearity with finite response time. Our analysis is based on the differential model proposed in Ref. [@Ikeda82]. In spite of the simplicity of such model, which makes it an ideal prototype for understanding the role of relaxation processes, a full characterization of SP and its competition with bistability in the full parameter space was never reported (to the best of our knowledge) after Ref. [@Ikeda82]. We propose such analysis, adopting a different normalization with respect to that employed in Ref. [@Ikeda82], better aimed at capturing the key role of the relaxation time. This is especially important nowadays in view of assessing how a given designed nano-cavity may be expected to behave by changing the characteristic relaxation time of the nonlinearity as a consequence of choosing different materials and/or adopting techniques for fine tuning their response time. We propose an analytical characterization of the SP instability and its competition with bistability in the full parameter space, pointing out the qualitative similarity with the features observed for a nonlinearity dominated by free-carrier dispersion [@Malaguti11]. Nonetheless, we further investigate also the destabilization mechanism of the oscillatory states initially described in Ref. [@Ikeda82], showing that the chaotic regime, being confined to far off-resonance cavities, is indeed mutually exclusive with bistable switching.
Model definition and linear stability analysis
==============================================
We start from the following dimensionless coupled-mode model that rules the temporal evolution of the normalized intra-cavity field $a(t)$ coupled to the frequency deviation $n(t)$, owing to the intensity-dependent refractive index change
\[eq:model\] $$\begin{aligned}
&\frac{{\mathrm{d}}a}{{\mathrm{d}}t} = \sqrt{P} + i (\delta + \chi n) a - a,\label{eq1}\\
\tau &\frac{{\mathrm{d}}n}{{\mathrm{d}}t} + n = |a|^2.\label{eq2}\end{aligned}$$
Noteworthy Eqs. describe a photonic crystal nano-cavity with high Q coupled to a line-defect waveguide [@Barclay05; @Uesugi06; @deRossi09]. They implicitly assume that the nonlinearity is dominated solely by the Kerr effect with relaxation time $\tau$, while other possible nonlinear contributions, e.g. two-photon absorption along with the free-carrier dispersion [@Barclay05; @Uesugi06; @deRossi09; @Malaguti11]), are neglected. Here $P$ is the normalized power injected in the cavity through coupling with the waveguide, and $|a|^2$ is the normalized intra-cavity energy, which can be easily rescaled into real-world units by comparison with widely used dimensional models (see, e.g., Ref. [@deRossi09]). It is worth emphasizing that a unit coefficient in front of the loss term in Eqs. implies that the time $t$ is measured in units of the inverse damping coefficient $1/\Gamma_0=2Q/\omega_0=2 t_c$, where $Q$, $\omega_0$, and $t_c$ stand for the overall quality factor, the resonant frequency, and the photon lifetime of the cavity, respectively. In these units, the two key (normalized) parameters are the detuning $\delta = (\omega_0 - \omega) / \Gamma_0$, and the time constant $\tau = \tau_p \Gamma_0$, where $\tau_p$ is the response time of the nonlinearity in real-world units, while $\chi=\pm 1$ accounts for the sign of the nonlinear Kerr coefficient. We point out that our model differs from [@Ikeda82], inasmuch as the time scale is referred to the cavity lifetime instead of the response time of the medium. Indeed Eqs. can be reduced to the model analyzed in Ref. [@Ikeda82] by means of the substitution $a, n, t \rightarrow a/\sqrt{\tau}, n/\tau, \tau t$. The effect of such transformation is however to rescale the detuning and the injected power in such a way that they become dependent on the response time of the medium, which is not suitable for our purpose of investigating the impact of the relaxation time on the dynamics of a given cavity with fixed characteristics.
For a cw driving $P=constant$, Eqs. have the following steady-state solution $a(t)=A$, $n(t)=N=|A|^2$, where $$\label{steady}
P = E \left[ (1 +(\delta +\chi E)^2 \right],$$ $E=|A|^2$ being the stationary intra-cavity energy. It is well known that bistability occurs for $\delta > \sqrt{3}$ when $\chi=-1$, and $\delta < -\sqrt{3}$ when $\chi=1$ [@LL87]. In the discussion below, we will focus on the latter case, where the cavity resonance is blue-shifted due to the nonlinearity, a case which is directly comparable with the net effect of free-carrier dispersion induced by two-photon absorption [@Malaguti11]. All the conclusions of this paper remain valid also for $\chi=-1$, provided $\delta \rightarrow -\delta$. The values of intracavity energies corresponding to the knees of the bistable response are $$\label{knees}
E_b^{\pm} = \frac{-2\chi \delta \pm \sqrt{ \delta^2-3}}{3},$$ and the corresponding input powers $P_b^{\pm}=P(E_b^{\pm})$ can be calculated by means of Eq. (\[steady\]).
The stability of the solution (\[steady\]) can be investigated by plugging into Eqs. the ansatz $a(t)=A+\delta a(t)$, $n(t)=N+\delta n(t)$, while retaining linear terms in the perturbations $\delta a, \delta n$.
The perturbation column array $\varepsilon \equiv (\delta a, \delta a^*, \delta n)^T$ is found to obey the following linearized equation
\[eq:lin\] $$\begin{aligned}
\frac{{\mathrm{d}}\varepsilon}{{\mathrm{d}}t} &=M \varepsilon; \label{lsa1}\\
M &=
\begin{pmatrix}
i\hat{\delta} -1 & 0 & i\chi A\\
0 & -i\hat{\delta} -1 & -i\chi A^* \\
A^*/\tau & A/\tau & -1/\tau
\end{pmatrix}, \label{lsa2}\end{aligned}$$
where $\hat{\delta} \equiv \delta + \chi E$.
The characteristic equation of $M$ reads as $$\lambda^3 + a_2 \lambda^2 + a_1 \lambda + a_0=0,$$ where the coefficients are $a_2=2 +\frac{1}{\tau}$, $a_1=\left( 1 + \hat{\delta}^2 + \frac{2}{\tau} \right)$, and $a_0 = \frac{1}{\tau} \left( 1 + \hat{\delta}^2 + 2 \chi E \hat{\delta} \right)$.
![(Color online) Steady-state response $E$ vs. $P$ for $\delta=-4$ (a-b-c) and $\delta=-10$ (d) and different $\tau$ ($\chi=1$). Stable and unstable branches are reported as solid and dotted lines, respectively. The blue and red curves superimposed on the right side show how the real part $Re(\lambda)$ and imaginary part $Im(\lambda)$ (of the dominant eigenvalue underlying the instability) change with $E$. Bifurcation points are highlighted over with the dashed line $Re(\lambda)=0$. The shaded regions labeled BI (light blue) and SP (light yellow) correspond to the negative slope branch of the bistable response (real eigenvalue) and SP instability (pair of conjugate eigenvalues with positive real part), respectively. Four different scenarios are shown: (a) SP occurs at $P=P_H^-$ values above the bistable knee value $P_b^+$, and is unbounded for increasing $P$; (b) SP occurs at $P=P_H^-$ values below the bistable knee $P_b^+$, still being unbounded; (c) as in (a), except SP occurs in a finite range below a given value $P=P_H^+$; (d) as in (b), except SP occurs in a finite range below a given value $P=P_H^+$. The shaded green regions yield the range of power where bistable up-switching to a stable state is permitted, while red ones identify the coexistence of a SP and an unstable saddle branch. []{data-label="fig:eigs"}](f1.eps){width="9cm"}
SP occurs when the system undergoes a Hopf bifurcation, i.e. a pair of complex conjugate eigenvalues $\lambda_R \pm i \lambda_I$ crosses into the right half complex plane, entailing an exponential growth of a pulsating perturbation with period $T=2\pi/ |\lambda_I |$. The bifurcation point ($\lambda_R=0$) corresponds to the constraint $a_1 a_2 = a_0$, which can be solved to yield the following explicit expression for the SP (Hopf) threshold values $E_H^{\pm}$ $$\label{eq:Hopf}
E_H^{\pm} = \frac{-\chi \delta \left( 2 - \frac{1}{\tau} \right) \pm
\sqrt{\frac{\delta^2}{\tau^2}-4 \left(1 + \frac{1}{\tau} \right)^2 \left(1 - \frac{1}{\tau}\right)}}{2\left(1 - \frac{1}{\tau} \right)},$$ and the corresponding injected power threshold $P_H^{\pm} = E_H^{\pm} \left[ (1 +(\delta +\chi E_H^{\pm} )^2 \right]$.
![(Color online) SP threshold energies $E_H^{\pm}$ as a function of $\tau$ for fixed detuning $\delta = -4$. The shaded area corresponds to the domain $E_H^{-} \le E \le E_H^{+}$ where SP occurs, which lies above the upper knee level of energy $E_b^+$ (red dashed line). []{data-label="fig:th"}](f2.eps){width="8cm"}
![(Color online) Color level plots of the (a) “on” $E_H^{-}$ and (b) “off” $E_H^{+}$ values of threshold energy for SP, in the parameter plane ($\tau, \delta$). Bistability occurs below the line $\delta =\delta_b$. The curve $P(E_H^-)=P(E_b^-)$ (black solid) divides the bistable region into a domain labeled BI+SP where SP sets in only for powers above the bistable knee for up-switching, and a domain labeled SP, where stable up-switching is not possible, being hampered by SP, which dominates the dynamics.[]{data-label="fig:map2D"}](f3.eps){width="45.00000%"}
The analysis reported above shows that the bistable response depends only on the detuning, while the time constant $\tau$ can affect qualitatively the onset of SP due to the Hopf bifurcation. In fact different scenarios are possible depending on the existence of one or both Hopf thresholds (in turn corresponding to the roots in Eq. ) being real), and whether such threshold occurs at powers above or below the bistable knee for up-switching. Four possible scenarios are displayed in Fig. \[fig:eigs\], where we show the bistable stationary response along with the unstable eigenvalues responsible for instabilities. First, Fig. \[fig:eigs\] shows the well-known fact that a purely real and positive eigenvalue of $M$ leads to instability of the steady state solution along the negative slope branch, which turns out to be a saddle point in phase-space. Conversely, SP is characterized by a pair of complex conjugate eigenvalues, and the relative threshold are highlighted by empty triangles. We find such threshold to occur always on the upper branch of the bistable response (or when the response is monotone, see below). However, depending on the value of $\tau$, the lower Hopf threshold $P(E_H^-)$ can take place above \[as in Fig. \[fig:eigs\](a,c)\] or below \[see Fig. \[fig:eigs\](b,d)\] the knee $P(E_b^-)$ which characterize the bistable up-switching. In the former case the cavity can exhibit bistable up-switching to a stable steady state, whereas in the latter case up-switching occurs inevitably towards a SP-unstable state: stabilization requires to move along the hysteresis cycle below the Hopf threshold. Moreover, also depending on the value of $\tau$, the upper branch can be indefinitely SP-unstable above the first threshold $P(E_H^-)$ \[see Fig. \[fig:eigs\](a,b)\], or, viceversa, exhibit a SP switch-off energy or secondary threshold beyond which SP ceases to take place \[see Fig. \[fig:eigs\](c,d); note that we change the detuning in Fig. \[fig:eigs\](d) only to make the picture clearer, though the same qualitative behavior occurs at $\delta=-4$\]. Indeed, if $\tau \geq 1$, Eq. may also admit a real solution $E_H^{+}$, and hence SP occurs only in a finite range of energies (and powers) $E_H^-<E<E_H^+$. When $\tau$ grows, this finite interval shrinks, and vanishes as a critical value $\tau=\tau_c$ is reached. For $\tau=\tau_c$, SP no longer takes place and the entire upper branch becomes stable. From Eq. , we find such critical value $\tau_c$ to be given by the positive real root of the cubic polynomial (its explicit expression is too cumbersome) $$\label{eq:tauc}
\tau_c^3 + \tau_c^2 - \left(1+\frac{\delta^2}{4} \right) \tau_c -1 = 0,
\,\,\tau_c\approx\frac{|\delta|}{2},\,\,|\delta|\gg1.$$
The behavior discussed above can be clearly seen by reporting the SP threshold energies $E_H^{\pm}$ versus $\tau$, at constant detuning. Such plot, displayed in Fig. \[fig:th\] for $\delta =-4$, shows a shaded region ($E_H^-<E<E_H^+$), which correspond to SP. We clearly see that no SP occurs for $\tau > \tau_c$. Conversely, decreasing $\tau$ below $\tau_c$ results into widening the portion of the upper branch that exhibits SP, until below $\tau=1$ the whole upper branch above the switch-on threshold $E_H^-$ turns out to be unstable. In this case, the SP switch-off energy $E_H^+$ diverges as the asymptote $\tau=1$ (dashed vertical line) is approached. Importantly for $\tau \rightarrow 0$ also the first threshold $E_H^{-}$ diverges, which means that a finite response time is a key ingredient for SP to be observable. This is consistent with the fact that Kerr instantaneous nonlinearities yield no SP at all. In fact, in this case, the eigenvalues are easily found to be $\lambda^{\pm}=-1 \pm \sqrt{E^2 - (\delta + 2\chi E)^2}$, which rule out the possibility to have a complex conjugate pair with positive real part.
In order to have a complete picture and further show how the onset of SP changes with detuning, we have drawn in Fig. \[fig:map2D\] a color map of the level curves of SP threshold energies $E_H^{\pm}$, in the parameter plane ($\tau, \delta$). A number of interesting observations can be drawn. The SP region is bounded by the border $\tau=\tau_c$, and in the bistable region ($\delta<\delta_b$) $\tau_c$ decreases with decreasing values of absolute detuning $|\delta|$. Interestingly enough, the scenario illustrated in Fig. \[fig:th\] remains valid also for detunings $\delta > \delta_b$, where bistability disappears. Finally in the region of positive detunings, we are left with the upper branch being fully unstable for all energies $E>E_H^-$. In this region, the threshold energy $E_H^-$ diverges, not only in the instantaneous limit $\tau=0$, but also for $\tau=1$, whereas relatively low values of $E_H^-$ are found for $\tau \sim 1/2$, i.e. when the response time of the nonlinearity is nearly equal to the photon lifetime. Furthermore, the bistable region is divided into two distinct domains by the (solid black) curve which arises from the condition $P(E_H^-)=P(E_b^-)$ (its explicit expression is too cumbersome). In the domain BI+SP above such curve (bounded from above also by the line $\delta=\delta_b$), one has that the cavity can work as a bistable switch since the upper branch right above the knee for up-switching is stable \[as in Fig. \[fig:eigs\](a,c)\], whereas in the domain labeled SP below the curve, up-switching is no longer allowed, since the upper branch above the knee is SP-unstable \[as shown in Fig. \[fig:eigs\](b,d)\] The reader can easily recognize a qualitative similarity of the picture discussed here with the dynamics of SP ruled by free carrier dispersion, recently discussed in Ref. [@Malaguti11]. Although a detailed analytic investigation of the stability of the SP-oscillating state (limit cycle) is beyond the scope of this paper, similarly to the case discussed in Ref. [@Malaguti11], our numerical simulations of Eqs. suggest that stable limit cycles, working as attractors from a large basin, exist in a wide domain of the parameter plane (witnessing the supercritical nature of the Hopf bifurcation). An example of such stable dynamics is shown in Fig. \[dynamics\].
[Let us comment on the observability of the SP dynamics. In nanocavities with high $Q$ ($Q=10^3-10^5$) the photon lifetime in the near infrared ranges from few picoseconds to tens of picoseconds. While the constraint to have a response time of the nonlinearity of the same order of magnitude is naturally met in semiconductors with nonlinear response dominated by free-carrier dispersion, the same constraint in the framework of the Kerr model rule out the possibility to observe SP dynamics in media with nonlinearities of electronic origin since they are too fast (fs range). Nevertheless the predictions of our Kerr model become interesting for Kerr-like materials with response time in the ps range such as, e.g., soft matter, metal films [@Conforti11], or more traditional liquids with reorientational nonlinearity, which are still the object of recent studies [@Fanjoux08; @Conti10]. In particular, for instance, highly nonlinear liquids such as nitrobenzene [@nitrobenzene] or CS$_2$ could be easily employed to fill a photonic crystal matrix (as also recently proposed for microstructured fibers [@Conti10]), while metal films could be employed in conjunction with dielectrics to form a single cavity or cavity arrays [@Sipe99]. Assuming, for instance, a response time $\tau_{phys} \sim 30$ ps [@nitrobenzene], which yield $\tau =(\tau_{phys}/2t_c)\sim 0.75$ in a cavity with $Q \sim 25000$ ($t_c \sim 20$ ps at $\lambda=1.55 \mu$m), assuming $n_{2I} \sim 10^{-17} m^2/W$ and a nonlinear modal volume $V=3 (\lambda/n)^3$, the threshold power $P=10$ in Fig. \[dynamics\] corresponds to a real-world power $P_{in}=(\gamma/\Gamma_0^2) P \sim 10$ mW in the waveguide coupled to the nanocavity, where $\gamma=\omega_0 n_{2I} c/(n_{eff}nV)$ is the overall nonlinear coefficient [@deRossi09]. Here we have assumed a refractive index $n \sim n_{eff} \sim 1.5$ and $Q$ to be essentially determined by the coupling itself. ]{}
Having characterized so far the threshold for SP and its competition with bistability, since Ikeda and Akimoto have shown that the limit cycles destabilize, leading eventually to chaos [@Ikeda82], in the next section we deepen this point with the aim of determining the domain of the parameter plane where the transition to chaos could be observed.
![(Color online) Dynamics of SP ruled by Eqs. (\[eq1\]-\[eq2\]), with $\delta = -4$, and $\tau =0.75$: (a) temporal evolution of the intra-cavity energy and carrier density corresponding to the rightmost blue circle in (b); (b) steady response with superimposed peak energy of the periodic oscillations (blue open circles); The red filled circle marks the Hopf bifurcation point. The black filled circle marks the SP-unstable steady state which gives rise to the dynamics shown in (a,c); (c) phase-space picture of the optical field showing the attracting limit cycle from two different initial conditions (open circles); []{data-label="dynamics"}](f4.eps){width="8.0cm"}
The turbulent regime
====================
In Ref. [@Ikeda82] Ikeda and Akimoto have studied the transition to chaos, identifying a period doubling cascade up to $2^2P$ (i.e. oscillation with period-four) at a fixed value of detuning. Here we report further details about the emergence of chaos in a wide domain of parameters. We employ different tools, ranging from Poicar[é]{} section and its corresponding bifurcation diagram to the calculation of Lyapunov exponents. Our principal purpose is to assess whether a nano-cavity described by the model can work as a reliable bistable switch, and hence whether the onset of chaos should be expected when the cavity operates progressively off-resonance, especially in the region labeled BI+SP in Fig. \[fig:map2D\].
![(Color online) Dynamical evolution ruled by Eqs. as the forcing term $\sqrt{P}$ varies adiabatically in time (dashed line). Here $\tau = 0.45$, and (a) $\delta=-4$; (b) $\delta=-10$. The vertical lines mark the time instants at which the input power cross the main bifurcation points: bistable knees $P_b^\pm$ (dotted blue and black, respectively) and Hopf threshold $P_H^-$ (dashed red). The insets zoom over characteristic time intervals, indicated by arrows. Notice that in (b) the two thresholds $P_b^\pm$ and $P_H^-$ almost overlap.[]{data-label="fig:Ebifurcation_tau045_deltam4"}](./f5a.eps "fig:"){width="23.50000%"} ![(Color online) Dynamical evolution ruled by Eqs. as the forcing term $\sqrt{P}$ varies adiabatically in time (dashed line). Here $\tau = 0.45$, and (a) $\delta=-4$; (b) $\delta=-10$. The vertical lines mark the time instants at which the input power cross the main bifurcation points: bistable knees $P_b^\pm$ (dotted blue and black, respectively) and Hopf threshold $P_H^-$ (dashed red). The insets zoom over characteristic time intervals, indicated by arrows. Notice that in (b) the two thresholds $P_b^\pm$ and $P_H^-$ almost overlap.[]{data-label="fig:Ebifurcation_tau045_deltam4"}](./f5b.eps "fig:"){width="23.50000%"}
![(Color online) As in Fig. \[fig:Ebifurcation\_tau045\_deltam4\], with $\delta=-15$. The chaotic region is highlighted in yellow.[]{data-label="fig:Ebifurcation_tau045_deltam15"}](./f6.eps){width="45.00000%"}
To begin with, it is instructive to report about the dynamics ruled by Eqs. when, starting above the threshold $P_H^-$, the input power $P$ is adiabatically decreased. In fact, this is the situation where the onset of chaos is expected according to Ref. [@Ikeda82]. We start at moderately low detuning ($\delta=-4$) and for $\tau=0.45$, which corresponds to SP being unbounded on the upper branch. As shown in Fig. \[fig:Ebifurcation\_tau045\_deltam4\](a), in this case, the Hopf bifurcation is clearly supercritical, since the system settles on a limit cycle, whose amplitude vanishes approaching the bifurcation point (approximately as $[E-E_{H}^-]^{1/2}$). However, at larger (in modulus) detunings, we observe a jump in $|a|^2$ as the limit cycle ($1P$) looses its stability and the system settles on a period-two ($2P$) oscillation, as shown in Fig. \[fig:Ebifurcation\_tau045\_deltam4\](b) for $\delta=-10$. The $2P$ solution does not visit anymore the simplest 1P limit cycle. Conversely it abruptly switches to the stable low-branch steady state. Remarkably this happens still above the Hopf bifurcation point $E_H^-$. A more complex sequence of period doubling bifurcations and chaotic motion are detected at higher detunings when the input power approaches the knee value $P_b^-$. In Fig. \[fig:Ebifurcation\_tau045\_deltam15\], where $\delta=-15$, oscillations with several different periods are evident. Moreover a chaotic regime appears, in the range $\sqrt{P} \approx17-20$ ($P\approx280-400$). In phase space this corresponds to the appearance of a strange attractor (not shown because its structure is already illustrated in Ref. [@Ikeda82]).
Since following simply the adiabatic dynamics could be possibly misleading (e.g. because of critical slowing down), as it neglects the rich variety of phenomena that occurs over the small scale, we have drawn also bifurcation diagrams calculated by collecting trajectory points on a Poincar[é]{} section for different powers. A typical example, using the same parameters as in Fig. \[fig:Ebifurcation\_tau045\_deltam15\], and defining the Poincar[é]{} section on the fixed phase $\angle{a}-\pi=0$ of the intra-cavity field, is reported in Fig. \[fig:bifdiagram\]. We can clearly identify a period doubling cascade (up to $2^3P$), as well as chaotic regimes. The onset of chaos follows a non-trivial scenario where narrow windows of period-three solutions ($3P$) are interspersed between two ranges of powers where the motion turns out to be chaotic. The vertical dashed lines in Fig. \[fig:bifdiagram\] mark indeed a $3P$ window. This is analogous to the Feigenbaum’s route to chaos and confirms the observation of chaos for $P\approx220-380$, already drawn above from Fig. \[fig:Ebifurcation\_tau045\_deltam15\].
![(Color online) Bifurcation diagram, $\tau=0.45,\,\,\delta=-15$. The vertical dashed lines highlight a $3P$ window and the collapse at small $P$. Inset shows a detail in which period doubling and periodic windows can be identified more clearly.[]{data-label="fig:bifdiagram"}](./f7.eps){width="45.00000%"}
The bifurcation diagram is computed up to $P\approx 150$ because lower power levels do not result into any limit cycle. Vice-versa the solution is observed to collapse toward the stable node represented by the lower branch of the bistable response. This phenomenon is independent from chaos, as mentioned above with reference to detuning $\delta = -10$. It occurs at large negative detunings for input powers above the SP threshold ($P>P_H^-$). This can be qualitatively explained by the coexistence of a stable fixed point (lower branch solution), a saddle (negative-slope branch), and an unstable limit cycle. As the $2P$ limit cycle spans the phase space with wide oscillations around the upper branch, it can approach the saddle point (which near the first bistable knee is closer in phase space to the center of the oscillations) being forced away, until eventually it can be captured by the lowest energy (stable) solution.
While the bifurcation map is a useful visual tool to characterize the onset to chaos and the full dynamics at fixed parameters, in order to explore in which region of the parameters one should expect to observe the chaotic dynamics, we have resorted to compute the dominant (maximal) Lyapunov exponent. We have explored a wide region of the parameter plane $(\tau,\delta)$, where, in each point of such plane, we have iterated over the values of power $P$ to find the largest exponent. We recall that a positive Lyapunov exponent (within numerical inaccuracies) entails that the system exhibits a chaotic behavior (here quasi-periodic motion is excluded by the dissipative character of our model ). From the map displayed in Fig. \[fig:Lyapunov\], we notice that chaotic motion manifests itself only when the cavity is detuned far off-resonance (i.e. at very large values of $|\delta|$), provided that the SP-unstable range is not finite, or in other words that the Hopf bifurcation is not bounded from above ($E_H^+\to \infty$) which requires $\tau<1$. Therefore we can draw the important conclusion that the region where the cavity could work as a bistable switch is mutually exclusive with chaos. Therefore the onset of chaos cannot spoil the behavior of the cavity as a switch, once the latter is used in the region labeled BI+SP in Fig. \[fig:map2D\]. [ We point out that, in terms of power, the observation of chaos is much more challenging than stable SP since power levels leading to the former turn out to be much larger than those leading to the latter; indeed at very large detuning, which corresponds to several times the cavity linewidth, bistability is observed at much higher power level (compare the horizontal axis scale in Fig. \[dynamics\] and Fig. \[fig:bifdiagram\]). ]{}
![(Color online) Color level map of maximal Lyapunov exponent in the parameter plane $(\tau,\delta)$. []{data-label="fig:Lyapunov"}](./f8.eps){width="45.00000%"}
As a final remark about the existence of $2^nP$ periodic solutions, we point out that they can be detected only when $P_H^-<P_b^-$ \[as in the examples shown in Fig. \[fig:eigs\](b,d)\]. As discussed with reference to Figs. \[fig:eigs\]-\[fig:map2D\], this may occur not only for both $\tau<1$ (as shown explicitly above), but also for $\tau>1$, where the Hopf bifurcation is bounded from above. More specifically $2P$ solutions are easily seen in a small subset of the region marked as SP in Fig. \[fig:map2D\]. In this case two unstable solutions, namely a repulsive (negative slope) branch and a SP branch, coexist. This seems to be a key ingredient for the limit cycles to loose their stability.
Conclusions
===========
In this work we have revisited the model that rules the behavior a passive [*small*]{} cavity with Kerr delayed response, pioneered in Ref. [@Ikeda82]. We have reported a full characterization of SP instabilities and their competition with bistability, outlining the existence of different possible scenarios. Importantly we have found a maximal critical value for the relaxation time that allows SP to occur, and have shown that SP can have two bifurcation points, while it can occur also in the absence of bistability. We have further characterized the destabilization mechanism of the limit cycle in the full parameter space, finding that chaos is mutually exclusive with the domain where the cavity can be employed as a bistable switching element. [In particular the chaotic regime predicted by Ikeda [@Ikeda82] in this system requires to be detuned strongly off-resonance, in turn implying the use of extremely high powers, thus making its observation rather challenging. Viceversa, in contrast with Kerr instantaneous nonlinearities, the observation of stable SP appear feasible in high-Q nanocavities filled with Kerr-like media with response time in the range of picoseconds. ]{} Future work will be devoted to study the effect of coupled cavity systems and the interplay of different nonlinear mechanisms. This work was supported by the European Commission, in the framework of the Copernicus project (no. 249012).
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title: 'Tunneling effect of the spin-2 Bose condensate driven by external magnetic fields '
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[2.5 ]{}
Introduction
============
Recent advance of experimental techniques on Bose-Einstein condensate (BEC) prompts us to closely and seriously look into theoretical possibilities which were mere imagination for theoreticians in this field. This is particularly true for spinor BEC where all hyperfine states of an atom Bose-condensed simultaneously, keeping these “spin” states degenerate and active. Recently, Barrett et al \[1\] have succeeded in cooling $Rb^{87}$ with the hyperfine state $F=1$ by all optical methods without resorting to a usual magnetic trap in which the internal degrees of freedom is frozen. Since the spin interaction of the $Rb^{87}$ atomic system is ferromagnetic, based on the refined calculation of the atomic interaction parameters by Klausen et al \[2\], we now obtain concrete examples of the three-component spinor BEC ($F=1$, $m_{F}=1,0,-1$) for both antiferromagnetic ($Na^{23}$) \[3\] and ferromagnetic interaction cases. In the present spinor BEC the degenerate internal degrees of freedom play an essential role to determine the fundamental physical properties. There is a rich variety of topological defect structures, which are already predicted in the earlier studies \[4,5\] on the spinor BEC. Law et al \[6\] constructed an excellent algebraic representation of the $F=1$ BEC Hamiltonian to study the exact many-body states, and found that spin-exchange interactions cause a set of collective dynamic behavior of BEC. Since the spinor BEC appears feasible by using the $F=2$ multiplet of bosons, it is necessary to investigate the ground-state structure and magnetic response of $F=2$ spinor BEC. Recently, Ciobanu et al \[7\] generalized the approach for the $F=1$ spinor BEC to study the ground state structure of the $F=2$ spinor BEC. They found that there are three possible phases in zero magnetic field, which are characterized by a pair of parameters describing the ferromagnetic order and the formation of singlet pairs. From current estimates of scattering lengths, they also found that the spinor BEC’s of $Rb^{87}$ and $Na^{23}$ have a polar ground state, whereas those of $Rb^{85}$ and $Rb^{83}$ are cyclic and ferromagnetic, respectively. Koashi et al \[8\] studied the exact eigenspectra and eigenstates of $F=2$ spinor BEC. They found that, compared to $F=1$ spinor BEC, the $F=2$ spinor BEC exhibits an even richer magnetic response due to quantum correlations among three bosons. Recently, Zhang et al \[9\] studied dynamic response of the $F=2$ spinor BEC under the influence of external magnetic fields, they found that when the frequency and the reduced amplitude of the longitudinal magnetic field are related in a specific manner, the population of the initial spin-0 state will be dynamically localized during time evolution. In this paper, we shall investigate the tunneling effect of the spin-2 Bose condensate driven by external magnetic fields.
Model
======
We consider the $F=2$ spinor BEC subject to a spatial weak uniform magnetic fields which consist of longitudinal and transverse components. Without loss of generality, the transverse direction of the field is chosen to be along the $x$ axis, i.e., $\hat{B}(t)=B_l(t)\hat{z}+B_x\hat{x}$. In such a case, the second-quantized Hamiltonian of the system is \[9\] $$H=H_0+H_B,$$ $$H_0=\frac{c_1}{2}\hat{F}\cdot\hat{F}+\frac{2c_2}{5}\hat{S}_+\hat{S}_-,$$ $$H_B=-\mu_Bg_fB_l(t)(\hat{a}_2^\dag\hat{a}_2+\hat{a}_1^\dag\hat{a}_1-
\hat{a}_{-1}^\dag\hat{a}_{-1}-\hat{a}_{-2}^\dag\hat{a}_{-2})$$ $$-\mu_Bg_fB_x(\hat{a}_2^\dag\hat{a}_1+\sqrt{\frac{3}{2}}\hat{a}_1^\dag\hat{a}_0+
\sqrt{\frac{3}{2}}\hat{a}_{0}^\dag\hat{a}_{-1}+\hat{a}_{-1}^\dag\hat{a}_{-2}+H.c.),$$ here, the $5\times5$ spin matrices $
\hat{F}_i=\hat{a}_\alpha^\dag(F_i)_{\alpha\beta}\hat{a}_\beta
(i=x, y, z),
\hat{S}_+=\hat{S}_-^\dag=(\hat{a}_0^\dag)^2/2-\hat{a}_1^\dag\hat{a}_{-1}^\dag+\hat{a}_2^\dag\hat{a}_{-2}^\dag$, and $c_i=(\bar{c_i}\int d\vec{r}|\phi|^4)$. $\bar{c_0}$, $\bar{c_1}$, and $\bar{c_2}$ are related to scattering lengths $a_0$, $a_2$, and $a_4$ of the two colliding bosons.
Similar to Ciobanu et al \[7\], mean-field approximation is used such that the field operators $\hat{a}_\alpha$ are replaced by $c$ numbers $a_\alpha=\sqrt{P_\alpha}e^{i\theta_\alpha}$, where $P_\alpha=N_\alpha/N$ is the population in spin $\alpha$, and $\theta_\alpha$ the phase of wave function $a_\alpha$. Furthermore, since this paper deals with the quantum coherent behavior of the system under the influence of the external magnetic fields, we assume that the initial spin state of the BEC is the eigenstate in the absence of external fields, then contribution from Hamiltonian (2) is a constant energy shift and can be neglected in the study of the dynamics. The semiclassical equations of motion in Heisenberg representation can be derived from the Hamiltonian $H_B$ (here, the state vector $a=(a_2,...,a_{-2})^T $ is introduced) $$i\dot{a}=H_{eff}(t)a,$$ where $$H_{eff}(t)=-b_l(t)F_{z}-b_xF_x.$$ Here $b_l(t)=\mu_Bg_fB_l(t)$ and $b_x=\mu_Bg_fB_x.$ In this paper, we consider the case that the system begins with unperturbed spin-0 state $a(0)=(0,0,1,0,0)^T$. If considering the case that the transverse magnetic field $b_x$ is weak, one can get the time evolution of the population of spin-$\alpha$ state as \[9\] $$P_\alpha(t)=|a_\alpha(t)|^2=|\sum_{\beta=-2}^{2}d_{\beta\alpha}^2(\pi/2)d_{\beta0}^2
(\pi/2)exp(-i\alpha\int_0^tb_l(\tau)d\tau)e^{i\beta\lambda}|^2,$$ where $$\lambda=\sqrt{[\int_0^tb_x\cos[\int_0^\tau
b_l(\tau)d\tau]d\tau]^2+[\int_0^tb_x\sin[\int_0^\tau
b_l(\tau)d\tau]d\tau]^2},$$ $$d_{\beta\alpha}^2(\theta)=<2\beta|e^{-i\theta
F_y}|2\alpha>
(\beta,\alpha=2,...,-2).$$
From figures 1-3, we find that the population transfers among spin-0 and spin-$\pm1$, spin-0 and spin-$\pm2$ exhibit the step structure under the external cosinusoidal magnetic field $b_l(t)=b\cos(\omega t)$ respectively, but there do not exist step structure among spin-$\pm1$ and spin-$\pm2$.
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Tunneling effect of the spin-2 Bose condensate driven by external magnetic fields
===================================================================================
In this section, we shall study the tunneling effect between two different spin components. The tunneling currents between spin-$\alpha$ and spin-$\beta$ $(\alpha\neq\beta,\alpha,\beta=-2,-1,0,1,2)$ can be defined by $$I_{\alpha\rightarrow\beta}(t)=\frac{d}{dt}(P_\alpha-P_\beta),$$ It is easy to get $$I_{0\rightarrow\pm1}(t)=(3-\frac{15}{2}\cos^2\lambda)\sin(2\lambda)\frac{d\lambda
}{dt},$$ $$I_{0\rightarrow\pm2}(t)=(\frac{3}{4}-\frac{15}{4}\cos^2\lambda)\sin(2\lambda)\frac{d\lambda
}{dt},$$ $$I_{\pm1\rightarrow\pm2}(t)=\frac{3}{8}(10\cos^2\lambda-6)\sin(2\lambda)\frac{d\lambda
}{dt},$$ where $\lambda$ is determined by Eq. (7).\
Figures 4-6 give the evolutions of the tunneling current with the external magnetic field $b_l(t)=b\cos(\omega t)$. The tunneling current among spin-$\pm1$ and spin-$\pm2$ may exhibit periodically oscillation behavior, but among spin-0 and spin-$\pm1$, spin-0 and spin-$\pm2$, the tunneling currents exhibit irregular oscillation behavior.
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Conclusions
===========
In this paper, we have studied tunneling effect of the spin-2 Bose condensate driven by external magnetic field $b_l(t)=b\cos(\omega
t)$. We find that the population transfers among spin-0 and spin-$\pm1$, spin-0 and spin-$\pm2$ exhibit the step structure under the external cosinusoidal magnetic field respectively, but there do not exist step structure among spin-$\pm1$ and spin-$\pm2$. The tunneling current among spin-$\pm1$ and spin-$\pm2$ may exhibit periodically oscillation behavior, but among spin-0 and spin-$\pm1$, spin-0 and spin-$\pm2$, the tunneling currents exhibit irregular oscillation behavior.
Acknowledgments
===============
This work was supported by the NSF of Shandong Province.
[150]{}
M. Barrett et al., Phys. Rev. Lett. [**87**]{}(2001)010404. N. N. Klausen et al.,Phys.Rev. A[**64**]{}(2001)053602. J. Stenger et al., Nature [**369**]{}(1998)345. T. Ohmi and K.Machida, J.Phys. soc. Jpn. [**67**]{}(1998)1822. T.-L. Ho, Phys. Rev. Lett. [**81**]{}(1998)742. C. K. Law, H. Pu, and N. P. Bigelow, Phys. Rev. Lett. [**81**]{}(1998)5257. C. V. Ciobanu, S.-K. Yip, and T.-L. Ho, Phys. Rev. A [**61**]{}(2000)033607. M. Koashi, and M. Ueda, Phys. Rev. Lett. [**84**]{}(2000)1066. P. Zhang, A. Z. Zhang, S. Q. Duan, X. G. Zhao., Phys. Rev. A[**66**]{}(2002)043606.
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abstract: 'Urbanization is a common phenomenon in developing countries and it poses serious challenges when not managed effectively. Lack of proper planning and management may cause the encroachment of urban fabrics into reserved or special regions which in turn can lead to an unsustainable increase in population. Ineffective management and planning generally leads to depreciated standard of living, where physical hazards like traffic accidents and disease vector breeding become prevalent. In order to support urban planners and policy makers in effective planning and accurate decision making, we investigate urban land-use in sub-Saharan Africa. Land-use dynamics serves as a crucial parameter in current strategies and policies for natural resource management and monitoring. Focusing on Nairobi, we use an efficient deep learning approach with patch-based prediction to classify regions based on land-use from 2004 to 2018 on a quarterly basis. We estimate changes in land-use within this period, and using the Autoregressive Integrated Moving Average (ARIMA) model, our results forecast land-use for a given future date. Furthermore, we provide labelled land-use maps which will be helpful to urban planners.'
---
Introduction {#submission}
============
There have been rapid infrastructural developments in major cities in sub-Saharan Africa (especially in Nairobi) in terms of significant increase in residential houses, factories, and office buildings. These developments have led to a large increase in population and, in turn, to overcrowding. Africa’s population is projected to almost triple by the year 2050 [@donnelly]. This increase will occur primarily in urban areas so that by 2025 about 800 million people will live in urban communities [@donnelly].
While urban development was generally believed to improve living lifestyle, and reduce risk of disease vector breeding, rapid increase in population, if not properly managed, usually presents series of challenges. These challenges include reduced standard of living, poor housing, inadequate environmental sanitation facilities which increase vector breeding and human contact, and road traffic congestion which may lead to road accidents.
In order to inform the decisions of investors and policy makers in Nairobi on urbanization, we study urbanization in Nairobi by analyzing historical satellite imagery of certain regions in Nairobi and forecast future land-use in these regions. We provide labelled land-use maps that will be helpful in urban planning.
The approach involves using a deep Convolutional Neural Network (CNN) classifier to do patch-based land-use predictions in order to classify land-use into the following ten classes: commercial area, dense residential, medium residential, spatial residential, parking lot, freeway, round-about, meadow, chaparral, and open space(bareland, wetland). We cannot apply object-based classification, for instance detecting each type of plant as in [@daniel] to determine a vegetation class. This consumes more compute resources and takes more time.
We detect changes in each land-use class (in square kilometers) and show changes visually through image differencing. We also compute percentage change per land-use class for each successive (temporal) pair of images. Using Autoregressive Integrated Moving Average (ARIMA) [@arima], we forecast land-use for future dates.
Importance and Prior Work
=========================
In this section, we briefly review previous approaches that used various deep learning techniques in land-use classification and we examine possible training datasets for land-use classification.
Land-Use Classification
-----------------------
Efforts have been made over the years in developing various methods for the task of remote sensing image scene classification. Image scene classification is concerned with categorizing scene images into a discrete sets of meaningful Land-Use and Land Cover (LULC) classes according to the image contents [@cheng]. This has the potential of helping natural hazards detection and disaster recovery processes [@martha; @cheng2; @stumpf], LULC determination [@wu2], geospatial object detection [@cheng3], geographic image retrieval [@wang; @liY], vegetation mapping [@kim; @li], environment monitoring, and urban planning.
Albert [@albert] used computer vision techniques based on deep CNNs for the classification of satellite imagery in order to identify patterns in urban environments using large-scale satellite imagery data. Ground truth land-use class labels were sampled from the Urban Atlas land classification dataset acquired through open-source surveys. The urban atlas dataset comprises of 20 land-use classes from about 300 European cities. This data was used to train and compare deep architectures with ResNet-50 [@he2] exhibiting the best performance. ResNet-50 is a residual learning framework which is substantially deeper than previous networks. Unfortunately, models trained with the Urban Atlas dataset are unlikely to generalize well to other part of the world since training samples were acquired from only European cities. Consequently, the training dataset must be carefully selected to ensure that it can be applied successfully in sub-sahara Africa.
Training Dataset
----------------
Data is required for training classification networks. We require an urban land-use dataset that is representative of the target geography, that has a sufficient geographical coverage, that has rich variation of image data, and that possesses high within class diversity and between class similarity.
### DeepSat Dataset
The DeepSat [@basu] dataset was released in 2015 and it contains two benchmarks which include the Sat-4 data of 500,000 images over 4 land use classes (barren land, trees, grassland, other), and the Sat-6 data of 405,000 images over 6 land use classes (barren land, trees, grassland, roads, buildings, water bodies). All the samples have a size of $28 \times 28$ pixels at a 1m per pixel spatial resolution with 4 channels (red, green, blue, and NIR - near infrared). CNN models trained on this dataset have perform well during classification. While useful as input for pre-training more complex models, (e.g., image segmentation), the dataset lack diverse land-use classes, hence, not suitable for detailed land use analysis and comparison of urban environments across cities.
### NWPU-RESISC45 Dataset
The North-West Polytechnic University (NWPU-RESISC45) dataset [@cheng] consists of 31,500 remote sensing images divided into 45 scene classes. Each class is made up of 700 images with a size of $256 \times 256$ pixels in the Red Green Blue (RGB) color space. The spatial resolution of the images varies from about 30 m to 0.2 m per pixel except for the images that belong to the island, lake, mountain, and snowpack class. The 31,500 images cover more than 100 countries and regions all over the world, including developing, transition, and highly developed economies. In contrast to other existing datasets such as UC Merced Land-Use dataset, WHU-RS19 dataset [@sheng], SIRI-WHU dataset [@zhao3], RSSCN7 dataset [@zou], RSC11 dataset [@zhao2], and DeepSat dataset [@basu], the NWPU-RESISC45 dataset is large scale, it possesses rich image variation, and it has high within class diversity and between class similarity. This dataset therefore provides the best coverage of the required characteristics.
Data Collection
---------------
Nairobi is one the African countries with high urbanization rate. In fact, Nairobi was ranked among the top 20 most dynamic cities in the world in the 2018 JLL’s Global Real Estate Transparency Index, it was the only city in Africa considered within this position. This is why have chosen Kilimani region of Nairobi as our region of interest in this research. In order to forecast urban land-use, some data about the region of interest is required. To obtain land-use dataset of Kilimani, we reviewed 3 aerial imagery sources, which include LandSat8 satellite, Sentinel-2 satellite, and Google Earth Software. We chose Google Earth’s software as our source of aerial images because it possesses free high resolutution images and it also has rich collection of aerial images from 2014 to 2019. We retrieved images of Kilimani region of interest from 2004 to 2018 on a quarterly basis. This amounted to 60 aerial images.
Data Pre-Processing
-------------------
Several factors affect the quality of satellite imagery. Some of these factors include cloud cover in the area of interest at the time of capture, inconsistent gaps in image time-series dataset, and low hardware capacity. All these factors can cause acquisition of low quality images. Pre-processing dataset to remove clouds and interpolating dataset to fill gaps is important for deep learning tasks. We removed images totally covered by cloud, from our dataset and we interpolated to make up for missing dates.
Unsuitability of Google Earth’s Images for Direct Training
----------------------------------------------------------
The nature of the images from Google Earth’s desktop software dismisses the idea of directly training a simple deep learning model on them. This is because they are of high spatial and pixel resolutions. Furthermore, they do not contain labels for training, thus, we examined 8 possible training datasets for land-use classification using the metrics in Section 2.2.2 for best fit. The North West Polytechnic University land-use (NWPU-RESISC45) dataset [@cheng]) was the best fit for the classification tasks at hand. We hypothesized that a model trained on the NWPU-RESISC45 dataset should be able to predict land-use on the Google Earth’s dataset using our proposed patch-based prediction technique, because of the similarities in the land-use classes in both datasets.
Suitability of NWPU-RESISC45 Dataset
------------------------------------
We discuss the suitability of the NWPU-RESISC45 dataset for training a model for predicting aerial images of Kilimani retrieved from Google Earth’s software. The following points will be considered: dataset geographical distribution, dataset size, image variation, high within class diversity and between class similarity, and the dataset distribution.
#### Dataset Geographical Distribution
The region under consideration is Kiliimani in Kenya. Kenya happens to be a developing country, therefore it is important that our dataset for training the classification network contains samples from developing region in order to get obtain a good classification accuracy. NWPU-RESISC45 dataset contains 31,500 images from more than 100 countries and regions all over the world, including developing, transition, and highly developed economFies. This makes the dataset representative.
#### Dataset Size
A contributing factor to the good performance of deep learning algorithms is the presence of large-scale dataset. NWPU-RESISC45 dataset is 15 times larger than the well popular UC Merced land-use dataset, it has about the largest scale on the number of scene classes. Also, each image has a dimension of 256 x 256 pixels, this makes provision for more information that will enhance the learning process of a deep CNN network.
#### Image Variation
Tolerance to image variations is an important and well desired property of any scene classification system, be it human or machine [@cheng]. Unfortunately, many of the existing dataset are not rich with varieties of samples for same class. In contrast, the NWPU-RESISC45 images were carefully selected under all kinds of weathers, seasons, illumination conditions, imaging conditions, and scales [@cheng]. Thus, for each scene class, the dataset possesses richer variations in translation, viewpoint, object pose and appearance, illumination, background, spatial resolution, and occlusion.
#### High within Class Diversity and between Class Similarity
due to low class diversity and low between class similarity in dataset, some top-performing methods built upon deep neural networks have achieved saturation of classification accuracy. In contrast, NWPU-RESISC45 dataset contains high within class diversity and between class similarity making it good for land-use classification tasks [@cheng].
#### Data Distribution
The 31,500 image dataset contains 45 classes with 700 images in each class. We selected 10 classes out if the 45 classes based on the common important land-use classes obtainable in developing regions. The selection of only 10 classes also helped to avoid class overlaps. The selected 10 land-use classes are: commercial area, dense residential, medium residential, spatial residential, parking lot, freeway, road-junction, meadow, chaparral and open-space(dryland, wetland or vegetation). Thus, our dataset contains 7000 images $(700 \times 10)$.
Patch-Based Prediction
======================
We propose a patch-based prediction technique for classification and segmentation. A patch in this case is a rectangular super-pixel from an image. For a given patch, we apply a classification network, we obtain the maximum probability class, and color the tile according to that class; this results into a segmentation map. Each patch has the pixel dimension of the training images ($256 \times 256 $ pixel in our case). A patch is selected by convolving each image at a fixed stride. The stride is the number of pixels by which we shift at each instance to select a new patch from an image. The smoothness of the segmentation map depends on the value of the stride value used (the smaller the better). We state the patch-based prediction algorithm in Algorithm \[alg:example\]. In convolving, we start from the pixel at co-ordinate $(0,0)$ and select a patch based on a specified dimension, predict the class of the selected patch, color the pixels (based on its class) in the corresponding location of this patch in a copy of the original image.
$test\_img$, $empty\_matrix$, $size$, $stride$ Initialize $A \Leftarrow test\_image$ Initialize $E \Leftarrow empty\_matrix$ $patch \Leftarrow A[x$ [**to**]{} $ x+size-1][y$ [**to**]{} $y+size-1][0$ [**to**]{} $2]$ $preprocess\_and\_reshape(patch)$ $class \Leftarrow predict(patch)$ $color(E[x$ [**to**]{} $x+size-1][y$ [**to**]{} $y+size-1][0$ [**to**]{} $2], class\_color)$
Aerial images retrived from Google Earth’s software are of high pixel and spatial resolution. Performing semantic segmentation on high resolution images requires masked images as labels during training. Creating mask labels for land-use images is expensive and takes much time. In fact, such labelled dataset are not easily accessible and do not even exist for some developing countries. Also, model architectures for semantic segmentation may be complex and may therefore take more training time with high resolution images ($3200 \times 4800$ pixels in our case). Patch-based prediction exhibits better efficiency in this case. We therefore train a simple deep CNN network with a regular land-use dataset (NWPU-RESISC45 dataset with $256 \times 256 $ pixels images, and text labels rather than masked image labels) and perform patch-based prediction on the high resolution images after training, to obtain segmentation maps.
Implementation of the Classification System
============================================
In this section, we discuss the chosen performance benchmark, model architecture design and the training procedure.
Benchmark Model
---------------
In order to provide a performance benchmark for the intended classification model, we require a benchmark model performance which will inform us of the relative performance of our model. We use VGGNet-16 performance recorded in [@cheng] as a benchmark. Cheng et. al [@cheng] evaluated few pre-trained deep CNN models such as AlexNet, VGGNet-16, and GoogLeNet on the NWPU-RESISC45 dataset after fine-tuning the models. The authors used 20% of the entire dataset to train these networks. It turns out that VGGNet-16 (one of the top performing image pre-trained classification model on imagenet) produced the highest overall accuracy of 90.36%. The overall accuracy is defined as the number of correctly classified samples, regardless of which class they belong to, divided by the total number of samples. We expect our model to reach or surpass this performance.
Model Architecture
------------------
We used two approaches: transfer learning with ResNet-50 and a convolutional neural network designed from scratch. In the transfer learning approach, we performed fine-tuning by removing the top layer of ResNet-50 network (i.e the fully connected and the classification layer), then added GlobalAveragePooling layer, Dense layer, a 50% drop out, another dense layer, another 50% drop out, and finally, a fully connected layer with softmax. Although, the network’s training accuracy was 93%, the model validation and test accuracy were less than 50%. This motivated us to design a new architecture which adapts and improves on the architecture from [@daniel]. This neural network architecture possesses 6 convolution layers. Output from each convolution layer is passed through a rectified linear unit (ReLU). The first and the second convolution layers possess 64 filters, the third and the fourth has 128 filters while the fifth and the sixth convolution layer has 256 filters. Each convolution layer has zero padding. We have a max pooling layer after each pair of convolution layer with 10% dropout for dimensionality reduction and over-fitting reduction respectively. At the end of the six convolution layers are 3 fully connected layers. The last fully connected layer has a softmax activation function which outputs probability distribution for each of the 10 classes.
Model Training and Performance
------------------------------
We used Adam optimizer, weighted categorical cross-entropy loss function, batch size of 20 for each step, and an epoch of 50. Training took about 1 hour, 45 minutes on a NVIDIA Tesla P100 Workstation GPU with 16G RAM. The performance of the model is discussed in Section 6.
Image Segmentation through Patch-Based Prediction
-------------------------------------------------
Each satellite image has a dimension of 3200 x 4800 pixels. This can be too large for directly training a CNN network and will therefore require very extensive computational resources and more training time. In addition, there exist no labelled data from Nairobi region for training patches of this image. We therefore pre-processed the dataset by removing cloudy images and performing linear interpolation to fill gaps, and then applied our patch-based prediction technique on each image using our trained model. The final output is an image, segmented on land-use basis. The visual smoothness of the output image depends on the stride value chosen. We chose a stride value of 32 in this paper (see Figure \[seg\_map\]), as a balance between acceptable visual quality and acceptable computational time.
Post Classification
===================
In this section, we explain our approach to land-use change computation, image differencing, build-up index (BUI) formulation, and finally forecasting.
Land-Use Square Area Change
---------------------------
The source of the Google Earth’s images for the region we considered is Digital Globe’s QuickBird satellite. QuickBird satellite captures the earth’s surface at high resolution of 0.65 meters. This information provides a means for translating our pixel values to the actual ground values. Thus, we simply convert to actual ground values by multiplying by $(0.65 \times 0.65)m^2$. $$AC = PC * (0.65 \times 0.65)m^2$$ where AC is actual change value and PC is pixel change value.
Percentage Change
-----------------
We compute the percentage change for the segmentation maps using the formulation below: $$PC = \frac{C_t - C_{t-1}}{ C_{t-1}} \times 100$$ $PC$ is percentage change, $C_t$ is change on the image at time $t$, and $C_{t-1}$ is change on the image preceding the first (image at time $t$).
Image Differencing
------------------
To determine visual changes between segmented maps, we use image differencing technique. This involves computing the difference between two images by finding the difference between each pixel in each image and generating an image based on the result.
Built-Up Index (BUI)
--------------------
In order to see the degree of land spaces being used up through the construction of buildings and other infrastructures, we compute BUI for each image. We do this by finding the ratio of the sum of land-use classes with buildings or roads to the sum of the entire land-use classes. $$BUI = \frac{\sum{CWB}}{\sum{C}}$$ where CWB is land-use classes with buildings and roads. They include road junction/roundabout, commercial area, dense residential, freeway, parking lot, sparse residential, and medium residential. C is the entire land-use classes.
Land-Use Forecasting
--------------------
We implemented an ARIMA model, we set parameters as follows: lag value of 4 (since we got data on quarterly basis) for autoregression, difference order of 1 to make the time series stationary, and then a moving average value of 0. Parameters were chosen after studying the nature of the series using auto-correlation plots.
ARIMA is a popular and widely used statistical method for time series forecasting. It is capable of capturing different standard temporal structures in time series data. Due to the limited data points, we fitted the ARIMA model on 60 images and forecast till Dec. 2025; See model performance in Figure \[forecast\].
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![Image difference map between March 2004 and December 2018, the black pixel indicates no change.[]{data-label="fig:image_diff"}](images/image_diff.jpg){width="9cm" height="6cm"}
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![Forecast for Commercial Area and Road Junctions, with evidences of seasonality.[]{data-label="forecast"}](images/com_road.jpg){width="8cm" height="5cm"}
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![Forecast for few other classes.[]{data-label="other"}](images/other.jpg){width="8cm" height="5cm"}
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Table \[tab:table1\] and Table \[tab:table2\] give the change values and forecast values respectively.
[ |p[5cm]{}||p[2cm]{}|p[2cm]{}|p[2cm]{}|p[2cm]{}|]{}\
Land-Use& Dec. 2011 Area $(km^2)$ & Dec. 2018 Area $(km^2)$& Change in Area $(km^2)$ & Change %\
-Chaparrel & 0.000 &0.000& 0.000& 0.000\
-Commercial Area & 3.811& 3.063& -0.748& -19.645\
-Dense Residential & 0.064& 0.139& 0.075& 116.779\
-Freeway &0.006& 0.088& 0.082& 1356.867\
-Open Space (bare land, wetland, plantations)&2.308& 0.755& -1.552& -67.254\
-Meadow& 0.000 & 0.005 &0.005& $\infty$\
-Medium Residential& 0.000 & 0.350 &0.350& $\infty$\
-Parking lot & 0.073& 0.023& -0.050& -68.236\
-Road Junctions &0.117& 1.930& 1.812& 1540.102\
-Sparse Residential &0.117& 0.025& -0.091& -77.938\
[ |p[5cm]{}||p[2cm]{}|p[2cm]{}|p[2cm]{}|p[2cm]{}|]{}\
Land-Use& Dec. 2018 Area $(km^2)$ & Dec. 2025 Area $(km^2)$& Change in Area $(km^2)$ & Change %\
-Chaparrel & 0.000 &0.000& 0.000& 0.000\
-Commercial Area & 3.063& 3.175& 0.112& 3.675\
-Dense Residential &0.139& 0.076& -0.063& -45.394\
-Freeway &0.088& 0.184& 0.096& 109.600\
-Open Space (bare land, wetland, plantations)& 0.755& 0.374& -0.381& -50.445\
-Meadow& 0.005 & 0.018& 0.012& 221.167\
-Medium Residential& 0.350& 0.446& 0.096& 27.604\
-Parking lot & 0.023& 0.124& 0.101& 434.633\
-Road junction &1.930& 2.131& 0.201& 10.424\
-Sparse Residential &0.025& 0.045& 0.019& 76.329\
Results and Discussion
======================
We evaluated our model on the NWPU-RESISC45 test dataset which is 15% of the original dataset and contains 1050 samples. The accuracy of the model on the test dataset was 91.03% with overall average AUC ROC score of 0.99 and an F1 score of 0.90. This performance is an improvement over the chosen benchmark which employed transfer learning approach using VGGNet-16 and achieved a test accuracy of 90.36%; See Figure \[confusion\].
The output of our patch-based prediction, which are labelled segmentation maps of land-use, is visually smooth, and provides qualitative evidence of the correctness of the results based on visual inspection; See Figure \[seg\_map\]. Thus, our hypothesis that a model trained on NWPU-RESISC45 dataset would perform well on Google Earth’s image patches for regions with similar land-use classes with NWPU-RESISC45 dataset is valid. The visual smoothness of the output images depend on the stride value chosen. For instance, patch-based prediction with stride value of 8 produces better segmentation map in terms of visual smoothness, when compared with segmentation map generated using stride values of 32 or 256.
Figure \[fig:image\_diff\] shows the image differencing output, it highlights regions that have changed and those that have remained unchanged over the years. The black regions indicate regions that did not change, while other colours show that changes occurred. This can help expose rapidly changing environments and stable environments. Figure \[forecast\] and Figure \[other\] show the forecast plots for land-use. Visual inspection of these graphs shows evidences of seasonality in some of the land-use classes.
Table \[tab:table1\] compares land-use in 2011 and 2018. The table shows that commercial area’s size reduced, this may be because of the increase in dense residential regions, some commercial buildings must have been modified and now converted to residential houses. It could also be that as the population increased, people built residential houses in the commercial environment. BUI increased from within 2011 to 2018. Table \[tab:table2\] shows the predicted values for 2025 December. In this case, commercial area and road junctions/round-abouts are predicted to increase. The predicted BUI for 2025 also indicates an increase. This shows an evidence of infrastructural development.
Conclusion and Future Work
==========================
An efficient machine learning technique for large-scale urban land-use classification has potential to help policy makers and urban planners make decisions. In this paper, we presented a new efficient deep convolutional neural network for urban land-use classification, which applies patch-based prediction to classify satellite images and generate segmentation maps. The NWPU-RESISC45 dataset was used to train the classification network. This network was used to do a patch-based prediction on a dataset that contained 60 satellite images of Kilimani area of Nairobi, acquired from 2004 to 2018 on a quarterly basis. Our model exceeds the performance of the benchmark model by 1%, but using a significantly simpler CNN which can be trained much more quickly where transfer learning is not suitable. We detected changes in land-use over the period and also computed build-up index (BUI) for each aerial image scene. Forecasting was done using ARIMA model fitted on the 60 data points. Forecast result shows evidences of seasonality. Since developing countries are well represented in our training dataset, we conclude that our model can be applied in any region in sub-Saharan Africa.
As one of the strengths of this research, our approach saves much time when an optimal value of stride is chosen during patch-based prediction. This is because the we applied the patch-based prediction approach with a simple deep CNN model. This approach is novel for land-use forecasting in sub-Saharan Africa and it can even be extended to some other developing countries in the world. One of the limitations of this work is that when a very small value of stride (for instance 1) is used during the prediction of a very large aerial image, much time is consumed. Therefore, there is a problem in deciding a stride value that will produce good visual quality in good time. Future works include generating synthetic forecast land-use maps based on historical land-use maps using Generating Adversarial Networks (GAN). We also want to formulate an equation to determine the optimal value of stride for each occasion. For compatibility and reuse reasons, the shape file format of the land-use classification outputs of this research will be created. This will help researchers and remote sensing engineers easily reuse our outputs in other projects. Finally, we would extend this work to other regions in Africa, and investigate the societal impacts of this land-use forecast in the coming years.
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---
abstract: 'In this paper we review the classification of isolated quotient singularities over the field of complex numbers due to H. Zassenhaus, G. Vincent, and J. A. Wolf. As an application we describe Gorenstein isolated quotient singularities over ${\mathbb{C}}$, generalizing a result of K. Kurano and S. Nishi.'
address: |
The Department of Mathematical Modelling\
Bauman Moscow State Technical University\
2-ya Baumanskaya ul. 5, Moscow 105005, Russia
author:
- 'D. A. Stepanov'
title: |
Gorenstein isolated quotient\
singularities over ${\mathbb{C}}$
---
[^1]
Introduction {#S:intro}
============
Let $\Bbbk$ be a field, and $R$ a finitely generated $\Bbbk$-algebra. Recall that $R$ is called *Gorenstein* if it has a canonical module generated by one element (see, e. g., [@Eisenbud], 21.1). Geometrically, the affine algebraic variety $X=\operatorname{Spec}R$ is Gorenstein if the canonical divisor $K_X$ of $X$ is Cartier. Now let $\Bbbk={\mathbb{C}}$ be the field of complex numbers, $S={\mathbb{C}}[x_1,\dots,x_N]$ the polynomial ring in $N$ variables, $G$ a finite subgroup of $GL(N,{\mathbb{C}})$, and $R=S^G$ the algebra of invariants. By Hilbert Basis Theorem $R$ is finitely generated. We say that $X=\operatorname{Spec}R$ is an *isolated singularity*, if the algebraic variety $X$ is singular and its singular set has dimension $0$.
Our paper is devoted to the following
\[Pm:classgisqsing\] Classify all Gorenstein isolated quotient singularities of the form ${\mathbb{C}}^N/G$, where $G$ is a finite subgroup of $GL(N,{\mathbb{C}})$, up to an isomorphism of algebraic varieties.
Recently K. Kurano and S. Nishi obtained the following result.
\[T:KN\] Let $G$ be a finite subgroup of $GL(p,{\mathbb{C}})$, where $p$ is an odd prime, and assume that ${\mathbb{C}}^p/G$ is an isolated Gorenstein singularity. Then ${\mathbb{C}}^p/G$ is isomorphic to a cyclic quotient singularity, i. e., to a singularity ${\mathbb{C}}^p/H$, where $H<GL(p,{\mathbb{C}})$ is a cyclic group.
Kurano and Nishi give a nice direct proof of their theorem. On the other hand, there exists a complete classification of isolated quotient singularities over ${\mathbb{C}}$, and it is natural to try to derive results of this sort and even a complete description of Gorenstein isolated quotient singularities over ${\mathbb{C}}$ from this classification.
Isolated quotient singularities (IQS in the sequel) over ${\mathbb{C}}$ were classified mainly in works of H. Zassenhaus [@Zassenhaus] (1935), G. Vincent [@Vincent] (1947), and J. A. Wolf [@Wolf] (1972). These authors were motivated neither by singularity theory nor by algebraic geometry. In his pioneer work Zassenhaus studied nearfields, whereas Vincent and Wolf classified compact Riemannian manifolds of constant positive curvature. The final results were obtained by Wolf and stated in [@Wolf] as a classification of such manifolds. The fact that Zassenhaus-Vincent-Wolf classification gives also a classification of IQS has already been known for a long time (see, e. g., [@Popov]), but it seems that this topic has not attracted much of the attention of algebraic geometers yet.
An element $g\in GL(N,{\mathbb{C}})$ is called a *quasireflection*, if $g$ fixes pointwise a codimension $1$ linear subspace of ${\mathbb{C}}^N$. Gorenstein IQS over ${\mathbb{C}}$ are characterized by the following theorem of K.-I. Watanabe.
\[T:Watanabe\] Let $G$ be a finite subgroup of $GL(N,{\mathbb{C}})$ free from quasireflections. Then ${\mathbb{C}}^N/G$ is Gorenstein if and only if $G$ is contained in $SL(N,{\mathbb{C}})$.
It follows that in order to get a classification of Gorenstein IQS one has to look through Zassenhaus-Vincent-Wolf groups giving IQS and check which of them are contained in the special linear group. We perform this program in our paper and present the results in Theorem \[T:classgiqsing\].
Chapters 5, 6, and 7 of Wolf’s book [@Wolf] are our main reference. The presentation there is so clear, detailed and self-contained that it is hard to imagine it to be improved. But the classification is done there from the point of view of the Riemannian geometry. Also, anyway we have to introduce a lot of notation and this requires a considerable place. Therefore we decided to include to our work a review of Zassenhaus-Vincent-Wolf classification written from the point of view of the theory of singularities.
Our paper is organized as follows. In Section \[S:prelim\] we list some preliminaries, observations and general results on IQS and their classification. In particular, we explain the connection of IQS with classification of compact Riemannian manifolds of constant positive curvature. In Section \[S:class\] we present the classification of isolated quotient singularities over ${\mathbb{C}}$ as it is given in [@Wolf]. The results are summarized in Theorem \[T:classisqsing\]. The only new thing we add is the computation of determinants of irreducible representations which we use later in description of Gorenstein singularities. We also discuss which part of the classification extends more or less obviously to other fields. In Section \[S:Gorenstein\] we show how the result of Kurano and Nishi can be deduced and generalized using the classification. As an example we give a description of Gorenstein isolated quotient singularities over ${\mathbb{C}}$ in dimensions $N\leq 7$.
We hope that the beautiful classification of Zassenhaus, Vincent, and Wolf will be completed over as many different fields $\Bbbk$ as possible and will find other interesting applications in algebraic geometry.
This paper originates from the talk which was given by the author at the workshop “Quotient singularities” organized by Ivan Cheltsov in June 2010 at the University of Edinburgh. We thank the organizer and all the participants of the workshop for warm and stimulating atmosphere.
Preliminaries {#S:prelim}
=============
The results of this section are either well known or easy, so even if we state some of them without reference, there is no claim of originality.
Generalities on isolated quotient singularities. Groups without fixed points
----------------------------------------------------------------------------
We start with formulating general problems on classification of IQS. Let $V$ be an algebraic variety defined over a field $\Bbbk$. Let $G$ be a finite group acting on $V$ by automorphisms and $P\in V$ a closed nonsingular fixed point of this action. Denote by $\pi\colon V\to V/G$ the canonical projection to the quotient variety and let $Q=\pi(P)$. Assume that $Q$ is an isolated singular point of $V/G$.
\[Pm:classisqsing\] Classify all singularities $Q\in V/G$ of the form described above up to a formal or, when $\Bbbk={\mathbb{C}}$, up to an analytic equivalence.
At a nonsingular point $P$ any algebraic variety is formally equivalent to $0\in \Bbbk^N$, $N=\dim V$, i. e., we have an isomorphism of complete local rings $\widehat{\mathcal{O}_{P,V}}\simeq
\widehat{\mathcal{O}_{0,\Bbbk^N}}$ ([@AMD], Remark 2 after Proposition 11.24). With the additional condition that characteristic of $\Bbbk$ does not divide order of $G$, the action of $G$ can be formally linearized at $P$ (see Lemma \[L:linearization\] below). In the analytic case $\Bbbk={\mathbb{C}}$, $V$ is a manifold at $P$ and the action of $G$ can be linearized in some local analytic coordinates at $P$, see, e. g., [@Akhiezer], p. 35. Thus we see that if $\operatorname{char}\Bbbk \nmid |G|$, Problem \[Pm:classisqsing\] is equivalent to the following
\[Pm:classaffquotients\] Classify all isolated quotient singularities of the form $\Bbbk^N/G$, where $G$ is a finite subgroup of $GL(N,\Bbbk)$.
Let us prove a lemma on formal linearization.
\[L:linearization\] Let $R=\Bbbk[[x_1,\dots,x_N]]$ be the local ring of formal power series in $N$ variables over a field $\Bbbk$, and $G$ a finite group acting on $R$ by local automorphisms. Assume that characteristic of the field $\Bbbk$ does not divide order of $G$. Then one can choose new local parameters $y_1,
\dots,y_N$ in $R$ such that $G$ acts by linear substitutions in $y_1,
\dots,y_N$.
Recall that $R=\Bbbk[[x_1,\dots,x_N]]$ can be identified with the inverse limit $\varprojlim R/\mathfrak{m}^n$, where $\mathfrak{m}$ is the maximal ideal $(x_1,\dots,x_N)$. Since $G$ acts by local automorphisms, it preserves all the ideals $\mathfrak{m}$, $\mathfrak{m}^2,\dots,
\mathfrak{m}^n,\dots$, and therefore acts linearly on all quotients $\mathfrak{m}/\mathfrak{m}^n$, $n\geq 1$, which in turn are finite dimensional vector spaces over $\Bbbk$. Moreover, if $\pi_{nn'}$, $n\geq n'$, denotes the natural truncation map $\mathfrak{m}/
\mathfrak{m}^{n+1}\to\mathfrak{m}/\mathfrak{m}^{n'+1}$, the action of $G$ is compatible with $\pi_{nn'}$. We shall construct $y_1,\dots,y_N$ as limits of Cauchy sequences $y_{1}^{(n)},\dots$, $y_{N}^{(n)}$, where $\forall i=1,\dots,N$, $y_{i}^{(n)}\in \mathfrak{m}/\mathfrak{m}^{n+1}$ and $\pi_{nn'}(y_{i}^{(n)})=y_{i}^{(n')}$.
Let $y_{1}^{(1)},\dots,y_{N}^{(1)}$ be any basis of $\mathfrak{m}/
\mathfrak{m}^2$. Note that $G$ acts on $y_{1}^{(1)},\dots,y_{N}^{(1)}$ by linear substitutions. Now suppose that $y_{1}^{(n)},\dots,y_{N}^{(n)}$ have already been constructed. Due to our assumption on characteristic of $\Bbbk$, the representation of $G$ on $\mathfrak{m}/\mathfrak{m}^{n+2}$ is completely reducible. Thus we have a decomposition $\mathfrak{m}/
\mathfrak{m}^{n+2}\simeq \mathfrak{m}^{n+1}/\mathfrak{m}^{n+2}\bigoplus V$, where $V$ projects isomorphically onto $\mathfrak{m}/\mathfrak{m}^{n+1}$ under the map $\pi_{n+1,n}$. Now we can uniquely lift $y_{1}^{(n)},\dots,y_{N}^{(n)}$ to some elements $y_{1}^{(n+1)},\dots,y_{N}^{(n+1)}$ of $\mathfrak{m}/\mathfrak{m}^{n+2}$. By construction $G$ acts on $y_{1}^{(n+1)},\dots,y_{N}^{(n+1)}$ by linear substitutions, hence the same holds for the limits $y_1,\dots,y_N$ of the sequences $y_{1}^{(n)},\dots,y_{N}^{(n)}$.
From now on, if not stated otherwise, we work only with the case when the field $\Bbbk$ is algebraically closed and $\operatorname{char}\Bbbk\nmid |G|$. Let us consider Problem \[Pm:classaffquotients\]. Recall that an element $g\in GL(N,\Bbbk)$ is called a *quasireflection*, if $g$ fixes pointwise a codimension $1$ subspace of $\Bbbk^N$. By Chevalley-Shephard-Todd Theorem ([@Benson], Theorem 7.2.1), a quotient $\Bbbk^N/H$, where $H<GL(N,\Bbbk)$ is finite and $\operatorname{char}\Bbbk\nmid |H|$, is smooth if and only if the group $H$ is generated by quasireflections. Quasireflections contained in a finite group $G<GL(N,\Bbbk)$ generate a normal subgroup $H\vartriangleleft G$. It follows that from the point of view of Problem \[Pm:classaffquotients\] we can restrict ourselves to groups $G<GK(N,\Bbbk)$ free from quasireflections. We require also the singularity $\Bbbk^N/G$ to be isolated. This imposes additional strong restrictions on the group $G$.
\[L:eigen1\] Let $G$ be a finite subgroup of $GL(N,\Bbbk)$. Assume that $G$ is free from quasireflections and $\operatorname{char}\Bbbk\nmid |G|$. Then $\Bbbk^N/G$ has isolated singularities if and only if $1$ is not an eigenvalue of any element of $G$ except the unity.
Sufficiency of the condition given in the lemma is clear, so let us prove necessity. Suppose on the contrary that $G$ has some non-unit element with eigenvalue $1$. Let $\mathcal{U}$ be the set of linear subspaces of $\Bbbk^N$ with nontrivial stabilizers in $G$, and let $U$ be a maximal element in $\mathcal{U}$ with respect to inclusion. Denote by $H$ the stabilizer of $U$. If $U$ is positive dimensional, the given representation of $H$ is reducible, so let us consider the splitting $\Bbbk^N=U\bigoplus U'$, where $U'$ is the complementary invariant subspace of $H$. Let $\overline{U}$ be the image of $U$ under the canonical projection $\pi\colon \Bbbk^N\to \Bbbk^N/G$. At a general point $P$ of $\overline{U}$ the quotient $P\in \Bbbk^N/H$ is formally isomorphic to the direct product $(0,\overline{0})\in U\times (U'/H)$. On the other hand, $\Bbbk^N/G$ has to be nonsingular at $P$, thus, by Shephard-Todd Theorem, the group $H$ acting on $U'$ is generated by quasireflections. But then also $H$ acting on $\Bbbk^N$ and thus $G$ contain quasireflections, which contradicts the conditions of the lemma.
\[D:grpswithoutfps\] Following [@Wolf], we call a group $G$ satisfying the conditions of Lemma \[L:eigen1\] (i. e. $1$ is not an eigenvalue of any element of $G$ except the unity) a *group without fixed points*.
Clifford-Klein Problem. $pq$-conditions
---------------------------------------
To introduce the context in which a solution for Problem \[Pm:classaffquotients\] was first obtained, set $\Bbbk={\mathbb{C}}$. It is a standard fact of the representation theory that any finite subgroup of $GL(N,{\mathbb{C}})$ is conjugate to a subgroup of the unitary group $U(N)$. On the other hand conjugate groups $G$ and $G'$ obviously give isomorphic quotient singularities ${\mathbb{C}}^N/G$ and ${\mathbb{C}}^N/G'$. Thus we may assume from the beginning that $G$ is a subgroup without fixed points of $U(N)$. If we equip ${\mathbb{C}}^N$ with the standard Hermitian product and the corresponding metric, we see that $G$ acts by isometries on the unit sphere $S^{2N-1}$ of ${\mathbb{C}}^N$. Moreover, $G$ acts on $S^{2N-1}$ without fixed points in the usual sense, which justifies Definition \[D:grpswithoutfps\]. Further, the quotient $S^{2N-1}/G$ with the induced metric is a compact Riemannian manifold of constant positive curvature. We immediately get
Let $P\in X$ be an isolated quotient singularity over the field ${\mathbb{C}}$. Then the link of $P$ in $X$ can be equipped with a Riemannian metric of constant positive curvature.
Classification of Riemannian manifolds of constant curvature was an important field of research in XX century. A key result about manifolds of positive curvature is the following fundamental theorem.
Let $M$ be a Riemannian manifold of dimension $n\geq 2$, and $K>0$ a real number. Then $M$ is a compact connected manifold of constant curvature $K$ if and only if $M$ is isometric to a quotient space of the form $$S_{K}^{n}/\Gamma\,,$$ where $S_{K}^{n}$ is the standard $n$-sphere of radius $1/K^2$ in ${\mathbb{R}}^{n+1}$, and $\Gamma$ is a finite subgroup of $O(n+1)$ acting on $S_{K}^{n}$ without fixed points.
The problem of classification of such spaces $S_{K}^{n}/\Gamma$ was called by Killing the *Clifford-Klein Space Form Problem*. In the sequel we simply call it the *Clifford-Klein Problem*.
It follows from the discussion above that Problem \[Pm:classaffquotients\] for $\Bbbk={\mathbb{C}}$ is a part of the Clifford-Klein Problem. But in fact they are almost equivalent. Indeed, it is shown in [@Wolf], Section 7.4, that for even $n$ we have only $2$ solutions for the Clifford-Klein Problem: the sphere $S^n$ ($\Gamma=1$) and the real projective space $\mathbb{RP}^n$ ($\Gamma={\mathbb{Z}}/2$). For odd $n$ all needed subgroups $\Gamma$ of $O(n+1)$ (orthogonal representations) are obtained from the subgroups of $U((n+1)/2)$ (unitary representations) without fixed points by means of the standard representation theory (Frobenius-Schur Theorem, see [@Wolf], Theorem 4.7.3).
Certainly, the problem of classification of finite subgroups $G<GL(N,\Bbbk)$ is wider than the problem of classification of IQS themselves, because different subgroups $G$ and $H$ of $GL(N,\Bbbk)$ can give isomorphic quotients $\Bbbk^N/G$ and $\Bbbk^N/H$. This happens, for example, if $G$ and $H$ are conjugate in $GL(N,\Bbbk)$. At least for $\Bbbk={\mathbb{C}}$ the converse statement also holds.
\[L:conjgrps\] Let $G$ and $H$ be two finite subgroups of $GL(N,{\mathbb{C}})$, $N\geq 2$. Assume that $G$ and $H$ act without fixed points, so that ${\mathbb{C}}^N/G$ and ${\mathbb{C}}^N/H$ have isolated singularities. Then the quotients ${\mathbb{C}}^N/G$ and ${\mathbb{C}}^N/H$ are isomorphic if and only if $G$ and $H$ are conjugate in $GL(N,{\mathbb{C}})$.
Denote by $g$ and $h$ the canonical projections $g\colon
{\mathbb{C}}^N\to {\mathbb{C}}^N/G$ and $h\colon {\mathbb{C}}^N\to C^N/H$, by $0$ the origin in ${\mathbb{C}}^N$, and let $P=g(0)$, $Q=h(0)$. The sufficiency condition of the lemma is obvious, so let us prove necessity.
Let $f\colon {\mathbb{C}}^N/G\to {\mathbb{C}}^N/H$ be an isomorphism. Since $G$ and $H$ act freely on ${\mathbb{C}}^N\setminus\{0\}$, $g$ and $h$ restrict to universal coverings ${\mathbb{C}}^N\setminus\{0\}\to ({\mathbb{C}}^N/G)\setminus\{P\}$ and ${\mathbb{C}}^N\setminus\{0\}\to ({\mathbb{C}}^N/G)\setminus\{Q\}$. Clearly $f(P)=Q$, thus $f$ lifts to an analytic isomorphism $\bar{f}\colon {\mathbb{C}}^N\setminus\{0\}\to
{\mathbb{C}}^N\setminus\{0\}$. But $f$ is also continuous at $P$, and it follows that $\bar{f}$ extends continuously to $0$ by setting $\bar{f}(0)=0$. This means that $0$ is a removable singular point of the analytic map $\bar{f}$, and actually $\bar{f}$ is an analytic isomorphism in the diagram $$\begin{xymatrix}{
&{\mathbb{C}}^N\ar[r]^{\bar{f}}\ar[d]_g &{\mathbb{C}}^N\ar[d]^h \\
&{\mathbb{C}}^N/G\ar[r]^f &{\mathbb{C}}^N/H
}
\end{xymatrix}$$ After fixing a reference point $O$ in $({\mathbb{C}}^N/G)\setminus\{P\}$, $f$ induces an isomorphism $$f_*\colon \pi_1(({\mathbb{C}}^N/G)\setminus\{P\},O)\to
\pi_1(({\mathbb{C}}^N/H)\setminus\{Q\},f(O))$$ between fundamental groups, which in turn are isomorphic to $G$ and $H$ respectively. Fixing the lifting $\bar{f}$, we fix also an isomorphism between $G$ and $H$ such that $\bar{f}$ becomes equivariant with respect to it. Now observe that $G$ and $H$ also act naturally on the tangent space $T_0{\mathbb{C}}^N$ which is canonically isomorphic to ${\mathbb{C}}^N$. Let $A\colon T_0{\mathbb{C}}^N\to T_0{\mathbb{C}}^N$ be the differential of $\bar{f}$ at $0$. Then $$H=AGA^{-1}\,.$$
\[L:automorphism\] Let $\varphi,\psi\colon G\to GL(N,\Bbbk)$ be two exact linear representations of a finite group $G$ over a field $\Bbbk$. Then the images $\varphi(G)$ and $\psi(G)$ are conjugate in $GL(N,\Bbbk)$ if and only if there exists an automorphism $\alpha$ of the group $G$ such that representations $\psi\circ\alpha$ and $\varphi$ are equivalent.
The proof is straightforward and left to the reader. See also [@Wolf], Lemma 4.7.1.
Let $p$ and $q$ be prime numbers, not necessarily distinct. We say that a finite group $G$ satisfies the *$pq$-condition*, if every subgroup of $G$ of order $pq$ is cyclic.
The following theorem is the cornerstone of the whole classification of groups without fixed points. Note that it does not impose any conditions on the characteristic of the base field $\Bbbk$.
\[T:pq\] Let $G$ be a finite subgroup of $GL(N,\Bbbk)$ without fixed points, where the field $\Bbbk$ is arbitrary. Then $G$ satisfies the $pq$-conditions for all primes $p$ and $q$.
See [@Wolf], Theorem 5.3.1.
Now we can formulate a program which should give a classification of IQS $\Bbbk^N/G$ in the nonmodular case $\operatorname{char}\Bbbk \nmid |G|$. This program was first suggested by Vincent [@Vincent] for $\Bbbk={\mathbb{C}}$ as a method of solution of the Clifford-Klein Problem. First one classifies all finite groups satisfying all the $pq$-conditions. This is a purely group-theoretic problem completely settled for solvable groups by Zassenhaus, Vincent and Wolf. Next, for every such group $G$ one studies its linear representations over the given field $\Bbbk$, $\operatorname{char}\Bbbk \nmid |G|$, and determines all exact irreducible representations without fixed points. Arbitrary representation without fixed points is a direct sum of irreducible ones. Finally, for every $G$ one calculates its group of automorphisms and determines when two irreducible representations without fixed points are equivalent modulo an automorphism. The last two steps were performed by Vincent and Wolf for $\Bbbk={\mathbb{C}}$. It follows from Lemma \[L:conjgrps\] and other results of this section that in this way we indeed get a complete classification of isolated quotient singularities over ${\mathbb{C}}$ up to an analytic equivalence. For other fields some further identifications may be needed, i. e., different output classes of the Vincent program might still give isomorphic quotient singularities. However, we do not have any examples of such phenomenon.
Consequences of $pq$-conditions. Some representations
-----------------------------------------------------
In the particular case $p=q$, the $pq$-condition is called the *$p^2$-condition*.
\[T:Sylowsgrps\] If $G$ is a finite group, then the following conditions are equivalent:
1. $G$ satisfies the $p^2$-conditions for all primes $p$;
2. every Abelian subgroup of $G$ is cyclic;
3. if $p$ is an odd prime, then every Sylow $p$-subgroup of $G$ is cyclic; Sylow $2$-subgroups of $G$ are either cyclic, or generalized quaternion groups.
We recall the definition of generalized quaternion groups below (see ).
Groups $G$ such that all their Sylow $p$-subgroups are cyclic constitute the simplest class of groups possessing representations without fixed points (provided also that $G$ satisfies all the $pq$-conditions). All such groups are solvable ([@Wolf], Lemma 5.4.3) and even metacyclic ([@Wolf], Lemma 5.4.5). Nonsolvable groups without fixed points necessarily contain Sylow $2$-subgroups isomorphic to generalized quaternion groups. Classification of nonsolvable groups without fixed points is available only over ${\mathbb{C}}$ and is based on the following result of Zassenhaus.
The binary icosahedral group $I^*$ is the only finite perfect group possessing representations without fixed points over ${\mathbb{C}}$.
The proof of this theorem is very technical and occupies about 15 pages of [@Wolf] (the original proof of Zassenhaus was not complete).
Now let us describe some groups and their complex representations which we use in the sequel.
*Generalized quaternion group* $Q2^a$, $a\geq 3$, is a group of order $2^a$ with generators $P$, $Q$, and relations $$\label{E:Qgroup}
Q^{2^{a-1}}=1\,,\; P^2=Q^{2^{a-2}}\,,\; PQP^{-1}=Q^{-1}\,.$$ We get the usual quaternion group $Q8=\{\pm 1,\pm i,\pm j,\pm k\}$ for $a=3$.
\[L:Qgrpreprs\] Let $Q2^a$, $a\geq 3$, be a generalized quaternion group and let $k$ be one of $2^{a-3}$ numbers $1$, $3$, $5,\dots,2^{a-2}-1$. Any irreducible complex representation without fixed points of the group $Q2^a$ is equivalent to one of the following $2$-dimensional representations $\alpha_k$: $$\alpha_k(P)=
\begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix},\;
\alpha_k(Q)=
\begin{pmatrix}
e^{2\pi ik/2^{a-1}} & 0 \\
0 & e^{-2\pi ik/2^{a-1}}
\end{pmatrix}.$$ The representations $\alpha_k$ are pairwise nonequivalent.
*Generalized binary tetrahedral group* $T_{v}^{*}$, $v\geq 1$, has generators $X$, $P$, $Q$, and relations $$X^{3^v}=P^4=1\,,\; P^2=Q^2\,,$$ $$XPX^{-1}=Q\,,\; XQX^{-1}=PQ\,,\; PQP^{-1}=Q^{-1}\,.$$ The usual binary tetrahedral group $T^*$ is $T_{1}^{*}$. $T_{v}^{*}$ has order $8\cdot 3^v$.
\[L:Tgrpreprs\] The binary tetrahedral group $T^*$ has only one irreducible complex representation without fixed points. This is the standard representation $\tau\colon T^*\to SU(2)$ obtained from the tetrahedral group $T<SO(3)$ and the double covering $SU(2)\to SO(3)$. If $v>1$, then $T_{v}^{*}$ has exactly $2\cdot 3^{v-1}$ pairwise nonequivalent irreducible complex representations without fixed points $\tau_k$, where $1\leq k< 3^v$, $(k,3)=1$, given by $$\tau_k(X)=-\frac{1}{2} e^{2\pi ik/3^v}
\begin{pmatrix}
1+i & 1+i \\
-1+i & 1-i
\end{pmatrix},$$ $$\tau_k(P)=
\begin{pmatrix}
i & 0 \\
0 & -i
\end{pmatrix},\;
\tau_k(Q)=
\begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix}.$$ The representation $\tau$ of $T^*$ is also given by these formulae if we formally set $v=0$.
The lemma is proved in [@Wolf], only the precise matrices are not given there. But their computation is an easy exercise which we omit.
*Generalized binary octahedral group* $O_{v}^{*}$, $v\geq 1$, has generators $X$, $P$, $Q$, $R$, and relations $$X^{3^v}=P^4=1\,,\; P^2=Q^2=R^2\,,$$ $$PQP^{-1}=Q^{-1}\,,\; XPX^{-1}=Q\,,\; XQX^{-1}=PQ\,,$$ $$RXR^{-1}=X^{-1}\,,\; RPR^{-1}=QP\,,\; RQR^{-1}=Q^{-1}\,.$$ The usual binary octahedral group $O^*$ is $O_{1}^{*}$. $O_{v}^{*}$ has order $16\cdot 3^v$.
\[L:Ogrpreprs\] The group $O^*$ has exactly $2$ irreducible complex representations without fixed points, say $o_1$ and $o_2$. Images of $o_1$ and $o_2$ are conjugate subgroups of $SU(2)$. For $o_1$ we may take the standard representation obtained from the octahedral group $O<SO(3)$ and the double covering $SU(2)\to SO(3)$. We may set $o_1(X)=\tau(X)$, $o_1(P)=\tau(P)$, $o_1(Q)=\tau(Q)$ (see Lemma \[L:Tgrpreprs\]), and $$o_1(R)=\frac{1}{\sqrt{2}}
\begin{pmatrix}
1+i & 0 \\
0 & 1-i
\end{pmatrix}.$$ If $v>1$, then $O_{v}^{*}$ has exactly $3^{v-1}$ pairwise nonequivalent irreducible complex representations without fixed points $o_k$, where $1\leq k<3^v$, $k\equiv 1({\mathrm{mod}\:}3)$, induced by the representations $\tau_k$ (see Lemma \[L:Tgrpreprs\]) of the subgroup $T_{v}^{*}=\langle X,P,Q \rangle$, i. e., $o_k$ has dimension $4$ and is given by $$o_k(X)=
\begin{pmatrix}
\tau_k(X) & 0 \\
0 & \tau_k(X^{-1})
\end{pmatrix},\;
o_k(P)=
\begin{pmatrix}
\tau_k(P) & 0 \\
0 & \tau_k(QP)
\end{pmatrix},$$ $$o_k(Q)=
\begin{pmatrix}
\tau_k(Q) & 0 \\
0 & \tau_k(Q^{-1})
\end{pmatrix},\;
o_k(R)=
\begin{pmatrix}
0 & 1 \\
\tau_k(R^2) & 0
\end{pmatrix}.$$
Recall that the *binary icosahedral group* $I^*$ is obtained from the group $I$ of rotations of icosahedron by the double covering $SU(2)\to SO(3)$. For generators of $I$ we may take one of the rotations by angle $2\pi/5$, which we denote by $V$, a rotation $T$ of order $2$, whose axis has angle $\pi/3$ with the axis of $V$, and another rotation $U$ of order $2$ whose axis is perpendicular to aces of $V$ and $T$. We denote the corresponding generators of $I^*$ by $\pm V$, $\pm T$, and $\pm U$.
\[L:Igrpreprs\] The group $I^*$ has only $2$ irreducible complex representations without fixed points $\iota_1$ and $\iota_{-1}$. They both have dimension $2$ and their images are conjugate subgroups of $SU(2)$. For $\iota_1$ we may take the standard representation $$\iota_1(\pm V)=
\begin{pmatrix}
\pm\varepsilon^3 & 0 \\
0 & \pm\varepsilon^2
\end{pmatrix},
\iota_1(\pm U)=
\begin{pmatrix}
0 & \mp 1 \\
\pm 1 & 0
\end{pmatrix},$$ $$\iota_1(\pm T)=\frac{1}{\sqrt{5}}
\begin{pmatrix}
\mp(\varepsilon-\varepsilon^4) & \pm(\varepsilon^2-\varepsilon^3) \\
\pm(\varepsilon^2-\varepsilon^3) & \pm(\varepsilon-\varepsilon^4)
\end{pmatrix},$$ where $\varepsilon=e^{2\pi i/5}$.
The matrices of $\iota_1$ can be found in [@Klein], Chapter II, §6.
Classification of isolated quotient singularities over ${\mathbb{C}}$ {#S:class}
=====================================================================
This is the central section of our paper. Here we state the main theorems on classification of IQS over the field ${\mathbb{C}}$ following [@Wolf], Chapters 6 and 7.
As it was said in Section \[S:prelim\], first we have to classify finite groups with $pq$-conditions. According to [@Wolf], a complete abstract classification exists only for solvable groups. It is given below in Theorem \[T:solvgrps\]. This is a purely group-theoretic result which can be useful in classification of IQS over any field. Theorem \[T:nonsolvgrps\] classifies nonsolvable groups with $pq$-conditions possessing representations without fixed points over ${\mathbb{C}}$, and the last condition is essentially used.
\[T:solvgrps\] Let $G$ be a finite solvable group satisfying all the $pq$-conditions. Then $G$ is isomorphic to one of the groups listed in Table \[Tb:solvgrps\]. Moreover, $G$ satisfies an additional condition: if $d$ is order of $r$ in the multiplicative group of residues modulo $m$ coprime with $m$, then every prime divisor of $d$ divides $n/d$ ($r$, $m$, and $n$ are defined in Table \[Tb:solvgrps\]).
Type Generators Relations Conditions Order
------ ------------------------- --------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------- --------
I $A$, $B$ $A^m=B^n=1$, $BAB^{-1}=A^r$ $m\geq 1$, $n\geq 1$, $(n(r-1),m)=1$, $r^n\equiv 1(m)$ $mn$
II $A$, $B$, $R$ As in I; also $R^2=B^{n/2}$, $RAR^{-1}=A^l$, $RBR^{-1}=B^k$ As in I; also $l^2\equiv r^{k-1}\equiv 1(m)$, $n=2^u v$, $u\geq 2$, $k\equiv -1(2^u)$, $k^2\equiv 1(n)$ $2mn$
III $A$, $B$, $P$, $Q$ As in I; also $P^4=1$, $P^2=Q^2=(PQ)^2$, $AP=PA$, $AQ=QA$, $BPB^{-1}=Q$, $BQB^{-1}=PQ$ As in I; also $n\equiv 1(2)$, $n\equiv 0(3)$ $8mn$
IV $A$, $B$, $P$, $Q$, $R$ As in III; also $R^2=P^2$, $RPR^{-1}=QP$, $RQR^{-1}=Q^{-1}$, $RAR^{-1}=A^l$, $RBR^{-1}=B^k$ As in III; also $k^2\equiv 1(n)$, $k\equiv -1(3)$, $r^{k-1}\equiv l^2\equiv 1(m)$ $16mn$
: Solvable groups with $pq$-conditions[]{data-label="Tb:solvgrps"}
\[T:nonsolvgrps\] Let $G$ be a nonsolvable finite group. If $G$ admits a representation without fixed points over ${\mathbb{C}}$, then $G$ belongs to one of the following two types.
Type V. $G=K\times I^*$, where $K$ is a solvable group of type I (see Theorem \[T:solvgrps\]), order of $K$ is coprime with $30$, and $I^*$ is the binary icosahedral group.
Type VI. $G$ is generated by a normal subgroup $G_1$ of index $2$ and an element $S$, where $G_1=K\times I^*$ is of type V and $S$ satisfies the following conditions. If we identify $I^*$ with the group $SL(2,5)$ of $2\times 2$ matrices of determinant $1$ over the field ${\mathbb{Z}}/5$, then $S^2=
-I\in SL(2,5)$, and for any $L\in SL(2,5)$ $SLS^{-1}=\theta(L)$, where $\theta$ ia an automorphism of $I^*$ given by $$\theta(L)=
\begin{pmatrix}
0 & -1 \\
2 & 0
\end{pmatrix} L
{\begin{pmatrix}
0 & -1 \\
2 & 0
\end{pmatrix}}^{-1}\,.$$ Moreover, any such $L$ belongs to the normalizer of the subgroup $K$. If $A$ and $B$ are the generators of the group $K$, then $SBS^{-1}=B^k$, $SAS^{-1}=A^l$, where integers $k$ and $l$ satisfy $l^2\equiv 1(m)$, $k^2\equiv 1(n)$, $r^{k-1}\equiv 1(m)$ (see Table \[Tb:solvgrps\]).
Conversely, every group of type V or VI is nonsolvable and admits representations without fixed points over ${\mathbb{C}}$.
\[R:qgroup\] All groups listed in Theorems \[T:solvgrps\] and \[T:nonsolvgrps\] with the exception of groups of type I contain a generalized quaternion group $Q2^a$ as a subgroup. This follows from Theorem \[T:Sylowsgrps\] and Theorem 5.4.1 of [@Wolf], the last stating that if all Sylow subgroups of a finite group $G$ are cyclic, then $G$ is a group of type I.
The Kleinian subgroups of $SL(2,{\mathbb{C}})$ are certainly among the groups of types I – VI. The cyclic group ${\mathbb{Z}}/n$ belongs to type I ($m=1$). The binary dihedral group $D_{b}^{*}$ belongs to type I for odd $b$ ($m=b$, $n=4$, $r=-1$), and to type II for even $b$ ($m=1$, $n=2b$, $k=-1$). The binary tetrahedral group $T^*$ belongs to type III ($m=1$, $n=3$). The binary octahedral group $O^*$ belongs to type IV ($m=1$, $n=3$, $k=-1$), and the binary icosahedral group to type V ($K=\{1\}$).
It would be useful to write the action of the automprphism $\theta$ from Theorem \[T:nonsolvgrps\] in terms of matrices $\iota_1(\pm V)$, $\iota_1(\pm U)$, and $\iota_1(\pm T)$ of the standard representation $\iota_1$ of the group $I^*$. Let us identify the image of this representation with $I^*$. Note that matrices $-V$ and $-T$ alone generate $I^*$. We can fix an isomorphism with $SL(2,5)$ by setting, for example, $$-V \leftrightarrow
\begin{pmatrix}
-1 & -1 \\
0 & -1
\end{pmatrix},\:
-T \leftrightarrow
\begin{pmatrix}
2 & 0 \\
-1 & 3
\end{pmatrix}.$$ Then $\theta$ is given by $$\theta(-V)=\frac{1}{5}
\begin{pmatrix}
1-\varepsilon+2\varepsilon^2-2\varepsilon^4 & -2+2\varepsilon+
\varepsilon^2-\varepsilon^4 \\
2+\varepsilon-\varepsilon^3-2\varepsilon^4 & 1-2\varepsilon+
2\varepsilon^3-\varepsilon^4
\end{pmatrix},\:
\theta(-T)=
\begin{pmatrix}
0 & -\varepsilon \\
\varepsilon^4 & 0
\end{pmatrix},$$ where $\varepsilon=e^{2\pi i/5}$.
Next we need a classification of irreducible complex representations without fixed points of groups from Theorems \[T:solvgrps\] and \[T:nonsolvgrps\]. The reference for this classification is [@Wolf], Section 7.2. Let us give a list of such representations for all types I – VI. The set of all irreducible complex representations without fixed points of a given group $G$ is denoted by $\mathfrak{F}_{{\mathbb{C}}}(G)$, $\varphi$ denotes here the Euler function. We also use the notation introduced in Theorems \[T:solvgrps\], \[T:nonsolvgrps\], and Table \[Tb:solvgrps\].
**The list of irreducible complex representations\
without fixed points**
*Type I.* Let $G$ be a group of type I. Recall that $d$ is order of $r$ in the group of residues modulo $m$ coprime with $m$. Then the set $\mathfrak{F}_{{\mathbb{C}}}(G)$ consists of the following $\varphi(mn)/d^2$ representations $\pi_{k,l}$ of dimension $d$: $$\pi_{k,l}(A)=
\begin{pmatrix}
e^{2\pi ik/m} & 0 & \cdots & 0 \\
0 & e^{2\pi ikr/m} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & e^{2\pi ikr^{d-1}/m}
\end{pmatrix},$$ $$\pi_{k,l}(B)=
\begin{pmatrix}
0 & 1 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & 1 \\
e^{2\pi il/n'} & 0 & \cdots & 0
\end{pmatrix}.$$ Here $(k,m)=(l,n)=1$, $n'=n/d$.
*Type II.* Let $G$ be a group of type II. Then it contains a subgroup $\langle A,B\rangle$ of type I. All representations without fixed points of the group $G$ are induced from the representations of the subgroup $\langle A,B\rangle$. Namely, $\mathfrak{F}_{{\mathbb{C}}}(G)$ consists of the following $\varphi(2mn)/(2d)^2$ representations $\alpha_{k',l'}$ of dimension $2d$: $$\alpha_{k',l'}(A)=
\begin{pmatrix}
\pi_{k',l'}(A) & 0 \\
0 & \pi_{k',l'}(A^l)
\end{pmatrix},\;
\alpha_{k',l'}(B)=
\begin{pmatrix}
\pi_{k',l'}(B) & 0 \\
0 & \pi_{k',l'}(B^k)
\end{pmatrix},$$ $$\alpha_{k',l'}(R)=
\begin{pmatrix}
0 & I \\
\pi_{k',l'}(B^{n/2}) & 0
\end{pmatrix}.$$ Here $(k',m)=(l',n)=1$, $k$ and $l$ are defined in row II of Table \[T:solvgrps\], and $I$ is the unit $d\times d$ matrix.
*Type III.* Let $G$ be a group of type III. Then it contains a subgroup $\langle A,B\rangle$ which is a group of type I of odd order. We have to distinguish $3$ cases.
<span style="font-variant:small-caps;">Case 1.</span> $9\nmid n$, in particular, $3\nmid d$. In this case $$G=\langle A,B^3\rangle \times \langle B^{n''},P,Q\rangle\,,$$ where $n''=n/3$, $\langle B^{n''},P,Q\rangle$ is the binary tetrahedral group $T^*$, and $\langle A,B^3\rangle$ is a group of type I with the same value of $d$ as the group $\langle A,B\rangle$. Then $\mathfrak{F}_{{\mathbb{C}}}(G)$ consists of the following $\varphi(mn)/2d^2$ representations $\nu_{k,l}$ of dimension $2d$: $$\nu_{k,l}=\pi_{k,l}\otimes\tau\,,$$ where $\pi_{k,l}\in \mathfrak{F}_{{\mathbb{C}}}(\langle A,B^3\rangle)$, and $\tau$ is the only irreducible representation without fixed points of the group $T^*$, see Lemma \[L:Tgrpreprs\].
<span style="font-variant:small-caps;">Case 2.</span> $9| n$, $3\nmid d$. Let $n=3^vn''$, where $(3,n'')=1$ and $v>1$. Then $$G=\langle A,B^{3^v}\rangle \times \langle B^{n''},P,Q\rangle=
\langle A,B^{3^v}\rangle \times T_{v}^{*}\,,$$ where $\langle A,B^{3^v}\rangle$ is a group of type I with the same value of $d$ as $\langle A,B\rangle$, and $T_{v}^{*}$ is a generalized tetrahedral group. Then $\mathfrak{F}_{{\mathbb{C}}}(G)$ consists of the following $\varphi(mn)/d^2$ representations $\nu_{k,l,j}$ of dimension $2d$: $$\nu_{k,l,j}=\pi_{k,l}\otimes\tau_j\,,$$ where $\pi_{k,l}\in\mathfrak{F}_{{\mathbb{C}}}(\langle A,B^{3^v}\rangle)$, and $\tau_j$ are defined in Lemma \[L:Tgrpreprs\].
<span style="font-variant:small-caps;">Case 3.</span> $9|n$ and $3|d$. Then $G$ contains a normal subgroup of index $d$ $$\langle A,B^d,P,Q\rangle=\langle A\rangle\times\langle B^d\rangle\times
\langle P,Q\rangle\,,$$ where $\langle P,Q\rangle$ is the quaternion group $Q8$. Irreducible representations without fixed points of $G$ are induced from irreducible representations without fixed points of the subgroup $\langle A,B^d,P,Q
\rangle$. The set $\mathfrak{F}_{{\mathbb{C}}}(G)$ consists of the following $\varphi(mn)/d^2$ representations $\mu_{k,l}$ of dimension $2d$: $$\mu_{k,l}(A)=
\begin{pmatrix}
e^{2\pi ik/m}I_2 & 0 & \cdots & 0 \\
0 & e^{2\pi ikr/m}I_2 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & e^{2\pi ikr^{d-1}/m}I_2
\end{pmatrix},$$ $$\mu_{k,l}(B)=
\begin{pmatrix}
0 & I_{2d-2} \\
e^{2\pi il/n'}I_2 & 0
\end{pmatrix},$$ $$\mu_{k,l}(P)=
\begin{pmatrix}
\alpha(P) & 0 & \cdots & 0 \\
0 & \alpha(Q) & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \alpha(B^{d-1}PB^{1-d})
\end{pmatrix},$$ $$\mu_{k,l}(Q)=
\begin{pmatrix}
\alpha(Q) & 0 & \cdots & 0 \\
0 & \alpha(PQ) & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \alpha(B^{d-1}QB^{1-d})
\end{pmatrix}.$$ Here $(k,m)=(l,n)=1$, $n'=n/d$, $I_2$ and $I_{2d-2}$ are the unit matrices of dimensions $2\times 2$ and $(2d-2)\times(2d-2)$ respectively, and $\alpha$ is the representation of the quaternion group $Q8$ defined in Lemma \[L:Qgrpreprs\].
*Type IV.* Let $G$ be a group of type IV. Note that elements $A$, $B$, $P$, $Q$ of $G$ generate a subgroup of type III. Here we again have to consider several cases.
<span style="font-variant:small-caps;">Case 1.</span> $9\nmid n$, in particular $3\nmid d$.
<span style="font-variant:small-caps;">Subcase 1</span>a. Assume that there exists an element $\pi\in\mathfrak{F}(\langle A,B^3 \rangle)$ ($\langle A,B^3\rangle$ is a group of type I), which is equivalent to the representation $\pi'\colon
g\mapsto \pi(RgR^{-1})$ of the group $\langle A,B^3\rangle$. Then $G$ is a direct product $$G=\langle A,B^3\rangle\times\langle R,B^{n/3},P,Q\rangle=
\langle A,B^3\rangle\times O^*\,,$$ where $O^*$ is the binary octahedral group. Then $\mathfrak{F}_{{\mathbb{C}}}(G)$ consists of the following $\varphi(mn)/d^2$ representations $\psi_{k,l,j}$ of dimension $2d$: $$\psi_{k,l,j}=\pi_{k,l}\otimes o_j\,,$$ where $\pi_{k,l}\in\mathfrak{F}_{{\mathbb{C}}}(\langle A,B^3\rangle)$ and $o_j=o_1$ or $o_{-1}$ is a representation of the binary octahedral group $O^*$ defined in Lemma \[L:Ogrpreprs\].
<span style="font-variant:small-caps;">Subcase 1</span>b. If the assumption of Subcase 1a is not satisfied, the group $G$ is not the direct product of subgroups $\langle A,B^3\rangle$ and $O^*$. Then $\mathfrak{F}_{{\mathbb{C}}}(G)$ consists of $\varphi(mn)/4d^2$ representations $\gamma_{k',l'}$ of dimension $4d$ induced by representations $\nu_{k',l'}$ of the subgroup $\langle A,B,P,Q\rangle$. We have $$\gamma_{k',l'}(A)=
\begin{pmatrix}
\nu_{k',l'}(A) & 0 \\
0 & \nu_{k',l'}(A^l)
\end{pmatrix},\;
\gamma_{k',l'}(B)=
\begin{pmatrix}
\nu_{k',l'}(B) & 0 \\
0 & \nu_{k',l'}(B^k)
\end{pmatrix},$$ $$\gamma_{k',l'}(P)=
\begin{pmatrix}
\nu_{k',l'}(P) & 0 \\
0 & \nu_{k',l'}(QP)
\end{pmatrix},\;
\gamma_{k',l'}(Q)=
\begin{pmatrix}
\nu_{k',l'}(Q) & 0 \\
0 & \nu_{k',l'}(Q^{-1})
\end{pmatrix},$$ $$\gamma_{k',l'}(R)=
\begin{pmatrix}
0 & I_{2d} \\
\nu_{k',l'}(P^2) & 0
\end{pmatrix}.$$ Here $(k',m)=(l',n)=1$, $k$ and $l$ are defined in row IV of Table \[T:solvgrps\], and $I_{2d}$ is the unit matrix of dimension $2d\times 2d$.
<span style="font-variant:small-caps;">Case 2.</span> $9 | n$ but $3\nmid d$. Let $n=3^vn''$, $(3,n'')=1$. Then the subgroup $\langle R,B^{n''},P,Q\rangle$ of $G$ is isomorphic to the generalized binary octahedral group $O_{v}^{*}$.
<span style="font-variant:small-caps;">Subcase 2</span>a. Assume that there exists an element $\pi\in\mathfrak{F}(\langle A,B^{3^v} \rangle)$ ($\langle A,B^{3^v}\rangle$ is a group of type I), which is equivalent to the representation $\pi'\colon g\mapsto \pi(RgR^{-1})$ of the group $\langle A,B^{3^v}\rangle$. Then $G$ is a direct product $$G=\langle A,B^{3^v}\rangle\times\langle R,B^{n''},P,Q\rangle=
\langle A,B^{3^v}\rangle\times O_{v}^{*}\,,$$ and the set $\mathfrak{F}_{{\mathbb{C}}}(G)$ consists of the following $\varphi(mn)/2d^2$ representations $\xi_{k,l,j}$ of dimension $4d$: $$\xi_{k,l,j}=\pi_{k,l}\otimes o_j\,,$$ where $\pi_{k,l}\in\mathfrak{F}_{{\mathbb{C}}}(\langle A,B^{3^v}\rangle)$ and $o_j$ is a representation of the generalized binary octahedral group $O_{v}^{*}$ defined in Lemma \[L:Ogrpreprs\].
<span style="font-variant:small-caps;">Subcase 2</span>b. If the assumption of Subcase 2a is not satisfied, the group $G$ is not the direct product of subgroups $\langle A,B^{3^v}\rangle$ and $O_{v}^{*}$. Then $\mathfrak{F}_{{\mathbb{C}}}(G)$ consists of $\varphi(mn)/2d^2$ representations $\gamma_{k',l',j}$ of dimension $4d$ induced by representations $\nu_{k',l',j}$ of the subgroup $\langle A,B,P,Q\rangle$. We have $$\gamma_{k',l',j}(A)=
\begin{pmatrix}
\nu_{k',l',j}(A) & 0 \\
0 & \nu_{k',l',j}(A^l)
\end{pmatrix},$$ $$\gamma_{k',l',j}(B)=
\begin{pmatrix}
\nu_{k',l',j}(B) & 0 \\
0 & \nu_{k',l',j}(B^k)
\end{pmatrix},$$ $$\gamma_{k',l',j}(P)=
\begin{pmatrix}
\nu_{k',l',j}(P) & 0 \\
0 & \nu_{k',l',j}(QP)
\end{pmatrix},$$ $$\gamma_{k',l',j}(Q)=
\begin{pmatrix}
\nu_{k',l',j}(Q) & 0 \\
0 & \nu_{k',l',j}(Q^{-1})
\end{pmatrix},\;
\gamma_{k',l',j}(R)=
\begin{pmatrix}
0 & I_{2d} \\
\nu_{k',l',j}(P^2) & 0
\end{pmatrix}.$$ Here $(k',m)=(l',n)=1$, $j$ numerates representations of the group $O_{v}^{*}$ (see Lemma \[L:Ogrpreprs\]), $k$ and $l$ are defined in row IV of Table \[T:solvgrps\], and $I_{2d}$ is the unit matrix of dimension $2d\times 2d$.
<span style="font-variant:small-caps;">Case 3.</span> $3 | d$, in particular $9 | n$. The set $\mathfrak{F}_{{\mathbb{C}}}(G)$ consists of $\varphi(mn)/2d^2$ representations $\eta_{k',l'}$ of dimension $4d$ induced by representations $\mu_{k',l'}$ of the subgroup $\langle A,B,P,Q\rangle$. We have $$\eta_{k',l'}(A)=
\begin{pmatrix}
\mu_{k',l'}(A) & o \\
0 & \mu_{k',l'}(A^l)
\end{pmatrix},\;
\eta_{k',l'}(B)=
\begin{pmatrix}
\mu_{k',l'}(B) & 0 \\
0 & \mu_{k',l'}(B^k)
\end{pmatrix},$$ $$\eta_{k',l'}(P)=
\begin{pmatrix}
\mu_{k',l'}(P) & 0 \\
0 & \mu_{k',l'}(QP)
\end{pmatrix},\;
\eta_{k',l'}(Q)=
\begin{pmatrix}
\mu_{k',l'}(Q) & 0 \\
0 & \mu_{k',l'}(Q^{-1})
\end{pmatrix},$$ $$\eta_{k',l'}(R)=
\begin{pmatrix}
0 & I_{2d} \\
\mu_{k',l'}(P^2) & 0
\end{pmatrix}.$$ Here $(k',m)=(l',n)=1$, $k$ and $l$ are defined in row IV of Table \[T:solvgrps\], and $I_{2d}$ is the unit matrix of dimension $2d\times 2d$.
*Type V.* Let $G$ be a group of type V. Then $G$ has the form $G=K\times I^*$ and the set $\mathfrak{F}_{{\mathbb{C}}}(G)$ consists of $2\varphi(mn)/d^2$ representations $\iota_{k,l,j}$ of dimension $2d$: $$\iota_{k,l,j}=\pi_{k,l}\otimes \iota_j\,,$$ where $\pi_{k,l}$ are described in entry Type I of our List and $\iota_j=\iota_1$ or $\iota_{-1}$ are described in Lemma \[L:Igrpreprs\].
*Type VI.* Let $G$ be a group of type VI. Then the set $\mathfrak{F}_{{\mathbb{C}}}(G)$ consists of $\varphi(mn)/d^2$ representations $\varkappa_{k',l',j}$ of dimension $4d$ induced by representations $\iota_{k',l',j}$ of the subgroup $K\times I^*$. We have, in particular, $$\varkappa_{k',l',j}(A)=
\begin{pmatrix}
\iota_{k',l',j}(A) & 0 \\
0 & \iota_{k',l',j}(A^l)
\end{pmatrix},\;
\varkappa_{k',l',j}(B)=
\begin{pmatrix}
\iota_{k',l',j}(B) & 0 \\
0 & \iota_{k',l',j}(B^k)
\end{pmatrix},$$ $$\varkappa_{k',l',j}(S)=
\begin{pmatrix}
0 & I_{2d} \\
-I_{2d} & 0
\end{pmatrix},$$ where $k$ and $l$ are defined in Theorem \[T:nonsolvgrps\], and $I_{2d}$ is the unit matrix of dimension $2d\times 2d$. Matrices $\varkappa_{k',l',j}(\pm V)$, $\varkappa_{k',l',j}(\pm T)$, $\varkappa_{k',l',j}(\pm U)$ for the generators of $I^*$ can be obtained using Lemma \[L:Igrpreprs\] and Theorem \[T:nonsolvgrps\].
\[T:irreprs\] Let $G$ be a finite group possessing a complex representation without fixed points. Then $G$ is one of the groups of types I – VI and all irreducible complex representations of $G$ without fixed points are given in Table \[Tb:reprs\]. The columns of the table have the following meaning. The first column gives type of the group $G$, the second additional conditions on the group $G$, the third irreducible representations of $G$ without fixed points, the fourth the dimension of representations, the fifth column gives the determinant of the matrix of representation corresponding to the generator $B$ (sometimes some power of $B$) of the group $G$, and the sixth the conditions when the image of a representation is contained in $SL(d,{\mathbb{C}})$ ($SL(2d,{\mathbb{C}})$, $SL(4d,{\mathbb{C}})$). All the other generators of the group $G$ have determinant $1$. We use in the table the notation introduced above in Theorems \[T:solvgrps\], \[T:nonsolvgrps\], and the List of irreducible representations without fixed points. $D_{n}^{*}$ denotes the binary dihedral group of order $4n$.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Type Case Repr. $\dim$ $\det(B)$ $G<SL$
------ --------------------------------------------------------- ----------------------- -------- ------------------------------------------------------------------ ---------------------------------------------------------
I $\pi_{k,l}$ $d$ $(-1)^{d-1}e^{2\pi il/n'}$ $n=2^{s+1}$, $d=2^s$, $s\geq 1$, or $d=1$ and $G=\{1\}$
II $\alpha_{k',l'}$ $2d$ $e^{2\pi il'(k+1)/n'}$ $d=1$ and $G=D_{n}^{*}$, or $d=2$ and $k=-1$
III $9\nmid n$ $\nu_{k,l}$ $2d$ $|\nu_{k,l}(B^3)|= \newline $d=1$ and $G=T^*$
e^{12\pi il/n'}$
III $9|n$, $3\nmid d$ $\nu_{k,l,j}$ $2d$ $|\nu_{k,l,j}(B^{n''})|= never
\newline e^{4\pi ijd/3^v}$, $|\nu_{k,l,j}(B^{3^v})|=\newline
e^{4\pi il3^v/n'}$
III $3|d$ $\mu_{k,l}$ $2d$ $e^{4\pi il/n'}$ never
IV $9\nmid n$, $G=\langle A,B^3\rangle\times O^*$ $\psi_{k,l,j}$ $2d$ $|\psi_{k,l,j}(B^3)|= \newline $d=1$ and $G=O^*$
e^{4\pi il/(n'/3)}$
IV $9\nmid n$, $G\ne\langle A,B^3\rangle\times O^*$ $\gamma_{k',l'}$ $4d$ $|\gamma_{k',l'}(B^3)|= \newline $d=1$, $A=1$, and $k=-1$
e^{4\pi il'(k+1)/(n'/3)}$
IV $9|n$, $3\nmid d$, $G=\langle A,B^{3^v}\rangle\times $\xi_{k,l,j}$ $4d$ $|\xi_{k,l,j}(B^{3^v})|= \newline $d=1$ and $G=O_{v}^{*}$
O_{v}^{*}$ e^{2\pi il/(n''/d)}$
IV $9|n$, $3\nmid d$, $G\ne\langle A,B^{3^v}\rangle\times $\gamma_{k',l',j}$ $4d$ $|\gamma_{k',l',j}(B^{3^v})|= $d=1$, $A=1$
O_{v}^{*}$ \newline e^{2\pi il'(k+1)/(n''/d)}$
IV $3|d$ $\eta_{k',l'}$ $4d$ $e^{4\pi il'(k+1)/n'}$ never
V $\iota_{k,l,j}$ $2d$ $e^{4\pi il/n'}$ $d=1$ and $G=I^*$
VI $\varkappa_{k',l',j}$ $4d$ $e^{4\pi il'(k+1)/n'}$ $d=1$, $A=1$, $k=-1$
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Irreducible representations without fixed points[]{data-label="Tb:reprs"}
Everything with the exception of computation of determinants is done in [@Wolf], Section 7.2. So let us concentrate on columns 5 and 6 of Table \[Tb:reprs\].
*Type I.* From the matrices given in the corresponding entry of the List of irreducible representations without fixed points one easily finds $$|\pi_{k,l}(A)|=\exp\left(\frac{2\pi ik}{m}(1+\dots+r^{d-1})\right)=1\,,$$ because $m|(r^d-1)$ but $(r-1,m)=1$, see Table \[Tb:solvgrps\]. Further, $$|\pi_{k,l}(B)|=(-1)^{d-1}e^{2\pi il/n'}\,.$$ Since $(l,n')=1$, $e^{2\pi il/n'}$ is a primitive root of unity of degree $n'$. It follows that the last determinant can be equal to $1$ either when $d=n'=n=m=1$, that is when the group $G$ is trivial, or when $n'=2$. Since any prime divisor of $d$ divides $n'$, we have $d=2^s$, $n=n'd=2^{s+1}$, $s\geq 1$. The case $s=0$, i. e. $d=1$, again leads to the trivial group $G$. We have proved the theorem for groups of type I.
*Type II.* From the matrices of the representation $\alpha_{k',l'}$ and from row I of Table \[Tb:reprs\] which we just proved one easily finds $$|\alpha_{k',l'}(A)|=|\pi_{k',l'}(A)||\pi_{k',l'}(A^l)|=1\,,$$ $$|\alpha_{k',l'}(R)|=(-1)^d|\pi_{k',l'}(B^{n/2})|=(-1)^d e^{\pi il'd}=1,$$ and $$|\alpha_{k',l'}(B)|=|\pi_{k',l'}(B)||\pi_{k',l'}(B^k)|=
e^{2\pi i(k+1)l'/n'},$$ where one has to use that $n$ and $k+1$ are divisible by $4$. If determinant of $B$ is equal to $1$, then $n'|k+1$, and thus any prime divisor of $d$ divides also $k+1$. On the other hand, $d$ divides $k-1$ (see row II of Table \[Tb:solvgrps\]). It follows that $d=1$ or $d=2$. In the first case we get $m=1$ and $A=1$, thus $$G=\langle B,R\,|\,R^2=B^{n/2},\; RBR^{-1}=B^{-1},\; B^n=1\rangle$$ is the binary dihedral group and $\alpha_{k',l'}$ is its standard representation. In the second case we get $n'=2^{u-1}v$. On the other hand, $(k+1)/n'$ must be integer, but $k$ is defined modulo $n$. Thus we may take $k=-1$ and the group reduces to $G=\langle A,B,R \,|\,
A^m=B^n=1,\; R^2=B^{n/2},\; BAB^{-1}=A^r,\; RAR^{-1}=A^l,\;
RBR^{-1}=B^{-1}\rangle$.
*Type III.* Recall that if $S$ and $T$ are matrices of size $m\times m$ and $n\times n$ respectively, then $$|S\otimes T|=|A|^n |B|^m\,.$$
<span style="font-variant:small-caps;">Case 1.</span> $9\nmid n$. From the definition of the representation $\nu_{k,l}$ we get $$|\nu_{k,l}(A)|=|\pi_{k,l}\otimes\tau(A)|=1\,,$$ $$|\nu_{k,l}(B^3)|=|\pi_{k,l}(B^3)|^2=e^{12\pi il/n'}\,.$$ Matrices of $B^{n/3}$, $P$, and $Q$ all have determinant $1$ since they generate the binary tetrahedral group. Recall that now $n'$ is odd, thus $|\nu_{k,l}(B^3)|=1$ only if $n'=1$ or $n'=3$. But the first case is impossible, because it would imply that $n=1$, but $n$ is divisible by $3$. In the second case we have $d=1$ or $d=3$. Again the second case is impossible because it would imply $9|n$. We conclude that $d=1$, $G=T^*$, and $\nu_{k,l}=\tau$.
<span style="font-variant:small-caps;">Case 2.</span> $9|n$, but $3\nmid d$. In this case the group $G$ is the direct product of the subgroups $\langle A,B^{3^v}\rangle$ and $\langle B^{n''},P,Q\rangle\simeq T_{v}^{*}$, where $n=3^vn''$, $v\geq 2$, and $B^{n''}$ corresponds to the generator $X$ of the group $T_{v}^{*}$. It is easy to check that $|\tau_j(X)|=e^{4\pi ij/3^v}$ (see Lemma \[L:Tgrpreprs\]), which is never equal to $1$. It follows that $|\nu_{k,l,j}(B^{n''})|=e^{4\pi ijd/3^v}$ and the image of the group $G$ is never contained in the group $SL(2d,{\mathbb{C}})$. It is also a straightforward verification that $|\nu_{k,l,j}(A)|=|\nu_{k,l,j}(P)|=|\nu_{k,l,j}(Q)|=1$ and $|\nu_{k,l,j}(B^{3^v})|=e^{4\pi il3^v/n'}$.
<span style="font-variant:small-caps;">Case 3.</span> $3|n$. From the definition of the representation $\mu_{k,l}$ one finds that determinants of the matrices corresponding to $A$, $P$, and $Q$ are all equal to $1$, whereas $|\mu_{k,l}(B)|=
e^{4\pi il/n'}$. Note that $n'$ is divisible by $3$, $(n',l)=1$, thus the last determinant is never equal to $1$ and the image of $\mu_{k,l}$ is never contained in $SL(2d,{\mathbb{C}})$.
*Type IV.*
<span style="font-variant:small-caps;">Case 1</span>a. $9\nmid n$, $G$ is the direct product $\langle A,B^3\rangle\times O^*$, where $O^*=\langle B^{n/3},P,Q,R\rangle$. From the definition of the representation $\psi_{k,l,j}$ one finds $$|\psi_{k,l,j}(A)|=|\psi_{k,l,j}(B^{n/3})|=|\psi_{k,l,j}(P)|=
|\psi_{k,l,j}(Q)|=|\psi_{k,l,j}(R)|=1\,,$$ $$|\psi_{k,l,j}(B^3)|=|\pi_{k,l}\otimes o_j(B^3)|=e^{4\pi il/(n'/3)}\,.$$ Note that $n'$ is an odd number, so the only possibility for the last determinant to be equal to $1$ is $d=1$, $n=n'=3$, thus $G=O^*$.
<span style="font-variant:small-caps;">Case 1</span>b. Conditions are as in Case 1a, but $G$ is not the direct product. From the definition of the representation $\gamma_{k',l'}$ it easily follows that determinants of matrices corresponding to $A$, $B^{n/3}$, $P$, $Q$, $R$ are all equal to $1$. For $B^3$ we have $$|\gamma_{k',l'}(B^3)|=|\nu_{k',l'}(B^3)||\nu_{k',l'}(B^{3k})|=
e^{4\pi il'(k+1)/(n'/3)}\,.$$ By an argument analogous to that of Type II we prove that either $d=2$ or $d=1$. But here $n$ is odd, thus $d=1$. This implies $A=1$. Further, $n/3$ divides $k+1$, and at the same time $3$ divides $k+1$. Since $k$ is determined modulo $n$, we may take $k=-1$.
<span style="font-variant:small-caps;">Case 2</span>a. $9|n$, $3\nmid d$, and $G$ is the direct product $\langle A,B^{3^v}\rangle\times O_{v}^{*}$, where $O_{v}^{*}$ is generated by $B^{n''}$, $P$, $Q$, and $R$, $n=3^vn''$, $(3,n'')=1$. First please check that $|o_j(X)|=1$, where $X$ is a generator of the generalized binary octahedral group. This follows easily from Lemma \[L:Ogrpreprs\]. Then it follows from the definition of the representation $\xi_{k,l}$ that the matrices corresponding to $A$, $B^{n''}$, $P$, $Q$, $R$ all have determinant $1$. Further, $$|\xi_{k,l}(B^{3^v})|=e^{8\pi il/(n''/d)}\,.$$ This determinant equals $1$ only if $n''=d$. But any prime divisor of $d$ divides $n''/d$, hence $n''=d=1$, and $G=O_{v}^{*}$.
<span style="font-variant:small-caps;">Case 2</span>b. The conditions are as in Case 2a, but $G$ is not the direct product. From the definition of the representation $\gamma_{k',l',j}$ one easily computes that all determinants are equal to $1$ with the exception of $$|\gamma_{k',l',j}(B^{3^v})|=e^{2\pi il'(k+1)/(n''/d)}\,.$$ If this determinant equals $1$ and $p$ is a prime divisor of $d$, we again see that $p$ must divide both $k-1$ and $k+1$, thus $p=2$. But in this case $n$ is odd, so the only possibility is $d=1$ and $A=1$.
<span style="font-variant:small-caps;">Case 3.</span> $3|d$. From the definition of the representation $\eta_{k',l'}$ one directly finds that all the determinants involved equal $1$, with the exception of $$|\eta_{k',l'}(B)|=e^{4\pi il'(k+1)/n'}.$$ This determinant can be equal to $1$ only if $d=1$, but this would contradict $3|d$. Therefore $\eta_{k',l'}(G)$ is never contained in $SL(4d,{\mathbb{C}})$ in this case.
*Type V.* Now $G$ a direct product $K\times I^*$, where $K$ is a group of type I and order of $G$ is coprime with $30$. It follows from the definition of the representation $\iota_{k,l,j}$ and Lemma \[L:Igrpreprs\] that matrices corresponding to generators of $G$ all have determinant $1$ with the exception of $|\iota_{k,l,j}(B)|$ which equals $e^{4\pi il/n'}$. The number $n'$ is odd, thus this determinant can be equal to $1$ only if $n'=1$. This implies $d=n=1$ and $K$ is trivial. Hence $G=I^*$ is the binary icosahedral group and $\iota_{k,l,j}$ is one of its representations described in Lemma \[L:Igrpreprs\].
*Type VI.* Again an easy computation that we omit shows that the question can be only in the determinant $|\varkappa_{k',l',j}(B)|$. From the definition of $\varkappa_{k',l',j}$ one finds $$|\varkappa_{k',l',j}(B)|=e^{4\pi il'(k+1)}\,.$$ The same type of argument as for Type II shows that this can be equal to $1$ only if $d=1$, $A=1$, and $k=-1$.
Note that if one of the irreducible representations without fixed points of a group $G$ is contained in the special linear group, then the others are also. Thus this seems to be a property of the group $G$ rather than of a particular representation.
For a given group $G$ from Theorems \[T:solvgrps\] and \[T:nonsolvgrps\], all its irreducible representations have the same dimension $d$, $2d$, or $4d$. Thus any representation without fixed points of $G$ has dimension multiple to $d$, $2d$, or $4d$ respectively.
Note also that all representations from Theorem \[T:irreprs\] are imprimitive and induced from primitive representations of dimension $1$ or $2$.
Finally we have to determine the automorphisms of groups of types I–VI and the action of the automorphisms on the irreducible representations of these groups by the rule $\varphi\to\varphi\circ\alpha$, where $\varphi$ is a representation and $\alpha$ is an automorphism. This will give the conditions when the images of two representations are conjugate in $GL$.
\[T:aut\] Let $G$ be a finite group possessing a representation without fixed points, i. e., one of the groups listed in Theorems \[T:solvgrps\] and \[T:nonsolvgrps\]. Then the action of the automorphisms on the irreducible representations of $G$ is described in Table \[Tb:aut\]. We use here the notation introduced in Table \[Tb:solvgrps\], Theorem \[T:nonsolvgrps\], and the List of irreducible complex representations without fixed points.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Type Case Action Conditions
------ --------------------------------------------------------- -------------------------------------------------------- -----------------------------------------------------
I $A_{a,b}\colon \pi_{k,l}\mapsto \pi_{ak,bl}$ $(a,m)=1$, $(b,n)=1$, $b\equiv 1(d)$
II $A_{a,b}\colon \alpha_{k',l'}\mapsto $(a,m)=1$, $(b,n)=1$, $b\equiv 1(d)$
\alpha_{ak',bl'}$
III $9\nmid n$ $A_{a,b}\colon \nu_{k,l}\mapsto \nu_{ak,bl}$ $(a,m)=1$, $(b,n/3)=1$, $b\equiv 1(d)$
III $9|n$, $3\nmid d$ $A_{a,b,c}\colon\nu_{k,l,j}\mapsto $(a,m)=1$, $(b,n/3^v)=1$, $(c,3)=1$, $b\equiv 1(d)$
\nu_{ak,bl,cj}$
III $3|d$ $A_{a,b}\colon \mu_{k,l}\mapsto \mu_{ak,bl}$ $(a,m)=1$, $(b,n)=1$, $b\equiv 1(d)$
IV $9\nmid n$, $G=\langle A,B^3\rangle\times O^*$ $A_{a,b,c}\colon \psi_{k,l,j}\mapsto \psi_{ak,bl,cj}$ $(a,m)=1$, $(b,n/3)=1$, $b\equiv 1(d)$, $c=\pm 1$
IV $9\nmid n$, $G\ne\langle A,B^3\rangle\times O^*$ $A_{a,b}\colon \gamma_{k',l'}\mapsto \gamma_{ak',bl'}$ $(a,m)=1$, $(b,n/3)=1$, $b\equiv 1(d)$
IV $9|n$, $3\nmid d$, $G=\langle A,B^{3^v}\rangle\times $A_{a,b,c}\colon \xi_{k,l,j}\mapsto \xi_{ak,bl,cj}$ $(a,m)=1$, $(b,n/3^v)=1$, $(c,3)=1$, $b\equiv 1(d)$
O_{v}^{*}$
IV $9|n$, $3\nmid d$, $G\ne\langle A,B^{3^v}\rangle\times $A_{a,b,c}\colon \gamma_{k',l',j}\mapsto $(a,m)=1$, $(b,n/3^v)=1$, $(c,3)=1$, $b\equiv 1(d)$
O_{v}^{*}$ \gamma_{ak',bl',cj}$
IV $3|d$ $A_{a,b}\colon \eta_{k',l'}\mapsto \eta_{ak',bl'}$ $(a,m)=1$, $(b,n)=1$, $b\equiv 1(d)$
V $A_{a,b,c}\colon \iota_{k,l,j}\mapsto $(a,m)=1$, $(b,n)=1$, $b\equiv 1(d)$, $c=\pm 1$
\iota_{ak,bl,cj}$
VI $A_{a,b,c}\colon \varkappa_{k',l',j}\mapsto $(a,m)=1$, $(b,n)=1$, $b\equiv 1(d)$, $c=\pm 1$
\varkappa_{ak',bl',cj}$
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Action of automorphisms on representations[]{data-label="Tb:aut"}
Now we are able to formulate the main theorem on classification of IQS over the field ${\mathbb{C}}$.
\[T:classisqsing\] Let $Q\in X$ be an isolated quotient singularity of a variety $X$ defined over the field ${\mathbb{C}}$, as it is described in Section \[S:prelim\]. Then locally analytically at $Q$ the variety $X$ is isomorphic to the quotient ${\mathbb{C}}^N/\varphi(G)$, where $N=\dim X$, $G$ is one of the groups described in Theorems \[T:solvgrps\] and \[T:nonsolvgrps\], and $\varphi$ is a direct sum of irreducible representations of $G$ described in Theorem \[T:irreprs\].
Conversely, any group from Theorems \[T:solvgrps\] and \[T:nonsolvgrps\] acting on ${\mathbb{C}}^N$ via a direct sum $\varphi$ of irreducible representations described in Theorem \[T:irreprs\] gives an isolated singularity $O\in{\mathbb{C}}^N/\varphi(G)$. If $\varphi$ and $\psi$ are two such representations, then singularities ${\mathbb{C}}^N/\varphi(G)$ and ${\mathbb{C}}^N/\psi(G)$ are isomorphic if and only if $\varphi$ and $\psi$ can be transformed into each other by an action of an automorphism of the group $G$ as described in Theorem \[T:aut\].
The conditions for singularities ${\mathbb{C}}^N/\varphi(G)$ and ${\mathbb{C}}^N/\psi(G)$ to be isomorphic can be stated more explicitely. An interested reader can consult [@Wolf], Sections 7.3 and 7.4.
This classification generalizes almost literally to other algebraically closed fields of characteristic $0$.
Let $\Bbbk$ be an algebraically closed field of characteristic $0$. Let $Q\in X$ be an isolated singularity of an algebraic variety $X$ defined over $\Bbbk$. Then at the point $Q$ the variety $X$ is formally isomorphic to $\Bbbk^N/\varphi(G)$, where $N=\dim X$, $G$ is one of the groups described in Theorems \[T:solvgrps\] and \[T:nonsolvgrps\], and $\varphi$ is a direct sum of irreducible representations without fixed points of $G$ described in Theorem \[T:irreprs\], where in matrices of representations one has to replace the complex roots of unity with roots of unity of corresponding degree contained in the field $\Bbbk$.
First let us recall some terminology. Let $\varphi\colon G\to GL(V_\Bbbk)$ be a representation of a group $G$ on a finitely dimensional $\Bbbk$-vector space $V_\Bbbk$. We say that $\varphi$ is defined over a subfield $L\subseteq\Bbbk$, if $\varphi$ is obtained from a representation $\varphi'\colon G\to GL(V_L)$ over $L$ by extension of scalars: $V_\Bbbk=
V_L\otimes\Bbbk$.
Since $\Bbbk$ is algebraically closed field of characteristic $0$, we may assume that it contains the field $\overline{{\mathbb{Q}}}$ of algebraic numbers. By Brower’s Theorem ([@Serr], 12.2) any representation of a finite group $G$ over $\Bbbk$ is actually defined over $\overline{{\mathbb{Q}}}$ (in fact over a smaller field). It follows that everything that we need about classification of groups without fixed points and their linear representations and that we have proved over ${\mathbb{C}}$ holds also over $\Bbbk$.
We can not yet tell much on IQS over fields of positive characteristic. The only result we have is the following.
Let $\Bbbk$ be an algebraically closed field of characteristic $0$ or $p>0$. Let $G$ be a finite group of order not divisible by $p$, $N$ an odd number, and assume that $G$ acts linearly without quasireflections on the affine space $\Bbbk^N$. Then $\Bbbk^N/G$ is an isolated singularity if and only if $G$ is a metacyclic group satisfying the conditions of Theorem \[T:solvgrps\], type I.
It follows from the conditions of the theorem that $G$ acts on $\Bbbk^N$ without fixed points, and thus its Sylow subgroups are either cyclic or generalized quaternion (Theorem \[T:Sylowsgrps\]). But every exact $\Bbbk$-linear irreducible representation of a generalized quaternion group is $2$-dimensional ([@CR], §47). If $Q2^a<G$ is such a subgroup, then $\varphi$ restricted to $Q2^a$ is an exact representation, but then $N$ must be even. This shows that every Sylow subgroup of $G$ is cyclic. Such groups are classified in Theorem \[T:solvgrps\], type I (see also [@Wolf], Theorem 5.4.1), all of them are metacyclic.
Not all irreducible representations of metacyclic groups have the form given in our List above, type I (where one has to take roots of unity of the field $\Bbbk$ instead of complex roots), see [@CR], §47. An example of a group of type I satisfying all the $pq$-conditions but admitting irreducible representations different from type I of the List is given by $m=7\cdot 19$, $n=27$, $r=4$, so that $d=9$.
On the other hand, representations of metacyclic groups over $\Bbbk$, $\operatorname{char}\Bbbk\nmid |G|$, have been completely described in the literature ([@Tucker]), and it should be possible to extract a classification of representations without fixed points from there.
Gorenstein isolated quotient singularities {#S:Gorenstein}
==========================================
In this section we work over the field ${\mathbb{C}}$ of complex numbers. First let us deduce Theorem \[T:KN\] of Kurano and Nishi from the classification of IQS over the field ${\mathbb{C}}$.
*Proof of Theorem \[T:KN\].* Let $p$ be an odd prime, and let $G$ be a finite group acting on ${\mathbb{C}}^p$ via a representation $\varphi\colon
G\to GL(p,{\mathbb{C}})$. We may assume that $G$ acts without quasireflections. If ${\mathbb{C}}^p/\varphi(G)$ is a Gorenstein isolated singularity, by Theorem \[T:classisqsing\] $G$ is one of the groups of types I – VI, $\varphi$ is a direct sum of irreducible representations of dimension $d$, $2d$, or $4d$ from Theorem \[T:irreprs\], and $\varphi(G)\subset
SL(p,{\mathbb{C}})$ by Theorem \[T:Watanabe\]. Since $p$ is odd, $G$ has type I. Thus either $d=1$ and $G$ is cyclic, or $d=p$. But it follows from Theorem \[T:irreprs\], type I, that the last case is impossible.
[$\Box$]{}
If we want to classify all Gorenstein IQS over the field ${\mathbb{C}}$, we have only to refine the classification of Theorem \[T:classisqsing\] by taking into account the determinants of representations.
\[T:classgiqsing\] Let $X={\mathbb{C}}^N/G'$ be a Gorenstein isolated quotient singularity, where $G'$ is a finite subgroup of ${\mathbb{C}}^N$. Then the variety $X$ is isomorphic to the quotient ${\mathbb{C}}^N/\varphi(G)$, where $G$ is one of the groups described in Theorem \[T:solvgrps\] and Theorem \[T:nonsolvgrps\], and $\varphi$ is a direct sum $\varphi=\varphi_1\oplus\dots\oplus
\varphi_s$, $s\geq 1$, of irreducible representations described in Theorem \[T:irreprs\] and satisfying the additional condition $$\det(\varphi_1(B))\cdot\det(\varphi_2(B))\cdots\det(\varphi_s(B))=1\,,$$ where the values $\det(\varphi_i(B))$, $1\leq i\leq s$, are given in the 5th column of Table \[Tb:reprs\]. More explicitely this condition is stated in Table \[Tb:gorcond\], where we use the same notation as in Table \[Tb:reprs\]. The representations $\varphi_i$ have one and the same dimension $d$, $2d$, or $4d$, $d|N$ ($2d|N$, $4d|N$).
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Type Case Representation Condition
------ --------------------------------------------------------- ------------------------------------- -----------------------------------------------------------------------------------------------------------------------
I $\varphi_i=\pi_{k_i,l_i}$ $2\sum\limits_{i=1}^{s} l_i\equiv sn'(d-1)({\mathrm{mod}\:}2n')$
II $\varphi_i=\alpha_{k_i',l_i'}$ $(k+1)\sum\limits_{i=1}^{s} l_i'\equiv 0({\mathrm{mod}\:}n')$
III $9\nmid n$ $\varphi_i=\nu_{k_i,l_i}$ $\sum\limits_{i=1}^{s} l_i\equiv 0({\mathrm{mod}\:}n'/3)$
III $9|n$, $3\nmid d$ $\varphi_i=\nu_{k_i,l_i,j_i}$ $\sum\limits_{i=1}^{s} j_i\equiv 0({\mathrm{mod}\:}3^v)$, $\sum\limits_{i=1}^{s} l_i\equiv 0({\mathrm{mod}\:}n'/3^v)$
III $3|d$ $\varphi_i=\mu_{k_i,l_i}$ $\sum\limits_{i=1}^{s} l_i\equiv 0({\mathrm{mod}\:}n')$
IV $9\nmid n$, $G=\langle A,B^3\rangle\times O^*$ $\varphi_i=\psi_{k_i,l_i,j_i}$ $\sum\limits_{i=1}^{s} l_i \equiv 0({\mathrm{mod}\:}n'/3)$
IV $9\nmid n$, $G\ne\langle A,B^3\rangle\times O^*$ $\varphi_i=\gamma_{k_i',l_i'}$ $(k+1)\sum\limits_{i=1}^{s} l_i' \equiv 0({\mathrm{mod}\:}n'/3)$
IV $9|n$, $3\nmid d$, $G=\langle A,B^{3^v}\rangle\times $\varphi_i=\xi_{k_i,l_i,j_i}$ $\sum\limits_{i=1}^{s} l_i \equiv 0({\mathrm{mod}\:}n''/d)$
O_{v}^{*}$
IV $9|n$, $3\nmid d$, $G\ne\langle A,B^{3^v}\rangle\times $\varphi_i=\gamma_{k_i',l_i',j}$ $(k+1)\sum\limits_{i=1}^{s} l_i' \equiv 0({\mathrm{mod}\:}n''/d)$
O_{v}^{*}$
IV $3|d$ $\varphi_i=\eta_{k_i',l_i'}$ $(k+1)\sum\limits_{i=1}^{s} l_i' \equiv 0({\mathrm{mod}\:}n')$
V $\varphi_i=\iota_{k_i,l_i,j_i}$ $\sum\limits_{i=1}^{s} l_i \equiv 0({\mathrm{mod}\:}n')$
VI $\varphi_i=\varkappa_{k_i',l_i',j}$ $(k+1)\sum\limits_{i=1}^{s} l_i' \equiv 0({\mathrm{mod}\:}n')$
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: When the image of a representation is contained in $SL(N,{\mathbb{C}})$[]{data-label="Tb:gorcond"}
The theorem is a direct consequence of Theorems \[T:Watanabe\], \[T:irreprs\], \[T:classisqsing\], and conditions on the parameters $n$, $d$, etc. Fore instance, assume that $G$ is a group of type I, and $\varphi$ is a direct sum of $s$ $d$-dimensional irreducible representations $\pi_{k_j,l_j}$ described in the List of irreducible representations without fixed points, $1\leq j\leq s$, $s=N/d$. From Table \[Tb:reprs\] we get the value $|\pi_{k_j,l_j}(B)|=(-1)^(d-1) e^{2\pi il_j/n'}$, $n'=n/d$. It follows that $$|\varphi(B)|=\prod_{j=1}^{s}(-1)^(d-1) e^{2\pi il_j/n'}=
(-1)^{s(d-1)}\exp\left({\frac{2\pi i}{n'}\sum_{j=1}^{s} l_j}\right)\,.$$ This number is equal to $1$ if and only if $(1/n')\sum l_j-(sn'(d-1))/2$ is integer, which is equivalent to the condition in the first row, the fourth column of Table \[Tb:gorcond\].
Let us consider also the condition in the third row of Table \[Tb:gorcond\]. It follows from Table \[Tb:reprs\] that $\varphi(B)$ has determinant $1$ if and only if $$6\sum_{i=1}^{s} l_i\equiv 0({\mathrm{mod}\:}n')\,.$$ But recall that here $n$ is odd, divisible by $3$, but not divisible by $9$, and $d$ is not divisible by $3$. Thus $n'$ is odd and divisible by $3$. Hence we can rewrite the condition in the form given in Table \[Tb:gorcond\]. The rest of Table \[Tb:gorcond\] can be checked in the same way.
By Lemma \[L:conjgrps\] two Gorenstein IQS ${\mathbb{C}}^N/\varphi(G)$ and ${\mathbb{C}}^N/\psi(G)$ are isomorphic if and only if $\varphi(G)$ and $\psi(G)$ are conjugate subgroups of $SL(N,{\mathbb{C}})$. One can get explicite conditions from Theorem \[T:aut\]. Also note that since the numbers $n''$ and $3^v$ are coprime, two conditions of row III, Case $9|n$, $3\nmid d$ of Table \[Tb:gorcond\] can actually be reduced to one.
To illustrate Theorem \[T:classgiqsing\], let us describe complex Gorenstein IQS in dimensions $N$, $2\leq N\leq 7$.
$N=2$. Here we have to classify up to conjugacy finite subgroups of $SL(2,{\mathbb{C}})$. This case is classical, and it is well known that all such groups are either cyclic (they belong to type I in Wolf’s terminology), binary dihedral (type I and II), binary tetrahedral (type III), binary octahedral (type IV), or binary icosahedral (type V). The corresponding quotients ${\mathbb{C}}^2/G$ are the Kleinian singularities $A_n$, $D_n$, $E_6$, $E_7$, $E_8$.
$N=3$. By Theorem \[T:KN\] of Kurano and Nishi, all Gorenstein IQS of dimension $3$ are cyclic, i. e., they are isomorphic to quotients ${\mathbb{C}}^3/({\mathbb{Z}}/n)$, $n\geq 2$, where the cyclic group is generated by the matrix $$\begin{pmatrix}
e^{2\pi il_1/n} & 0 & 0 \\
0 & e^{2\pi il_2/n} & 0 \\
0 & 0 & e^{2\pi il_3/n}
\end{pmatrix},$$ and $(l_1,n)=(l_2,n)=(l_3,n)=1$, $l_1+l_2+l_3\equiv 0({\mathrm{mod}\:}n)$ (in fact it follows from this that $n$ must be odd).
$N=4$. Let ${\mathbb{C}}^4/\varphi(G)$ be a $4$-dimensional Gorenstein isolated quotient singularity, as described in Theorem \[T:classgiqsing\]. The representation $\varphi$ is a direct sum of irreducible representations of dimension $q$, $q=1$, $2$, or $4$. The case $q=1$ corresponds to cyclic IQS analogous to case $N=3$. Now let $q=2$. If $G$ is a group of type I, then $\varphi=\pi_{k_1,l_1}\oplus\pi_{k_2,l_2}$, the corresponding condition of Table \[Tb:gorcond\] takes the form $l_1+l_2\equiv 0({\mathrm{mod}\:}n/2)$, and $n$ must be divisible by $4$. The corresponding singularities can be described in the following way. Consider all ordered collections $(m,n,r,k_1,l_1,k_2,l_2)$ of $7$ positive integers satisfying the conditions $4|n$, $r^2\equiv 1({\mathrm{mod}\:}n)$, $((r-1)n,m)=1$, $k_i$, $l_i$ are defined modulo and coprime to $m$ and $n$ respectively, $i=1,2$, and $l_1+l_2\equiv 0({\mathrm{mod}\:}n/2)$. Consider a group $G=G(m,n,r)$ of type I defined in Table \[Tb:solvgrps\] and its irreducible $2$-dimensional representations $\pi_{k_i,l_i}$, $i=1,2$, defined in the List of irreducible representations without fixed points (simply the List in the sequel). Then for any such collection we have a Gorenstein isolated quotient singularity $$X_I(m,n,r,k_1,l_1,k_2,l_2)=
{\mathbb{C}}^4/\pi_{k_1,l_1}(G)\oplus\pi_{k_2,l_2}(G)\,.$$
Let $G$ be a group of type II. Then $\varphi=\alpha_{k_1',l_1'}\oplus
\alpha_{k_2',l_2'}$. In order to get a $2$-dimensional irreducible representation of it, we have to take $d=1$, thus also $A=1$ (see Table \[Tb:solvgrps\]). The corresponding condition of Table \[Tb:gorcond\] takes the form $(k+1)(l_1'+l_2')\equiv
0({\mathrm{mod}\:}n)$. The corresponding singularities can be described in the following way. Consider all ordered collections $(n,k,l_1',l_2')$ of $4$ positive integers such that $n=2^u v$, $u\geq 2$, $k\equiv -1(2^u)$, $k^2\equiv 1(n)$, $(l_1,n)=(l_2,n)=1$ are defined modulo $n$, $(k+1)(l_1'+l_2')\equiv 0(n)$. Consider a group $G=G(m=1,n,r=1,l=1,k)$ of type II defined in Table \[Tb:solvgrps\] and its irreducible representations $\alpha_{k_i'=1,l_i'}$, $i=1,2$, defined in the List. Then for any such collection we have a Gorenstein isolated quotient singularity $$X_{II}(n,k,l_1',l_2')={\mathbb{C}}^4/\alpha_{1,l_1'}(G)\oplus\alpha_{1,l_2'}(G)\,.$$
Let $G$ be a group of type III. From the condition $q=2$ we again conclude that $d=1$ and $A=1$. We have $3$ possibilities for the group $G$. If $9\nmid n$, then $\varphi=\nu_{k_1,l_1}\oplus\nu_{k_2,l_2}$. The corresponding condition of Table \[Tb:gorcond\] takes the form $l_1+l_2\equiv 0(n/3)$. The corresponding singularities can be described in the following way. Consider all triples $(n,l_1,l_2)$ of positive integers such that $n$ is odd, $3|n$ but $9\nmid n$, $l_1$ and $l_2$ are defined modulo and coprime to $n$, and $l_1+l_2\equiv 0(n/3)$. Consider a group $G=G(m=1,n,r=1)$ of type III defined in Table \[Tb:solvgrps\] and its irreducible representations $\nu_{k_i=1,l_i}$, $i=1,2$, defined in the List. Then for any such triple we have a Gorenstein isolated quotient singularity $$X_{III}^{(1)}(n,l_1,l_2)={\mathbb{C}}^4/\nu_{1,l_1}(G)\oplus\nu_{1,l_2}(G)\,.$$
If $9|n$, but $3\nmid d$, then $\varphi=\nu_{k_1,l_1,j_1}\oplus
\nu_{k_2,l_2,j_2}$. Here we get the following description of the corresponding singularities. Consider all ordered collections $(n,l_1,l_2,j_1,j_2)$ of $5$ positive integers satisfying the conditions $n=3^v n''$, $v\geq 2$, $(l_i,n)=1$ are defined modulo $n$, $(j_i,3)=1$ are defined modulo $3^v$, $i=1,2$, $l_1+l_2\equiv
0(n/3^v)$, $j_1+j_2\equiv 0(3^v)$. Consider a group $G=G(m=1,n,r=1)$ of type III defined in Table \[Tb:solvgrps\] and its irreducible representations $\nu_{k_i=1,l_i,j_i}$, $i=1,2$, defined in the List. Then for any such collection we have a Gorenstein isolated quotient singularity $$X_{III}^{(2)}(n,l_1,l_2,j_1,j_2)={\mathbb{C}}^4/\nu_{1,l_1,j_1}(G)\oplus
\nu_{1,l_2,j_2}(G)\,.$$
The third possibility $3|d$ obviously does not realize here.
Assume that $G$ is a group of type IV. $2$-dimensional irreducible representations exist only in the case $G\simeq O^*\times
\langle A,B^3\rangle$. Again there must be $d=1$ and $A=1$. We deduce the following description of the corresponding singularities. Consider all ordered collections $(n,k,l_1,l_2,j_1,j_2)$ of $6$ positive integers such that $n$ is odd, $3|n$ but $9\nmid n$, $k$ is defined modulo $n$, $k\equiv -1(3)$, $k^2\equiv 1(n)$, $l_i$ are defined modulo $n$, $(l_i,n)=1$, $j_i=\pm 1$, $i=1,2$, and $l_1+l_2\equiv 0(n/3)$. Consider a group $G=G(m=1,n,r=1,l=1,k)$ of type IV defined in Table \[Tb:solvgrps\] and its irreducible representations $\psi_{k_i=1,l_i,j_i}$, $i=1,2$, defined in the List. Then for any such collection we have a Gorenstein isolated quotient singularity $$X_{IV}(n,k,l_1,l_2,j_1,j_2)={\mathbb{C}}^4/\psi_{1,l_1,j_1}(G)\oplus
\psi_{1,l_2,j_2}(G)\,.$$
Assume that $G$ is a group of type V. We have $2$-dimensional irreducible representations only in the case $d=1$, $A=1$. The corresponding singularities can be described in the following way. Consider all ordered collections $(n,l_1,l_2,j_1,j_2)$ of $5$ positive integers such that $(n,30)=1$, $(l_i,n)=1$ are defined modulo $n$, $j_i=\pm 1$, $i=1,2$, and $l_1+l_2\equiv 0(n)$. Consider a group $G=G(m=1,n,r=1)$ of type V defined in Theorem \[T:nonsolvgrps\] and its irreducible representations $\iota_{k_i=1,l_i,j_i}$, $i=1,2$, defined in the List. Then for any such collection we have a Gorenstein isolated quotient singularity $$X_V(n,l_1,l_2,j_1,j_2)={\mathbb{C}}^4/\iota_{1,l_1,j_1}(G)\oplus
\iota_{1,l_2,j_2}(G)\,.$$
There are no $2$-dimensional irreducible representations of groups of type VI.
Now let $q=4$, i. e., the group $G$ acts on ${\mathbb{C}}^4$ via an irreducible representation $\varphi$. Irreducible representations of groups of types I – VI with determinants $1$ are described in Theorem \[T:irreprs\] and Table \[Tb:reprs\]. We obtain the following singularities. Let $G$ be a group of type I. Consider all triples $(m,r,k)$ of positive integers such that $m$ is odd, $r^4\equiv 1(m)$, $(r-1,m)=1$, and $k$ is defined modulo $m$, $(k,m)=1$. Consider a group $G=G(m,n=8,r)$ of type I defined in Table \[Tb:solvgrps\] and its irreducible representation $\pi_{k,l=2}$ defined in the List. Then for any such triple we have a Gorenstein isolated quotient singularity $$X_I(m,r,k)={\mathbb{C}}^4/\pi_{k,2}(G)\,.$$
Let $G$ be a group of type II. Consider all ordered collections $(m,n,r,l,k',l')$ of $6$ positive integers such that $(m,n,r,l)$ satisfy the conditions of Table \[Tb:solvgrps\] Type II with $d=2$, $k=-1$, $k'$ is defined modulo $m$, $l'$ is defined modulo $n$, $(k',m)=(l',n)=1$. Consider a group $G=G(m,n,r,l,k=-1)$ of type II defined in Table \[Tb:solvgrps\] and its irreducible representation $\alpha_{k',l'}$ defined in the List. Then for any such collection we have a Gorenstein isolated quotient singularity $$X_{II}(m,n,r,l,k',l')={\mathbb{C}}^4/\alpha_{k',l'}(G)\,.$$
There are no irreducible representations of dimension $4$ with determinant $1$ of groups of type III.
Let $G$ be a group of type IV. We get the following singularities. Consider a pair $(n,l')$ of positive integers such that $n$ is odd, $3|n$ but $9\nmid n$, $(l',n)=1$ is defined modulo $n$. Consider a group $G=
G(m=1,n,r=1,l=1,k=-1)$, $G\ne\langle A,B^3\rangle\times O^*$, of type IV defined in Table \[Tb:solvgrps\] and in the List, and its irreducible representation $\gamma_{k'=1,l'}$ defined in the List. Then for any such pair we have a Gorenstein isolated quotient singularity $$X_{IV}^{(1)}(n,l')={\mathbb{C}}^4/\gamma_{1,l'}(G)\,.$$
For any $v>1$ and $k$, $1\leq k<3^v$, $k\equiv 1(3)$, we have a Gorenstein isolated quotient singularity $$X_{IV}^{(2)}(v,k)={\mathbb{C}}^4/o_k(O_{v}^{*})\,,$$ where $O_{v}^{*}$ is the generalized octahedral group and $o_k$ is its irreducible representation defined in Lemma \[L:Ogrpreprs\].
Consider all ordered collections $(n,k,l',j)$ of $4$ positive integers such that the pair $(n,k)$ satisfies the conditions of Table \[Tb:solvgrps\], Type IV with $m=1$, $r=1$, $l=1$, $n=3^v n''$, $v\geq 2$, $l'$ is defined modulo $n$, $(l',n)=1$, $1\leq j<3^v$, $j\equiv 1(3)$. Consider a group $G=G(m=1,n,r=1,l=1,k)$, $G\ne
\langle A,B^{3^v}\rangle\times O_{v}^{*}$, of type IV defined in Table \[Tb:solvgrps\] and in the List, and its irreducible representation $\gamma_{k'=1,l',j}$ defined in the List. Then for any such collection we have a Gorenstein isolated quotient singularity $$X_{IV}^{(3)}(n,k,l',j)={\mathbb{C}}^4/\gamma_{1,l',j}(G)\,.$$
There are no irreducible representations of dimension $4$ with determinant $1$ of groups of type V.
Let $G$ be a group of type VI. Consider all triples $(n,l',j)$ satisfying the conditions $(n,30)=1$, $l'$ is defined modulo $n$, $(l',n)=1$, $j=\pm 1$. Consider a group $G=G(m=1,n,r=1,l=1,k=-1)$ of type VI defined in Theorem \[T:nonsolvgrps\] and its irreducible representation $\varkappa_{k'=1,l',j}$ defined in the List. Then for any such triple we have a Gorenstein isolated quotient singularity $$X_{VI}(n,l',j)={\mathbb{C}}^4/\varkappa_{1,l',j}(G)\,.$$
$N=5$. By Theorem \[T:KN\], in this case we have only cyclic quotient singularities ${\mathbb{C}}^5/({\mathbb{Z}}/n)$, where the group is generated by a diagonal matrix with $e^{2\pi il_j/n}$, $(l_j,n)=1$, $j=1,\dots,5$, on the diagonal, $l_1+\dots+l_5\equiv 0({\mathrm{mod}\:}n)$.
$N=6$. Again we have to consider all divisors $q$ of $6$. If $q=1$, we get cyclic quotient singularities. Let $q=2$. In this case we get singularities of the form ${\mathbb{C}}^6/\varphi(G)$, where $G$ is a group of one of the types I – VI, and $\varphi$ is a direct sum of $3$ $2$-dimensional irreducible representations of $G$ without fixed points, satisfying the Gorenstein condition of Table \[Tb:gorcond\]. For example, let $G$ be a group of type I. Consider all ordered collections $(m,n,r,k_1,l_1,k_2,l_2,k_3,l_3)$ of $9$ positive integers, where $(m,n,r)$ satisfy the conditions of Table \[Tb:solvgrps\], type I, $r^2\equiv 1(m)$, $k_i$ are defined modulo $m$, $(k_i,m)=1$, $l_i$ are defined modulo $n$, $(l_i,n)=1$, and $2(l_1+l_2+l_3)\equiv 3(n/2)({\mathrm{mod}\:}n)$. It is not difficult to see that the last condition is equivalent to the following: $n/4$ is odd and divides $l_1+l_2+l_3$. Consider a group $G=G(m,n,r)$ of type I defined in Table \[Tb:solvgrps\] and its irreducible representations $\pi_{k_i,l_i}$, $i=1,2,3$, defined in the List. Then for any such collection we have a Gorenstein isolated singularity $$X_I(m,n,r,k_1,l_1,k_2,l_2,k_3,l_3)={\mathbb{C}}^6/\pi_{k_1,l_1}(G)\oplus
\pi_{k_2,l_2}(G)\oplus\pi_{k_3,l_3}(G)\,.$$ We leave to an interested reader to state a more precise description of the singularities corresponding to types II – V. Note that groups of type VI does not give any singularities in this case.
Assume that $q=3$. In this case all singularities are produced by groups of type I, since only they have irreducible representations without fixed points of odd dimension. Consider all ordered collections $(m,n,r,k_1,l_1,k_2,l_2)$ of $7$ positive integers, where $(m,n,r)$ satisfy conditions of Table \[Tb:solvgrps\], type I, $r^3\equiv 1(m)$, $k_i$ are defined modulo $m$, $(k_i,m)=1$, $l_i$ are defined modulo $n$, $(l_i,n)=1$, $i=1,2$, and $l_1+l_2\equiv 0(n/3)$. Consider a group $G=G(m,n,r)$ of type I defined in Table \[Tb:solvgrps\] and its irreducible representations $\pi_{k_i,l_i}$, $i=1,2$, defined in the List. Then for any such collection we have a Gorenstein isolated quotient singularity $$X_I(m,n,r,k_1,l_1,k_2,l_2)={\mathbb{C}}^6/\pi_{k_1,l_1}(G)\oplus
\pi_{k_2,l_2}(G)\,.$$
There are no irreducible representations without fixed points of dimension $6$ of groups of types I – VI.
$N=7$. By Theorem \[T:KN\], there are only cyclic Gorenstein IQS in dimension $7$.
[13]{} Akhiezer, D. N. Lie Groups Actions in Complex Analysis, Aspects of Mathematics, Vieweg-Verlag, Braunschweig/Wiesbaden, 1995. Atiah, M. F., MacDonald, I. G. Introduction to Commutative Algebra, Addison-Wesley, Reading, Massachusetts, 1969. Benson, D. J. Polynomial Invariants of Finite Groups, London Mathematical Society Lecture Note Series **190**, Cambridge University Press, Cambridge, 1993. Curtis, C., Reiner, J. Representation Theory of Finite Groups and Associative Algebras, Intersc. Publ., New York, 1962. Eisenbud, D. Commutative Algebra with a View Toward Algebraic Geometry, Springer, 2004. Klein, F. Vorlesungen [ü]{}ber das Ikosaeder und die Aufl[ö]{}sung der Gleichungen von f[ü]{}nften Grade, Leipzig, 1884. Kurano, K., Nishi, S. *Gorenstein isolated quotient singularities of odd prime dimension are cyclic*, arXiv:0903.327, 2009. Popov, V. L. *When are the stabilizers of all nonzero semisimple points finite?*, in “Operator algebras, unitary representations, enveloping algebras and invariant theory”, Progress in Mathematics **92** (1990), 541–559. Serr, J.-P. Repr[é]{}sentations Lin[é]{}aires des Groupes Finis, Hermann, Paris, 1967. Tucker, P. *On the reduction of induced representations of finite groups*, Am. J. Math. **84** (1962), 400–420. Vincent, G. *Les groupes lin[é]{}ares finis sans points fixes*, Comment. Math. Helv. **20** (1947), 117–171. Watanabe, K.-I. *Certain invariant subrings are Gorenstein I, II*, Osaka J. Math. **11** (1974), 1–8, 379–388. Wolf, J. A. Spaces of Constant Curvature, University of California, Berkeley, California, 1972. Zassenhaus, H. *[Ü]{}ber endliche Fastk[ö]{}rper*, Abh. Math. Sem. Univ. Hamburg **11** (1935), 187-220.
[^1]: The research was supported by the Russian Grant for Scientific Schools 1987.2008.1 and by the Russian Program for Development of Scientific Potential of the High School 2.1.1/227
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---
abstract: 'This paper develops a power management scheme that jointly optimizes the real power consumption of programmable loads and reactive power outputs of photovoltaic (PV) inverters in distribution networks. The premise is to determine the optimal demand response schedule that accounts for the stochastic availability of solar power, as well as to control the reactive power generation or consumption of PV inverters *adaptively* to the real power injections of all PV units. These uncertain real power injections by PV units are modeled as random variables taking values from a finite number of possible scenarios. Through the use of second order cone relaxation of the power flow equations, a convex stochastic program is formulated. The objectives are to minimize the negative user utility, cost of power provision, and thermal losses, while constraining voltages to remain within specified levels. To find the global optimum point, a decentralized algorithm is developed via the alternating direction method of multipliers that results in closed-form updates per node and per scenario, rendering it suitable to implement in distribution networks with large number of scenarios. Numerical tests and comparisons with an alternative deterministic approach are provided for typical residential distribution networks that confirm the efficiency of the algorithm.'
author:
- 'Mohammadhafez Bazrafshan, , and Nikolaos Gatsis, [^1] [^2] [^3]'
bibliography:
- 'IEEEabrv.bib'
- 'biblio.bib'
title: |
Decentralized Stochastic Optimal Power Flow\
in Radial Networks with Distributed Generation
---
Optimal power flow, distribution networks, photovoltaic inverters, stochastic optimization, alternating direction method of multipliers, distributed algorithms
Introduction
============
-scale solar photovoltaic (PV) systems are paving their way into today’s distribution systems, affecting higher incorporation of distributed generation into modern power systems. The chief advantage is that energy is generated closer to the point of consumption, thereby helping to reduce the transmission network congestion. A major challenge in incorporating PV systems in distribution networks is the uncertain availability of solar energy due to changes in irradiance conditions. These changes lead to insufficient or at times excess electricity generation, and if unaccounted for, can result in reduced user satisfaction, poor voltage regulation, and eventually equipment failure.
To deal with these issues, programmable loads that enable control of their real power consumption provide an opportunity for distribution system operators (DSO) to reduce the peak load in periods of inadequate generation. Moreover, reactive power generation or consumption by PV inverters, which provide the AC interface between the PV system and the grid, can be leveraged to improve voltage regulation. Although current standards prohibit these inverters to operate at a variable power factor [@ieee1547], the potential advantages of using these capabilities for voltage regulation have been extensively reported in literature; see e.g., [@TuSuBaCh-ProcIEEE11; @baran2007; @adaptivelearning].
Building upon the aforementioned capabilities, this paper proposes a decentralized real and reactive power management framework in distribution systems with high levels of PV generation. To account for the inherent uncertainty in solar power generation, stochastic programming tools [@Conejo-Uncert] are used to achieve common objectives such as loss minimization, operational cost reduction, and acceptable voltage regulation.
Prior Art
---------
Power management in distribution networks amounts to an optimal power flow (OPF) problem that minimizes certain objectives subject to power flow equations that are generally nonconvex. See [@BaWu89; @baranWu3; @bwu89LoadBalance], for the canonical form of power flow equations in radial distribution networks. Due to the nonconvexity of power flow equations, many relaxations and approximations have been recently proposed. A comprehensive study summarizing the recent advances in convex relaxations of OPF can be found in [@low-co-ex] and [@low-co]. In particular, conic relaxation techniques are used in [@Jabr2006] for radial distribution load flow, while [@Lavaei2012], [@SoLa2012], and [@GaLiToLo2015] provide conditions to guarantee optimality of relaxations to the original OPF.
Deterministic approaches to reactive power management in distribution networks have recently been investigated in many studies where user power consumption and PV power generation are known. The reactive power management problem is approached in [@TSBC10a], [@TSBC10b], and [@TuSuBaCh-ProcIEEE11] using a linear approximation of the power flow equations, called `LinDistFlow` equations, and local reactive power control policies. Although these local policies are computationally attractive and perform well in practical scenarios, they do not provide optimality guarantees.
An optimal decentralized algorithm for solving the reactive power control problem under the `LinDistFlow` model is developed in [@SuBaCh-arx13a] using the alternating direction method of multipliers (ADMM). An adaptive VAR control scheme is pursued in [@YGL12] based on the `LinDistFlow` where the adaptation law switches between minimizing power losses or maintaining voltage regulation. Centralized reactive power control and PV inverter loss minimization using the second-order cone programming (SOCP) relaxation is the theme of [@FNCL12].
Decentralized solvers for real and reactive power optimization using the SOCP relaxations are developed in [@LiGaChLo12] and [@LiChLo12] using the Predictor Corrector Proximal Method of Multipliers (PCPM). Leveraging the SOCP relaxation and the ADMM, a decentralized solver for OPF with closed-form updates is designed in [@PeLo14] where the user-consumed reactive power is modeled as independent of users’ real power consumption. Decentralized real power control using ADMM with convex envelop approximations is developed in [@KrChLaBo13]. Leveraging semidefinite programming (SDP) relaxations, decentralized algorithms are designed in [@DaZhGi14] and [@DaDhJoGi14] using ADMM, and in [@LaZhDoTs12] via a dual subgradient method. Distributed reactive power control is performed in [@Robbins13] with the purpose of maintaining nodal voltages within specification limits. A nonconvex formulation for the reactive power control problem is presented in [@voltvar2012], and a solver based on sequential convex programming is developed, but without global optimality guarantees. An adaptive local learning algorithm is devised in [@adaptivelearning] which provides a fast approximate solution for voltage regulation using the solutions of past optimizations.
Up to this point, all previously mentioned approaches assume that real power injections in buses with renewable generation are known, and hence are deterministic. The works in [@distcont] and [@bolognani2015] propose distributed online algorithms for optimal reactive power compensation and loss minimization based on feedback control from local voltage measurements. By modeling local voltages as functions of reactive power in microgenerators, the loss terms are casted as a quadratic form in reactive powers, and a modified dual method is used to minimize the losses subject to power constraints. These works model the real power injection of distributed generators as unmeasured disturbances.
In this light, the work in [@nali2014conf] also assumes no knowledge of the real power injection of distributed generators. It is shown that a purely local reactive power control that leverages only voltage measurements and does not rely on communication, does not by itself guarantee acceptable voltage regulation. Therefore, to perform voltage regulation and provide optimality guarantees, additional information, such as previous control inputs is incorporated in their proposed algorithm.
The work in [@conejo-giannakis14] models loads and real power injections on nodes as stochastic processes, collects noisy and delayed estimates of those, and decides the reactive power injection by solving a centralized optimization problem using a stochastic approximation algorithm. Uncertainty-aware optimal real and reactive power management from PV units is analyzed in [@DaDhJoGi15] where the conditional-value-at-risk is utilized to minimize the risk of overvoltages.
Contributions and outline
-------------------------
The contributions of this paper are as follows:
1. An optimal power flow problem for distribution networks accounting for the uncertainty in solar generation is formulated. In particular, the real powers generated by PV units at different nodes are modeled as random variables that take values from a finite set of scenarios. Real power of user controllable loads is optimized jointly with reactive power injection or absorption by PV inverters and network power flows per scenario. This stochastic model captures the uncertainty in solar generation which is not accounted for in previous approaches [@TuSuBaCh-ProcIEEE11]–[@adaptivelearning] and [@TSBC10a]-[@voltvar2012], which assume known (deterministic) PV injections. In contrast to [@distcont]–[@nali2014conf], in which voltage regulation through reactive power control is based on feedback from local voltage measurements, our methodology maintains voltages within specified limits while accounting for the underlying uncertainty of the PV generation. Recently, uncertainty in distributed PV generation has been addressed in [@DaDhJoGi15] and [@conejo-giannakis14]. Reactive power control by PV units and power flows are decided in the present work adaptively to solar power outputs, as opposed to a static fashion in [@DaDhJoGi15]. User controllable loads are not modeled in [@conejo-giannakis14], which in addition features a centralized optimization scheme, in contrast to the decentralized solution algorithm developed here (featured as the second contribution next).
2. This paper develops a decentralized solver with the following desirable attributes, motivated by scalability considerations: (a1) updates are decomposed per node and per scenario; (a2) all updates are in closed-form and (a3) communication only between neighboring nodes is required. The decentralized algorithm is based on ADMM. Even though ADMM naturally lends itself to distributed computation, there are two challenges that need to be addressed, in order to successfully arrive to an optimization algorithm with the previously mentioned attributes: (c1) introduce properly designed auxiliary variables, and (c2) appropriately split the set of constraints into coupling and individual constraints. Different choices for (c1) and (c2) may lead to entirely different algorithms; one of this paper’s contributions is to address those challenges towards a fully decentralized solver with closed-form updates. The decentralization methodology in this paper is extending the ADMM approach in [@PeLo14] in a stochastic programming setup. In addition, the closed-form solution of an SOCP in 4 variables with upper bounds on certain variables is derived, motivated by the incorporation of line current limits in the present formulation. This paper also expands previous work in [@BG-Sg2014]—which featured only single-line networks and the `LinDistFlow` approximations—in two major ways: (1) SOCP relaxation of power flow equations is incorporated, which is a more complex but also more accurate model; and (2) the formulation incorporates tree networks.
3. Considering realistic PV generation models and only a small number of representative scenarios for the uncertainty, numerical tests highlight the benefits of the proposed stochastic formulation with regards to user satisfaction and thermal loss minimization. Improved voltage regulation and reduced thermal losses are demonstrated in comparison to alternative distributed control schemes in [@TuSuBaCh-ProcIEEE11], and also in networks that include shunt capacitors.
The remainder of this paper is organized as follows. Section \[sec:netmod\] introduces the network model, the decision variables, and pertinent constraints. The optimization problem is formulated in Section \[sec:opf\]. Section \[sec:algorithm\] develops the equivalent formulation suitable for decentralized solution by the ADMM, and derives the closed-form updates. Section \[sec:commreq\] deliberates on the algorithm implementation and the communication requirements. Numerical tests are provided in Section \[sec:numtest\] as well as comparisons with competing approaches. Section \[sec:conc\] concludes the paper.
Network Model and Decision Variables {#sec:netmod}
====================================
Consider a radial distribution network as depicted in Fig. \[fig:trne\] modeled by a tree graph, where the set of all nodes is denoted as $\mathcal{N}=\{0,1,2,...,N\}$. Node $0$ (i.e., root of the graph) is the substation connected to the transmission network, and the remaining $N$ nodes represent users.
Following the tree model for distribution networks, each node $i \in \mathcal{N} \setminus \{0\}$ has a unique ancestor denoted by $A_i$. The line connecting node $A_i$ to node $i$ is labeled as line $i$, and is considered to have resistance $r_i$ and reactance $x_i$. Each node $i \in \mathcal{N}$ has an associated set of child nodes denoted by $\mathcal{C}_i$. For the terminal nodes (i.e., leaves of the tree, or nodes without children), it holds that $\mathcal{C}_i=\emptyset$.
![A radial distribution network modeled as a tree graph.[]{data-label="fig:trne"}](Figures/fig1.eps)
User load model
---------------
User $i$ consumes a non-elastic real and reactive load denoted by $P_{L_i}$ and $Q_{L_i}$ respectively. Moreover, users are supposed to have demand response capabilities, and their elastic consumption $p_{c_i}$ is permitted to vary in a certain range: $$\begin{aligned}
\label{eqn:rebo}
0 \le p_{c_i} \le p_{c_i}^{\max}, \quad i \in \mathcal{N} \setminus \{0\}.\end{aligned}$$ The elastic reactive power consumption $q_{c_i}$ has a linear relationship with the real power $p_{c_i}$: $$\begin{aligned}
\label{eqn:pf}
q_{c_i}=\left(\sqrt{\frac{1}{\mathrm{PF}_i^2}-1}\right)~p_{c_i}, \quad i \in \mathcal{N} \setminus \{0\}.\end{aligned}$$ where $\mathrm{PF}_i$ is the power factor, a dimensionless number in the interval $(0,1]$.
PV generation model {#sub:stochmodel}
-------------------
User nodes may also be enabled with PV generation. Attributable to the stochastic nature of solar power, the real power injections of PV systems are modeled as random variables. The real power injections across the network take values from a finite set of $M$ possible scenarios, each with probability $\pi^m$, with $m \in \mathcal{M}= \{1,2,\ldots,M\}$. It is thus assumed that the real power generated by the PV unit at node $i$ and in scenario $m$ is given by $w_i^m$. A typical probabilistic model for generating scenarios is the beta distribution, see e.g., [@WaChWaBe2014] and [@NiZaAg2012]. The mean value of the distribution can be set to a forecasted generation for the next time period that could range from e.g., 15 minutes to 1 hour.
The DC electrical output generated by the PV modules are translated into an AC output through the use of PV inverters. These PV inverters are also capable of generating or consuming reactive power by themselves; see e.g., [@TuSuBaCh-ProcIEEE11]. Let $q_{w_i}^m$ denote the reactive power generated by the PV inverter at node $i$ in scenario $m$. Then, $q_{w_i}^m$ is a decision variable constrained by $$\begin{aligned}
\label{eqn:reabo}
-q_{w_i}^{\max} \le q_{w_i}^m \le q_{w_i}^{\max}, \quad i \in \mathcal{N} \setminus \{0\}, m \in \mathcal{M}\end{aligned}$$ where $q_{w_i}^{\max}=\sqrt{s_{w_i}^2-(w_i^m)^2}$ and $s_{w_i}$ is the maximum apparent power capacity of the PV at node $i$, that is, the nameplate capacity of the PV inverter at that node.
Power flow equations
--------------------
The scenario-dependency of real and reactive power injections of PV units renders the power flow equations across the network to be scenario dependent as well. At scenario $m$, real and reactive power flows on line $i$ are denoted by $P_i^m$ and $Q_i^m$ respectively; the squared magnitude of the voltage phasor at node $i$ and the squared magnitude of the current phasor on line $i$ are represented by $v_i^m$ and $l_i^m$, respectively. The power flow equations leveraging the SOCP relaxation are as follows, where all the constraints hold for $m\in\mathcal{M}$ and $i\in \mathcal{N}$:
[rCl]{} P\_i\^m &=& \_[j\_i]{}(P\_j\^m+r\_jl\_j\^m)+P\_[L\_i]{}+p\_[c\_i]{}-w\_i\^m \[eqn:repofl\]\
Q\_i\^m &=& \_[j \_i]{}(Q\_j\^m+x\_jl\_j\^m)+Q\_[L\_i]{}+q\_[c\_i]{}-q\_[w\_i]{}\^m-q\_[s\_i]{}v\_i\^m\[eqn:reapofl\]\
v\_[A\_i]{}\^m&=&v\_[i]{}\^m+2(r\_iP\_i\^m+x\_iQ\_i\^m)+(r\_i\^2+x\_i\^2)l\_i\^m\[eqn:vopofl\], i 0,
[rCl]{} (P\_i\^m)\^2+(Q\_i\^m)\^2 && v\_i\^ml\_i\^m, i 0,\[eqn:sopofl\]\
v\_i\^m && 0, i 0. \[eqn:vocupofl\]
In the equations above, $r_i$ and $x_i$ have units of $\Omega$. Real powers have units of $\mathrm{MW}$ and reactive powers are in $\mathrm{MVars}$. Square currents are in $(kA)^2$ while square magnitude of voltages $v_i^m$ have units of $(\mathrm{kV})^2$. Note that $v_i^m=(V_i^m)^2$ where $V_i^m$ is the magnitude of the voltage phasor at node $i$ and scenario $m$. The substation voltage is fixed at $v_0=v_{0}^m=(V_0^m)^2$ and since there is no user at the substation, $p_{c_0}, w_0^m, q_{w_0}^m$ are all zero.
The power flow equations - clearly show that fluctuations in solar real power injection ($w_i^m$) ultimately lead to variations in voltage levels ($v_i^m$) across nodes in the network. In order to guarantee that node voltages remain within safety levels, the following voltage regulation constraint is enforced at every node $i \in \mathcal{N} \setminus \{0\}$ per scenario $m \in \mathcal{M}$
[rCl]{} \[eqn:vore\] (1-)\^2 && (1+)\^2
where $\epsilon$ could be chosen to be 0.05. The current magnitudes for every line are capped as per the following constraint:
[rCl]{} 0 && l\_i\^m l\_i\^. \[eqn:linelimit\]
A summary of decision variables along with the user and network parameters is given in Table \[table:variabledescription\].
[|c|c|]{}\
Scenario-independent & $p_{c_i}$, $q_{c_i}$\
Scenario-dependent & $P_i^m$, $Q_i^m$, $v_i^m$, $l_i^m$, $q_{w_i}^m$\
\
User & $P_{L_i}$, $Q_{L_i}$, $\mathrm{PF}_i$, $p_{c_i}^{\max}$, $w_i^m$, $s_{w_i}$\
Network & $r_i$, $x_i$, $q_{s_i}$, $\epsilon$, $l_i^{\max}$\
Having described the optimization variables and constraints, the next section elaborates on the relevant objective function and completes the problem formulation.
Optimal power flow formulation {#sec:opf}
==============================
One of the main objectives for a DSO is to meet customer demand. In order to quantify user satisfaction for demand response decisions, a concave utility function denoted as $u_i(p_{c_i})$ is adopted for each user $i$. Maximizing the sum of user utilities hence constitutes the first objective.
Power flows in the network are provided through the substation that is connected to the transmission grid. The DSO undergoes a cost to obtain this power from the transmission network. Although any cost function $C(P_0^m)$ that is convex can be used to evaluate the cost of power provision at scenario $m$, a particular cost function of interest could be one that distinguishes between the buying price (import) and the selling price (export). One such example is a piece-wise linear function of the form: $$\label{eqn:coP0}
C(P_0^m)=
\left\{
\begin{array}{ll}
aP_0^m & \mbox{if } P_0^m \geq 0 \\
bP_0^m & \mbox{if } P_0^m < 0
\end{array}
\right.$$ with $a > b \ge 0$ to preserve convexity. The expected value of $C(P_0^m)$ over all scenarios, that is $\sum\limits_{m=1}^M \pi^mC(P_0^m)$ is the second term in the objective. The third term in the objective is the expected incurred thermal losses in the lines over all the scenarios, that is $\sum\limits_{m=1}^M\pi^m\sum\limits_{i=1}^Nr_il_i^m$.
Let $\mathbf{P}, \mathbf{Q}, \mathbf{v}, \mathbf{l}, \mathbf{p_{c}}, \mathbf{q_w}$ collect the respective variables per node and per scenario (if the variable is scenario dependent). The optimization problem amounts to
[rl]{} \[eqn:mainQP\] \_ & -\_[i=1]{}\^N u\_i(p\_[c\_i]{}) + \_[m=1]{}\^M \^m C(P\_0\^m)\
&+K\_\_[m=1]{}\^M\^m\_[i=1]{}\^Nr\_il\_i\^m\
& -
where $K_{\mathrm{Loss}} \ge 0$ is a weight which can be selected by the system operator to reflect the relative priority of loss minimization with respect to the two other objective terms.
Problem forms a two-stage stochastic convex program with first-and-second-stage decisions. First-stage decisions are determined independently of the uncertainty and comprise elastic load consumptions $p_{c_i}$’s (and ultimately $q_{c_i}$’s). Second-stage decisions are determined adaptively to the uncertainty and include reactive power injection/absorption provided by the PV inverters ($q_{w_i}^m$), power flows ($P_i^m, Q_i^m$), and squared magnitude of voltages and currents ($v_i^m$ and $l_i^m$ for every scenario).
Auxiliary variables can be used to relieve the difficulty of directly working with piecewise linear (nondifferentiable) functions such as in the objective. For example, $C(P_0^m)$ can be written as $$\begin{aligned}
\label{eqn:specialcost}
C(P_0^m)=aP_{0+}^m-bP_{0-}^m \end{aligned}$$ and the following constraints are added (for $m \in \mathcal{M}$): $$\begin{aligned}
\label{eqn:P0originalconstr}
P_0^m=P_{0+}^m-P_{0-}^m, \quad P_{0+}^m \ge 0 , P_{0-}^m\ge 0 .\end{aligned}$$
Solving with $C(P_0^m)$ given by and with added constraints of is equivalent to solving with $C(P_0^m)$ given by . The advantage of the former approach is that it includes a smooth objective. Appendix \[sec:appendixa\] proves the equivalence by showing that solving with $C(P_0^m)$ given by ensures that only one of the variables $(P_{0+}^m, P_{0-}^m)$ is nonzero per scenario $m$. This implies that either $P_0^m= P_{0+}^m>0$, or $P_0^m=-P_{0-}^m<0$. For the algorithm that is to be presented, this particular cost function is considered as an example since it is more challenging to deal with. For a differentiable convex function, only one variable ($P_0^m$) is needed to be considered, which yields simpler updates.
The second-order cone relaxation of OPF has been proved to be exact for tree networks under certain assumptions [@GaLiToLo2015]. By applying the techniques developed in [@GaLiToLo2015], sufficient conditions under which the inequality in holds as equality for the convex stochastic program are derived in Appendix \[sec:appendixb\].
The two-stage convex stochastic program developed in this section can be solved by centralized algorithms such as interior point methods. In the next section, a decentralized solver for based on ADMM is developed featuring closed-form updates per node and per scenario.
Solution Algorithm
==================
The detailed design of the decentralized solution algorithm is presented in this section. An equivalent problem to which is of the general form amenable to application of ADMM is derived in Subsection \[sub:eqpr\]. This equivalent problem includes judiciously designed auxiliary variables that allow decomposition per node and per scenario. Subsection \[admm\], briefly outlines the ADMM. Finally, in Subsection \[sub:updates\], the closed-form updates per node and per scenario are detailed. \[sec:algorithm\]
Equivalent problem {#sub:eqpr}
------------------
The only obstacle in fully decomposing into separate nodes is the coupling in power flow equations -. For instance, in and , node $i$ will need to know $P_j^m$, $Q_j^m$, and $l_j^m$ from child nodes $j \in \mathcal{C}_i$. In order to decouple each node from its child nodes, $N$ respective copies of these variables, $\hat{P}_j^m$, $\hat{Q}_j^m$, and $\hat{l}_j^m$ are introduced per scenario. Moreover, since all nodes except for the root have ancestors, constraint is also coupling the nodes. Therefore, per scenario, another set of $N$ variables $\hat{v}_i^m$ copies $v_{A_i}^m$ at node $i$. Finally, an additional set of copies per scenario, namely, $\tilde{P}_i^m$, $\tilde{Q}_i^m$, $\tilde{l}_i^m$, and $\tilde{v}_i^m$ for all $n \in \mathcal{N}\backslash\{0\}$ and the variables $\tilde{P}_{0+}^m$, $\tilde{P}_{0-}^m$ for the root, are also introduced, the purpose of which will be evident shortly.
Let the set of boldface variables $\{\mathbf{P},\mathbf{\hat{P}},\mathbf{\tilde{P}}, \mathbf{Q},\mathbf{\hat{Q}},\mathbf{\tilde{Q}}, \mathbf{v},\mathbf{\hat{v}},\\ \mathbf{\tilde{v}},\mathbf{l},\mathbf{\hat{l}},\mathbf{\tilde{l}},\mathbf{p_{c}}, \mathbf{\tilde{p}_c}, \mathbf{q}_{w},\mathbf{\tilde{q}}_{w},\mathbf{P_{0+}},\mathbf{P_{0-}},\mathbf{\tilde{P}_{0+}},\mathbf{\tilde{P}_{0-}}\}$ represent vectors collecting the corresponding variables in all scenarios and nodes. As listed in Table \[table:variables\], these variables are further collected in vectors $\mathbf{x}=\{\mathbf{x}_i^m\}_{i \in \mathcal{N}, m \in \mathcal{M}}$ and $\mathbf{z}=\{\mathbf{z}_i^m\}_{i \in \mathcal{N}, m \in \mathcal{M}}$.
The problem takes the following form:
\[eqn:equivalent\]
[rCl]{} \_[,]{} -\_[i=1]{}\^Nu\_i(\_[c\_i]{})+\_[m=1]{}\^M \^m (aP\_[0+]{}\^m-bP\_[0-]{}\^m)\
+ K\_\_[m=1]{}\^M\^m\_[i=1]{}\^Nr\_il\_i\^m
subject to:\
**Coupling Constraints** ($i \in \mathcal{N}, m\in\mathcal{M}$):
[lllll]{} \[eqn:copqlv\] i 0: &P\_i\^m=\_i\^m & Q\_i\^m=\_i\^m & l\_i\^m=\_i\^m &\
v\_i\^m=\_i\^m & \_i\^m=\_[A\_i]{}\^m & p\_[c\_i]{}\^m=\_[c\_i]{} & q\_[w\_i]{}\^m=\_[w\_i]{}\^m\
\[eqn:copqlpq\] j \_i:& \_j\^m=\_j\^m & \_j\^m=\_j\^m& \_j\^m=\_j\^m\
\[eqn:cop0\] &P\_[0+]{}\^m=\_[0+]{}\^m& P\_[0-]{}\^m=\_[0-]{}\^m
**Individual Equality Constraints** ($i \in \mathcal{N}, m\in\mathcal{M}$):
[lll]{} \[eqn:eqpqpc\] P\_i\^m&=&\_[j \_i]{}(\_j\^m+r\_j\_j\^m)+P\_[L\_i]{}+p\_[c\_i]{}\^m-w\_i\^m\
Q\_i\^m&=&\_[j \_i]{}(\_j\^m+x\_j\_j\^m)+Q\_[L\_i]{}+q\_[c\_i]{}\^m-q\_[w\_i]{}\^m-q\_[s\_i]{}v\_i\^m\
\[eqn:eqv\] \_i\^m&=&v\_i\^m+2(r\_iP\_i\^m+x\_iQ\_i\^m)+(r\_i\^2+x\_i\^2)l\_i\^m , i0
where $q_{c_i}^m=\left(\sqrt{\frac{1}{\mathrm{PF}_i^2}-1}\right)p_{c_i}^m$.
[lll]{} \[eqn:eqp0\] P\_0\^m&=&P\_[0+]{}\^m-P\_[0-]{}\^m
**Individual Inequality Constraints** ($i \in \mathcal{N}, m\in\mathcal{M}$)
[lll]{} &(\_i\^m)\^2+(\_i\^m)\^2 (\_i\^m)(\_i\^m), i 0 \[eqn:ineqpqvl\]\
&(1-)\^2 (1+)\^2, i0 \[eqn:vtildebounds\]\
&0 \_i\^m l\_i\^, i 0 \[eqn:vltilde\]\
\[eqn:ineqpc\] &p\_[c\_i]{}\^ \_[c\_i]{} p\_[c\_i]{}\^\
\[eqn:ineqq\] &-q\_[w\_i\^m]{}\^ \_[w\_i]{}\^m q\_[w\_i\^m]{}\^\
\[eqn:ineqp0\] &\_[0+]{}\^m 0, \_[0-]{}\^m 0.
Clearly, is equivalent to .
Nodes involved Variables
-------------------- ----------------------- ---------------------------------------------------------------------------------------------------
$\mathbf{x}_0^m$ Root $\{P_0^m,Q_0^m,P_{0+}^m, P_{0-}^m$
$\{\hat{P}_j^m,\hat{Q}_j^m,\hat{l}_j^m\}_{j \in \mathcal{C}_0}\}$
$\mathbf{x}_i^m$ Neither root nor leaf $\{P_i^m,Q_i^m,v_i^m,l_i^m,$
$\{\hat{P}_j^m,\hat{Q}_j^m,\hat{l}_j^m\}_{j \in \mathcal{C}_i},\hat{v}_i^m,p_{c_i}^m,q_{w_i}^m\}$
$\mathbf{x}_{i}^m$ Leaf $\{P_i^m,Q_i^m,v_i^m,l_i^m,\hat{v}_i^m,p_{c_i}^m,q_{w_i}^m\}$
$\mathbf{x}_{i}$ All nodes $\{\mathbf{x}_i^m\}_{m=1}^M$
$\mathbf{z}_0^m$ Root $\{\tilde{P}_{0+}^m,\tilde{P}_{0-}^m\}$
$\mathbf{z}_i^m$ Not root $\{\tilde{P}_i^m,\tilde{Q}_i^m,\tilde{v}_i^m,\tilde{l}_i^m,\tilde{q}_{w_i}^m\}$
$\mathbf{z}_i$ Not root $\{\{\mathbf{z}_i^m\}_{m=1}^M, \tilde{p}_{c_i}\}$
: $\mathbf{x}$ and $\mathbf{z}$ variables for the ADMM algorithm[]{data-label="table:variables"}
Review of ADMM {#admm}
--------------
With the previous definitions of $\mathbf{x}$ and $\mathbf{z}$, problem is of the general form [@BoPaChPeEc-FnT11] $$\begin{aligned}
\min_{\mathbf{x}\in \mathcal{X},\mathbf{z}\in \mathcal{Z}}~~ & f(\mathbf{x})+g(\mathbf{z}) \quad
\text{subj.~to}~~~~ & A\mathbf{x}+B\mathbf{z}=\mathbf{c} \label{admm-coupling} \end{aligned}$$ where $f$ and $g$ are convex functions. The set $\mathcal{X}$ corresponds to the individual equality constraints - and $\mathcal{Z}$ captures all the inequality constraints -.
The augmented Lagrangian function is defined as: $$\begin{aligned}
\label{eqn:augmented}
L_{\rho}(\mathbf{x},\mathbf{z},\mathbf{y})=f(\mathbf{x})+g(\mathbf{z})+\mathbf{y}^T(A\mathbf{x}+B\mathbf{z}-\mathbf{c})\notag \\+\frac{\rho}{2}\|A\mathbf{x}+B\mathbf{z}-\mathbf{c}\|_2^2\end{aligned}$$ where $\mathbf{y}$ is the Lagrange multiplier vector for the linear equality constraints in , and $\rho>0$ is a parameter. The primal and dual iterations of ADMM are as follows, where $k$ is the iteration index.
$$\begin{aligned}
\mathbf{x}{(k+1)} & :={\operatornamewithlimits{argmin}}_{\mathbf{x}\in\mathcal{X}}{L_{\rho}(\mathbf{x},\mathbf{z}(k),\mathbf{y}(k))}\\
\mathbf{z}{(k+1)} & :={\operatornamewithlimits{argmin}}_{\mathbf{z}\in\mathcal{Z}}{L_{\rho}(\mathbf{x}(k+1),\mathbf{z},\mathbf{y}(k))}\\
\mathbf{y}{(k+1)} & :=\mathbf{y}(k)+\rho\left[A\mathbf{x}(k+1)+B\mathbf{z}(k+1)-\mathbf{c}\right]. \label{lagrangeUpdateGeneral}\end{aligned}$$
The purpose of introducing the *tilde* variables in is so that the individual inequality constraints in $\mathcal{Z}$ can be handled separately in the $\mathbf{z}$-update. The $\mathbf{x}$-update on the other hand turns out to be an equality constrained quadratic program. This separation of variables are essential to finding closed-form solutions for the updates. The following primal and dual residuals are measured in every step, and the algorithm is stopped once these are below an acceptable threshold:
\[eqn:residuals\] $$\begin{aligned}
r(k)&:=||A\mathbf{x}(k)+B\mathbf{z}(k)-c|| \label{eqn:primalres} \\
s(k)&:=\rho||A^TB(\mathbf{z}(k)-\mathbf{z}(k-1))|| \label{eqn:dualres}.\end{aligned}$$
Updates {#sub:updates}
-------
The Lagrange multipliers corresponding to the coupling constrains of - are listed in Table \[table:lagrange\]. To perform ADMM, first the Augmented Lagrangian for problem needs to be formed. This Augmented Lagrangian is separable across variables $\mathbf{x}_i^m$ ($i \in \mathcal{N}$, $m \in \mathcal{M}$) with $\mathbf{z}$ fixed, or across variables $\mathbf{z}_i^m$ and $\tilde{p}_{c_i}$ ($i \in \mathcal{N}$, $m \in \mathcal{M}$) with $\mathbf{x}$ fixed. Each step of the ADMM will consist of minimizing the augmented Lagrangian with respect to either $\mathbf{x}$ or $\mathbf{z}$ and updating the Lagrange multipliers.
### $\mathbf{x}_i^m$-update
For node $i \in \mathcal{N} \setminus \{0\}$, per scenario $m$, the $\mathbf{x}$-update will be derived by minimizing the corresponding part of the augmented Lagrangian per node $i$ and scenario $m$ :
[lCl]{} \[eqn:xi\] \_[\_i\^m]{} K\_ \^m r\_il\_i\^m + \_i\^m(P\_i\^m-\_i\^m)+\_[j \_i]{}\_j\^m(\_j\^m-\_j\^m)\
+\_i\^m(Q\_i\^m-\_i\^m)+ \_[j \_i]{}\_j\^m(\_j\^m-\_j\^m) +\_i\^m(l\_i\^m-\_i\^m)\
+\_[j \_i]{}\_j\^m(\_j\^m-\_j\^m)+\_i\^m(v\_i\^m-\_i\^m) +\_[j\_i]{}\_j\^m(\_j\^m-\_i\^m)\
+\_i\^m(p\_[c\_i]{}\^m-\_[c\_i]{}) +\_i\^m(q\_[w\_i]{}\^m-\_[w\_i]{}\^m)+
subject to – .
For the special case of $i=0$ , the corresponding Lagrangian will include the term $
\pi^m (aP_{0+}^m-bP_{0-}^m)+ \frac{\rho}{2}[(P_{0+}^m-\tilde{P}_{0+}^m)^2+(P_{0-}^m-\tilde{P}_{0-}^m)^2] $ with the constraint $P_0^m=P_{0+}^m-P_{0-}^m$.
For all $i \in \mathcal{N}$ problem is of the following form: $$\begin{aligned}
\label{eqn:lineq-opt-closedform}
\frac{1}{2}(\mathbf{x}_{i}^m)^T\mathbf{A}_i^m\mathbf{x}_i^m+(\mathbf{b}_i^m)^T\mathbf{x}_i^m \quad \text{subj. to\ } \mathbf{C}_i^m\mathbf{x}_i^m=\mathbf{d}_i^m.\end{aligned}$$ The structure of the problem leads to a diagonal $\mathbf{A}_i^m$ and a full-rank $\mathbf{C}_i^m$, and therefore has a closed-form solution: $$\begin{aligned}
\label{eqn:xclose}
\mathbf{x}_i^{m^*}=
&\mathbf{A}_i^{m^{-1}}(-\mathbf{b}_i^m+\mathbf{C}_i^{m^T}\mathbf{F}_i^m) \\
\text{where} \quad &\mathbf{F}_i^m=(\mathbf{C}_i^m\mathbf{A}_i^{m^{-1}}\mathbf{C}_i^{m^T})^{-1}(\mathbf{d}_i^m+\mathbf{C}_i^m\mathbf{A}_i^{m^{-1}}\mathbf{b}_i^m). \notag\end{aligned}$$
### $\mathbf{z}_i$-update
In order to find the updates for $\mathbf{z}_i^m$–variables the augmented Lagrangian in can be minimized separately for $ \{ \tilde{P}_i^m,\tilde{Q}_i^m,\tilde{v}_i^m,\tilde{l}_i^m \}$, $\tilde{p}_{c_i}$, $\tilde{q}_{w_i}^m$, $\tilde{P}_{0+}^m$, and $\tilde{P}_{0-}^m$ per $i \in \mathcal{N}$ and $m \in \mathcal{M}$ subject to -.
The minimization with respect to $ \{ \tilde{P}_i^m,\tilde{Q}_i^m,\tilde{v}_i^m,\tilde{l}_i^m \}$ has a closed-form solution which is developed in Appendix \[sec:appendixc\] by generalizing [@PeLo14 Appendix I].
The variable $\tilde{p}_{c_i}$ is not dependent on the scenario, and its update is by solving the following program at every node: $$\begin{aligned}
\tilde{p}_{c_i}={\operatornamewithlimits{argmin}}_{0 \le \tilde{p}_{c_i} \le p_{c_i}^{\max}} \left[u_i(\tilde{p}_{c_i})+\sum\limits_{m=1}^{M}\eta_i^m(p_{c_i}^m-\tilde{p}_{c_i})\nonumber \right. \\ \left. +\frac{\rho}{2}\sum\limits_{m=1}^{M}(p_{c_i}^m-\tilde{p}_{c_i})^2\right].\end{aligned}$$ This problem is a scalar box-constrained convex optimization problem, and has a closed-form solution if e.g., the utility function is quadratic. The remaining $\mathbf{z}$–variables, namely $\tilde{q}_{w_i}^m$, $\tilde{P}_{0+}^m$, and $\tilde{P}_{0-}^m$, are scenario dependent, and their closed-form updates are given as follows (by minimizing the corresponding term in the Lagrangian):
[rCl]{} \[eqn:tildeqwupdate\] \_[w\_i]{}\^[m]{}(k+1)&=&\_[-q\_[w\_i]{}\^]{}\^[q\_[w\_i]{}\^]{}\
\[eqn:tildeP0plusupdate\] \_[0+]{}\^m(k+1)&=&\^+\
\[eqn:tildeP0minusupdate\] \_[0-]{}\^m(k+1)&=&\^+
where $[t]^+:=\max\{0,t\}$ and $[t]_{t_1}^{t_2}:=\max\left\{t_1,\min\{t,t_2\}\right\}$.
Equality Constraint Lagrange Multiplier
------------------------------------------------- ---------------------
$P_i^m=\tilde{P}_i^m$ $\lambda_i^m$
$Q_i^m=\tilde{Q}_i^m$ $\mu_i^m$
$l_i^m=\tilde{l}_i^m$ $\gamma_i^m$
$v_i^m=\tilde{v}_i^m$ $\omega_i^m$
$\hat{v}_j^m=\tilde{v}_{A_j}^m$ $\hat{\omega}_j^m$
$\hat{P}_j^m=\tilde{P}_{j \in \mathcal{C}_i}^m$ $\hat{\lambda}_j^m$
$\hat{Q}_j^m=\tilde{Q}_{j \in \mathcal{C}_i}^m$ $\hat{\mu}_j^m$
$\hat{l}_j^m=\tilde{l}_{j \in \mathcal{C}_i}^m$ $\hat{\gamma}_j^m$
$p_{c_i}^m=\tilde{p}_{c_i}$ $\eta_i^m$
$q_{w_i}^m=\tilde{q}_{w_i}^m$ $\theta_i^m$
$P_{0+}^m=\tilde{P}_{0+}^m$ $\zeta_{+}^m$
$P_{0-}^m=\tilde{P}_{0-}^m$ $\zeta_{-}^m$
: Lagrange Multipliers[]{data-label="table:lagrange"}
Following the detailed derivation of the ADMM-based algorithm in the present section, the next section deals with the implementation of the algorithm in a distributed fashion, and highlights its advantages.
Decentralized Implementation {#sec:commreq}
============================
In the implementation of this algorithm, each node $i \in \mathcal{N}$ is responsible for maintaining and updating variables $\mathbf{x}_i$, $\mathbf{z}_i$, and the corresponding Lagrange multipliers.
The ADMM algorithm works as depicted in Fig. \[fig:commReqTree\]. First, $\mathbf{z}$ and the Lagrange multipliers are initialized with arbitrary numbers. In each iteration, every node $i$ that has children receives $\tilde{P}_j^m$, $\tilde{Q}_j^m$, and $\tilde{l}_j^m$ from all $j \in \mathcal{C}_i$. Also, each node $i$ receives $\tilde{v}_{A_i}^m$ from its ancestor. Using these variables, $\mathbf{x}_i^m$ is updated according to the closed-form solution in .
Prior to the $\mathbf{z}$-update step, $\hat{P}_i^m, \hat{Q}_i^m$, and $\hat{l}_i^m$ are sent to node $i$ from ancestor $A_i$, and node $i$ collects $\{\hat{v}_j^m\}_{j \in \mathcal{C}_i}$ from its children. Upon receiving the required information, node $i$ performs the $\mathbf{z}$-update step. Upon completion of the $\mathbf{z}$-update step, the Lagrange multipliers are updated.
Note that, node 0 only communicates with its children, and the leaf nodes only communicate with their ancestors. All other nodes communicate both with their children and ancestors. Therefore in this algorithm only neighbors will need to communicate.
Algorithm \[alg:1\] summarizes the algorithm. Convergence in Step 2 of the algorithm is declared when the residuals $r(k)$, $s(k)$ in are sufficiently small and the value $\max_{i,m}\{v_i^ml_i^m-(P_i^m)^2-(Q_i^m)^2\}$ is smaller than $10^{-3}$ $(\mathrm{pu})^2$. The latter ensures exactness of the SOCP relaxation in .
This section is wrapped up by highlighting the merits of the developed algorithm:
1. The algorithm comprises closed-form updates per node and per scenario. The need for solving complex optimization problems per node is therefore bypassed which greatly simplifies implementation.
2. Computational effort per node does not change as the size of the network increases—that is, each node still needs to run the same closed-form updates.
3. This distributed algorithm is conducive to maintaining user privacy. Specifically, global optimality is achieved while parameters such as bounds on user consumption (i.e., $p_{c_i}^{\min}, p_{c_i}^{\max}$ ), power factor, and user utility are not transmitted to a central agent.
Initialize $\mathbf{z}$-variables and Lagrange multipliers with random numbers at every node $i$. For every node $i$ repeat steps \[step1\]-\[stepEnd\] until convergence. \[step1\] Receive $\tilde{P}_j^m$, $\tilde{Q}_j^m$, $\tilde{l}_j^m$, $\hat{\lambda}_j^m$, $\hat{\mu}_j^m$, and $\hat{\gamma}_j^m$ from all $j \in \mathcal{C}_i$ and for $m \in \mathcal{M}$. Also receive $\tilde{v}_{A_i}^m$ from node $A_i$ and $m \in \mathcal{M}$. Perform $\mathbf{x}_i$-update. Receive the updated $\mathbf{x}$-variables $\hat{P}_i^m$, $\hat{Q}_i^m$ and $\hat{l}_i^m$ for $m \in \mathcal{M}$ from $A_i$. Also receive $\hat{v}_j^m$ and $\hat{\omega}_j^m$ from all nodes $j \in \mathcal{C}_i$ and $m \in \mathcal{M}$. Perform $\mathbf{z}_i$-update. Update the Lagrange multipliers. \[stepEnd\]
Numerical Tests {#sec:numtest}
===============
Network setup {#sec:numtesta}
-------------
Numerical simulations are conducted on a sample tree distribution network as illustrated in Fig. \[fig:twolateral\]. Resistance and reactance values on line $i$, i.e., $r_i+jx_i$, are considered constant and equal to $0.33+j0.38 \frac{\Omega}{\mathrm{km}} \times d (\mathrm{km})$ where $d$ represents the distance between two nodes (fixed at $0.2 \mathrm{km}$). For the network, $S_{\mathrm{base}}=1$ MVA is selected, while the substation voltage is fixed at $V_0=7.2$ kV. Voltage regulation is performed with $\epsilon=0.05$ so that nodal voltages are allowed to vary within $5 \%$ of the nominal value $V_0$. The line flow limit is $l_i^{\max}=0.5 (\mathrm{kA})^2$. The cost $C(P_0^m)$ is set to zero, and similar to [@LiChLo2012], the utility function is selected as $u_i(p_{c_i})=-K_{u_i}(p_{c_i}-p_{c_i}^{\max})^2 $.The objective weights are $K_{u_i}=1$ and $K_{\mathrm{Loss}}=1$. Notice that if we set $K_{\mathrm{Loss}}=0$, then $C(P_0^m)$ can correctly account for loss terms.
For users $i=1,\ldots,N$, the non-elastic load is $P_{L_i}=0.1 \ \mathrm{MW}$, while $p_{c_i}$ is constrained to be in $[0,p_{c_i}^{\max}]=[0,0.05] \ \mathrm{MW}$. The power factor for both elastic and non-elastic loads is selected to be $\mathrm{PF}_i=0.94$.
![Radial distribution network used in the numerical tests.[]{data-label="fig:twolateral"}](Figures/fig3.eps)
Methodology for generating scenarios {#sec:numtestb}
------------------------------------
If there is a PV unit at node $i$, then the relationship between $s_{w_i}$ and the maximum real power capability of the inverter $w_i^{\max}$ is given by $w_i^{\max}=\frac{s_{w_i}}{1.1}$. Following [@WaChWaBe2014] and [@NiZaAg2012], the actual power generated by the PV unit, $w_i$, is a random variable that takes values from a beta distribution with mean $\bar{w}_i$ and variance $\sigma_i^2$, as follows:
\[eqn:beta\]
[rCl]{} &f\_[W\_i]{}(w\_i)=()\^[-1]{}(1-)\^[-1]{}\
&=|[w]{}\_i=w\_i\^ \[eqn:betaMean\]\
&=\_i\^2=(w\_i\^)\^2 \[eqn:betavar\]
where the relationship between the mean and the standard deviation is given as:
[rCl]{} =0.2+0.21.
In each of the ensuing case studies, it is assumed that $\bar{w}_i$ is known, and subsequently $\alpha$ and $\beta$ can be found numerically using and . Then, 1000 equiprobable scenarios are generated according to . This number is then reduced to $M=7$ representative scenarios using the fast forward reduction method [@Conejo-Uncert], [@WaChWaBe2014].
The scenario reduction methodology starts with an original scenario set $\Omega$ with cardinality $|\Omega|=1000$ and an empty set $\Omega_s=\emptyset$. Then, in each iteration, a scenario from $\Omega \backslash \Omega_s$ is selected which minimizes the Kantorovich Distance between $\Omega$ and $\Omega_s$ [@Conejo-Uncert]. The algorithm stops once the prescribed cardinality $|\Omega_s|=M=7$ is achieved. The probability of scenarios that are not included in the reduced representative set (i.e., $\Omega_s$) are aggregated on to the probability of the closest representative scenario in $\Omega_s$. The choice for $M=7$ is so that the smallest probability in the reduced set $\Omega_s$ is at least $0.01$.
In a network with $N$ generators whose power injections come from independent distributions, an exponential number of scenarios would potentially be needed to formulate the stochastic program. However, the stochastic program in this paper will not suffer from this exponential growth because of the following reasons: 1) Generators are all in a geographically limited area and under similar irradiance conditions, and hence generator outputs are spatially correlated; 2) the forecasted power output for the next time period—ranging from e.g., 15 minutes to 1 hour—is available, and is used to set the mean value of the distribution for each injection. The resulting distributions will have similar parameters, according to the type of day (e.g., sunny, cloudy, partly cloudy).
Different scenarios are generated for each of the case studies that follow, and problem is solved. It is numerically verified for all problems that the SOCP inequality of holds as equality for every scenario.
Case study for different day types {#subsec:numc}
----------------------------------
In this case study, the network features $N=50$ nodes with 30 nodes on the main branch and two laterals of 10 nodes each branching off at node 20. Nodes 31 to 40 correspond to the first branch while nodes 41 to 50 correspond to the second branch. All users are equipped with distributed PV generators and one large PV generation unit is located at the terminal node of the main branch.
For the larger PV installation in the terminal node of the main branch, $s_{w_{30}}=1$ MVA is selected, while for the remaining nodes we set $s_{w_i}=0.1$ MVA. The ratio of $\frac{\bar{w}_i}{w_i^{\max}}$ takes the values $0.3, 0.6$, and $0.9$, corresponding respectively to cloudy, partly cloudy and sunny days.
[|p[3.3cm]{}|c|c|c|]{} & Cloudy & Partly Cloudy & Sunny\
-------------------------------------------------
Negative Utility
$-\sum\limits_{i=1}^Nu_i(p_{c_i})$ monet. units
-------------------------------------------------
: Objective Value Breakdown for the Three Day Types
& 0.1142 & 0.0689 & 0.0232\
--------------------------------------------------------------------------
Expected Thermal Losses
$\sum\limits_{m=1}^M \pi^m \sum\limits_{i=1}^N r_il_i^m$ ($\mathrm{MW}$)
--------------------------------------------------------------------------
: Objective Value Breakdown for the Three Day Types
& 0.3181 & 0.0664 & 0.0271\
-----------------------
Objective Value Total
($\times 10^6$)
-----------------------
: Objective Value Breakdown for the Three Day Types
& 0.4323 & 0.1357 & 0.0503\
\[table:daytypeComparisons\]
Table \[table:daytypeComparisons\] shows the breakdown of objective values for problem solved for the three different day types. As conditions range from cloudy to sunny, it is observed that the level of satisfied elastic demand increases, while the expected thermal losses decrease. In particular, Fig. \[fig:pcday\] shows the total load scheduled for each user in these three day types. On cloudy days, i.e., when PV generation is low, the proposed stochastic program ensures meeting the non-elastic demand. As PV generation increases during partly cloudy and sunny days, larger portions of the elastic demand are also guaranteed.
Fig. \[fig:averageabs\] depicts the worst-case voltage profile across all scenarios for the three different day types. There are three break points for each voltage plot. The voltage rise at node 30 corresponds to the larger generation of the terminal node. Voltage drop at node 20 corresponds to the branching of the network. On a cloudy day, the reduced power generation in the network results in a higher voltage drop. This voltage drop is smaller for improved solar conditions.
Comparison with distributed local control of [@TSBC10b]
-------------------------------------------------------
The second set of numerical tests are conducted on a larger network with $N=100$ nodes comprising $60$ nodes on the main branch and two laterals each of length 20 branching off at node $40$. In this setup, $50 \%$ of the user nodes (randomly selected) are capable of PV generation. The terminal node on the main branch also provides large injection with $s_{w_{60}}=1$ MVA, while $s_{w_i}=0.4$ MVA is selected for the smaller generations. The simulations are performed for a sunny day. All other parameters are as listed in Subsections \[sec:numtesta\] and \[sec:numtestb\].
In this case, problem is solved first with $N=100$, $M=7$ scenarios and the optimal demand response schedules (i.e., $p_{c_i}$’s) are obtained. Then, in the online phase, problem with $p_{c_i}$’s fixed to the previously found values is solved for a newly generated set of 100 scenarios. The results are compared to the ones obtained by the local reactive power control policy proposed in [@TSBC10b]. In this scheme, only the local variables $w_i^m$, $p_{c_i}$, and $q_{c_i}$, are used to set $q_{w_i}^m=F_i(w_i^m,P_{c_i},Q_{c_i})$, where
[lll]{} F\_i(w\_i\^m,P\_[c\_i]{},Q\_[c\_i]{})= \_[-q\_[w\_i]{}\^]{}\^[q\_[w\_i]{}\^]{} \[locContrK\]\
F\_i\^[(L)]{}=\_[-q\_[w\_i]{}\^]{}\^[q\_[w\_i]{}\^]{}\
F\_i\^[(V)]{}=\_[-q\_[w\_i]{}\^]{}\^[q\_[w\_i]{}\^]{}
with $P_{c_i}=P_{L_i}+p_{c_i}$ and $Q_{c_i}=Q_{L_i}+q_{c_i}$.
This local control policy considers $p_{c_i}$ and $q_{c_i}$ to be specified and does not optimize the user consumption. Therefore, the optimal $p_{c_i}$ and $q_{c_i}$ values previously obtained by the solution of problem are set as inputs to the local algorithm . Finally, upon setting $q_{w_i}^m$, the power flows $P_i^m$ and $Q_i^m$ as well as the voltages $V_i^m$ can be found through solving the nonlinear power flow equations using Newton’s method [@loadflow]. Notice that the parameter $K \in \mathbb{R}$ in also needs to be experimentally set via trial and error.
Table \[table:compareVsTuritsyn\] lists the thermal losses and the maximum voltage deviation resulting from the stochastic programming approach and the local control policy for different values of $K$.
Method Loss (MW) $\max_{i,m}\frac{|V_i^m-V0|}{V0}$ (pu)
--------------- ----------- ----------------------------------------
Stoch. Progr. 0.0940 0.0500
$K=1.1$ 0.1927 0.3682
$K=1.2$ 0.1229 0.3098
$K=1.3$ 0.1005 0.2647
$K=1.4$ 0.1155 0.2266
$K=1.5$ 0.1616 0.1931
$K=1.6$ 0.2341 0.2018
$K=1.7$ 0.2674 0.2154
$K=1.8$ 0.2622 0.2166
$K=1.9$ 0.2539 0.2166
$K=3$ 0.2440 0.2160
: Objective Values, Max Voltage Deviation and Average Voltage Deviation
\[table:compareVsTuritsyn\]
![Empirical cumulative distribution function of the maximum voltage deviation, i.e., $\max_i \frac{|V_i-V_0|}{V_0}$, for the proposed stochastic model as well as local control policy with several values of $K$. Ideally, it is preferred to have the CDF plots to be on the left side of the solid $\epsilon$ line which corresponds to voltage deviations below $\epsilon$. []{data-label="fig:comp"}](Figures/fig6.eps)
The table reveals that for certain values of $K$ (such as $K=1.3$) the local control policy performs well in terms of thermal losses—partially due to the fact that the inputs to are the optimal real power consumptions. However, the local control policy fails to guarantee that voltage levels are within the $\epsilon$ range. The best $K$ in terms of voltage regulation was found with a grid search to be $K=1.54$, resulting in a voltage deviation of 0.18 pu, which violates the voltage constraint, and in thermal losses of 0.18 MW—which is two times greater than that of the proposed stochastic programming approach.
To get a closer look at the voltage regulation, the empirical cumulative distribution function (CDF) of the maximum voltage deviation across nodes, i.e., $\max_{i}\frac{|V_i-V_0|}{V_0}$, is plotted in Fig. \[fig:comp\] for the stochastic programming approach as well as for different values of $K$. The CDF is obtained by counting the number of scenarios in the online phase for which $\max_i \frac{|V_i-V_0|}{V_0}$ is less than $\delta V$ and dividing this number by the total number of 100 test scenarios. It is seen that the stochastic program, even by only considering just a few scenarios, guarantees the maximum voltage deviation to be less than the required threshold in all instances whereas the CDF induced by the local control policy reveals that voltage deviations may exceed the required threshold.
PV Penetration (%) 30 55 60 65 70
---------------------- ----- ---- ---- ---- ----
No. Infeasible cases 100 92 61 29 8
: Number of infeasible scenarios when reactive power compensation of PV inverters is not allowed
\[table:varcontrol\]
Case study on a feeder with shunt capacitors
--------------------------------------------
To show effectiveness of inverter reactive power control, the 56-node network of [@FNCL12 Fig. 2] that already includes shunt capacitors is selected for additional numerical tests. The details of the network are given in [@FNCL12 Table I]. Only non-elastic load is considered in this case study (i.e., $p_{c_i}=0$). The values for $P_{L_i}$’s and $Q_{L_i}$’s are calculated using the apparent peak load in [@FNCL12 Table I] increased by 50 % because the original network is lightly loaded, and assuming $\mathrm{PF}_i=0.94$ per node. All nodes are PV-enabled. By defining PV penetration level as the ratio of total PV apparent power capability to the total load, $s_{w_i}$’s per node are varied so that different PV penetration levels can be simulated.
Problem is first solved for 100 PV generation scenarios when reactive power compensation by PV inverters is not allowed (i.e., $q_{w_i}^m=0$). The number of infeasible scenarios is recorded in Table \[table:varcontrol\]. For the same scenarios, problem is solved with reactive power compensation capability as in . In this case, all scenarios were feasible.
When inverters are not allowed to compensate for reactive power, voltages may drop below the required threshold, while reactive power provided by the inverters prevents large voltage drops and increases system reliability. The voltage profile for a scenario with $w_i=\bar{w}_i$ and $\sum_i\bar{w}_i=0.5\sum_i P_{L_i}$ (i.e., 50 % penetration) is also plotted in Fig. \[fig:varControlcomp\] to illustrate this effect.
![Voltage profile $\frac{|V_i-V_0|}{V_0}$ for a fixed scenario. When inverters do not provide reactive power compensation, the voltage drop may exceed the threshold; while allowing reactive power compensation can prevent this drop.[]{data-label="fig:varControlcomp"}](Figures/fig7.eps)
Effect of stepsize $\rho$ in convergence of ADMM
------------------------------------------------
This subsection numerically investigates the effect of the stepsize $\rho$ on the convergence of the ADMM algorithm. In particular, the network setup in Subsection \[sec:numtesta\] is considered here, where $50 \%$ of nodes are capable of PV generation with $s_{w_i}=0.15$ MVA, and $s_{w_{50}}=1.5$ MVA for the terminal node at one of the laterals. Moreover, $\frac{\bar{w}_i}{w_i^{\max}}=0.75$ is selected. The remaining parameters are selected as described in Subsections \[sec:numtesta\] and \[sec:numtestb\]. Initially, 1000 scenarios are generated, which are subsequently reduced to $M=7$ scenarios. Problem is then solved using the ADMM algorithm of Section \[sec:algorithm\] for three constant stepsizes, namely $\rho=2, 20, 100$, and one adaptive stepsize ($\rho_{\mathrm{adaptive}}$) according to the following rule [@BoPaChPeEc-FnT11]:
[rCl]{} (k+1):=
2 (k) & ||r(k)|| > 10 ||s(k)||\
& ||s(k)|| > 10 ||r(k)||\
(k) &
where $\rho(1)=100$, and the primal and dual residuals, i.e., $r(k)$ and $s(k)$, are respectively calculated via and .
![Primal residual per ADMM iteration for various stepsizes ($\rho=2,20,100$ and $\rho_{\mathrm{adaptive}}$).[]{data-label="fig:diffrhoRerr"}](Figures/fig8.eps)
![Dual residual per ADMM iteration for various stepsizes ($\rho=2,20,100$ and $\rho_{\mathrm{adaptive}}$).[]{data-label="fig:diffrhoSerr"}](Figures/fig9.eps)
The resulting primal and dual residuals per iteration are respectively given in Fig. \[fig:diffrhoRerr\] and \[fig:diffrhoSerr\] for the various values of $\rho$. All choices of $\rho$ (including the adaptive one) perform well in terms of reducing the primal residual, while $\rho=20$ and $\rho=100$ perform poorly in minimizing the dual residual \[potentially due to the multiplication in \]. Moreover, $\rho=2$ and $\rho_{\mathrm{adaptive}}$ have a similar performance, noting that the $\rho_{\mathrm{adaptive}}$ reaches 1.56 upon convergence.
The objective value per iteration is shown in Fig. \[fig:diffrhoObj\] for the various values of $\rho$. For $\rho=100$, an accurate objective value is not found within the 3000 iterations. The exactness of the SOCP relaxation, i.e., $\max_{i,m} |(P_i^m)^2+(Q_i^m)^2-v_i^ml_i^m|$, is depicted in Fig. \[fig:diffrhoSOCP\]. This value eventually approaches zero for all values of $\rho$; however, the progress is rather slow after 1000 iterations in the case of $\rho=2$ or $\rho_{\mathrm{adaptive}}$.
![Comparison of objective value per ADMM iteration for various stepsizes ($\rho=2,20,100$ and $\rho_{\mathrm{adaptive}}$).[]{data-label="fig:diffrhoObj"}](Figures/fig10.eps)
![Exactness of the SOCP relaxation, i.e., $\max_{i,m} |(P_i^m)^2+(Q_i^m)^2-v_i^ml_i^m|$, per ADMM iteration for various stepsizes ($\rho=2,20,100$ and $\rho_{\mathrm{adaptive}}$).[]{data-label="fig:diffrhoSOCP"}](Figures/fig11.eps)
Effect of number of scenarios on convergence
--------------------------------------------
This section investigates the convergence of ADMM under larger number of scenarios. Problem is solved for the network of Fig. \[fig:twolateral\]. The penetration level is set to $50 \%$, while $s_{w_i}=0.15$ (MVA) is selected for PV-enabled nodes, and $s_{w_{50}}=1.5$ (MVA) is chosen for the terminal node at one lateral. Problem is solved using $M=100$ and $M=500$ randomly generated equiprobable scenarios with $\frac{\bar{w}_i}{w_i^{\max}}=0.75$. Table \[table:scalability\] lists the parameters indicating the convergence in these two test cases. The algorithm scalability is not necessarily dependent on the number of scenarios, but rather on the network structure and the specific power injections per node. As long as each node is capable of performing individual updates for the specified number of scenarios, the algorithm will converge.
M 100 500
----------------------------------------------- ----------------------- -----------------------
Number of iterations 3059 4100
Primal residual $r(k)$ $9.9 \times 10^{-6}$ $2.23 \times 10^{-5}$
Dual residual $s(k)$ $1.24 \times 10^{-6}$ $4.4 \times 10^{-5}$
$\max_{i,m} |(P_i^m)^2+(Q_i^m)^2-v_i^ml_i^m|$ $8.24 \times 10^{-4}$ $9.38 \times 10^{-4}$
: Convergence of ADMM for Increased Number of Scenarios
\[table:scalability\]
Conclusion {#sec:conc}
==========
This paper developed a stochastic power management framework for radial distribution networks with high levels of PV penetration. Decision variables included real power consumption of programmable loads in user nodes and the reactive power generation or consumption of the PV inverters. The uncertain real power injections of the user buses were modeled as random variables taking values from a finite number of scenarios. A convex stochastic optimization program was formulated to minimize the sum of negative utility, the expected value of cost of power provision, and the expected thermal losses subject to the SOCP relaxation of the power flow equations, power consumption constraints, and voltage regulation specifications. A decentralized method using the ADMM was developed to solve the stochastic program, in which the updates per node and per scenario turn out to be in closed form.
Proof that at most one of the two variables $P_{0+}^m$ and $P_{0-}^m$ is nonzero {#sec:appendixa}
================================================================================
Let $\tilde{P}_{0+}^m$ and $\tilde{P}_{0-}^m$ be the solution of with $C(P_0^m)$ replaced by $aP_{0+}^m-bP_{0-}^m$. Suppose that $\tilde{P}_{0+}^m > 0$ and $\tilde{P}_{0-}^m > 0$. Then, $\tilde{P}_{0+}^m-\epsilon$ and $\tilde{P}_{0-}^m- \epsilon$ are feasible for sufficiently small $\epsilon > 0$ and give an objective $a\tilde{P}_{0+}^m-b\tilde{P}_{0-}^m-(a-b)\epsilon$ which is strictly smaller than $a\tilde{P}_{0+}^m-b\tilde{P}_{0-}^m$ since $a > b$. This is a contradiction.
On the exactness of the SOCP Relaxation {#sec:appendixb}
=======================================
Based on [@GaLiToLo2015], this appendix presents conditions under which the optimal solution to problem satisfies with equality. In order to state the results, some notations are introduced next.
Given net nodal consumptions $(P_{L_i}+p_{c_i}-w_i^m, Q_{L_i}+q_{c_i}-q_{w_i}^m)$, the solution to the `LinDistFlow` approximation of the power flow equations presented in – for scenario $m$ is given by:
[rCl]{} \_i\^m(\_[c]{}) &=&\_[j: i \_j]{} (P\_[L\_j]{} +p\_[c\_j]{}-w\_j\^m) \[eqn:phatpcipl\]\
\_i\^m(\_[c]{}, \_[w]{}\^m) &=& \_[j: i \_j]{}(Q\_[L\_j]{}+q\_[c\_j]{}-q\_[w\_j]{}\^m)\[eqn:qhatqciql\]\
\_i\^m(\_[c]{},\_[w]{}\^m) &=& v\_0 -2 \_[j \_i\^+]{} \[r\_j\_j\^m+x\_j\_j\^m\]
where $q_{c_j}=\left(\sqrt{\frac{1}{\mathrm{PF}_j^2}-1}\right) p_{c_j}$, $\mathcal{P}_j$ is the unique path from the root node to node $j$ (including node $j$), and $\mathcal{P}_j^+$ is $\mathcal{P}_j \backslash \{0\}$. Also, $\mathbf{p}_{c}$ and $\mathbf{q}_{w}^m$ collect the corresponding values of all nodes.
Now, consider the following modifications of problem :
1. Assume $C(P_0^m)$ to be strictly increasing in $P_0^m$;
2. remove the upper bounds on the current magnitudes;
3. consider shunt capacitors to be modeled by fixed reactive power injections and independent of voltage magnitudes;
4. set $K_{\mathrm{Loss}}=0$; and
5. enforce the constraint $\breve{v}_i^m(p_{c_i},q_{w_i}^m) \le (1+\epsilon) ^2v_0$, instead of $v_i^m \le (1+\epsilon)^2 v_0^2$.
In addition, following [@GaLiToLo2015], define for every scenario $m$, $$\underline{A}_i^m:= I + \frac{2}{(1-\epsilon)^2v_0} \begin{bmatrix} r_i \\ x_i \end{bmatrix} \begin{bmatrix} \breve{P}_i^{m-} (\mathbf{p}_{c}^{\min}) & \breve{Q}_i^{m-} (\mathbf{p}_{c}^{\min}, \mathbf{s}_{w}) \end{bmatrix}$$ where $a^-=\min \{a,0\}$. Also $\mathbf{p}_{c}^{\min}$ and $\mathbf{s}_{w}$ collect all the corresponding values per node. Then, the main result is that the modified SOCP problem is exact if the following condition holds for every scenario $m$:
[rCl]{} \_[j=s+1]{}\^[t-1]{} \_[d\_j]{}\^m
r\_[d\_t]{}\
x\_[d\_t]{}
> 0, 2 t n\_d, 0 s t-2 \[eqn:sufficient\]
where $n_d=|\mathcal{P}_{i}|$ for all leaf nodes $i$ (i.e., nodes such that $\mathcal{C}_i=\emptyset$) and $d_j \in \mathcal{P}_{i}$ .
A sketch of proof based on [@GaLiToLo2015] is presented next. For given $p_{c_i}$ and $q_{w_i}^m$, we can establish that if $P_i^m$, $Q_i^m$ and $v_i^m$ satisfy –, then the following holds (equivalent to [@GaLiToLo2015 Lemma 1]): $$\begin{aligned}
\breve{P}_i^m(\mathbf{p}_{c}) \le P_i^m \\
\breve{Q}_i^m(\mathbf{p}_{c}, \mathbf{q}_{w}^m) \le Q_i^m \\
\breve{v}_i^m(\mathbf{p}_{c},\mathbf{q}_{w}^m) \ge v_i^m. \end{aligned}$$ Assume we solve problem and it turns out that is not satisfied with equality in at least one node and one scenario. Here we show that we can construct a feasible solution with a lower objective value. Call that scenario $m$ and label the node $K$. Further, without loss of generality, we can assume that the node $K$ is the $k+1$’th node in $\mathcal{P}_{d_n}$ where $d_n$ is a leaf node and $|\mathcal{P}_{d_n}|=n+1$. Path $\mathcal{P}_{d_n}$ is illustrated as follows: $$\begin{aligned}
0 \xrightarrow{1}d_1 \xrightarrow{2} d_2 \ldots \xrightarrow{k} d_k=K\xrightarrow{k+1} \ldots \xrightarrow{n} d_n. \end{aligned}$$ The current solution will be called $s=(\mathbf{P}, \mathbf{Q}, \mathbf{v}, \mathbf{l}, \mathbf{p}_{c}, \mathbf{q}_{w})$, where $\mathbf{P}, \mathbf{Q}, \mathbf{v},\mathbf{l},\mathbf{p}_{c}, \mathbf{q}_{w}$ are vectors collecting all the corresponding values per node and per scenario. Solution $s$ has the the following property: $$\begin{aligned}
\frac{(P_K^m)^2 + (Q_K^m)^2}{v_K^m} & < l_K^m, \\
\frac{(P_{d_i}^m)^2 + (Q_{d_i}^m)^2}{v_{d_i}^m} &= l_{d_i}^m \quad \text{ for } i=1,\ldots, k-1.\end{aligned}$$
Algorithm \[alg:2\] constructs a new feasible solution $s'=(\mathbf{P}',\mathbf{Q}',\mathbf{v}',\mathbf{l}',\mathbf{p}_{c}',\mathbf{q}_{w}')$, which will be proved to have a lower objective. In particular, the new solution $s'$ has the following properties: $$\begin{aligned}
&l_{K}^{'m} < l_{K}^m \Rightarrow \Delta l_{K}^m= l_{K}^{'m}- l_{K}^m < 0, \\
&\frac{ (P_i^{'m})^2 +(Q_i^{'m})^2}{v_i^m} \le l_i^{'m} \text{ for all } i \in \mathcal{N} \backslash \{0\} \label{eqn:newfeasibleproperty2}.\end{aligned}$$
Initialization: $s' \leftarrow s$ , $ v'_0 \leftarrow v_0$ , $\mathcal{N}_{\mathrm{visit}}=\{0\}$ . Backward sweep: **For** $i=k,\ldots,1$ **do**
[rCl]{} &l\_[d\_i]{}\^[’m]{} & \[eqn:l’\]\
&P\_[d\_[i-1]{}]{}\^[’m]{} &\_[j \_[d\_[i-1]{}]{}]{} P\_[j]{}\^[’m]{}+r\_[j]{}l\_[j]{}\^[’m]{}+P\_[L\_[d\_i]{}]{}\^m\
&& + p’\_[c\_[d\_i]{}]{}-w\_[d\_i]{}\^m\
&Q\_[d\_[i-1]{}]{}\^[’m]{} &\_[j \_[d\_[i-1]{}]{}]{} Q\_[j]{}\^[’m]{} + x\_[j]{}\^[m]{}l\_[j]{}\^[’m]{}+ Q\_[L\_[d\_i]{}]{}\^m\
&&+ () p’\_[c\_[d\_i]{}]{}-q\_[w\_[d\_i]{}]{}\^[’m]{}.
Forward sweep: **While** $\mathcal{N}_{\mathrm{visit}} \neq \mathcal{N}$ **do** $$\begin{aligned}
&\text{find } j \notin \mathcal{N}_{\mathrm{visit}}, i \in \mathcal{N}_{\mathrm{visit}} \text{ such that } j \in \mathcal{C}_i,\\
&v_{j}^{'m}=v_{i}^{'m}-2r_iP_{j}^{'m}-2x_iQ_{j}^{'m}-(r_i^2+x_i^2)l_{j}^{'m}, \\
&\mathcal{N}_{\mathrm{visit}} \leftarrow \mathcal{N}_{\mathrm{visit}} \cup \{j\}. \end{aligned}$$
At this point, by proving that $v_i^{'m} \ge v_i^m$, we can use to show that $s'$ is feasible. Furthermore, by additionally proving that $P_0^{'m} < P_0^{m}$, the new solution will have a smaller objective value, which yields a contradiction. These two facts are proved next.
Define $\Delta P_i^m = P_i^{'m} - P_i^{m}$ and $\Delta Q_i^m = Q_i^{'m} - Q_i^{m}$. We can establish the following on path $\mathcal{P}_{d_k}$: $$\begin{aligned}
\begin{bmatrix}
\Delta P_{d_{i-1}}^{m} \\
\Delta Q_{d_{i-1}}^{m}
\end{bmatrix} = B_{d_i}^m\begin{bmatrix} \Delta P_{d_i}^m \\ \Delta Q_{d_i}^m \end{bmatrix}\end{aligned}$$ where for $i=k, \ldots, 1$: $$\begin{aligned}
B_{d_i}^m=( I + \frac{2}{v_{d_i}^m} \begin{bmatrix} r_{d_i} \\ x_{d_i} \end{bmatrix} \begin{bmatrix} \frac{P_{d_i}^{'m}+P_{d_i}^m}{2} & \frac{Q_{d_i}^{'m}+ Q_{d_i}^m}{2} \end{bmatrix} )\end{aligned}$$ and $$\begin{aligned}
\begin{bmatrix} \Delta P_{d_k}^m \\ \Delta Q_{d_k}^m\end{bmatrix}= \begin{bmatrix} r_{d_k} \\ x_{d_k} \end{bmatrix} \Delta l_K^m.\end{aligned}$$ Thus we can write for $s=k-2, \ldots, 0$ $$\begin{aligned}
\begin{bmatrix}
\Delta P_{d_s}^{m} \\
\Delta Q_{d_s}^{m}
\end{bmatrix}= \prod\limits_{i=s+1}^{k-1} B_{d_i}^m \begin{bmatrix} r_{d_k} \\ x_{d_k} \end{bmatrix} \Delta l_{d_k}^m.\end{aligned}$$ Observe that $B_{d_i}^m - \underline{A}_{d_i}^m = \begin{bmatrix} r_{d_i} \\ x_{d_i} \end{bmatrix} b_{d_i}^T$, where $$\begin{aligned}
b_{d_i}=\begin{bmatrix}\frac{P_{d_i}^{'m}+P_{d_i}^m}{2v_{d_i}^m} -\frac{\breve{P}_{d_i}^{m-} (\mathbf{p}_{c}^{\min})}{(1-\epsilon)^2v_0} \\ \frac{Q_{d_i}^{'m}+ Q_{d_i}^m}{2v_{d_i}^m}-\frac{\breve{Q}_{d_i}^{m-} (\mathbf{p}_{c}^{\min}, \mathbf{s}_w)}{(1-\epsilon)^2v_0} \end{bmatrix} \ge 0,\end{aligned}$$ for $i=k-1, \ldots, 1$. Therefore, [@GaLiToLo2015 Lemma 3] can be used to show that $P_{d_i}^{'m} < P_{d_i}^m$ and $Q_{d_i}^{'m} < Q_{d_i}^m$ for $i=k-1, \ldots,0$. Notice that $d_0=0$ and hence the following holds: $$\begin{aligned}
P_0^{'m} < P_0^m \label{eqn:p0p0prime}.
\end{aligned}$$
Next, we show that $v_i^{'m} \ge v_i^m$. Define $\Delta v_i^m = v_i^m-v_i^{'m}$. For $A_i \notin \mathcal{P}_{K}$, we have that $$\begin{aligned}
\Delta v_{i}^m-\Delta v_{A_i}^m = -2 r_i \Delta P_i^m- 2 x_i \Delta Q_i^m -(r_i^2+x_i^2) \Delta l_i^m=0.\end{aligned}$$ For $A_i \in \mathcal{P}_K$, it holds that $$\begin{aligned}
\Delta v_{i}^m-\Delta v_{A_i}^m =- 2 r_i \Delta P_i^m- 2 x_i \Delta Q_i^m - (r_i^2+x_i^2) \Delta l_i^m \ge 0. \end{aligned}$$ Adding these inequalities over path $\mathcal{P}_{d_k}$ yields $$\begin{aligned}
\Delta v_{i}^m - \Delta v_{0} \ge 0 \Rightarrow \Delta v_{i}^m \ge 0, \text{ for } i \in \mathcal{P}_{d_k},\end{aligned}$$ which proves that $v_i^{'m} \ge v_i^m$. We have shown that solution $s'$ is feasible. Due to and the assumption that $C(P_0)$ is strictly increasing, the new solution $s'$ has a smaller objective; this is a contradiction.
We have derived sufficient condition under which the SOCP relaxation is exact for a modified problem close to problem . This sufficient condition can be checked a priori. Notice that modifications 1–3 and 5 are exactly the same as the ones proposed in [@GaLiToLo2015] where the problem is not stochastic.
Condition is stated per scenario. It is also possible to state a single sufficient condition that does not depend on $m$. Specifically, replace $w_i^m$’s with $\max_{m} w_i^m$ in ; then, the resulting matrix $\underline{A}_i$ does not depend on $m$, and condition is stated with $\underline{A}_i$ instead of $\underline{A}_i^m$. This latter condition is a more stringent sufficient condition that accounts for the lowest-consumptions. We have numerically verified that this latter sufficient condition holds for the network in the numerical tests.
Closed-form updates for $\tilde{P}_i^m,\tilde{Q}_i^m, \tilde{v}_i^m, \tilde{l}_i^m$ {#sec:appendixc}
===================================================================================
Let $z_1=\tilde{P}_i^m$, $z_2=\tilde{Q}_i^m$, $z_3=\sqrt{\frac{|\mathcal{C}_i|+1}{2}}\tilde{v}_i^m$ and $z_4=\tilde{l}_i^m$. The z-update for these variables will be equivalent to solving the following optimization problem:
\[eqn:pqvlupdate\]
[rl]{} &\_ \_[i=1]{}\^4 (z\_i\^2+c\_iz\_i)\
&\
&z\_3\^ z\_3 z\_3\^ \[eqn:z3bounds\]\
& k\^2z\_4 \[eqn:socpbound\]\
&z\_4 z\_4\^ \[eqn:z4bound\]
where
[lll]{} k\^2=\
z\_3\^=(1+)\^2v\_0\
z\_3\^=(1-)\^2v\_0\
c\_1=-(P\_i\^m+\_i\^m+)\
c\_2=-(Q\_i\^m+\_i\^m+)\
c\_3=-(v\_i\^m+\_[j \_i]{}\_j\^m+ )\
c\_4=-(l\_i\^m+\_i\^m+).
Problem without considering constraint is solved in closed-form in [@PeLo14 Appendix I]. Using a similar approach, we develop a methodology to obtain a closed-form solution to problem , when it includes constraint .
Let $\bar{\lambda}, \underline{\lambda}, \mu$ and $\gamma \ge 0$ be Lagrange multipliers corresponding to , and respectively. The KKT conditions for problem are:
\[eqn:KKT\]
[lll]{} 2z\_1+c\_1+2=0\
2z\_2+c\_2+2=0\
2z\_3+c\_3-+|- =0\
2z\_4+c\_4-k\^2+ =0\
|(z\_3-z\_3\^)=0\
(z\_3\^-z\_3)=0\
(-k\^2z\_4)=0\
(z\_4-z\_4\^)=0 \[eqn:z4comp\].
The closed-form solution for the KKT conditions in is obtained by enumerating the cases for $\gamma$:
Case 1: $\gamma=0$
-------------------
In this case, as detailed in [@PeLo14 Appendix I], there exists a unique solution to which can be obtained in closed form. If the obtained closed-form solution satisfies $z_4 \le z_4^{\max}$ then it is also optimal for . Otherwise, we need to proceed to the next case.
Case 2: $\gamma > 0$
--------------------
In this case, using , we can establish that $z_4^*=z_4^{\max}$. Now we examine possible choices for $\mu$.
### $\mu=0$
[lll]{} z\_1\^\*=-, z\_2\^\*=-, z\_3\^\*=-\_[z\_3\^]{}\^[z\_3\^]{}.
### $\mu >0$, $z_3^{\min} < z_3 < z_3^{\max}$
[lll]{} z\_3\^\*=(az\_3\^2+bz\_3+c=0)\
\
a=k\^2z\_4\^(2+)\^2\
b=c\_3(2+)k\^2z\_4\^-(c\_1\^2+c\_2\^2)\
c=c\_3\^2\
z\_1\^\*=-\
z\_2\^\*=z\_1\^\*.
### $\mu >0$, $z_3^{\min} =z_3$
[lll]{} z\_3\^\*=z\_3\^\
z\_1\^\*=-\
z\_2\^\*=-z\_1\^\*\
\^\*=-z\_3\^.
### $\mu >0$, $z_3^{\max} =z_3$
[lll]{} z\_3\^\*=z\_3\^\
z\_1\^\*=-\
z\_2\^\*=-z\_1\^\*\
\^\*=-z\_3\^.
[^1]: Manuscript received February 9, 2015; revised June 12, 2015 and October 30, 2015; accepted January 4, 2016.
[^2]: The authors are with the department of electrical and computer engineering, the University of Texas at San Antonio, San Antonio, TX 78249, USA (emails: aju084@my.utsa.edu, nikolaos.gatsis@utsa.edu)
[^3]: This material is based upon work supported by the National Science Foundation under Grant No. CCF-1421583.
|
---
abstract: 'Vacuum polarization in a strong field, such as Hawking radiation from black holes, is constrained by the directional metric of strong field vectors. Relative localization of the quanta and the Schwarzschild radius is examined. Concept of thermal equilibrium of excited vacuum is critically analyzed and judged as dis-equilibrium, because virtual modes or particles are mutually independent. The result shows that primordial micro-size black holes ([P$\mu$BHs]{}) cannot produce much Hawking radiation via fermions, bosons or thermal photons. Primordial black holes produced in the early epoch of the Big Bang should be very stable over Hubble time and consequently, may constitute a significant part of the cold dark matter in the structural formation of the universe. [P$\mu$BHs]{} should be gravitationally highly-condensed as missing mass around galaxies and galaxy clusters, and remain very difficult to observe.'
author:
- Yoshiyuki Takahashi
date: 'May 6, 2004'
title: Stability of primordial black holes
---
Introduction
============
Understanding the nature of dark matter as well as the missing mass around galaxies and galaxy clusters is one of the most serious and well-publicized problems in contemporary astronomy, cosmology, and particle physics. This paper provides a revision of the predominant quantum physical description of primordial black holes (PBHs), and proposes that PBHs should be considered in a renewed light as possible constituents of missing mass and of dark matter in general. In the past, massive neutrinos were thought to be the prospective candidates for constituting missing mass and hot dark matter. However, upon closer experimental examination of solar neutrinos, the mass of neutrinos became bound to a very small value on the order of 0.01-1 eV/c$^2$, rendering the massive neutrino hypothesis much less tenable. Many hypothetical alternatives have since been proposed with both baryons (dark stellar objects) and non-baryons. The former includes brown dwarfs, extended thin gases, stellar mass black holes and dead quasars. The latter includes axions, weakly interacting massive particles (WIMPs), super-symmetric particles, and a plethora of exotic hypothetical particles.
The luminous baryonic mass density being observed for universe ($\rho_b$) is only 3% of the critical mass density $\rho_c=3 h^2 / 8 \pi G$ required for the flat universe model, where H denotes the Hubble constant and G is the gravitational constant. About $\sim$35% of $\rho_c$ is accounted for by gravitational missing mass and is regarded as Cold Dark Matter (CDM) bound to galaxies and galaxy clusters. Dark energy is considered for up to the remaining 65% of the critical mass. It has recently become popular due to observational inference of an accelerating universe model from the data analyses of receding supernovae and of small-scale fluctuations of cosmic microwave background radiation (CMBR) that also indicate the flatness of universe. Dark energy is a new name for the cosmological constant used in Lemaitre solutions of Einstein equations and in inflation models, and for the critical mass of the cosmology for the Einstein-de Sitter flat Universe. However, it has negative pressure ($P_{\Lambda} = -\rho_{\Lambda} < 0$) \[or in a more original definition: negative mass density\] and has a characteristic repulsive force. This is opposite to what is gravitationally required in order to account for the local missing mass in actual galaxies and Clusters.
Primordial black holes (PBHs) have not been so seriously considered in these last few decades as candidates for either the galactic missing mass, or for dark matter in general. The reason for this vague status is due mostly to a belief that they should have evaporated very quickly after the Big Bang as a result of Hawking radiation \[1\], and therefore, that they can not be around us still. However, there is no observational evidence \[2, 3\] for the type of explosive gamma ray or radio \[4\] emission from the rapid decay of micro black holes which Hawking radiation mechanisms predict as being observable \[5, 2\]. Moreover, as shown below, a relevant distinction of quantum requirements for the metric of the Schwarzschild radius and the evaporation mechanism at a black hole event horizon reveals that Hawking radiation cannot significantly occur from PBHs, or from primordial micro-sized black holes ([P$\mu$BHs]{}) in particular.
Hawking thermal radiation
=========================
The fundamental quantum nature of radiation from a vacuum in strong gravitational fields of BHs was first pointed out in a theoretical fashion by Hawking \[1\], with an intrinsic and unstated assumption that acceleration occurs uni-directionally. This radiation mechanism was then applied to primordial micro-size black holes ([P$\mu$BHs]{}) without much concern about the size of the BH itself. However, the metric changes from a unidirectional field geometry to an omni-directional radial field when the Schwarzschild radius decreases to the size of a Compton wavelength ($\lambda$) of quanta; in this situation, the acceleration vector differs significantly from that of the unidirectional case. Hawking gave the mass-loss formula from his thermal treatment \[1\] based on a unidirectional geometry without a categorical separation of the mass (M) of black holes,
$$\frac{dM}{dt}=10^{-26}M^{-2}\; [g s^{-1}]
\label{eq1}$$
According to this, the lifetime of a black hole with mass ($M$), measured in units of grams ($M_g$), would consequently be: $$\tau_{Hawking} = 3 \times 10^{-27} M_g^3 \; [s]
\label{eq2}$$
This is the well-known basis for the concept that [P$\mu$BHs]{} with a mass less than $10^{14}$ g should have evaporated much earlier than the Hubble time ($\approx 3\times10^{16}$ s), during which the universe is believed to have evolved from the Big Bang to the present. Microscopic analysis of the metric of the [P$\mu$BHs]{} shows a different picture: [P$\mu$BHs]{} do not evaporate as fast as what had been previously thought. The responsible mechanism we provide in this paper is a well-known, simple quantum requirement: it is, namely, the unidirectional length scale of one Compton wavelength, required for pair materialization from a virtual state in a vacuum. Hawking indeed cautioned in the end of his first paper \[1\] that his scheme was ignoring quantum fluctuations on the metric and that these might alter the picture.
The original Hawking paper \[1\] considered the case of thermal black body radiation from a plasma of many particles at a thin circumference around the Schwarzschild radius ($R_S$). The area of this thin shell is compared to entropy ($S = (c^3 k/2 G \hbar) 4\pi R_S^2$ in Schwarzschild metric), and the temperature ($T$) is defined by a thermodynamical relationship, $T = E/S \propto Mc^2/4\pi R_S^2$, leading to temperature $kT =\hbar c^3/8\pi k G M$ and the luminosity $L = 4\pi R^2\sigma T^4\propto M^{-2}$. The temperature-equivalent Planckian spectrum was later verified by quantum field theories in curved spacetime \[6, 7, 8\]. However, there is a prerequisite in order for this thermal equilibrium scheme and Eqs. \[eq1\]-\[eq2\] to be valid: the Compton wavelength ($\lambda$) of radiated quanta has to be smaller than the hole radius ($R_S$) so that entropy can be defined for the black hole geometry \[9\]. Moreover, the established quantum physics principles and microscopic considerations of the cutoff sphere do not support the idea of itself around the Schwarzschild radius of small BHs, because only one fermion of each species is allowed at any given time within a sphere of the Compton wavelength radius, which entirely engulfs a [P$\mu$BHs]{} when Schwarzschild radius ($R_S$) is smaller than $\lambda$ . If not radiated as a thermal black body, photons must originate from the electromagnetic annihilation of particle pairs. This process is negligibly small when the real-particle population of fermions or bosons is limited to one or much less (as described later).
We shall analyze the nature of virtual particles in particle picture. Before going into that description, we need to examine the approach from quantum field theory and the applicability of thermal concepts for vacuum states of wave modes or particles. This analysis includes the essential condition of equilibrium, which is founded on the basis of (1) [*mutual exchange of energies*]{} and (2) [*the existence of a large number of particles (i.e. numerous enough to be statistically meaningful) over a significant time duration*]{} that allows such exchanges. Vacuum states or virtual particles, though satisfying the second condition, have extremely too short a duration of real modes, and cannot satisfy the first condition. These bodies interact only with external gravity but not with other bodies, and should be regarded as so many free bodies in dis-equilibrium .
Support by quantum field theory
-------------------------------
The postulated mechanism of Hawking radiation was refined later by quantum field theory \[6, 8\], which used the curved spacetime of general relativity and Bogoliubov transformations \[10\] of the Fock representation \[11\] of the wave modes in a vacuum bounded by an event horizon. The boiling of vacuum states at the Schwarzschild radius has been considered by many to be a general consequence of the vacuum modes of a very strong field in curved spacetime; it occurs when a vacuum is geometrically cut off from the universe by the infinitely large plane of an event horizon so that a wave function does not propagate to one side of the universe. However, this geometry ignores the fact that the surface is actually spherical with finite radius, which does not cut off vacuum modes entirely from universe. The quantum field refinements \[6, 7, 8\] treat this actual 3D spatial geometry of the cutoff boundary for 3D (r,t) waves as if it were a 1-D wall, and as a uni-directionally accelerating configuration in space, irrespective of the size or the mass of a black hole. This goes against the basic notion that a curved spacetime configuration is supposed to be really used for all the spatial coordinate vectors of motion and gravitation. Consequently, these past schemes may yield a significant over-estimation in the mass-loss rate by a factor $\sim 1/M^2$.
“Thermal” concept of vacuum – dis-equilibrium and independence
--------------------------------------------------------------
The authentic (or standard) thermal interpretation of Planckian spectrum of quantum field theory is not free from imperfection. Despite a lot of sophisticated wave transformations being used in quantum field theoretical treatments, the whole scheme has more of a geometric nature in terms of fitting the wavelength ($\lambda$) modes of massless quanta (photons) to the size of the circumference ($\lambda_R=2 \pi R_S$) of the Schwarzschild sphere.
Is it really heat that is generated by moving bodies in vacuum Other than virtual massive quanta, there are no real bodies or photons in negative energies. Temperature of vacuum $T_0$ is always zero in classical physics when there are no real bodies in motion. $T_0$ in quantum-field picture is different. It has a positive non-zero value ($T_0 = \sqrt{<E_e^2>}/k > 0$) due to the transient energy-square average of [**virtual electrons**]{}. Because they are not real but in virtual states and the time is too brief to appear as real particles, they can not exchange energies with other virtual particles. In this sense, although they have non-zero (temperature-like) energy variance, they do not form any equilibrium state like that of an ordinary thermal body. They may have negative-energy Fermi motion, but this vanishes when they go back in vacuum states. Virtual particles are thus independent, and they are not really collectively and thermally connected with any neighbouring quanta.
The field energy (conventionally interpreted as temperature) can indeed increase with decreasing $R_S$ until $R_S \sim \lambda_e/2 \pi = \lambdabar_e$. Gravitational fields are effective in terms of causing a uni-directional acceleration for a brief transient time, so long as $R_S \gg \lambdabar_e$. However, when $R_S \leq \lambdabar_e$, virtual electrons are no more effectively excitable by a gravitational acceleration of a strong field due to the 3-D random fluctuation of the motional vector of electrons relative to the field vector. Thus, vacuum quantum temperature ($T_e$) ceases to increase when $R_S$ reaches $\lambdabar_e$. Consequently, thermal emission of photons by black body saturates to a constant value ($\sigma T_{e0}^4$, with $k T_0 = m_e c^2$ = 511 keV), as discussed in 2.1 (as over-estimation by a factor of $\sim M^{-2}$). We will discuss more in detail in the later Section 7.
The temperature ($T_m$) of different species of quanta with mass (m) at the black hole horizon is inversely proportional to m, $T_m \propto T_e (me/m)$ (see Appendix A). Virtual quanta with mass larger than that of electrons have a much lower temperature $T_m$ in the same gravitational field due to their heavier mass. The energy loss by their black-body radiation is much lower than their energy loss by electrons by a factor of $(me/m)^4$. In addition, even if we accept a sense of thermal picture, they have to be in dis-equilibrium (i.e., independent and remain at their own low temperature). For example, luminosity ($L_{Top}$) by the temperature of a top-quark (whose mass is $3.4 \times 10^5 m_e$) emits $1.3 \times 10^{22}$ times less energy than that ($L_e$) of electrons at $M_g = 10^{17}$ due to dis-equilibrium of the thermal reality of virtual fields. The largest of the emission rate by top-quarks is ($L_{Top}/L_e) = 3 \times 10^{-6}$ at $M_g = 7.1 \times 10^{11}$, for which the Compton wavelength of the top quark equals the Schwarzschild radius, and the top-quark temperature saturates, too. Hence, temperature of higher-mass quanta cannot play a major role for the lifetime of [P$\mu$BHs]{} in dis-equilibrium of vacuum temperatures.
Gravitational potential energy
------------------------------
Furthermore, as will be shown in Section 5, the initial assumption of total field energy being used by Hawking was not gravitational field energy: the total mass-energy $M c^2$ was used instead. This is approximately fine for large black holes, but is not accurate for small ones. This alone causes a large difference in temperature definition by a factor of M for [P$\mu$BHs]{}. As a result, the mass-loss rate by black body luminosity ($\propto T^4$) in the Hawking interpolation scheme is extremely overestimated by a factor of $M^{-4}$ for small [P$\mu$BHs]{}.
Geometrical overview and limitation by Pauli blocking for electrons
===================================================================
We examine the geometry first in light of quantum physics constraints. In a semi-classical Dirac-Schwinger particle picture, a virtual electron-pair is in its negative energy state, and it can become on-shell only after gaining enough energy by (coherent) interactions over a volume defined by $\lambdabar_e^3$ and a duration $t =\lambdabar_e /c \approx 1.3 \times10^{-21}$ s. We recall in the first place that there is an important quantum principle that strictly prohibits the rapid evaporation of sub-[P$\mu$BHs]{}. Pauli blocking applies to Fermi-Dirac quanta within one Compton-wavelength 4-volume. Any quantum fluctuation within one Compton-wavelength cannot produce any more than one electron ($10^{27}$ g) per $10^{-21}$ s. A [P$\mu$BH]{} with a mass less than $10^{17}$ g has a Schwarzschild radius, $R_S =2GM/c^2$, less than $\lambdabar_e \sim10^{11}$ cm. Only one electron can exist in 10$^{-21}$ s over its entire event horizon. This leads to the conclusion that there is in the [P$\mu$BH]{} horizon no plasma that can be regarded as a blackbody for radiation. This strictly sets a on the mass-loss rate for [P$\mu$BHs]{} to
$$-\frac{dM}{dt}\mid_{MAX} = m_e (m_e c^2 /\hbar) < 10^{-6}\; [g s^{-1}]
\label{eq3}$$
regardless of mass M (for $M < 10^{17}$ g). The corresponding absolute minimum lifetime is,
$$\tau^{Pauli} > 10^6 M_g \;[s]
\label{eq4}$$
which obviously gives a much longer lifetime than eq. (2) for any [P$\mu$BH]{}. Hawking’s interpolation of eq. \[eq2\] to small-mass black holes gave the lifetime for a Planck-mass ($10^{-5}$ g) black-hole on the order of $10^{-42}$ s; close to Planck time itself. The Pauli Principle, however, does not allow such an extremely short quantum evaporation time (at least for electrons), constraining it to be longer than $\sim$10 s. This huge discrepancy between a major quantum physics principle and Hawking’s original interpolation requires more critical scrutiny, particularly when considering the geometric size of the Schwarzschild radius. In fact, Bekenstein’s articles \[9\] that preceded the Hawking radiation papers pre-constrained the validity of the entropy concept itself in terms of the surface area of an event horizon to $\lambdabar < R_S$ [*so that entropy can be defined for quanta to fit into the hole* ]{} (Ref \[7\], p. 274). The formulae used by Hawking (and almost all the later follow-up treatments by quantum field theories) generally used the energy content of the quantum modes, but their thermal treatment is not valid for [P$\mu$BHs]{}; $\lambdabar > R_S$.
In a more analytical particle picture, a very strong acceleration and energy gain ($> 2 m_e c^2$) is required during a brief duration of less than the light-crossing time of sub-Compton wavelength $\epsilon \lambdabar$:
$$\Delta \tau_C = \epsilon \Delta \tau, \;(where \Delta \tau_C\equiv \lambdabar/c),
\label{eq5}$$
in order for a virtual particle ($E = - mc^2$) to acquire the rest-mass energy ($+mc^2$). General relativity around the Schwarzschild radius has a lapse function of curved spacetime,
$$\zeta \equiv (1-R_S/r)^{-1/2},
\label{eq6}$$
for the Schwarzschild line element: radial ($s=\zeta$dr) and temporal ($\tau=\zeta^{-1}$dt) coordinates in geodesics, $ds^2=(1-\frac{R_S}{r})c^2dt^2 - \frac{dr^2}{(1-\frac{R_S}{r})} - r^2(d\theta^2+sin^2\theta d\phi^2)$. This provides the condition required for a critical acceleration
$$a_C \geq c/\Delta \tau
\label{eq7}$$
where $\Delta \tau$ denotes the proper time of a particle. The required acceleration time $\Delta \tau$ (equivalent to $\epsilon$ times the light-crossing time of the Compton wavelength in proper time $\Delta \tau_C$) is $\zeta^{-1} \Delta t=\epsilon \lambdabar/c$ . Particle producing critical acceleration is possible in a thin shell surrounding the Schwarzschild radius with the condition that the particle is linearly localizable within this strong gravitational field and within the limited time. The length parameter ($\epsilon$) for the particle’s location ($r = R + \epsilon \lambdabar$ ) is restricted by the lapse function and the critical acceleration:
$$a_C \geq c/\Delta t = \epsilon^{-1} \zeta^{-1} (c^2 / \lambdabar) = \epsilon^{-1} \zeta^{-1} a_0
\label{eq7prime}$$
where $$a_0 \equiv c^2/\lambdabar = 3.2 \times 10^{31} \;cm\; s^{-2} = 3.3 \times 10^{28} {\bf g}
\label{eq8}$$
with [**g**]{} = 980 cm $s^{-2}$ denoting the gravitational acceleration on the Earth’s surface.
Two domains of black hole mass for the quanta’s Compton Wavelength
==================================================================
Hawking radiation may occur just outside the event horizon but only within a spherical shell ($\Delta r < \epsilon \lambdabar$) and within a short period ($\Delta t \sim \epsilon \lambdabar/c$; $\epsilon < 1$). If a virtual particle receives such a [**continuous unidirectional**]{} strong gravitational force, it can be accelerated within $\Delta t$ to the energy of a real particle. This strong acceleration ($a_S$) is characteristic of the lapse function of the radius ($r = R_S + \epsilon \lambdabar$) around the event horizon,
$$a_S(r)=\frac{G M /r^2}{\sqrt{1 - R_S/r}} = \frac{c^2 R_S}{2 (R_S + \epsilon \lambdabar)^{3/2} \sqrt{\epsilon \lambdabar}}
\label{eq9}$$
There is a clear distinction between two regimes of mass M (or R), separated by a Compton wave-length:
\(i) Relatively-large black holes ($R_S \gg \lambdabar$), where gravity is unidirectional.
\(ii) Micro-black holes ($R_S \ll \lambdabar$), where gravity is no longer unidirectional.
Gravitational field energy of black holes
=========================================
We will examine in the first place the energy content of a vacuum used in the Hawking scheme for these two different mass regimes. The thermal bath, if it is ever generated by gravity, is the result of the gravitational field at $R_S$. Its energy content in the spherical shell bounded by the radii ($R_S < r < R_S + \lambdabar$) is the most meaningful part in vacuum polarization, and it should not be the total mass energy ($M c^2$) of a black hole but its external gravitational field energy. It is evaluated by self-gravitating energy as:
$$E_g=\int^{\infty}_{R_S+\epsilon \lambdabar} M a_S ds = \int^{\infty}_{R_S+\epsilon \lambdabar} \frac{G M^2 dr}{r^2 (1-\frac{R_S}{r})}
=GM^2 ln (1+\frac{\epsilon \lambdabar}{R_S})=\frac{M c^2}{2} ln(1+\frac{2 G M}{c^2 \epsilon \lambdabar})
\label{eq10}$$
This potential energy for $R_S \gg \lambda (\epsilon=1)$ deviates (by a factor of ln(M)) from the Bekenstein-Hawking scheme that used $E_g = Mc^2$.
However the potential energy for case (ii) is $E_g = \frac{M c^2}{2} ln(1+\frac{2 G M}{c^2 \lambdabar})$ $\simeq \frac{G M^2}{\lambdabar}$. Therefore, temperature $T= (G M^2/ \lambdabar)(4 \pi R_S^2 c^3 k/ 2 G \hbar) = c \hbar/ (8 \pi k G \lambdabar)$ becomes a constant, irrespective of the mass of the black holes. This dramatically deviates from the Hawking temperature $T=c^3 \hbar/ 8 \pi k G M$ by a factor of M. Consequently, energy loss rate is $-dM/dt = S \sigma T^4 \propto M^{+2}$, significantly samller than the Hawking formula, Eq. \[eq1\], $dM/dt = S \sigma T^4 \propto M^{-2}$. This alone already suggests that the lifetime of [P$\mu$BHs]{} ($R_S < \lambdabar$) should be much, much longer than the Hawking formula, even though the concept of temperature is unquestioned.
The above result is consistent with our basic results of the absolute bound constrained by Pauli blocking for fermions, Eqs. \[eq3\] and \[eq4\]. Further analysis of particle production from vacuum, as described in the following, confirms that this discrepancy for $R_S < \epsilon \lambdabar$ indeed represents a failure of the Hawking scheme to characterize accurately the lifetime of [P$\mu$BHs]{}.
Sub-Compton coordinate parameter
================================
The potential energy \[$E_g = G M^2 ln (1 + R_S/\epsilon \lambdabar)$\] depends on ($\epsilon$) as well as on the ratio $R_S/\lambdabar \equiv \xi$. Correspondingly, temperature (T) in the thermal picture of waves, if defined by $E_g/S$, depends weakly on $\xi/\epsilon$; namely, $T = [c \hbar ln (1 + \xi/\epsilon) / 8 \pi k]$. The energy loss rate in the thermal model for $R_S < \epsilon \lambdabar$ is $dM/dt = S \sigma T^4 = (4 \pi M^2) \sigma
(c^4 \hbar/ 8 \pi k)^4 ln^4 (1 +\xi/\epsilon)$, and is very different from the Hawking’s thermal scheme, Eq. \[eq1\]. The valid accelerating region constrains the sub-Compton parameter ($\epsilon$) to be smaller than some value in a particle picture. We examine below the constraints on ($\epsilon$) under the condition that sufficient acceleration of vacuum particles can occur.
Regime (i): $R_S > \epsilon \lambdabar$ and a particle picture of large black holes
-----------------------------------------------------------------------------------
The lapse function $\zeta \cong \sqrt{\frac{R_S}{\epsilon \lambdabar}}$ for a stellar-size black hole ($R_S \gg \lambdabar$) is large: $\zeta^2 \approx 10^{-16} M_g$ for $\epsilon = 1$, and the acceleration of a particle is sub-critical for on-shell production unless $\epsilon \ll 1$:
$$a_S^{R_S > \lambdabar} = (\frac{c^2}{2}) \sqrt{\frac{1}{\epsilon_r \lambdabar R_S}} =
(\frac{1}{2 \sqrt{\epsilon}}) \sqrt{\frac{\lambdabar}{R_S}} a_0 \approx (\frac{a_0}{2 \sqrt{\epsilon}}) \sqrt{\frac{10^{16}}{2 M_g}}
\label{eq11}$$
This requirement for a sufficient acceleration $a_S > a_0$ restricts the coordinate coefficient $\epsilon < 10^{16} /8 M_g$, which is about unity for a black hole with mass $M_g$ around $10^{15}$ (g). For a stellar mass black hole ($M_g > 3\times 10^{33}$), this acceleration is so weak at $r = R_S + \lambdabar$ that the critical position coefficient $\epsilon$ must necessarily be limited to small values, $\epsilon < 10^{-18}$. The 3D position of a particle with such a small value of $\epsilon$ ($10^{-18}$ times sub-Compton for stellar-mass black holes) becomes extremely uncertain, but acceleration is nonetheless unidirectional and the integration $a_S \Delta t$ (=c) can provide the required energy for a virtual particle to achieve higher energy states. With the exception of the case where the black hole mass is around $10^{15}$ g, or $\epsilon \ll 1$, particles remain sub-thershold for real and go back into virtual states, contributing only to the variance of the vacuum energy ($T_0 = \sqrt{<E_e^2>}/k > 0$). Equilibrium thermal states do not occur and massless photons cannot be emitted from an equilibriated thermal bath. Photons can come out, if at all, only as secondaries of bremsstarhlung \[$dn/dp_{\gamma} \propto p_{\gamma}^{-1} \theta(p_e - p_{\gamma}) d p_{\gamma}$\] directed only toward the horizon, while electrons are only accelerated before going back to virtual states The mass-loss rate is practically null, and therefore, it has to be less than Eq. \[eq1\]. \[Quantum field theory does not use a particle picture, and mass-less particles (photons) can still be emitted at long wavelengths so long as the field is thermally excited by gravity of any strength. However, thermal equilibrium is unlikely to be an applicable concept for virtual states of vacuum particles.\] (\*)
Regime (ii): $R_S > \epsilon \lambdabar$ and non-thermal description of micro black holes
-----------------------------------------------------------------------------------------
In the case of a small black hole, (ii), $R_S \ll \lambdabar$, the lapse function tends toward unity like Newtonian gravity and it turns out by Eq. \[eq9\] that the gravitational acceleration is sufficiently strong, but again, only for small values of $\epsilon < 1/\sqrt{2}$.
$$a_S^{R_S < \lambdabar} = 2^{-1} \epsilon^{-2} (R_S /\lambdabar) a_0 \leq 2^{-1} \epsilon^{-2} a_0
\label{eq12}$$
These metric conditions must be applied to derive the total effective volume where acceleration for vacuum materialization becomes sufficient.
The product of the vacuum polarization density ($\rho_4$), effective 3-volume ($V_3$) for acceleration, and the quantum tunneling probability (P) gives the total materialization rate ($\varpi$) of particle-pairs per unit time in the strong gravitational field of a black hole: $$\varpi = \rho_4 V_4 (c/\lambdabar) P \;[s^{-1}]
\label{eq13}$$
The vacuum density ($\rho_4$) of virtual electrons is derived by Schwinger \[12\] by using a free Lagrangian, which gives $\rho_4^e \sim (3/4 \pi) (\lambdabar_e^4/c)^{-1} = 1.4 \times 10^{52} cm^{-3} s^{-1}$. To evaluate the total effective volume we now focus our attention to the case (ii). The materialization is constrained by $a_S > a_C$ as
$$a_S=2^{-1} \epsilon_r^{-2} (R_S / \lambdabar) a_0 > \epsilon_r^{-1} a_0
\label{eq14}$$
It follows that $\epsilon_r < R_S / (2 \lambdabar)< 0.5$ and $a_S > (2 \lambdabar/R_S)(c^2 /\lambdabar)=2 c^2/R_S = c^4/G M$. Correspondingly, the [**total**]{} effective four-volume is reduced to
$$\rho_4 V_4 \equiv \rho_4 \Delta_{\epsilon} V_3 \Delta_{\epsilon} t = \epsilon^4 < (R_S/2 \lambdabar)^4 =
(G M/c^2 \lambdabar)^4 < 1/16
\label{eq15}$$
This signifies the fact, on one hand, that the linear acceleration does not achieve the required energy for materialization in outer sphere of $r > \epsilon_r \lambdabar + R_S$. On the other hand, the inner sphere $r < \epsilon_r \lambdabar + R_S$ is not quite localizable for linear acceleration, and the materializable region in the case of (ii) $R_S < \lambdabar$ tends to null. We examine this point further in terms of tunneling probability.
Quantum limitation of unidirectional acceleration
=================================================
The probability of vacuum polarization for tunneling into on-shell state by a uni-directional field is e-fold as represented by the Gamow factor:
$$P \cong e^{-\frac{2}{\hbar} \int^{x_{+}}_{x_{-}} q(x) dx}
\label{eq16}$$
where q(x) denotes the imaginary momentum,
$$q(x)=\sqrt{m^2 c^2 - (W - [m a_S x]^2)/c^2}
\label{eq17}$$
The scalar integral $a_S x$ is not valid for omni-directionally and randomly fluctuating quanta within $x < \lambda$, and Eq.\[eq17\] should be written in its proper vector form,
$$q(x) = \sqrt{ m^2 c^2 -(W- [ \int m a_s \bullet d x]^2)/c^2}
\label{eq17b}$$
This clarifies the fact that the total gravitational work (scalar product) of isotropically fluctuating two vectors is null:
$$\int a_s \bullet d x = \int \int^{+2 \pi}_{-2 \pi} a_S dx cos \theta \frac{1}{\sqrt{2 \pi} \sigma_{\theta}} e^{-\frac{\theta^2}{2 \sigma_{\theta}^2}} d\theta = 0
\label{eq18}$$
Provided that the linear approximation, $a_S(x) \bullet x = a_S x$, is allowed for the integrand of the tunneling penetration factor, we get an imaginary upperbound of tunneling probability for fermions ($P_f$),
$$P_f \cong e^{-\frac{2}{\hbar c} \int_{x_-}^{x_+}\sqrt{m^2 c^4 - (W - m a_S x)^2/c^2} dx} =
e^{-\frac{2}{\hbar c} \frac{m^2 c^4}{m a_S} \int^{+1}_{-1} \sqrt{1 - u^2} du}
\label{eq19a}$$
$$P_f \cong e^{-\frac{\pi m c^3}{a_S \hbar}} \ll 1
\label{eq19b}$$
The total emission rate per second by tunneling electrons from the vacuum that surrounds a [P$\mu$BH]{} is
$$\varpi_{f}^{R_S} \leq (\frac{G M}{c^2})^4 (\frac{m c^2}{\hbar}) \varpi_f \;[s^{-1}]
\label{eq20}$$
where $\varpi_f$ denotes the particle creation rate per unit volume and time derived by the effective Lagrangian ($L^{\prime}$) of quantum field theory for a unidirectional gravitational acceleration ($a_S$) field,
$$\varpi_{f} = 2 Im(L^{\prime}) = \frac{1}{4 \pi^3} (\frac{m a_s \hbar}{m^2 c^3})^2 \frac{m c^2}{\hbar} (\frac{m c}{\hbar})^3
\Sigma^{\infty}_{n=1} \frac{1}{n^2} e^{-n \frac{\pi m c^3}{a_S \hbar}}
= \frac{1}{4 \pi^3} (\frac{m^2 a_S^2}{c \hbar^2}) \Sigma^{\infty}_{n=1} \frac{1}{n^2} e^{-n \frac{\pi m c^3}{a_S \hbar}}
\label{eq21}$$
$$\leq \frac{1}{4 \pi^3} (\frac{1}{c \hbar^2}) (\frac{m c^4}{G M})^2 \Sigma^{\infty}_{n=1} \frac{1}{n^2} e^{-n \frac{\pi G M m}{c \hbar}}
\label{eq21a}$$
The steep exponential factor as a function of mass ($m$) of quanta in Eq. \[eq21\] assures that high-mass quanta are produced negligibly less than the low-mass quanta. Each term of the sum of the exponential series (called the Spence Function) represents n-loop coupling and is very small, unless acceleration is higher than the critical value $a_C$. Thus, the total particle emission rate per second is bounded by,
$$\varpi^{R_S}_{f} \leq (\frac{G M}{c^2})^4 (\frac{1}{c}) (\frac{m c^2}{\hbar}) \frac{1}{4 \pi^3} (\frac{1}{c \hbar^2})
(\frac{m c^4}{G M})^2 \Sigma^{\infty}_{n=1} \frac{1}{n^2} e^{-n \frac{\pi G M m}{c \hbar}}
\label{eq22a}$$
$$\varpi^{R_S}_{f} \leq \frac{1}{4 \pi^3} (G M)^2 (\frac{m^3}{\hbar^3}) \Sigma^{\infty}_{n=1} \frac{1}{n^2} e^{-n \frac{\pi R_S}{2 \lambdabar}}
< \frac{c}{16 \pi^3 \lambdabar} (\frac{R_S}{\lambdabar})^2 < \frac{c}{16 \pi^3 \lambdabar}
\label{eq22b}$$
which is consistent with the $\varpi$ obtained earlier by the volume analysis: Eqs. \[eq13\] and \[eq15\].
The production probability of bosons ($P_b$) in a strong field resembles that of fermions \[13\], with a coefficient smaller by only a factor of two for the Spence function.
$$P_b \cong 2^{-1} \times e^{-\frac{\pi m c^3}{a_S \hbar}}
\label{eq23}$$
$$\varpi_b = \frac{1}{8 \pi^3} (\frac{m^2 a_S \hbar}{c}) \Sigma^{\infty}_{n=1} \frac{1}{n^2} e^{-n \frac{\pi m c^3}{a_S \hbar}}
\label{eq24}$$
The most critical problem for a tunneling integral for $x < \lambdabar$ is that the coordinate (x) fluctuates randomly over the path integral, changing the direction of the gravitational vector $a_S(x,t)$ [**so that it is totally uncertain**]{} for the effective one-directional path integral of $a_S \bullet dx = a_S (dx) cos \theta$, due to the uncertainty of $\theta$ $(-2 \pi \leq \theta \leq 2 \pi)$ at any time and position during this integration. This makes the Gamow factor practically zero for a black hole whose RS is smaller than the particle’s Compton wavelength $\lambdabar$:
$$P\ll e^{-\frac{\pi m c^3}{a_S \hbar}}
\label{eq25}$$
Thus, the 3-D uncertainty for the 1-D unidirectional integral in Eqs. \[eq17\] and \[eq18\] entirely kills the Gamow factor and makes the probability effectively zero as shown in Eq. \[eq19b\]. It is the same situation as that of a strong electric field $U(r) = Z e^2/4 \pi \epsilon r$ inside a uranium nucleus, e.g., ($r < 1$ fm), where the effective coupling probability becomes $> 137 \alpha = 1.0$. Literally, the vacuum could have decayed in a high electric field $Z e^2/r$ by a singular multiple coupling probability, if the unidirectional tunneling were ever allowed and if the Uncertainty Principle that requires $r > \lambdabar$ were ignored. The case of [P$\mu$BH]{} vacuum polarization by strong gravity has the same problem as this uranium electric case. The 1-D integral in the Gamow factor does not yield any meaningfully small value for the exponent within $r < \lambdabar$, and hence no spontaneous vacuum decay can take place, because [P$\mu$BHs]{} require Eq. \[eq14\], namely, $\epsilon_r < R_S (2 \lambdabar) < 0.5$.
For large BHs with $r \gg \lambdabar$, 1-D directionality is sufficient, and Eqs. \[eq11\], \[eq22b\] and \[eq24\] can be used. We recall Eqs. \[eq22b\] and \[eq24\] as the upperbound for energy loss from a BH. The probability of acquiring $a_S > a_C$ is, however, very small for vacuum electron pairs with any gravitational acceleration in curved spacetime (for the [P$\mu$BHs]{} case) or electrostatic acceleration in flat spacetime (as in the uranium case where $r \gg 1$ fm), with the metric more than one-Compton wavelength away from the Schwarzschild radius or from a nucleus center, respectively. Thus, we are led to conclude that fermions or bosons can barely be emitted from [P$\mu$BHs]{}. Consequently, no thermal radiation can occur from such a site where far less than one particle at any time can exist in the quantum-materialization spacetime volume.
Mass-loss rate and half lifetime
================================
The mass-loss rate of [P$\mu$BHs]{} ($M < 10^{17}$ g) can be evaluated as an upperbound by using Eqs. \[eq11\] and \[eq22a\] with P =1, but it is indeed an extreme overestimation in the sense that we ignore the ineffectiveness (Eq. \[eq25\]) of a path integral in Gamow factor P $\ll$1. It leads to an artificially faster evaporation rate; yet, this rate already gives many orders of magnitude less than that interpolated by Hawking’s thermal scheme. Our upperbound is thus
$$-\frac{dM}{dt} \ll m_e (\varpi^{R_S}_f +\varpi^{R_S}_b) \;[g s^{-1}] =
3 \times 10^{-47} M_g^2 \;[g s^{-1}]
\label{eq26}$$
The half lifetime ($\tau_{1/2}$) of a [P$\mu$BH]{} ($M \leq 10^{17}$ g) in this scheme is
$$\tau_{1/2 (P\mu BH)} \gg 3 \times 10^{48} M_g^{-1}\; [s]
\label{eq27}$$
This suggests $\tau_{1/2} > 10^{31}$ s at $M \sim 10^{17}$ g ($R_S = \lambdabar$), which is consistent with the absolute lower bound for fermions ($\tau_{1/2}> 10^{23}$ s). Any [P$\mu$BHs]{} with mass less than $10^{17}$ g can live many orders of longer time than the Hubble time ($\sim 3 \times 10^{16}$ s). The smallest [P$\mu$BH]{} is that of the Planck mass, ($\sim 10^{-5}$ g), which can live $\gg 10^{47}$ yr, and can provide a basis for a finite size spacetime without decay.
Consideration of the smallest-mass quanta represents a significant departure from the Hawking scheme of Eq. \[eq2\]. We used electrons for its known firmness of the lightest-mass electromagnetic quanta in this physical world. Neutrinos have been treated as massless until recently, but they are supposed nowadays to have small finite masses 0.01 -1 $eV/c^2$ as inferred from the observed deficits of fluxes of solar electron neutrinos and atmospheric muon neutrinos. With these finite masses ($m_{\nu}$), neutrinos cannot be emitted from black holes with a mass smaller than the neutrino critical mass $M_C^{\nu} = 10^{23}$ (1 $eV/m_{\nu} c^2$) g. Other light-mass particles such as axions ($m_A < 10^{-14}$ eV) are hypothetical. They would make the thermal equilibrium basis of Hawking formula partially invalid for $M_C^A [< 10^{37} g\; (\sim 10^{4} M_{\odot})]$, if they really exist. Although these are fundamental quanta, they are neutrals and do not directly couple to photons. These quanta cease to contribute to the thermal bath of a vacuum for $M < M_C^{\nu} < M_C^A$ long before electrons do so at $M < M_C^e = 10^{17}$ g. The role of quanta at higher masses is negligible so long as they remain only as virtual states in vacuum, as already stated in 2.2, due to the disequilibrium nature of virtual particles. The largest mass of fundamental quanta presently-known for production is that of the top quark (174 $GeV/c^2$) \[14\], which gives its critical mass $M_C^{top} = 7.6 \times 10^{11}$ g. Energy loss for a BH with a mass less than $10^{17}$ g (and above the critical mass $M_C^{top} = 7.6 \times 10^{11}$) can still take place. The probability of pair production from vacuum fluctuation for such heavy particles is, however, much smaller than for electrons for $M > M_C^{top}$ due to the exponential factor in tunneling probability. The thermal bath of a vacuum for photon emission is correspondingly highly limited. Similarly, due to the more stringent quantum conditions that destroy the Gamow factor by 3D fluctuations for $R_S (M) < \lambdabar$ (Eqs. \[eq19a\] and \[eq25\]), any unknown quanta with a mass heavier than the top-quark mass, if any would be discovered in future, can still safely be ignored in the consideration of the lifetimes of [P$\mu$BHs]{}.
Conclusions and discussions
===========================
In summary, the radiation rate from [P$\mu$BHs]{} is evaluated. Several different considerations and analyses indicated that the Hawking radiation from [P$\mu$BHs]{} was substantially overestimated by a factor of 1/$M^2$ in the Hawking’s thermal scheme with or without quantum field theoretical approaches. Those aspects having been analyzed are (1) metric in the original Hawking scheme, (2) geometry in quantum field theoretical approaches, (3) gravitational energy for small black holes, (4) non-unidirectional tunneling probability, and (5) dis-equilibrium nature of virtual quanta.
We are now led to conclude that all the [P$\mu$BHs]{} should be regarded as very massive, cold, and dark (black) matter. They would not have been lost since their generation in the early big-bang era, were [P$\mu$BHs]{} ever produced by density fluctuations or other mechanisms. They could have only grown larger during their high-density phase by gravitational cannibalism. The task of explaining their role in the structural formation of the universe, and their later role in the genesis of stars and galaxies, poses a new challenge for further research. The fundamental cosmological equations must incorporate significantly abundant and locally-static elements ([P$\mu$BHs]{}) from the beginning. It would alter the Friedman and Lemaitre solutions (including Einstein-de Sitter) because the Robertson-Walker metric has to be replaced by a rejuvenated hybrid-Schwarzschild metric, just like the so-called Swiss-Cheese model, which was invoked by Einstein and Strauss in 1945 \[15\]. Even the relevant critical mass density of the universe, $\rho_C = 3 H^2/8 \pi G$, of Friedman models (and Einstein-de Sitter model), could be altered, and our understanding of the matter content in the universe might require a major revision. Although unfashionable in the present-day standard cosmology, this point can become significant if [P$\mu$BH]{} generations in the early epoch of Big-Bang turns out to be substantial for the entire energy content.
Black holes are extremely difficult objects to search for, to identify, or to study. Micro-sized black holes that do not radically emit Hawking radiation are supposed to be undetectable. Since their mass is much too small (smaller than a mediocre asteroid), the local gravitational-lens method is ineffective for finding them. Gravitational lensing by many such [P$\mu$BHs]{} in line of sight may or may not appear in the very small perturbations of red-shift fluctuations such as the type of Sachs-Wolfe \[16\] and Rees-Sciarma \[17\] effects for deep-space optical observations. Because the cross sections of [P$\mu$BHs]{} are extremely small, it is likely that any such effect by [P$\mu$BH]{} would be buried in or overshadowed by the fluctuations caused by other sources.
The total mass of all the [P$\mu$BHs]{} depends on the production mechanism in the earliest era of the Big Bang. It can be a small portion of the total energy of the early era of the Big-Bang universe, but it could become a significant portion in the later epochs. [P$\mu$BH]{} have been treated as if they were thermal particles at production in density perturbations. The mass spectrum depends on the growth rate of the perturbed densities, for which there are excellent early studies \[18\]. They show that the mass distribution exponentially falls very rapidly with increasing mass, favoring [P$\mu$BH]{} as constituting the majority of PBHs so long as they do not evaporate swiftly. We have shown in this paper that [P$\mu$BH]{} do not evaporate within Hubble time.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author thanks Drs. T. Kibble, N. Sanchez, Y. Shimizu, I. Axford, R. Preece and M. Bonamente for helpful comments and discussions.
Bibliography {#bibliography .unnumbered}
============
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Footnote: (\*) The effective total volume for the case $R \gg \lambda$ is $V_3 = 4 \pi R_S^2 (\epsilon \lambda)^2 (c/\lambda)\propto M^0$. The production rate with the quadratic acceleration factor in Eq. \[eq21\], $a_S^2 \propto (1/\epsilon M) a_0^2 \propto M^0$, yields $\varpi_f^{R_S} \propto M^0$= constant. The mass loss rate becomes constant of mass M and the lifetime is proportional to M, but not $M^3$ of the Eq. \[eq2\] indicated.
Correspondence and requests for materials should be addressed to Y. Takahashi (e-mail: yoshi@cosmic.uah.edu).
Appendix A:\
Acceleration and Kinematics {#appendix-a-acceleration-and-kinematics .unnumbered}
===========================
Virtual particles can be accelerated by gravity within the Uncertainty Principle (or more originally, the commutation relations) that governs the Gaussian uncertainty of proper distance (ds) and duration (d$\tau$ ) with dispersions $\sigma_{\tau} = \lambda /c$ and $\sigma_s = \lambda$, respectively. Gravitational acceleration is given by $a(r) = \frac{GM}{r^2\sqrt{1-R/r}}$, where r is measured from the event horizon (r = R + $\Delta r$). This acceleration provides velocity (v) and energy (E) of a virtual particle during (d$\tau$) at the radial region (ds), $E = 2m c^2 (1 - v^2/c^2)^{-1/2}mc^2 =(\sqrt{c^2 -a^2 t^2} -1)m c^2$. The radial coordinates where acceleration takes place is sub-Compton wavelength away from the event horizon. The coordinate coefficient ($\epsilon$) for the radial coordinate difference \[$\Delta r = (R +r_2) - (R + r_1)]$ can be used for $\Delta r = \epsilon \lambdabar_e$, where R and $\lambdabar_e$ denote the Schwarzschild radius and electron Compton wavelength, respectively. The probability function of energy in relativistic and non-relativistic formulae can be given by the following equations, Eq. \[eqA1\] and Eq. \[eqA2\], respectively.
$$P_R(E) dE = dE \int_{\infty}^{\infty} \int_{\infty}^{\infty} \frac{d \tau}{\sqrt{2 \pi} \sigma_{\tau}} \frac{d s}{\sqrt{2 \pi} \sigma_s}
e^{-\frac{\tau^2}{2 \sigma_{\tau}^2}} e^{\frac{s^2}{2 \sigma_s^2}} \delta[E-(m c^2\sqrt{c^2 - a^2 t^2} -1)]
\label{eqA1}$$
$$P_{NR}(E) dE = dE \int_{\infty}^{\infty} \int_{\infty}^{\infty} \frac{d \tau}{\sqrt{2 \pi} \sigma_{\tau}} \frac{d s}{\sqrt{2 \pi} \sigma_s}
e^{-\frac{\tau^2}{2 \sigma_{\tau}^2}} e^{\frac{s^2}{2 \sigma_s^2}} \delta[E-\frac{m_e a^2 t^2}{2}]
\label{eqA2}$$
Approximation with non-relativistic energy and constant acceleration makes the integral simple and easy. It is, however, rather inaccurate. In particular, it becomes increasingly inaccurate when R becomes closer to $\lambda$. We note also that the available acceleration time for heavier quanta (m) is shorter by a factor of ($m_e/m$) relative to that for electrons ($m_e$). Under these conditions one can easily rewrite eq. A2 as follows:
$$P_{NR}(E) dE = dE \int_{0}^{\infty} \int_{0}^{\infty} \frac{2 dt}{\sqrt{2 \pi} \sigma_t} \frac{2d\Delta(r)}{\sqrt{2 \pi} \sigma_r}
e^{-\frac{t^2}{2 \sigma_{t}^2}} e^{-\frac{(\Delta r)^2}{2 \sigma_r^2}} \delta[E-\frac{m a^2[t(m_e/m)]^2}{2}]
\label{eqA3}$$
$$=dE\int_{0}^{\infty} \frac{2E}{\hbar \sqrt{2 \pi}} \frac{2 E d(\Delta r)}{\hbar \sqrt{2 \pi c}}
e^{-\frac{(m/m_e)^2 (2/(m a^2)) E}{2 (\frac{\hbar}{m_e c^2})^2}} e^{-\frac{(\Delta r)^2}{2(\frac{\hbar}{m_e c^2})^2}}
\label{eqA4}$$
$$=dE\int_{0}^{\infty} (\frac{2 E^2 \lambdabar_e}{\pi \hbar^2}) e^{-\frac{(m/m_e)^2 (\frac{16 G M \epsilon \lambdabar_e}{m c^6}) E}{2 (\frac{\hbar}{m_e c^2})^2}}.e^{-\frac{(\Delta r)^2}{2(\frac{\hbar}{m_e c^2})^2}}
d (\Delta r)
\label{eqA5}$$
$$=dE\int_{0}^{\infty} (\frac{2 E^2 \lambdabar_e}{\pi \hbar^2}) e^{-\frac{(\frac{16 G M m \epsilon \lambdabar_e}{m_e^2 c^6})E}{2(\frac{\hbar}{m_e c^2})^2}}
e^{-\frac{\lambdabar_e^2 \epsilon^2}{2 \lambdabar_e^2}} \lambdabar_e (d \epsilon)
\label{eqA6}$$
$$=dE\int_{0}^{\infty} (\frac{2 E^2 \lambdabar_e}{\pi \hbar^2}) e^{-\frac{E}{c^3 \hbar m_e/ 8 G M (m/m_e) \epsilon}}
\label{eqA7}$$
If an assumption that $\epsilon=1$ is allowed, one gets a simple exponential distribution.
$$P(E) = dE(\frac{2 \lambdabar_e E^2 }{\pi \hbar^2}) e^{-\frac{E}{c^3 \hbar / 8 G M (m/m_e) \epsilon}}
\label{eqA8}$$
The original Hawking formula does not seem to have separated quanta of different species (i.e., $m = m_e$). In this case the denominator of the exponent misleads to appear having no mass factor of quanta. This familiar denominator is often interpreted as equivalent to kT of temperature (Hawking temperature):
$$T=c^3 \hbar/ 8 \pi G M
\label{eqA8}$$
This denominator has originated only from the randomness of quantum coordinates constrained by the Uncertainty Relationship. Furthermore, a quite significant simplification was made for acceleration as constant, while it varies drastically during acceleration in an extremely short distance. Hence, an interpretation of the exponent as exactly thermal by an appearance is hardly justifiable when considering its mathematical origin. Moreover, eq. (7), before the simplification of $m = m_e$ is made, indicates that the denominator (kT) is inversely proportional to the mass of quanta. If a thermal interpretation is made, it is in disequilibrium. The true origin of this mass dependence is in the shorter duration of acceleration available for heavier quanta for the same mass M and the same gravitational field strength in the same unit time.
Coordinates ds and $d \tau$ (or dr and dt) in curved spacetime are interwoven and the formulae become more complicated than Eqs. A3, A6. Furthermore, relativistic treatment of the particle energy with interwoven effects of two random distributions of ($\epsilon$) and (t) no longer makes P(E)dE close to a thermal-like simple exponential spectrum. It is not a simple analytic function of neither (t) or ($\epsilon$), and an interpretation by a thermal analogy for its appearance is not at all feasible. $$P_R(E) dE = dE \int_{t=0}^{+\infty} \int_{\epsilon=\epsilon_{min}}{\epsilon_{max}} \frac{2 c dt}{\pi \lambdabar_e}
e^{-\frac{(\frac{c^4 (R+\epsilon \lambdabar_e)^3 \epsilon \lambdabar_e}{G M})[1-(\frac{m c^2}{E + m c^2})^2]}{2(\frac{\hbar}{m_e c^2})}}$$ $$\times e^{-\frac{(\epsilon \lambdabar_e)^2}{2(\frac{\hbar}{m_e c})^2}} d (\epsilon)
\delta[E-(m c^2\sqrt{c^2-[\frac{4 G^2 M^2}{(R+\epsilon \lambdabar_e)^{3/2} \epsilon \lambdabar_e}]t^2} -1)]$$
Appendix B:\
Compton wavelength of the black hole itself {#appendix-b-compton-wavelength-of-the-black-hole-itself .unnumbered}
===========================================
\(1) Compton wavelength of the black hole is a very small value so long as its Schwarzschild radius is far greater than that of quanta. We define the radial coordinates dr = (R +dr) - R= ds = $\sqrt{1-R/(R+dr)} ds=\sqrt{1-R/(R+dr)} (\hbar/M c)$ In terms of mass, this is reduced to the following:
$$dr = - (GM/c^2) + [G^2M^2/c^4 +\hbar^2/(Mc)^2]^{1/2}$$ $$\simeq (2GM/c^2) (1/2) (1/2) \hbar^2c^2/G^2M^4) = (2GM/c^2) (1/2)(1/2)(M_P^4/M^4)$$ $$\geq (R_s/4) (M_P/M)^4 \;(at GM \gg \hbar c: M \gg M_P)$$
\(2) General relativistic Compton wavelength of an electron ($m_e$) around a Black Hole with mass (M) is $dr_e$ defined by $$dr_e = (R +dr_e) -R= \sqrt{1-R/(R+dr)}ds_e = \sqrt{1-/(R+dr)}(\hbar/m_ec)$$ $$(dr_e)^2 = (h/m c)^2 {dr_e/[(2GM/c^2)+dr_e]}$$ $$(dr_e)^2 + (2GM/c^2)(dr_e)-(h/mc)^2=0$$ $$dr_e = - (GM/c^2) \pm [G^2M^2/c^4 +\hbar^2/(mc)^2]^{1/2}$$ $$= (2GM/c^2) (1/2){[(1 + \hbar^2c^2/G^2M^2m^2)^{1/2}] -1}$$ $$\simeq (2GM/c^2) (1/2) (1/2) \hbar^2c^2/G^2M^2m^2 = (2GM/c^2) (1/2)(1/2)(M_P^4/M^2m^2)$$ $$\geq (R_s/4) (M_P/M)^2 (M_P/m_e)^2.\; (at GM \gg \hbar c: M \gg M_P \equiv \sqrt{c \hbar/G})$$
Compare this with the flat-spacetime Compton wavelength defined by $\lambda_M \equiv \hbar/M_c$, $$\Lambda_M/\lambda_M \geq (1/4) (Mc/\hbar) R_s (M_P/M)^4 = (1/4) (Mc/\hbar) (2MG/c^2) (M_P/M)^4$$ $$= (1/2) (M^2G/c \hbar) (M_P/M)^4 = (1/2) (M_P/M)^2.$$
|
---
abstract: 'In this paper, a novel approach for providing incentives for caching in small cell networks (SCNs) is proposed based on the economics framework of contract theory. In this model, a mobile network operator (MNO) designs contracts that will be offered to a number of content providers (CPs) to motivate them to cache their content at the MNO’s small base stations (SBSs). A practical model in which information about the traffic generated by the CPs’ users is not known to the MNO is considered. Under such asymmetric information, the incentive contract between the MNO and each CP is properly designed so as to determine the amount of allocated storage to the CP and the charged price by the MNO. The contracts are derived by the MNO in a way to maximize the global benefit of the CPs and prevent them from using their private information to manipulate the outcome of the caching process. For this interdependent contract model, the closed-form expressions of the price and the allocated storage space to each CP are derived. This proposed mechanism is shown to satisfy the sufficient and necessary conditions for the feasibility of a contract. Moreover, it is shown that the proposed pricing model is budget balanced, enabling the MNO to cover all the caching expenses via the prices charged to the CPs. Simulation results show that none of the CPs will have an incentive to choose a contract designed for CPs with different traffic loads.'
author:
- |
[Kenza Hamidouche$^{1,2}$]{}, Walid Saad$^{2}$, and Mérouane Debbah$^{1,3}$\
[^1]
bibliography:
- 'references.bib'
title: 'Breaking the Economic Barrier of Caching in Cellular Networks: Incentives and Contracts '
---
Introduction
============
The capacity limitations of the backhaul links that connect small base stations (SBSs) to the core network in a small cell network (SCN), make it difficult for the mobile network operators (MNOs) and content providers (CPs) to ensure the required data rates for emerging, bandwidth intensive users’ applications especially during peak hours [@andrews2014will]. To overcome these backhaul limitations and meet users’ requirements in terms of quality-of-service (QoS), distributed caching at the network edge has been recently proposed as a new promising solution [@poularakisexploiting; @blaszczyszyn2014optimal; @blasco2014content; @ji2014fundamental].
Caching typically relies on storing the most popular files at the levels of SBSs and devices, to reduce backhaul traffic [@poularakisexploiting]. To successfully deploy SCN caching solutions, MNOs require the cooperation of the CPs by sharing their content and providing this content’s global [@paschos2016wireless]. However, although CPs can improve the QoS of their users by caching, they might be reluctant to share their content with the MNOs. This can be due to reasons such as privacy since, once the content cached at the SBSs, the MNOs can get access to all the CP’s files as well as the traffic dynamics of the users subscribed to the CP [@paschos2016wireless]. Thus, the MNOs must provide incentives to the CPs to share their data and cache it at the SBSs by introducing suitable economic arrangements that can be beneficial for both the MNO and the CPs [@paschos2016wireless; @li2016pricing].
Most of the existing literature on caching [@poularakisexploiting; @blaszczyszyn2014optimal; @blasco2014content; @ji2014fundamental] has focused on determining the optimal caching policy under various network scenarios. In [@poularakisexploiting], the authors proposed a caching policy that optimizes the overall energy consumption of the SBSs while accounting for the multicast opportunities. The authors in [@blaszczyszyn2014optimal] proposed a caching policy that accounts for the geographical position of the SBSs and users. The work in [@blasco2014content] formulated the cache placement problem as a multi armed-bandit problem in which the SBSs do not know the popularity of the files. The authors in [@ji2014fundamental] proposed a coded caching strategy for wireless device-to-device (D2D) networks. Remarkably, none of these works have addressed the economic aspect of caching. Recently, the works in [@li2016pricing] and [@chen2016caching] provides some insights on the economics of caching using game-theoretic solutions. However, these works typically assume that the players are honest and have perfect knowledge of the cellular network information, which is not a reasonable assumption in practice. In fact, the CPs can modify the outcome of the incentive mechanism in their favor by attempting to manipulate the MNO’s agreements. For example, some CPs can declare inaccurate information about their traffic load so as to mislead the MNO into charging them much lower prices. Such possibilities can therefore incentivize some of the CPs to not share their truthful private information in which case the results in [@li2016pricing] and [@chen2016caching] no longer hold. The main contribution of this paper is to introduce a novel incentive mechanism for facilitating the deployment of caching in SCNs. We consider a model in which an MNO proposes agreements to the CPs to incite them to cache their content. We consider a practical scenario with *asymmetric information* within which the CPs can be of different, private types that are unknown to the MNO. This private information pertains to the level of generated CP traffic which corresponds to the popularity of its content. The proposed approach, based on contract theory [@borgers2015introduction], allows the MNO to define a contract for each CP in the presence of asymmetric information, by fixing a price for the allocated storage space to the CP. The goal of the MNO is to maximize the global reward of the CPs and cover the expenses of the caching process. Unlike classical contract-theoretic models [@gao2011spectrum; @duan2014cooperative; @zhang2015contract], we show that the proposed model exhibits strategic interdependence between the CPs. Consequently, for this interdependent contract model, we derive the closed-form expression of the price and prove that the resulting mechanism satisfies the feasibility constraints of a contract. The designed mechanism is then shown to be budget balanced as it allows the MNO to offload its backhaul while covering the cost of caching the CPs’ content. Simulation results show the performance advantage of the proposed scheme as well as its ability to incite truthfulness on the CPs.
System Model {#model}
============
Consider a SCN composed of a set $\mathcal{M}$ of macro base stations (MBSs) and a set $\mathcal{S}$ of SBSs deployed by an MNO. The SBSs are connected to the MBSs via capacity-limited backhaul links and serve a set of users $\mathcal{U}$ that are subscribed to multiple CPs from a set $\mathcal{C}$. The SBSs can cache content to offload the MNO’s backhaul. However, the MNOs needs the CPs’ cooperation to cache their content. To motivate the CPs to participate in the caching, the MNO must offer contracts that present significantly improved QoS for the CPs’ users. The contract terms between an MNO and a CP determine the price charged to the CP by the MNO and the amount of storage space offered.
We consider heterogeneous CPs with different traffic loads and content popularity. Based on this traffic load, the CPs have different incentive levels towards sharing their content with the MNOs. Naturally, there is an *information asymmetry* between the MNO and the CPs. The CP is aware of its users’ traffic as well as its preferences while the MNO may not have that information. Consequently, the CPs may have an incentive to not reveal their correct types so as to pay lower prices to the MNO. To overcome this challenge, the MNO must specify a suitable performance-reward *bundle contract* $(\pi,\rho)$, where $\pi$ is the monetary reward that is paid by the CP, and $\rho$ is the storage space allocated to the CP. The goal of each CP $k\in\mathcal{C}$ is to maximize the performance of its users which depends on the amount of content that other CPs in the set $\mathcal{C}\setminus k$ will be willing to cache at the SBSs. In fact, the higher the traffic load of the CPs in $\mathcal{C}\setminus k$, more storage space is allocated to cache their content. Thus, less storage space can be made available for the CP $k$. This will negatively impact the data rate of CP $k$’s users as more requests need to be served via the capacity-limited backhaul. Moreover, by introducing caching, the traffic load of each SBS increases when caching highly popular files. Thus, more power is required at the SBS to serve all the requests for a highly popular file. Consequently, other CPs’ users might experience large interference level from the SBSs that cache the files of CPs having a higher traffic load.
Transmission Data Rate
----------------------
The performance $r_k$ of a CP $k$ that results from caching its contents is measured by the transmission rate that its users experience from the serving MBS or SBS. When an SBS $i$ serves a user $j$, the data rate will be:
$$\alpha_{ij}(\rho(\boldsymbol{\theta}),\boldsymbol{\theta})=\mathbb{E}_t\left[w_{ij}\log{\Big(1+\frac{p_{ij}(\rho,\theta_i)|h_{ij}|^2}{\sigma^2+I(\rho,\boldsymbol{\theta})}\Big)}\right],
\label{first_equ}$$ where $I(\rho,\boldsymbol{\theta}) =\sum_{k\in\mathcal{S}\setminus i}{p_{kj}(\rho,\boldsymbol{\theta})|h_{kj}|^2}$ is the interference experienced by user $j$ from all the other SBSs. $w_{ij}$ is the channel bandwidth, $p_{ij}(\rho,\boldsymbol{\theta})$ is the transmit power from SBS $i$ to user $j$, $|h_{ij}|^2$ is the channel gain between SBS $i$ and user $j$, and $\sigma^2$ is the variance of the Gaussian noise. The vector $\boldsymbol{\theta}=[\theta_1,...,\theta_C]$ represents the traffic load of the CPs. Thus, the higher the traffic load of the CPs, the higher is the interference experienced by the users served from the neighboring SBSs. Since caching is done during off-peak periods, the transmit power is averaged over the considered time period.
If the file is cached at the SBSs, then the users will experience a relatively high data rate as the content is closer to them. However, if the data is not available at the associated SBS to serving a certain user, then the SBS must fetch the user content from the MBS over the capacity-limited and congested backhaul, yielding higher delays. The data rate of a user $j$ requesting file $f$ from its associated SBS $i$ can be given by:
$$\begin{gathered}
r_{ij}(\boldsymbol{\theta}) = (1-\beta_{if}(\rho(\boldsymbol{\theta}),\boldsymbol{\theta}))\min{\{\alpha_{ij}(\rho(\boldsymbol{\theta}),\boldsymbol{\theta}),\alpha^{\prime}_{mi}\}}\\+\beta_{if}(\rho(\boldsymbol{\theta}),\boldsymbol{\theta})\alpha_{ij}(\rho(\boldsymbol{\theta}),\boldsymbol{\theta}),\end{gathered}$$ where $\boldsymbol{\beta}\in \{0,1\}^{S\times F_k}$ is the outcome of the MNO’s storage allocation $\rho$ and $F_k$ is the cardinality of the set of files $\mathcal{F}_k$ provided by a CP $k$. $\boldsymbol{\beta}$ depends on the caching policy $\rho$ and the traffic load of the CPs. For instance, when a CP $k$ has highly popular files or its willingness level to cache its content is high, it will impact the storage allocation to the other CPs $\mathcal{C}\setminus k$. The larger the number of files that CP $k$ wants to cache, the lower is the storage space that will be allocated to other CPs $\mathcal{C}\setminus k$ and vice versa. Each entry $\beta_{if}$ is a binary variable that equals $1$ if file $f$ is cached at SBS $i$ and $0$ otherwise. $\alpha^{\prime}_{mi}$ is the data rate from the MBS $m$ to SBS $i$ and is given by:
$$\alpha^{\prime}_{mi}=\mathbb{E}_t\left[w_{mi} \log{\Big(1+\frac{p_{mi}|h_{mi}|^2}{\sigma^2+I^{\prime}}\Big)}\right],
\label{second_equ}$$
where $I^{\prime}=\sum_{l\in\mathcal{M}\setminus m}{p_{li}|h_{li}|^2}$ is the interference experienced by SBS $i$ from all the other transmitting MBSs.
Thus, the total rate of the users of CP $k$ can be given by:
$$r_k(\rho(\boldsymbol{\theta}),\theta_k) = \sum_{i\in\mathcal{S}}{\sum_{j\in\mathcal{U}_{ki}}{r_{ij}(\boldsymbol{\theta})}},$$
where $\mathcal{U}_{ki}\subseteq\mathcal{U}_k$ is the set of users that request at least one file from CP $k$ by using SBS $i$, and $\mathcal{U}_k$ is the set of users requesting files of CP $k$. Here, we note that, in our model, each CP will have private information that is modeled as a *type of CP* as discussed next.
Content Provider Type
---------------------
We define the CP’s type to be a representation of its traffic load and content popularity. In fact, when the MNO offers a contract, it must account for the generated traffic by the CPs. For example, by caching the contents of CPs with a high traffic, the MNO can serve more requests locally, thus decreasing its backhaul load considerably. Here, we consider that the number of CP types belongs to a discrete, finite space and grouped as follows:
*There are $C$ CPs that generate traffic over an MNO’s network. The CPs’ types are sorted in an ascending order and classified into $K$ types $\theta_1,...,\theta_K$ with $K\leq C$. Each type includes properties such as the willingness to cache and the global popularity of the CPs files. The types are ordered as follows: $\theta_1<...<\theta_k<...<\theta_K, ~~~k\in \{1,...,K\}.$*
Since the types are not known by the MNO, the CPs can announce wrong information about their types so that they improve the performance of their users. For example, by claiming that its content popularity is higher than it actually is, a CP $k$ can mislead the MNO to allocate more storage space. In such a case, the CP can end up paying lower prices while also lowering the interference experienced by its users. Indeed, the truthful popularity information is necessary for the MNO to define the contracts that optimize the benefit of the CPs and cover the implementation costs of caching. Such cost includes the expenses of deploying storage devices and the required power to download the content and refresh the storage units. Here, our goal is to design contracts that incentivize the CPs to reveal the true values of their types $\boldsymbol{\theta}$ to the MNO. To this end, the contracts will be designed such that no CP can profit by choosing a contract that is designed for other types.
Content Provider Model
----------------------
The utility function of a CP $k$ of type $\theta_k$ that decides to cache a set of files $\mathcal{F}_k$ at the operator’s network is: $$u_{k}(\boldsymbol{\theta})= r_k(\rho_k(\boldsymbol{\theta}),\theta_k)- \pi_{k}(\boldsymbol{\theta}) ,$$ where $r_k(\rho)$ is defined in (4) and represents the valuation function regarding the rewards, which is a strictly increasing concave function of $\rho_k$, with $r(0)=0$ and $r^{\prime}(\rho_k)>0$, $r^{\prime\prime}(\rho)<0$ for all $\rho_k$. $\pi_k$ represents the price charged by the MNO for a storage allocation $\rho_k$.
Mobile Network Operators Model
------------------------------
By caching the content of the CPs, the MNO will be able to reduce the traffic load on its backhaul. This benefit depends on the traffic load of the CPs as dictated by the popularity of their cached traffic. Thus, the MNO will generally prefer to cache the most popular files. By doing so, for the same storage capacity, the load can be reduced more for a CP whose files have a high popularity compared to other CPs. Thus, the cost of storage $c_s$ at the MNO can be given as a function of the traffic load of the considered CP as, $c(\theta)=\log{(1+\theta)}$. This storage cost function increases quickly up to a certain threshold and then increases slowly. It is suitable to model the storage cost as the MNO must allocate more storage space to serve a given traffic load, and this cost becomes insignificant when the traffic load increases as some requests become redundant. A proper utility function for the MNO can be defined as the monetary reward that is charged to the CPs minus the cost of the allocated resources by the MNO, including storage. $$v_{k}(\boldsymbol{\theta})= \pi_{k}(\boldsymbol{\theta})-c_k(\boldsymbol{\theta},\theta_k),%-c_e(\theta_k) ,$$ where $\pi_k$ is the price that the operator charges CPs of type $k$. The total expected utility of the operator can be given by $$v =\sum_{k\in\mathcal{C}} v_{k}(\boldsymbol{\theta}).$$
In the considered model, the MNO is assumed to get the CP’s types directly from the CPs. Based on this information, the goal of the MNO is to determine a contract for all possible CP types that maximizes the global benefit of the CPs. At the same time, the CP ensures that its utility is nonnegative by making the prices charged to the CPs to at least cover its cost. This optimization problem can be defined as follows: $$\label{PF1}
\begin{aligned}
& \underset{(\pi_k,\rho_k)}{\max}
& & \sum_{k\in\mathcal{C}}u_k(\rho_k(\boldsymbol{\theta}),\theta_k) \\
& \text{subject to}
& & v \geq 0.
\end{aligned}$$ In this formulation, we do not make any constraint on the participation of the CPs. Thus, when proposing the contracts resulting from solving (\[PF1\]), CPs may prefer not to select any of the contracts or select contracts that are not designed for their types. To analyze this economic incentive problem, next, we propose a solution based on the framework of *contract theory* for designing feasible contracts [@borgers2015introduction].
Proposed Incentive Mechanism for Caching {#form}
========================================
We consider a contract-theoretic problem in which an MNO seeks to motivate the CPs to participate and help it in the deployment of caching solutions. Due to the limited storage capacity of the SBSs as well as the limited capacity of the access links between the SBSs and the users, the allocation of storage space and power has an impact on the utilities of the CPs. In fact, the QoS achieved by the CPs’ users depends on the interference from the other SBSs as shown in (\[first\_equ\]) and (\[second\_equ\]). Moreover, the allocated storage capacity to a given CP depends on the number of CPs that have signed contracts with the same MNO. For instance, the more storage is allocated to a CP $k$ the lower is the available storage capacity for other CPs. Thus, in the considered model, there is *interdependence* between the signed contracts by the different agents.
Classical contract theory models that are used to model resource allocation problems in wireless networks such as in [@gao2011spectrum; @duan2014cooperative; @zhang2015contract] cannot be applied for the analysis of caching incentive problem between CPs and an MNO defined in (\[PF1\]). In fact, these works assume that the contract selected by a CP does not impact the utility of other CPs or focus only on models with one MNO and one CP. Thus, none of the existing works account for the interactions between the CPs. Moreover, the revelation of misleading information by a given CP in a multiple CPs model not only impacts the MNO but also impacts other CPs, which is not considered in [@gao2011spectrum; @duan2014cooperative; @zhang2015contract]. To define the most appropriate contract for the formulated problem (\[PF1\]), we consider the so-called truthful dominant strategy implementation. Under such contracts, the solution that maximizes the utility function of the CPs will require those CPs to reveal their private information which, in our model, pertains to the real popularity of their content. The goal of the MNO is to maximize the social welfare which effectively captures the global QoS that is experienced by the users of all CPs. Moreover, the MNO ensures that the cost of serving these users is at least covered by the price charged to the CPs. To incite the CPs to collaborate with the MNO via caching, the contract that a CP selects must be *feasible* in that it satisfies the following necessary and sufficient constraints:
Ex-post Individual Rationality (IR): *The contract that a CP selects should guarantee that the utility of the CP is nonnegative for any $\boldsymbol{\theta}_{-k}$ declared by the other CPs, $$\begin{gathered}
\label{IR}
r_k(\rho_k(\theta_k,\boldsymbol{\theta}_{-k}),\theta_k)- \pi_{k}\geq 0, ~ \forall k \in \{1,...,K\}.\end{gathered}$$*
Incentive Compatibility (IC): A *contract satisfies incentive compatibility constraint if each CP of type $\theta_k$ prefers to reveal its real type $\theta_k$ rather than another type $\hat{\theta}_k$, i.e., $$\begin{gathered}
\label{IC}
r_k(\rho_k(\theta_k,\boldsymbol{\theta}_{-k}), \theta_k)- \pi_{k}\geq r_k(\rho_k(\hat{\theta}_k,\boldsymbol{\theta}_{-k}),\theta_k)- \pi_{k}.\end{gathered}$$*
Incentive Mechanism Analysis
----------------------------
The goal of the MNO is to determine a pricing policy that motivates the CPs to declare their real type and simultaneously participate in the caching system through a budget balanced mechanism, i.e., the MNO would not experience a negative utility and its effort is covered by the price charged to the CPs. To this end, the CPs need to declare their types for the MNO that in turn optimizes their utility while accounting for the necessary conditions for contracts feasibility. The optimization problem of the MNO can be defined as follows:
$$\label{PF}
\begin{aligned}
& \underset{(\pi_k,\rho_k)}{\max}
& & \sum_{k\in\mathcal{C}}u_k(\rho_k(\boldsymbol{\theta}),\theta_k) \\
& \text{subject to}
& & (\ref{IR}),(\ref{IC}), v \geq 0.
\end{aligned}$$
The solution of this problem consists in the determination of the components of a contract that consist in the allocated storage space and the price charged to each CP. The closed-form of the contract is provided by the following theorem.
*The unique efficient solution of the optimization problem (\[PF\]) can be given by:*
$$\rho_k^* \in \arg\max\limits_{\rho_k}\sum_{i}\left[ r_i (\rho_i(\boldsymbol{\hat{\theta}}),\hat{\theta}_i)-c_i(\boldsymbol{\hat{\theta}})\right], \forall k,$$ $$\begin{gathered}
\label{sol}
\pi_k(\boldsymbol{\hat{\theta}})=\underbrace{\Big[\max\limits_{\rho_i} \sum_{i\neq k} r_i(\rho_i(\boldsymbol{\hat{\theta}}_{-k}),\hat{\theta}_i)-c_i(\boldsymbol{\hat{\theta}}_{-k})\Big]}_{(a)}\\ -\Big[\underbrace{\sum_{i \neq k} r_i(\rho_i^*( \boldsymbol{\hat{\theta}}),\hat{\theta}_i)-c_i(\boldsymbol{\hat{\theta}})\Big]}_{(b)},\end{gathered}$$ *where (a) represents the maximized social welfare when CP $k$ is not considered while in (b), CP $k$ is considered. Moreover, $\hat{\boldsymbol{\theta}}$ represents the revealed type by the CPs while $\boldsymbol{\theta}$ is the real type of the CPs.*
The proof is provided in the Appendix.
This result shows that, in order to determine the terms of a contract with a CP $k$, the MNO first, allocates the storage space to CP $k$ by solving the optimization problem (12). It is clear that the problem in (12) is NP-hard and thus it is challenging to find the optimal storage allocation. To solve (12), we use the framework of matching theory to analyze the assignment of storage space between the MNO that acts on behalf of its SBSs and the CPs [@gale1962college]. Matching theory is a suitable framework to solve NP-hard assignment problems such as in (12). As stated before, the allocated storage space to a CP $k$ depends on the allocated storage to the other CPs which is known as externalities. We solve the problem using a swap-based deferred acceptance algorithm which is guaranteed to converge to a stable outcome [@gale1962college]. Each CP $i$ starts by requesting from the MNO, a given storage space $\rho_i$ that maximizes $r_i(\rho_i(\boldsymbol{\hat{\theta}}_{-k}),\hat{\theta}_i)-c_i(\boldsymbol{\hat{\theta}}_{-k})$. After receiving all the requests from the CPs and based on the caching policy used by the MNO, it defines the accepted requests that maximize (12) and rejects the others. This is defined while accounting for the limited storage capacity of its SBSs. If the request of a given CP is rejected, the CP decreases the amount of the storage space it requests from the MNO. The MNO accepts new requests and rejects others based on the allocation configuration that maximizes (12). The procedure is repeated until there does not exist a CP $k$ that prefers to be assigned a given storage capacity $\rho_k$ and this allocation also maximizes (12) at the MNO.
Once the storage space is allocated to the CPs, the price paid by a CP $k$ is found from (13), which accounts for the impact of CP $k$ on the utility of other CPs. This price represents the difference between the global utility achieved by all the CPs when CP $k$ participates in the caching process, and the global utility achieved by the CPs when CP $k$ does not participate. Note that a CP $k$ can impact other CPs utilities in two ways. The first one is through the allocated storage space. In fact, when more storage space is allocated to CP $k$, less storage is available for other CPs and thus more requests of these CPs are served via the backhaul. The second is the traffic load of CP $k$ as the transmit power of the SBSs increases by increasing the number of served requests for that CP’s files. Thus, we can deduce that higher traffic load of a CP $k$ and large amounts of allocated storage to CP $k$ will result in an increase in the price charged by the MNO. The dependence of the price on the traffic load of the CPs, i.e., CPs type, is given next.
*When $\theta_k\geq\theta_l$ then we have $\pi_k\geq\pi_l$.*
This results follows directly from the monotonicity property of the rate function and the structure of (13).
Simulation Results {#sim}
==================
For our simulations, we consider five CPs with different traffic load levels from 1 to 5 with type 5 being the highest load. A type-1 CP is chosen with no traffic and is used as a baseline to compare the performance of our mechanism with the case in which there is no caching. We consider a set of 100 files whose popularity follows a Zipf distribution of parameter $\alpha=0.2$. The MNO has one MBS that serves all the requests that cannot be served from the SBSs’ cache. The number of SBSs is 10 and the total storage capacity of the SBSs is 1 Gbits. The transmit power of the SBSs is 1 W and the bandwidth capacity to 100 MHz.
In Fig. \[res1\], we show that the amount of content that is served via the backhaul for every CPs’ users when a CP selects the contracts designed by the MNO for each type. We account for the fact that the price that can be paid by each CP is limited and the limit increases by increasing the type of the CPs. From Fig. \[res1\], we can observe that when high type CPs select contracts that are designed for CPs with lower traffic load, the amount of content that is served via the backhaul increases until it reaches the maximum which corresponds to the lowest type contract. Similarly, when low type CPs select the contracts designed for higher types, the amount of traffic served from the backhaul increases. This is due to the high charged price by the MNO for high type CPs and thus CPs of lower type cannot afford that which results in a larger amount of served content from the backhaul. Thus, this result validates the fact the proposed approach for cache incentive compatible as is forces each CP to choose the contract designed for its own type.
In Fig. \[res2\], we compare the case in which all the CPs choose the contracts defined for their corresponding types by the MNO and the case in which the storage space is allocated equally by the MNO for all the CPs. Fig. \[res2\] show that the model of equal storage allocation outperforms the proposed mechanism for low type CPs. In this case, the allocated storage space for the CPs is higher than the required storage by the CPs which appears through the utilities of type-2 CP and type-3 CP that are only 2% to 10% higher than their utilities when following the proposed mechanism. On the other hand, the proposed mechanism outperforms the equal storage space allocation model for high type CPs. In fact, the utility of high level types is higher when selecting the offered contracts by the MNO for their specific types. The utilities of type-4 and type-5 CPs are 50% to 140% higher than their achievable utilities in the model of equally allocated storage space. The proposed mechanism is more beneficial for the CPs as it allows the MNO to offer to all CPs only the amount of storage space they need. In contrast, when using the equal allocation approach, the MNO allocates to the low type CPs more than their storage requirements and insufficient space for high level type CPs.
In Fig. \[res3\], we show the variation of the mean utility of the CPs when increasing the total number of CPs. We consider two different values for the parameter $\alpha$, $0.2$ which corresponds to the case in which the files have comparable popularity, and $2$ for the case in which some files are very popular while others have a very low popularity. We can observe that the mean utility for the CPs decreases by increasing the total number of CPs in the model. This is due to the data rate function that depends on the additional interference from the added CPs and the decrease of the available storage space that can be allocated to each CP. The achievable utility by a CP is up to 20% larger compared to the cases in which the CPs request a storage capacity that is larger or lower than the offered one by the MNO. Moreover, we can see that the popularity of the files impacts the mean utility of the CPs. In fact, the CPs can achieve a larger utility when the distribution of the popularity of the files is steep and thus by caching a file, a large amount of the requests can be served from the cache of the SBSs.
Conclusion {#conc}
==========
In this paper, we have proposed a new incentive framework to motivate the CPs to cooperate with an MNO and cache their content at the MNO’s SBSs. Based on contract theory, we have designed an incentive mechanism that allows the MNOs to offer a contract for each CP in which it sets the allocated storage for the CP and the charged price by the MNO for the caching service. This model accounts for both asymmetry of information and the interdependence between the different contracts. We have then derived the optimal pricing mechanisms and contracts that motivate the CPs to cache their content and reveal their private information. Simulations have shown the effectiveness of the proposed approach in inciting the participation of CPs in caching.
We prove Theorem 1 by showing the following Lemmas.
*The proposed mechanism is incentive compatible.*
We show this result by contradiction. Suppose that $\rho$ is an efficient decision rule but $(\rho,\pi)$ is not dominant strategy incentive compatible. Then, there exists $i, \boldsymbol{\theta},$ and $\boldsymbol{\hat{\theta}}$ such that: $$r_i(\rho_i(\hat{\theta}_i,\boldsymbol{\theta}_{-i}),\theta_i)-\pi_i(\boldsymbol{\theta}_{-i},\hat{\theta}_i)>r_i(\rho_i(\boldsymbol{\theta}),\theta_i)-\pi_i(\boldsymbol{\theta}).$$ From (\[sol\]), this implies that $$\begin{gathered}
r_i(\rho_i(\hat{\theta}_i,\boldsymbol{\theta}_{-i}),\theta_i)-(a)>r_i(\rho_i(\boldsymbol{\theta}),\theta_i)-(b),\end{gathered}$$ which is equivalent to
$$\begin{gathered}
r_i(\rho_i(\hat{\theta}_i,\boldsymbol{\theta}_{-i}),\theta_i)-\sum_{j\neq i}r_j(\rho_j(\hat{\theta}_i,\boldsymbol{\theta}_{-i}),\theta_j)-c_i(\hat{\theta}_i,\boldsymbol{\theta}_{-i})>\\r_i(\rho_i(\boldsymbol{\theta}),\theta_i)-\sum_{j\neq i}r_j(\rho_j(\theta_i,\boldsymbol{\theta}_{-i}),\theta_j)-c_i(\theta_i,\boldsymbol{\theta}_{-i}).\end{gathered}$$ This contradicts the efficiency of $\boldsymbol{\rho}$ based on (12) and thus, the assumption was incorrect.
*Truth telling is a dominant strategy under (\[sol\]).*
Consider the problem of choosing the best type $\hat{\theta}_i$ by a CP $i$. A best strategy for CP $i$ solves $$\begin{aligned}
& \underset{\hat{\theta}_i}{\max}
& & r_i(\rho_i(\hat{\boldsymbol{\theta}}),\hat{\theta}_i)-\pi_i(\hat{\boldsymbol{\theta}}).
\end{aligned}$$ Substituting the payment function by the proposed mechanism (12), we get
$$\begin{gathered}
\underset{\hat{\theta}_i}{\max} \Big[r_i(\rho_i(\hat{\boldsymbol{\theta}}),\hat{\theta}_i)-\underbrace{\sum_{j\neq i} r_j(\rho(\hat{\boldsymbol{\theta}}_{-i}),\theta_{j})-c_j(\theta_j,\hat{\boldsymbol{\theta}}_{-i})}_{(a)}\\+\sum_{j\neq i}r_j(\rho_j(\hat{\boldsymbol{\theta}}),\hat{\theta}_j)\Big].\end{gathered}$$ Since $(a)$ does not depend on $\hat{\theta}_i$, it is sufficient to solve
$$\begin{aligned}
& \underset{\hat{\theta}_i}{\max}
& & \bigg(r_i(\rho_i(\hat{\boldsymbol{\theta}}),\hat{\theta}_i)+\sum_{j\neq i}r_j(\rho_j(\hat{\boldsymbol{\theta}}),\hat{\theta}_j)\bigg).
\end{aligned}$$ Thus, CP $i$ would pick a declaration $\hat{\theta}_i$ that will lead the mechanism to pick a $\boldsymbol{\rho}$ which solves
$$\label{eq}
\begin{aligned}
& \underset{\boldsymbol{\rho}}{\max}
& & \bigg(r_i(\rho_i(\hat{\boldsymbol{\theta}}),\hat{\theta}_i)+\sum_{j\neq i}r_j(\rho_j(\hat{\boldsymbol{\theta}}_{-i}),\hat{\theta}_j)\bigg).
\end{aligned}$$ Under the proposed mechanism,
$$\boldsymbol{\rho}^*\in \arg\max\limits_{\boldsymbol{\rho}}~~\bigg(r_i(\rho_i(\hat{\boldsymbol{\theta}}),\theta_i)+\sum_{i\neq j}r_j(\rho_j(\hat{\boldsymbol{\theta}}),\hat{\theta}_j)\bigg).$$ The proposed mechanism (\[sol\]) will choose $\rho$ in a way that solves the maximization problem (\[eq\]) with $\hat{\theta}_i=\theta_i$. Thus, truth-telling is a dominant strategy for CP $i$.
*The proposed mechanism (\[sol\]) is ex-post individually rational.*
At the equilibrium, all the CPs are truthful and declare their real types. Thus, by replacing (\[sol\]) in the utility of a CP $i$, we have
$$\begin{gathered}
\label{IRP}
u_i=\sum_{i} r_i(\rho_i^*(\boldsymbol{\theta}),\theta_i)-c_i(\boldsymbol{\theta})-\sum_{j\neq i}r_j(\rho_j^*(\boldsymbol{\theta}_{-i}),\theta_j)-c_j(\boldsymbol{\theta}),\end{gathered}$$ where $\boldsymbol{\rho}^*$ is the outcome that maximizes the social welfare. The CPs could have picked $\rho_i(\boldsymbol{\theta}_{-i})$ instead of $\rho_i(\boldsymbol{\theta})$ as a solution of the optimization problem, as it is one of the possible strategies. This is possible because the set of strategies, i.e., the total storage capacity of the MNO, is fixed and does not change by changing the set of participating CPs. Thus,
$$\sum_{j}r_j(\rho_j^*(\boldsymbol{\theta}),\theta_j)\geq \sum_{j}r_j(\rho_j^*(\boldsymbol{\theta}_{-i}),\theta_j).$$ Furthermore, we know that the rate of a participating CP cannot be negative, i.e.,
$$r_j(\rho_j^*(\boldsymbol{\theta}_{-i}),\theta_j)\geq 0.$$ Therefore,
$$\sum_{i} r_i(\rho_i^*(\boldsymbol{\theta}),\theta_i)\geq\sum_{j\neq i}r_j(\rho_j^*(\boldsymbol{\theta}_{-i}),\theta_j).$$ Thus, (\[IRP\]) is non-negative and the proposed mechanism is ex-post individual rational.
*The proposed mechanism (\[sol\]) is weakly budget-balanced.*
Since the CPs are truth-telling at the equilibrium then we have
$$\begin{gathered}
\sum_{i} \pi_i(\boldsymbol{\theta})= \sum_{i}\bigg(\Big[\sum_{j\neq i}r_j(\rho_j(\boldsymbol{\theta}_{-i}),\theta_j)-c_j(\boldsymbol{\theta}_{-i},\theta_j)\Big]\\-\Big[\sum_{j\neq i}r_j(\rho_j(\boldsymbol{\theta}),\theta_j)-c_j(\boldsymbol{\theta},\theta_j)\Big]\bigg).\end{gathered}$$ Moreover, since the utility of a CP is a decreasing function of the number of the set of participating CPs, we have that, $\forall i$,
$$\sum_{j\neq i}r_j(\rho_j(\boldsymbol{\theta}_{-i}),\theta_j)-c_j(\boldsymbol{\theta}_{-i},\theta_j)\geq\sum_{j\neq i}r_j(\rho_j(\boldsymbol{\theta}),\theta_j)-c_j(\boldsymbol{\theta},\theta_j).$$ Thus, the proposed mechanism is weakly budget-balanced.
Next, we prove the provided result in Theorem 1.
Based on Lemma 1, Lemma 3 and Lemma 4, we can deduce that all the condition of the optimization problem (\[PF\]) are satisfied. Based on Lemma 2, we can deduce that the proposed mechanism (\[sol\]) is the unique efficient solution of the formulated problem (\[PF\]).
[^1]: This research was supported by ERC Starting Grant 305123 MORE (Advanced Mathematical Tools for Complex Network Engineering), the ANR project: WisePhy: Sécurité pour les communications sans fil à la couche physique, the U.S. National Science Foundation under Grants CNS-1513697 and CNS-1460316.
|
---
abstract: |
We present new CCD photometry of the distant old open star cluster Berkeley 32 in Johnson $V$ and Cousins $I$ passbands. A total of $\sim$ 3200 stars have been observed in a field of about $13^{'} \times 13^{'}$. The colour-magnitude diagram (CMD) in $V, (V-I)$ has been generated down to V = 22 mag. A broad but well defined main sequence is clearly visible. Some blue stragglers, a well developed subgiant branch and a Red Clump are also seen. By fitting isochrones to this CMD as well as to other CMDs available in the literature, and using the Red Clump location, the reddening, distance and age of the star cluster have been determined. The cluster has a distance of $\sim$ 3.3 kpc, its radius is about 2.4 pc; the reddening E(B-V) is 0.08 mag and the age is $\sim$ 6.3 Gyr. By comparison with theoretical isochrones, a metallicity of \[Fe/H\] $\approx -0.2$ dex has been estimated.
Theoretical isochrones have been used to convert the observed cluster luminosity function into a mass function in the mass range $\sim$ 0.6$-$1.1 $\Msun$. We find a much flatter mass function than what has been found for young clusters. If the mass function is a power law $dN \sim m^{\alpha} dm$, then we get $\alpha = -0.5 \pm 0.3$. This may be seen as a signature of the highly evolved dynamical state of the cluster.
Open star clusters: individual: Berkeley 32 - star: evolution - HR diagram - Mass functions - Galactic disk
---
[**A study of the old galactic star cluster Berkeley 32** ]{}
\
[*$1$ Departamento de Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile,\
e-mail: tom@coma.cfm.udec.cl*]{}\
[*$2$ U. P. State Observatory, Manora Peak, Naini Tal - 263 129, India*]{}\
[*$3$ Sternwarte der Universität Bonn, Auf dem Hügel 71, D-53121 Bonn, Germany*]{}\
Introduction
============
Berkeley 32 ($C0655+065\sim$ OCL 522, l=207.$^{\circ}$95, b=4.$^{\circ}$4), also known as Biurakan 8, is a small galactic (open) star cluster of angular diameter $\sim 6{'}$. It is located in the Galactic anticentre direction and has been classified as Trumpler class II2m (Lyngå 1987). This object was discovered by Iskudarjan (1960) and catalogued by Setteducati & Weaver (1960). On the sky survey maps, the cluster appears to be rich and likely of old age (cf. King 1964). The first photometric study of the cluster was carried out by Kaluzny & Mazur (1991) in the $UBV$ and Washington systems by CCD imaging of an area $\sim 6.^{'}6 \times 6.^{'}6$. They presented the colour-magnitude diagram (CMD) and discussed its morphology, derived cluster reddening and distance as $E(B-V) = 0.16$ mag and 3.1$\pm$0.2 kpc respectively, and estimated the metallicity as \[Fe/H\] = $-0.37\pm0.05$. They derived an age of $\sim$ 6 Gyr. Using morphological age parameters, Janes & Phelps (1994) estimated an age of 7.2 Gyr for Berkeley 32.
Scott et al. (1995) have determined radial velocities of 7 cluster members and found a mean heliocentric radial velocity of +101$\pm$3 km/sec, which among their cluster sample deviates most from a pure Galactic rotation after Berkeley 17.
Mass function studies of open star clusters indicated that the slopes of the mass functions of older (age $>$ 1 Gyr) clusters differ significantly from each other and also are not uniform over the entire mass range (cf. Sagar & Griffiths 1998b). However, the number of old objects studied so far is small. As such, a study of the old cluster Berkeley 32 can contribute interesting knowledge on the mass function of old, probably highly evolved clusters.
Our aim is to re-analyse Berkeley 32 with deeper photometry than Kaluzny & Mazur (1991) had at their disposal and study its mass function which is lacking. The observations, data reductions and comparison with earlier photometry are given in the next two sections. The cluster radius, other photometric results and mass function of the cluster are described in the subsequent sections of the paper.
**Observations and Reductions**
================================
The CCD observations have been obtained on 1999 March 22 with the 3.5m telescope at Calar Alto Observatory, Spain, run by the Max-Planck Institute for astronomy, Heidelberg. The focal reducer MOSCA attached at the RC Cassegrain focus provided an effective f/2.7 focal ratio ([**http://www.caha.es/caha/instruments/mosca/manual.html**]{}). The observations have been carried out in the Johnson $V$ and Cousins $I$ filters using the SITE 18b CCD chip. Each pixel (24$\mu$ square) corresponded to $0.^{''}53 \times
0.^{''}53$ on the sky. The non-vignetted area of the CCD was $ 1500 \times
1500 $ pixel$^2$ providing a field of about 13.$^{'}$25 $\times$ 13.$^{'}25$. The read-out noise was 5.4 electrons per pixel and the ratio electrons-per-ADU was $\sim$ 2.7. Figure 1 shows the finding chart for the imaged cluster region and Table 1 lists the log of the observations. For calibration purpose, we observed the standard star field SA 98 (Landolt 1992).
The present observations were carried out as a back-up programme. We therefore could not observe standard stars during the whole night. Instead, we observed the standard field SA 98 three times: before, between, and after the observations of Berkeley 32 (Table 1). The strategy was to observe Berkeley 32 at a similar air-mass as the standard fields, so that one can calibrate photometric data of the cluster region with an accuracy of a few percent without a precise determination of the atmospheric extinction coefficients.
[**Table 1.**]{} Log of CCD observations of the cluster Berkeley 32 and the calibration region SA 98 (Landolt 1992). The data have been obtained on March 22, 1999.
-------- ---------- -------- --------------- ----------
Object Time Filter Exposure time Air-mass
(UT) (seconds)
SA 98 19:23:45 $V$ 8 1.27
SA 98 19:29:00 $I$ 8 1.27
Be 32 19:40:04 $V$ 10 1.17
Be 32 19:44:57 $V$ 60 1.18
Be 32 19:49:05 $I$ 8 1.18
Be 32 19:51:47 $I$ 40 1.18
Be 32 19:54:36 $I$ 3 1.19
SA 98 20:00:04 $V$ 10 1.30
SA 98 20:02:37 $I$ 8 1.30
Be 32 20:05:29 $V$ 10 1.20
Be 32 20:07:44 $V$ 100 1.20
Be 32 20:11:37 $I$ 8 1.21
Be 32 20:14:04 $I$ 80 1.21
SA 98 20:43:19 $V$ 10 1.39
SA 98 20:45:37 $I$ 8 1.40
-------- ---------- -------- --------------- ----------
The basic processing of the data frames was done in the standard manner using the MIDAS data reduction package. The uniformity of flat fields is better than one percent in both filters.
Instrumental magnitudes have been measured using the DAOPHOT software (Stetson 1987, 1992) under MIDAS. The image parameters and errors provided by DAOPHOT were used to reject poor measurements. About 10% of the stars were rejected in this process. In those cases where brighter stars are saturated on deep exposure frames, their magnitudes have been taken only from the short exposure frames. Most of the stars brighter than V $\sim$ 12 mag could not be measured because they were saturated even on the shortest exposure frames.
[**Table 2.**]{} Colour coefficients and zero-points are for 1 second exposure time of the standard stars.
[ccc cc ]{}\
Air-mass & a0$\pm \sigma$ & a1$\pm \sigma$& b0$\pm \sigma$ & b1$\pm \sigma$\
1.27 & $-0.513\pm0.01$ & $0.07\pm0.01$ & $0.667\pm0.02$ & $0.03\pm0.003$\
1.30 & $-0.523\pm0.02$ & $0.07\pm0.01$ & $0.664\pm0.02$ & $0.03\pm0.003$\
1.39 & $-0.573\pm0.02$ & $0.07\pm0.01$ & $0.643\pm0.02$ & $0.03\pm0.003$\
The CCD instrumental magnitudes have been calibrated using the observations of the SA 98 field and the following relations $$V-v = a0 + a1*(V-I); \hspace{2cm} (V-I) - (v-i) = b0 +b1*(V-I)$$ where capital letters denote standard magnitudes and colours, and lower case letters denote instrumental values. The values refer to exposure time of 1 second. These equations along with the site mean atmospheric extinction values of 0.15$\pm$0.04 and 0.09$\pm$0.02 mag per unit air-mass in $V$ and $(V-I)$ respectively were used in determining the colour equations for the system as well as the zero-points. The effects of uncertainties in atmospheric extinction values are maximum on zero-points but least on the colour coefficients. We therefore averaged the colour coefficients from the individual standard observations. With these values fixed we calculated the zero-points. Table 2 lists the colour coefficients and zero-points derived in this way. The zero-points are uncertain by $\sim$ 0.02 mag in $V$ and $(V-I)$. The internal errors as a function of magnitude for each filter are given in Table 3. The errors become large ($>$ 0.15 mag) for stars fainter than $V = 22$ and $I = 21$ mag.
[**Table 3.**]{} Internal photometric errors as a function of brightness. $\sigma$ is the standard deviation ($\sigma$) per observations in magnitude.
Mag range $\sigma_V$ $\sigma_I$
--------------- ------------ ------------
12.0 $-$ 14.0 0.005 0.010
14.0 $-$ 16.0 0.005 0.010
16.0 $-$ 17.0 0.005 0.010
17.0 $-$ 18.0 0.006 0.013
18.0 $-$ 19.0 0.009 0.024
19.0 $-$ 20.0 0.017 0.051
20.0 $-$ 21.0 0.041 0.117
21.0 $-$ 22.0 0.106
22.0 $-$ 23.0 0.221
The $X$ and $Y$ pixel coordinates as well as the $V$ and $(V-I)$ magnitudes and DAOPHOT errors of the stars observed in Berkeley 32 are listed in Table 4. Stars observed by Kaluzny & Mazur (1991) have been identified in the last column. Table 4 is available only in electronic form at the open cluster database Web site at [**http://obswww.unige.ch/webda/.**]{} It can also be obtained from the authors.
**Comparison with previous photometry**
=======================================
In this section, we compare the present CCD photometry with the only previous CCD photometric observations of the cluster by Kaluzny & Mazur (1991) in the only common passband $V$. The transformation equations relating their ($X_{km}, Y_{km}$) coordinate system to ours ($X_{pres}, Y_{pres}$) were found to be $$X_{pres} = 1178.252 - 0.024 X_{km} - 1.512 Y_{km}; \hspace{1cm}
Y_{pres} = 259.383 + 1.511 X_{km} - 0.024 Y_{km}$$ There are 835 stars measured by Kaluzny & Mazur (1991) whose positions coincide within 1 pixel with the stars positions measured by us. The differences ($\Delta V$) between the present data and data obtained by them are plotted in Fig. 2, while the statistical results are given in Table 5. These show that except for a few outliers, which appear to be mostly stars that were treated as single in one study and as double (due to blending) in the other, the distribution of the photometric differences seems fairly random with almost no zero-point offset. As expected, the scatter increases with decreasing brightness and becomes more than $\sim$ 0.1 mag at fainter levels. Considering the uncertainties present in our and Kaluzny & Mazur’s (1991) measurements, we conclude that they are in very good agreement.
[**Table 5.**]{} Statistical results of the photometric comparison with data from Kaluzny & Mazur (1991). The difference ($\Delta$) is in the sense present minus comparison data. The mean and standard deviation ($\sigma$) are based on N stars. A few points discrepant by more than 3.5 $\sigma$ have been excluded from the analysis.
------------ ------------------ --------------- --------------- ------------------ -----
$ V$ range $(V-I)$ range
(mag) Mean$\pm \sigma$ N (mag) Mean$\pm \sigma$ N
12$-$14 $~0.006\pm$0.03 17 $-0.1$$-$0.65 $-0.021\pm$0.16 41
14$-$16 $-0.012\pm$0.04 67 0.65$-$0.8 $-0.013\pm$0.12 197
16$-$17 $-0.004\pm$0.04 129 0.8 $-$1.0 $-0.020\pm$0.12 242
17$-$18 $-0.015\pm$0.08 144 1.0 $-$1.5 $-0.026\pm$0.11 286
18$-$19 $-0.029\pm$0.09 159 1.5 $-$3.2 $ -0.011\pm$0.11 69
19$-$20 $-0.015\pm$0.14 157
20$-$21 $-0.023\pm$0.16 162
------------ ------------------ --------------- --------------- ------------------ -----
Radius of the cluster
=====================
We used radial stellar density profile for the determination of cluster radius. Such determinations can provide ambiguous results as it depends on the limiting magnitude of the star counts. The fainter the stars are, the larger becomes the cluster radius, if mass segregation due to two-body relaxation is present. Given these caveats, it is not our aim to derive a dynamically relevant radius, but to determine the region where the cluster population dominates over field stars so that it can be used for investigations of the cluster properties.
We derived the position of the cluster centre by iteratively calculating the average X and Y positions of stars within 150 pixels from an eye estimated centre, until it converged to a constant value. The (X,Y) pixel coordinates of the cluster centre are (700, 665) with an accuracy of few pixels. The radial stellar density profile determined up to $\sim 6^{'}$ from the cluster centre using stars brighter than V=18 mag is plotted in Fig. 3. The radius at which the star density flattens is considered as cluster radius which is $\sim 2.^{'}7$. This agrees fairly well with the value of 3$^{'}$ given by Lyngå (1987). We fit the following form of a King (1962) profile to the observed stellar density distribution
$$f(r)\propto C\cdot (\frac{1}{\sqrt(1+(r/r_c)^2)}-\frac{1}{\sqrt(1+(r_t/r_c)^2)})^2,$$
where $C$ is the central stellar density, $r_c$ and $r_t$ are the core and tidal radius respectively. A least square fitting of the profile to the observed points yielded $C = 33.9 \pm 8$ star/arcmin$^2, r_c = 1.0\pm 0.38,
r_t = 23 \pm 50$ arcmin.
Colour-magnitude diagrams (CMDs)
================================
The $V, (V-I) $ CMD
-------------------
We plot the $V, (V-I)$ CMD for all ($\sim$ 3200) measured stars in the region of Berkeley 32 in Fig.4(A). The CMD reaches down to $V = 22$ mag. The cluster main-sequence (MS) contaminated by field stars is clearly visible. Although it is clear that the stellar population of this region is of composite nature, the cluster population appears to be dominating. The only way to sharpen morphological features of the cluster sequence in the CMD is to select stars with small radial distances by compromising between a decreasing number of cluster stars and an increasing field population. The Fig. 4 (B) shows our best result. Here we have selected only stars with a radial distance up to $\sim 2.^{'}7$. The features of a very old open star cluster namely the distinct turn-off region and the subgiant branch are now very clearly visible. The giant branch (GB) is very sparsely populated and not well defined. Moreover, a group of stars can be seen which are brighter and bluer than the MS turn-off point suggesting that some of them are blue stragglers (BSs). Such objects have been found in most intermediate and old age open star clusters (see Kaluzny 1994; Phelps et al. 1994; Sagar & Griffiths 1999a). Many of them have been identified as close binary systems. The (X,Y) pixel coordinates, radius, magnitudes and colour of the stars located in the GB, red GB and BS regions of the CMDs are given in Table 6. The cluster membership of these stars is also indicated in the table. A star is considered as probable cluster member if it lies within $\pm$0.05 mag in colour and $\pm$0.1 mag in brightness with respect to the isochrone of the cluster age at least in two of the $V, (U-V); V, (B-V)$ and $V, (V-I)$ diagrams. In addition, brightening due to unresolved/optical binary stars has also been considered.
Fig. 4 (C) shows the $V, (V-I)$ diagram of stars with radial distances more than $\sim 4.^{'}4$ from the cluster centre. Overplotted are the fiducial points of the cluster sequence. There are a few red giants, which perhaps still belong to the cluster. A considerable part of the main sequence population has a turn-off similar to the cluster, but the bulk of the main sequence stars are clearly shifted towards the red, indicating higher reddening, and thus a background population. However, the interesting question whether there are evaporated stars surrounding the cluster can not be answered on the basis of the present data. For this, kinematic informations like proper motions and radial velocities of these stars are required.
The cluster age from the “Red Clump”
------------------------------------
It is well known that for intermediate and old open star clusters, the location of the Red Clump (RC) (the more massive analog of the horizontal branch in globular clusters) relative to the MS turn-off point is correlated with age (cf. Kaluzny 1994; Phelps et al. 1994; Carraro & Chiosi 1994; Pandey et al. 1997 and references therein). The two morphological parameters generally used for estimating cluster ages are the differences in magnitudes ($\triangle V$) and colours ($\triangle (B-V)$ or $\triangle (V-I)$) between the RG branch at the level of the clump and the MS turn-off point, with the advantage that no prior knowledge of cluster distance, reddening and accurate metallicity is required.
Following Kaluzny (1994), we find $\triangle V = 2.7\pm0.05, \triangle (V-I) =
0.45\pm0.03$ in the case of Berkeley 32. Using the relation given by Carraro & Chiosi (1994), we derive log (age) = $9.8\pm0.1$ for the cluster. A slightly modified version of $\triangle V$ has been introduced by Janes & Phelps (1994) who used the luminosity difference between the RC and the inflection point between the turn-off region and the subgiant branch. We confirm their value of 2.4 for Berkeley 32. From their Fig.1 one reads off the logarithm of age as 9.8-9.9, yielding an age of 6.3 - 8 Gyr. According to their Table 1, there are only a few clusters older than Berkeley 32, like NGC 6791, Berkeley 54, AM 2 and Cr 261. It is thus clear that Berkeley 32 belongs to the group of very old open clusters in our Galaxy.
Determination of the cluster parameters using theoretical isochrones
--------------------------------------------------------------------
We have determined the colour excess, the distance modulus, and also the age of the cluster by fitting theoretical stellar evolutionary isochrones from the set of Bertelli et al. (1994) to our $V, (V-I)$ diagram. These isochrones are derived from stellar models computed with updated radiative opacities and include the effects of mass loss and convective core overshooting. The models trace the evolution from the zero-age main-sequence (ZAMS) to the central carbon ignition for massive stars and to the beginning of the thermally pulsing regime of the asymptotic giant branch phase for low and intermediate mass stars.
As most of the factors responsible for the colour spread in the MS will redden the stars (differential reddening, binaries, rotation, star spots), we have used the blue envelope of the MS in the CM diagram for the estimation of the cluster parameters. We fit the isochrones by eye taking into account the observational error. It turns out that the isochrone with log(age) = 9.8, $X =
0.7, Y = 0.28$ and $Z = 0.008$ fits best to the cluster locus, including the RC, and thus is in good agreement with the age estimated from the morphological parameters of the cluster CMD. In order to also demonstrate upper limits of the effects of binaries in the CMD, the log (age) = 9.8 isochrone for the single stars has been brightened by 0.75 mag leaving the colour unchanged. A maximum reddening of $E(V-I) = 0.11$ (or $E(B-V) = 0.08$) mag can be applied to place the isochrone correctly on the cluster sequence observed in Fig. 4(B). Some stars above the turn-off point lie on the isochrones of binaries indicating the possibility of being indeed binary members of the cluster. The lower giant branch in the $V, (V-I)$ diagram appears marginally too blue, indicating that the cluster may have a slightly higher metallicity. However, a $Z=0.02$ isochrone is definitely too metal-rich. Moreover, a solar metallicity would decrease the cluster reddening even further, while Kaluzny & Mazur (1991) quote a reddening of $E(B-V) = 0.16$ mag. However, such a high reddening is supported only by their $V, (U-V)$ diagram (see section 5.4).
It can also be seen that the theoretical location of the RC fits rather well with the observed one for Berkeley 32 unlike in some other old open star clusters (see Sagar & Griffiths 1999a). For example, it is too faint for NGC 6603 and too bright for NGC 7044.
The value of the apparent distance modulus derived from Fig. 4(B) is 12.8 mag. Here we adopt the reddening law of Rieke & Lebofsky (1985), who give $A_V/E(V-I) = 1.94$. With the extinction $A_V = 0.21$ mag, we get for Berkeley 32, a true distance modulus of 12.6 with an uncertainty of $\sim$ 0.15 mag which includes errors in the photometric calibration, isochrone fitting and the reddening determination.
Isochrone fitting to the UBV data of Kaluzny & Mazur (1991)
-----------------------------------------------------------
Fig. 5 shows the $V, (U-V)$ and $V, (B-V)$ diagrams generated from the Kaluzny & Mazur (1991) photometric data. Overplotted are the isochrones of Bertelli et al. (1994), having the same ages, metallicity and helium abundance as we used for our $V, (V-I)$ diagram in Fig 4(B). In order to fit the isochrone to the cluster sequence, we had to employ a reddening of $E(B-V) = 0.08$ mag and $E(U-V) = 0.22$ mag in the $V, (B-V)$ and $V, (U-V)$ diagrams respectively. While for the reddening law of Rieke & Lebofsky (1985), the $E(B-V)$ value agrees well with our E(V-I) value ($E(V-I)/E(B-V)$ = 1.6), the $E(U-V)$ is too large ($E(U-V)/E(B-V)$=1.64). On the other hand, it is too small for $E(B-V) = 0.16$ mag, given by Kaluzny & Mazur (1991) and is thus not compatible with the reddening values derived from the $V, (V-I)$ and $V, (B-V)$ diagrams. This may suggest that the $U$-photometry is perhaps in error and we adopt $E(B-V) = 0.08$ mag as the value for the cluster reddening.
The cluster distance from the Red Clump
---------------------------------------
For a star cluster as old as Berkeley 32, an attractive method to determine its distance is using the location of the RC of intermediate-age helium core burning stars as a standard candle (e.g. Paczynski & Stanek 1998). The absolute I-magnitude of RC stars in the solar neighborhood has been calibrated by Hipparcos parallaxes, resulting in $M_I^0 = -0.23 \pm0.03$. Cole (1998) discusses the age and metallicity dependence of the RC-brightness and notes that for poulations older than 4-5 Gyr, the $M_I^0$ is independent of stellar mass, but still shows a metallicity dependence of the order $\delta
M_I^0 = (0.21\pm0.07) [Fe/H]$, where $\delta M_I$ is the brightness difference between the RC in the solar neighborhood and the population under consideration. As the value is small for Berkeley 32, we neglect this correction here.
The RC in Berkeley 32 has $V=13.67\pm0.03$ and $(V-I) = 1.16\pm0.03$, where the error is the uncertainty in the definition of the RC in the CMD. This yields $(m-M{_I}) = 12.74 \pm0.08$ as the apparent distance modulus, if we include the photometric calibration uncertainty in the error. The extinction in $I$ is determined by $A_I/E(V-I) = 0.93$ which is 0.10 for a value of $E(V-I) = 0.11$. If we assign an additional error of 0.05 to the extinction correction, we have $(m-M)_0 = 12.64\pm0.1$ as the value for true distance modulus. This agrees well with the value obtained using isochrone fitting. In the following, we therefore adopt $12.6\pm0.1$ as the value for the distance modulus of the cluster. The present distance determination of 3.3$\pm$0.2 kpc agrees very well with the value of 3.1 kpc given by Kaluzny & Mazur (1991). The cluster parameters derived by us are listed in Table 7.
[**Table 7.**]{} The age, metallicity $([Fe/H])$, reddening ($E(B-V)$ and $E(V-I)$), true distance modulus($(m-M)_0$), distance, galacto-centric distance ($R_{GC}$) (adopting galacto-centric distance of the Sun as 8 kpc) and z-distance for Berkeley 32.
----------------- ---------------
Age 6.3 Gyr
$[Fe/H]$ -0.2 dex
$E(V-I);E(B-V)$ 0.11;0.08
$(m-M)_0$ $12.60\pm0.1$
Distance 3.3 kpc
$R_{GC}$ 11.0 kpc
z 250 pc
----------------- ---------------
Location of Berkeley 32 in the Galaxy
-------------------------------------
The cluster Berkeley 32 occupies an important position for understanding the variation of metallicity in the Galactic disk, as the issue of the existence of a metallicity gradient is not yet settled. According to Friel (1995), the metallicities of open clusters indicate a gradient of $-0.09$ dex/kpc. On the other hand, Twarog et al. (1997) argue that the open cluster system can be divided in 2 radial groups, with a very flat or even vanishing gradient in each group. Their mean metallicities differ by 0.3 dex and there is a discontinuity at a radial distance of 10 kpc. As the galactocentric distance of Berkeley 32 puts it just near this discontinuity, Berkeley 32 could help to decide between these two metallicity patterns in the Galactic disk. However, a more accurate determination of the metallicity than we are able to do, is required. Also, more clusters/objects either in the vicinity of Berkeley 32 or at similar galacto-centric distances need to be observed before the the metallicity pattern can be unambiguously determined.
Mass function
=============
The study of the mass function (MF) of Berkeley 32 is based on a pair of deep $V$ and $I$ CCD frames only. This is done for evaluating the data completeness accurately. Guided by the radial stellar density profile in Fig. 3, we selected stars located within a circle of 165 arcsec radius (surface area 23.76 square arcmin) around the cluster centre for the MF study. With the aim of detecting possible radial MF variations, the data completeness has first been evaluated in an inner and outer region separately, but it turned out to be the same within the errors. Besides, the small number statistics prevented us from a detailed study. We therefore determined the MF for the entire region without any subdivision.
To suppress the field star contamination as far as possible, we selected cluster main sequence stars by using the following boundaries in the $V, (V-I)$ diagram: $$V > 8.91+9.45 \cdot (V-I) + 1.8 \cdot (V-I)^2\\$$ and\
$$V < 11.52 + 12.64 \cdot (V-I) + 3.86 \cdot (V-I)^2\\$$ The field star contamination has been determined using the remaining chip area outside a radius of 265 arcsec (surface area 123.1 square arcmin) from the cluster centre. Table 8 lists the field and cluster counts derived in this way along with their completeness factors. The completeness factors have been determined by using artificial stars along the clusters main sequence and recovering them in the CMD, not in one filter, to be as realistic as possible.
[**Table 8.**]{} The V-magnitude of the bin center, the raw counts for cluster ($N_C$) and field regions ($N_F$), and the corresponding completeness factors $f_C$ and $f_F$ are listed. Absolute $M_V$ and stellar mass (m) of the bin center, the mass interval ($\Delta m$) corresponding to the magnitude bin, the normalised counts ($N$) and their errrors ($\delta N$) are also listed.
V $N_C$ $N_F$ $f_C$ $f_F$ $M_V$ m $\Delta m$ $N$ $\delta N$
------- ------- ------- ------- ------- ------- ------- ------------ ------- ------------
16.25 41 53 1.00 1.00 3.40 1.113 0.0693 18.70 4.2
16.75 44 67 1.00 1.00 3.90 1.046 0.0658 19.88 4.6
17.25 47 86 1.00 1.00 4.40 0.982 0.0623 20.54 5.0
17.75 43 107 1.00 1.00 4.90 0.921 0.0588 16.00 5.1
18.25 50 135 0.95 1.00 5.40 0.864 0.0553 20.22 6.3
18.75 37 121 0.85 1.00 5.90 0.811 0.0518 16.39 6.3
19.25 35 127 0.80 0.93 6.40 0.761 0.0483 15.15 7.4
19.75 31 158 0.81 0.89 6.90 0.714 0.0449 3.76 7.4
20.25 37 154 0.80 0.90 7.40 0.671 0.0413 13.46 8.8
20.75 31 142 0.73 0.86 7.90 0.631 0.0379 11.73 8.9
21.25 24 116 0.60 0.70 8.40 0.595 0.0344 9.82 11.1
Table 8 also lists the numbers which are relevant for the MF determination. The transformation from apparent to absolute visual magnitude ($M_V$) has been done using the cluster parameters given in Table 7. The isochrone log(age) = 9.8 and $z=0.008$ provides the following parametrization of mass $(m)$ and $M_V$: $$m = 1.665 -0.186 \cdot M_V + 0.00698 \cdot M_V^2.$$ The values of the normalised counts ($N$) are in stars/arcmin$^2$. They are corrected for completeness and field star contamination, and divided by the mass interval of the magnitude bin. The errors of the normalised counts result from error propagation. This may be incorrect from a puristic viewpoint, as they are no longer small compared to the counts. However, we do not use them any further beyond a qualitative demonstration that they are large.
A common description of the stellar mass spectrum is a power law $dN \propto
m^\alpha dm$, where $dN$ is the number of stars in the mass interval $m+dm$, and $\alpha$ is the slope of the MF. However, a uniform exponent is at best realised within limited mass intervals. The universality of the slope of the initial mass function (IMF) is still a matter of discussion (for a recent review see Scalo 1998), but studies of a large number of young clusters in the Milky Way and the Large Magellanic Clouds do not speak evidently against an universal IMF at least above 1 $M_{\odot}$ (e.g. Sagar et al. 1986; Sagar & Richtler 1991; Janes & Phelps 1994; Fischer et al. 1998; Sagar 2000), with $\alpha$ around $-2.3$. Below 1 $M_{\odot}$, the data for young open clusters are sparse and any secure statement on a possible universal IMF is not yet possible.
However, in the case of a very old cluster like Berkely 32, we might anyway not expect the IMF to be still realized. As a cluster evolves dynamically, low mass stars evaporate out of the cluster potential faster than high mass stars. In a cluster much older than its relaxation time, the dynamical effect therefore can change an originally rising IMF into a flat or even declining MF.
Fig. 6 shows the MF for Berkeley 32. The logarithm of mass is plotted against the logarithm of the normalised counts. Note that the binning in mass is also logarithmic. Although the errors are too large for any deeper analysis, it is apparent that the MF is much flatter than of most young clusters. A fit to a power-law indeed returns the value $\alpha = -0.5 \pm 0.3$ while we expect in this mass domain an exponent around $-2$ for young clusters (Richtler 1994). Here it must be remarked that we assumed single stars for the applied mass-luminosity relation of the isochrone. Sagar & Richtler (1991) discussed how the presence of binaries flattens a “true” IMF. But even if we find a large binary fraction in Berkeley 32, as the CMD suggests, their effect would by far not be sufficient to steepen the observed MF to the level as it is observed for young clusters.
This behaviour, in agreement with theoretical expectations, has been found for other old open clusters as well. For example, Francic (1989), among his sample of 8 open clusters, found the old objects NGC 752 (2.5 Gyr) and M67 (5 Gyr) to show even declining mass functions with $\alpha > 0$. Further well studied open clusters like NGC 6791 (Kaluzny & Rucinski 1995), NGC 188 (von Hippel & Sarajedini 1998), NGC 2243 (Bergbusch et al. 1991) also have flatter MFs. But one also can find old clusters with MF not distinguishable from a Salpeter mass function, e.g., mass spectrum of Berkeley 99 (age 3.2 Gyr) has $\alpha \sim -2.4$ (Sagar & Griffiths 1998b). This demonstrates that open clusters do have distinctly different dynamical histories, which may depend on their structure, total mass, location, orbit characteristics etc.
Summary
=======
New $V$ and $I$ CCD photometry down to $V = 22$ mag is presented for about 3,200 stars in the region of the open cluster Berkeley 32. The present photometry serves as a data base for determining the cluster properties and to study the stellar mass function for the first time. The cluster’s radial density profile is well represented by a King (1962) profile. By fitting of theoretical isochrones and using the location of the Red Clump, we confirm earlier results that it is indeed a very old open cluster (6.3 Gyr). Its metallicity is between $Z = 0.008$ and $Z = 0.02$, distance is 3.3 kpc and galacto-centric distance is 10.8 kpc. Clusters/objects with these characteristics can play a very valuable role to distinguish between the two models of the metallicity variation in the Galactic disk, advocated by Friel (1995) and Twarog et al. (1997) respectively. However, the case of Berkeley 32 is ambiguous. The parameters of Berkeley 32 are compatible with both a smooth Galactic metallicity gradient as well as with its membership of the cluster population of the inner domain of Twarog et al. (1997).
We also investigated the mass spectrum of Berkeley 32 in the mass range 0.6-1.1 $M_{\odot}$. A power-law fit returns $\alpha = -0.5\pm0.3$ for the slope of the MF, which is much flatter than the slopes found in young open clusters. Berkeley 32 shares this behaviour with other old open clusters which indicates an evaporation of low-mass cluster stars.
[**ACKNOWLEDGEMENTS**]{}
The suggestions/comments given by the referee Randy L. Phelps improved the presentation and readablity of the paper. RS gratefully acknowledges the support from the Alexander von Humboldt Foundation. TR wants to thank the U. P. State Observatory, Nainital, the Deutsche Forschungsgemeinschaft, and the Indian National Science Academy, for financial support and warm hospitality. Thanks to Klaas de Boer for going through the manuscript critically. The [**Open Cluster Data Base**]{} maintained by J.-C. Mermilliod has been used in the present work.
[**REFERENCES**]{}
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Fig. 1
: Identification chart for the Berkeley 32 region. The (X,Y) coordinates are in CCD pixel units and one CCD pixel corresponds to 0.$^{''}$53 on the sky. North is up and East is to the left. Filled circles of different sizes represent the brightness of the stars. The smallest size denotes stars of $V = 17$ mag.
Fig. 2
: Comparison of the present $V$ magnitude with those of Kaluzny & Mazur (1991). The differences ($\triangle$) are in the sense of this study minus Kaluzny & Mazur. They are plotted against the present CCD photometry.
Fig. 3
: Plot of the radial density profile ($\bullet$) for stars brighter than $ V = 18$ mag in Berkeley 32 region. The length of the bar represents errors resulting from sampling statistics. Overplotted (solid curve) is a King (1962) profile with parameters given in the text. The arrow denotes the radius where the surface density of cluster stars becomes becomes comparable with the field star density.
Fig. 4
: The $V, (V-I)$ diagrams (A) for all stars observed by us, (B) for the cluster population (stars with radius $\le 2.^{'}7$) and (C) for the field population (stars with radius $\ge 4.{'}4$) in the Berkeley 32 region are plotted. The composite nature of the stellar population is apparent in (A). In the cluster population, the Bertelli at al. (1994) isochrones for $z=0.008$ and log (age) = 9.7 (short-dashed), 9.8 (continuous) and 9.9 (dot-short dashed) are shown. The isochrone of log (age) = 9.8 found to be best fitting to the observed cluster sequence with a reddening of $E(V-I) =
0.11$ (or $E(B-V) = 0.08$) and an apparent distance modulus of 12.8 mag. The dotted curve shows the extent that binaries of equal mass can brighten the isochrone of log (age) = 9.8. In the field population, the overplotted curve is displaying the cluster locus. There are red giants resembling cluster giant branch stars. A considerable part of the main sequence population has a turn-off similar to the cluster, but the bulk of the main sequence stars are clearly shifted towards the red, indicating higher reddening, and thus a background stellar population.
Fig. 5
: The $V, (U-V)$ and $V, (B-V)$ diagrams generated for the cluster population of Berkeley 32 from the photometric data of Kaluzny & Mazur (1991). The Bertelli et al. (1994) isochrones of the same metallicity and ages, shifted by the same value of the apparent distance modulus as in Fig. 4(B) are shown. The best eye fits to the cluster sequence for the reddening values shown in the plot. The $E(U-V)$ value is not compatible with $E(B-V)$, indicating a problem with the $U$ photometry.
Fig. 6
: The mass function of Berkeley 32 between 0.6 and 1.1 $M_\odot$ as derived from the present data. Although the error bars are large, the mass function is clearly flatter than what has been found for young star clusters.
[**Table 6.**]{} Spatial and $UBVI$ photometric values of the candidate giant branch and blue straggler stars are listed along with the identification of Kaluzny & Mazur (1991) prefixed with KM. The $(U-B)$ and $(B-V)$ values are taken from Kaluzny & Mazur (1991). The probable photometric members have been identified as PM in the last column.
[rrr rcc ccrc]{}\
\
\
Star &X & Y & Radius &$V$ &$(U-B)$ & $(B-V)$ & $(V-I)$ & Other &Membership\
&(pixel) &(pixel) &(pixel)&(mag)&(mag)&(mag)&(mag)&identification\
226 & 813.65& 509.20& 192.85& 16.00& 0.18 &0.64 & 0.74& KM86 & PM\
238 & 525.10 & 526.47& 223.12 & 15.57& 0.83 &1.05 & 1.14& KM59 &\
242& 667.64 & 532.82& 136.08& 13.86&0.85 & 1.10 & 1.19& KM21 & PM\
245& 720.59& 535.05& 131.57& 16.22&0.07 &0.59 & 0.71& KM120 & PM\
254& 679.15& 546.71& 120.11& 15.87&0.62 &0.97 & 1.10& KM74 & PM\
264& 426.71& 564.67& 291.12& 16.03&0.15 &0.67 & 0.79 & KM108 & PM\
269& 975.20& 571.96& 290.50& 14.28&0.88 &1.14& 1.25& KM24 & PM\
274& 947.00& 584.20& 259.88& 13.27&0.66 &1.01 & 1.13& KM10 & PM\
293& 760.87& 617.50& 77.21& 16.08&0.11 &0.62 & 0.74& KM103 & PM\
302& 603.93& 629.92& 102.27& 16.02&0.09 &0.57 & 0.69& KM91 & PM\
310& 498.03& 639.10& 203.62& 15.28&0.80 &1.01 & 1.06& KM46 & PM\
318& 880.21& 657.52& 180.37& 13.76&0.78 &1.08 & 1.18& KM19 & PM\
331& 584.61& 683.34& 116.84& 16.14 &0.30 &0.77 & 0.88& KM113 & PM\
332& 682.86& 684.53& 25.98& 16.00&1.10 &1.15 & 1.34&KM93 &\
347& 748.92& 706.51& 64.16& 12.90&0.68 &1.02 & 1.06& KM8 &\
352& 514.50& 720.01& 193.48& 14.98&0.12 & 0.68 & 0.78& KM38 &\
357& 812.84& 727.41& 128.95& 15.88&0.63 &0.96 & 1.04& KM75 & PM\
360& 534.13& 733.92& 179.62& 14.76&0.80 &1.16 & 1.25& KM33 &\
414& 759.17& 809.22& 155.89& 16.01&0.19 &0.69 & 0.82& KM87 & PM\
416& 904.29& 812.04& 251.70& 14.43&0.88 &1.12 & 1.24& KM27 & PM\
419& 859.71& 813.91& 218.36& 16.29&0.11 &0.60& 0.72& KM131 & PM\
449& 896.58& 866.32& 281.38& 16.44&0.12 &0.59 & 0.72& KM143 & PM\
456& 730.79& 884.57& 221.72& 13.71&0.79 &1.07 & 1.14& KM17 & PM\
465& 777.42& 897.74& 245.28& 16.07&0.45 &0.87 & 0.98& KM100 & PM\
488& 625.73& 944.14& 288.85& 16.14&0.40 &0.83& 0.88& KM116 & PM\
963& 699.83& 443.49& 221.51& 15.59&0.06 &0.57 & 0.69&KM58 & PM\
974& 570.29& 464.89& 238.47& 13.70&0.77 &1.05 & 1.15& KM18 & PM\
991& 824.76& 502.60& 204.79& 16.08&0.10 &0.60& 0.74& KM101 & PM\
1061& 532.84& 609.89& 176.01& 16.27&0.97 &1.02 & 1.11& KM137 &\
1077& 757.98& 637.12& 64.33& 16.24&0.10 &0.60 & 0.72&KM122 & PM\
1089& 861.68& 652.43& 162.17& 16.16&0.27 &0.81 & 0.95& KM110 & PM\
1101& 720.91& 668.72& 21.24& 16.15&0.12 &0.61& 0.71& KM112 & PM\
1104& 707.22& 671.58& 9.77& 16.09&0.13 &0.61 & 0.73&KM105 & PM\
1116& 751.69& 691.00& 57.86& 16.38&0.05 &0.64& 0.68& KM138 & PM\
1128& 663.60& 710.11& 57.96& 16.04&0.19 &0.65 & 0.76& KM92 & PM\
1132& 797.78& 710.48& 107.84& 16.35&0.11 &0.61 & 0.68& KM136 & PM\
1147& 448.68& 737.15& 261.47& 16.21&0.14 &0.63& 0.77& KM128 & PM\
1158& 649.41& 762.07& 109.46& 16.10&0.33 &0.71 & 0.82& KM117 & PM\
1171& 878.02& 779.25& 211.53& 16.37&1.19 &1.46 & 1.99& KM140 &\
1179& 821.92& 785.28& 171.27& 15.64&0.10 &0.59 & 0.70& KM57 & PM\
1626& 682.58& 373.01& 292.51& 15.70&0.61 &1.00 & 1.09& KM61 & PM\
1640& 828.25& 495.86& 212.26& 16.44&0.09 &0.59 & 0.77& KM149 & PM\
1642& 875.89& 505.16& 237.67& 15.68&0.59 &0.98 & 1.09&KM62 & PM\
1650& 753.47& 560.60& 117.30& 15.08&0.17 &0.65 & 0.70& KM39 &\
1654& 925.16& 583.53& 239.45& 14.55&0.61 &0.99 & 1.15& KM29 & PM\
1656& 725.49& 585.68& 83.32& 16.40&0.07 &0.61 & 0.71& KM146 & PM\
1657& 662.72& 594.85& 79.44& 15.98&0.18 &0.73 & 0.83& KM88 & PM\
[rrr rcc ccrc]{}\
\
Star &X & Y & Radius &$V$ &$(U-B)$ & $(B-V)$ & $(V-I)$ & Other &Membership\
&(pixel) &(pixel) &(pixel)&(mag)&(mag)&(mag)&(mag)&identification\
1661& 561.13& 604.88& 151.33& 16.06&0.42 &0.87 & 0.98& KM97 & PM\
1662& 418.84& 605.15& 287.46& 15.24&1.58 &1.50 & 1.61& KM47 &\
1663& 964.84& 611.62& 270.17& 16.10&0.24 &0.78 & 0.89& KM98 & PM\
1666& 666.79& 627.02& 50.45& 15.95&0.07 &0.62 & 0.69& KM84 & PM\
1668& 893.35& 645.43& 194.34& 15.35&0.33 &0.71 & 0.80& KM45 & PM\
1675& 831.23& 665.10& 131.23& 16.22&0.06 &0.61 & 0.71& KM119 & PM\
1685& 768.41& 695.85& 75.04& 13.64&0.72 &1.06 & 1.15&KM16 & PM\
1691& 604.59& 718.30& 109.29& 15.15&0.32 &0.82 & 0.95& KM41 &\
1701& 566.32& 766.05& 167.58& 16.17&0.10 &0.61 & 0.72& KM118 & PM\
1702& 616.26& 765.31& 130.67& 15.80&0.62 &0.97 & 1.07& KM71 & PM\
1714& 741.70& 830.99& 171.15& 15.82&0.12 &0.60 & 0.71& KM69 & PM\
1717& 566.13& 838.96& 219.51& 15.96&0.88 &1.01 & 1.11&KM90 & PM\
1718& 732.24& 853.39& 191.13& 16.27&0.14 &0.60 & 0.75& KM126 & PM\
1895& 578.41& 396.63& 294.63& 15.46&0.58 &0.86 & 0.89& KM52 & PM\
1937& 477.11& 518.17& 266.91& 16.43& & & 0.71& & PM\
1948& 847.47& 538.41& 194.35& 13.42&0.83 &1.11 & 1.21&KM12 & PM\
1986& 498.72& 632.23& 203.93& 16.35&0.34 &0.76 & 0.82&KM141 &\
1996& 977.95& 643.70& 278.76& 16.15&0.53 &1.08 & 1.04&KM127 & PM\
2009& 482.78& 667.41& 217.23& 16.44& & & 0.72& & PM\
2034& 687.78& 743.96& 79.90& 16.38&0.19 &0.66 & 0.74& KM145 & PM\
2239& 658.40& 647.67& 45.07& 15.36&0.65 &0.99 & 1.07& KM50 & PM\
2247& 733.91& 736.48& 79.12& 14.51&0.46 &0.89 & 0.98& KM32 &\
2260& 853.02& 906.84& 286.18& 16.14&0.28 &0.72 & 0.79& KM104 & PM\
2307& 492.18& 455.51& 295.09& 15.14&0.35 &0.72 & 0.76& KM42 &\
2313& 853.66& 539.06& 198.68& 16.14&0.10 &0.62 & 0.76& KM114 & PM\
\
Star &X & Y & Radius &$V$ &$(U-B)$ & $(B-V)$ & $(V-I)$ & Other &Membership\
&(pixel) &(pixel) &(pixel)&(mag)&(mag)&(mag)&(mag)&identification\
169& 718.22& 376.70& 288.88& 15.86&0.17 &0.40 &0.48&KM72 & PM\
292& 742.54& 615.41& 65.34& 15.97&0.16 &0.39 &0.41&KM82 & PM\
303& 743.82& 631.21& 55.33& 16.07&0.07 &0.50 &0.60&KM102 & PM\
373& 684.03& 753.60& 90.03& 16.36&0.09 &0.54 &0.58&KM144 & PM\
413& 823.05& 806.89& 187.81& 16.00&0.11 &0.54 &0.63&KM85 & PM\
433& 468.83& 844.78& 292.85& 13.55&0.08 &0.54 &0.60&KM14 &\
442& 638.28& 859.77& 204.32& 15.95&0.11 &0.51 &0.61&KM81 & PM\
446& 858.62& 863.88& 254.39& 14.44&0.12 &0.46 &0.52&KM26 &\
507& 712.97& 963.58& 298.86& 13.23&0.18 &0.27 &0.25&KM9 &\
1027& 545.34& 556.03& 189.19& 16.03&0.17 &0.49 &0.59&KM99 & PM\
1064& 849.30& 614.48& 157.62& 16.26&0.12 &0.49 &0.60&KM125 & PM\
1237& 564.34& 899.03& 270.51& 16.20&0.17 &0.47 &0.57&KM123 & PM\
1651& 616.62& 565.03& 130.18& 15.12&0.17 &0.22 &0.22&KM40 &\
1680& 650.47& 675.41& 50.61& 16.26&0.08 &0.49 &0.57&KM129 & PM\
1684& 742.58& 694.04& 51.54& 15.81&0.00 &0.60 &0.61& KM68 & PM\
1700& 732.72& 758.40& 98.97& 14.57&0.16 &0.21 &0.15&KM30 &\
1728& 728.99& 935.80& 272.35& 12.86&0.10 &0.60 &0.63&KM6 &\
1911& 513.95& 458.03& 278.30& 13.39&0.07 &0.49 &0.55&KM13 &\
1947& 733.43& 538.40& 130.94& 14.73&0.17 &0.29 &0.31&KM31 &\
2041& 549.93& 781.97& 190.27& 16.29&0.14 &0.47 &0.54&KM131 & PM\
2062& 699.50& 848.78& 183.78& 15.57&0.16 &0.42 &0.43&KM55 & PM\
2079& 621.45& 898.18& 246.05& 15.36&0.11 &0.54 &0.62&KM48 & PM\
2400& 647.52& 621.46& 68.19& 14.13&0.09 &0.55 &0.62&KM23 &\
|
---
abstract: 'We show that if $X\subseteq \mathbb{P}^{n-1}$, defined over $\mathbb{Q}$ by a cubic form that splits off two forms, with $n\geq 11$, then $X(\mathbb{Q})$ is non-empty. The same holds for an $(m_1,m_2)$-form with $m_1\geq 4$ and $m_2\geq 5$.'
author:
- Boqing Xue and Haobo Dai
---
Introduction {#intro}
============
Let $X\subseteq \mathbb{P}^{n-1}$ be a cubic hypersurface, defined by $C=0$ with $C\in \mathbb{Z}[x_1,\ldots,x_n]$ a cubic form. It is conjectured that $X(\mathbb{Q})\neq \emptyset$ whenever $n\geq 10$. Local obstacles may exist for cubic forms in less variables. Mordell[@Mor] gave a counterexample for $n=9$. For more general results, we refer the readers to Birch[@Birch61].
It was shown by Davenport [@Dav63] that an arbitrarily cubic surface has a $\mathbb{Q}$-point when $n\geq 16$. Heath-Brown [@HB07] improved this unconditional result to $n\geq 14$. Lewis and Birch also gave early results. Recently, Browning [@Browning1] showed that $X(\mathbb{Q})\neq \emptyset$ if $n\geq 13$ and $C$ splits off a form. Here $C$ splits off an $m_1$-form, or $C$ is an $(m_1,m_2)$-form, means that $$C(x_1,\ldots,x_n)=C_1(y_1,\ldots,y_{m_1})+C_2(z_1,\ldots,z_{m_2}), \label{split1}$$ with $m_1+m_2=n$, $m_1,m_2\geq 1$ and $C_1,C_2$ non-zero cubic forms with integer coefficients. And we say $C$ splits off a form if $C$ splits off an $m_1$-form for some $0<m_1<n$. He also suggested that cubic hypersurfaces that split off two forms be investigated, where we say $C$ splits off two forms if $C_1,C_2,C_3$ are non-zero cubic forms with integer coefficients and $$C(x_1,\ldots,x_n)=C_1(w_1,\ldots,w_{n_1})+C_2(y_{1},\ldots,y_{n_2})+C_3(z_{1},\ldots,z_{n_3}), \label{split2}$$ for appropriate $n_1, n_2, n_3\geq 1$ with $n_1+n_2+n_3=n$. We also call it an $(n_1,n_2,n_3)$-form. In this paper, we first establish the following theorem.
\[TH1\] Let $X\subseteq \mathbb{P}^{n-1}$ be a hypersurface defined by a cubic form that splits off two forms, with $n\geq 11$. Then $X(\mathbb{Q})\neq \emptyset$.
Requiring more on the hypersurface, the folklore conjecture has been verified true. We say a cubic hypersurface $X$ non-singular if over $\overline{\mathbb{Q}}^n$ the only solution to the system of equations $\nabla C(\mathbb{\mathbf{x}})=0$ is $\mathbf{x}=0$. Heath-Brown [@HB83] showed that for non-singular cubic forms $n\geq 10$ variables are enough to guarantee a $\mathbb{Q}$-point. An extended version by Browning and Heath-Brown[@Br-HB09] shows that the condition $n\geq 10$ can be replaced by $n\geq 11+\sigma_X$, where $\sigma_X$ denotes the dimension of the singular locus of $X$.
For non-singular cubic forms in no more than 9 variables, it is expected that the Hasse principle still holds as soon as $n\geq 5$, which means that $X(\mathbb{Q})\neq \emptyset$ provided that $X(\mathbb{R})\neq \emptyset$ and $X(\mathbb{Q}_p)\neq \emptyset$ for every prime $p$. Hooley studied nonary cubic forms in a series of papers [@Hoo88]-[@Hoo12]. He first proved that Hasse principle holds for non-singular $X$ whenever $n\geq 9$. And most recently, he showed the following theorem.
\[THH\] Let $X\subseteq \mathbb{P}^{8}$ be a cubic hypersurface defined over $\mathbb{Q}$. Suppose that $X$ possesses at most isolated ordinary (double) points as singularities and $X(\mathbb{Q}_p)\neq \emptyset$ for every prime $p$. Then $X(\mathbb{Q})\neq \emptyset$.
For singular cubic hypersurfaces $X$, Colliot-Thélène and Salberger [@Co-Sal89] proved that the Hasse principle holds if $X$ contains a set of three conjugate singular points.
\[THCS\] Let $X\subseteq \mathbb{P}^{n-1}$ be a cubic hypersurface defined over $\mathbb{Q}$, with $n\geq 4$. Suppose that $X$ contains a set of three conjugate singular points and $X(\mathbb{Q}_p)\neq \emptyset$ for every prime $p$. Then $X(\mathbb{Q})\neq \emptyset$.
The structure of hypersurfaces defined by cubic forms with few variables are not hard to determine. With some geometric lemmas, it is given in [@Browning1 Theorem 2] that $X(\mathbb{Q})\neq \emptyset$ if $C$ splits off an $m_1$-form with $m_1\geq 8$ and $n\geq 10$. And Browning[@Bro12] has shown us that the condition $m_1\geq 8$ can be replaced by $m_1\geq 5$. Based on his arguments, the following conclusions can be established.
\[TH2\] Let $X\subseteq \mathbb{P}^{n-1}$ be a cubic hypersurface defined by an $(m_1,m_2)$-form $C$, with $m_1+m_2=n$. Suppose that $C$ has shape (\[split1\]).
\(1) If $C_1$ is non-singular, $m_1\geq 4$, $n\geq 9$ and $(m_1,m_2)\neq (6,3)$, then $X(\mathbb{Q})\neq \emptyset$.
\(2) If $m_1\geq 4$ and $m_2\geq 5$, then $X(\mathbb{Q})\neq \emptyset$.
Let $X\subseteq \mathbb{P}^{n}$ be a cubic hypersurface defined by an $(n_1,n_2,n_3)$-form, with $n_1+n_2+n_3\in \{9,10\}, ~ n_1\leq n_2\leq n_3$, $(n_1,n_2)\notin \{(1,1),(1,2)\}$ and $(n_1,n_2,n_3)\neq (3,3,3)$. Then $X(\mathbb{Q})\neq \emptyset$.
The local conditions may fail for a $(3,3,3)$-form. Consider $$(x_1^3+2x_2^3+4x_3^3+x_1x_2x_3)+7(x_4^3+2x_5^3+4x_6^3+x_4x_5x_6)+49(x_7^3+2x_8^3+4x_9^3+x_7x_8x_9). \label{counterexample}$$ The only solution to $x_1^3+2x_2^3+4x_3^3+x_1x_2x_3\equiv 0 \, (\text{mod }7)$ is $x_1,x_2,x_3\equiv 0 \, (\text{mod }7)$. So (\[counterexample\]) does not represent zero non-trivially in $\mathbb{Q}_7$, and in $\mathbb{Q}$. See [@Mor] for more general counterexamples. Moreover, we say $C$ captures $\mathbb{Q}^\ast$ if $C$ represents all the non-zero $r=a/q \in \mathbb{Q}$, using rational values for the variables. Fowler [@Fow] showed that any non-degenerate cubic form, by which we mean that it is not equivalent over $\mathbb{Z}$ to a cubic form in fewer variables, in no less than 3 variables that represents zero automatically captures $\mathbb{Q}^\ast$. For $C$ that can’t represents zero non-trivially, we have $C$ captures $\mathbb{Q}^\ast$ if $$q C(x_1,\ldots,x_n)-a x_{n+1}^3=0$$ always has non-zero solutions for any integers $q$ and $a\neq 0$. Noting that the above equation involves a cubic form that splits off a $1$-form. Then Theorem \[TH1\] directly implies the following.
Let $C\in\mathbb{Z}[x_1,\ldots,x_n]$ be a non-degenerate cubic form that splits off a form, with $n\geq 10$. Then $C$ captures $\mathbb{Q}^\ast$.
We mainly follow the argument of [@Browning1]. Circle method is used. Note that the target forms can be reduced to forms in less variables if some of the variables take value 0, and forms of shape (\[split2\]) can also be regarded as forms of shape (\[split1\]). To prove Theorem \[TH1\] and \[TH2\], it is sufficient to handle forms with type $$(n_1,n_2,n_3)=(1,1,9),(1,2,8), \quad (m_1,m_2)=(4,5). \label{cases}$$ After studiously calculating, one can get a weaker version of Theorem \[TH1\], with $n\geq 11$ replaced by $n\geq 12$. To save another variable needs two additional ingredients. Since exponential sums in many variables are harder to understand than that in a single variable, minor arc estimates fail for cubic forms that split off an $m_1$-form with $m_1\geq 3$. Geometric points of view (especially Theorem \[TH2\]) do help in these cases. Another difficulty comes from some $(n_1,n_2,n_3)$-forms with $n_1,n_2\leq 2$. Although there exists many estimates that may be useful, the saving derived from $1$ or $2$-forms can’t be as much as we want due to the small number of variables. To make enough saving in the case (1,1,9), Brüdern’s result on a certain fourth moment of a cubic exponential sum is needed.
The remainder of this paper is laid out as follows. In §2 geometry of singular cubic hypersurfaces is quoted and the proof of Theorem \[TH2\] is given. In §3 the circle method is introduced. In §4 analytic results on exponential sums are listed. And two technical lemmas (Lemma \[LE7\] and Lemma \[LE8\]) are put into a type that can be computed and verified by computer easily. In §5, we bound the minor arc estimates and the rest of the proof is given. At last in §6, we make some further remarks that show the difficulty of improving $n\geq 11$ to $n\geq 10$ in Theorem \[TH1\].
Throughout this paper, parameters $\varepsilon,\delta,\Delta$ are carefully chosen small positive numbers satisfying $0<\varepsilon<\delta<\Delta$. For a point $\mathbf{x}=(x_1,\ldots,x_n)\in \mathbb{R}^n$, the norm $|\mathbf{x}|=\max_{1\leq i\leq n}|x_j|$. Symbols $\ll$, $\gg$. $\asymp$, $\textit{O}(\cdot)$, $\textit{o}(\cdot)$ are Vinogradov notations.
Geometric results on singular cubic hypersufaces
================================================
We use $C\in \mathbb{Z}[x_1,\ldots,x_{n}]$ to denote an arbitrary cubic form that defines $X\subseteq \mathbb{P}^{n-1}$. If $C$ has the shape (\[split1\]). We denote $X_i\subseteq \mathbb{P}^{n_i-1}$ the variety defined by $C_i$ $(i=1,2)$, respectively. If $C$ splits off a form $C_1$ and $C_1=0$ has a non-trivial rational solution, then obviously $X(\mathbb{Q})$ is non-empty. So in the rest of this section we always assume that any form split off by $C$ does not have non-trivial rational solutions.
Theorem H and Theorem CS have given conditions under which Hasse principle holds. Now we investigate when local conditions hold. It is shown by Heath-Brown[@HB83 Proposition 2] that any nonary cubic form defined over $\mathbb{Q}_p$ that splits off a $1$-form represents zero non-trivially in $\mathbb{Q}_p^9$. Some more effort leads to the following.
\[PR1\] Let $C(x_1,\ldots,x_9)$ be a nonary cubic form over $\mathbb{Q}_p$. If $C$ splits off an $m$-form, with $m\in [1,8]\setminus \{3,6\}$, then it represents zero non-trivially in $\mathbb{Q}_p^9$.
Suppose $C$ has shape (\[split1\]). By [@HB83 Proposition 1], either $C_1$ represents zero or there is a non-singular linear transformation sending $C_1$ to a form $$D_1(u_1,\ldots,u_{r_1},v_1,\ldots,v_{r_2},w_1,\ldots,w_{r_3})=F_1(\mathbf{u})+pF_2(\mathbf{v})+p^2 F_3(\mathbf{w})+p G(\mathbf{u},\mathbf{v},\mathbf{w}),$$ where $r_1,r_2,r_3\geq 0$ and $r_1+r_2+r_3=m$. And $F_1,F_2,F_3,G$ are forms with coefficients in $\mathbb{Z}$ with the following properties.
\(1) $F_1$ involves the variables $u_i$ only, and similar for $F_2,F_3$. Terms involving the variables $u_i$ only (or the $v_i$ only, or the $w_i$ only) are absent from $G$.
\(2) The congruence $F_1(\mathbf{u})\equiv 0 (\text{mod }p)$ has only the solution $\mathbf{u}\equiv \mathbf{0} (\text{mod }p)$, and similarly for $F_2,F_3$.
\(3) In $G$, the coefficients of the monomials $u_iw_jw_k, v_iw_jw_k,w_iv_jv_k$ (where $i,j,k$ need not be distinct) are multiples of $p$.
Similarly, if $C_2$ does not represent zero, we write $C_2$ as $$D_2(u_1^\prime,\ldots,u_{s_1}^\prime,v_1^\prime,\ldots,v_{s_2}^\prime,w_1^\prime,\ldots,w_{s_3}^\prime)=F_1^\prime(\mathbf{u}^\prime)+pF_2^\prime(\mathbf{v}^\prime)+p^2 F_3^\prime(\mathbf{w}^\prime)+p G^\prime(\mathbf{u}^\prime,\mathbf{v}^\prime,\mathbf{w}^\prime),$$ where $s_1,s_2,s_3\geq 0$ and $s_1+s_2+s_3=9-m$. And $F_1^\prime,F_2^\prime,F_3^\prime,G$ are forms with coefficients in $\mathbb{Z}$ with similar properties. Hence $C$ is equivalent to $$\left(F_1(\mathbf{u})+F_1^\prime(\mathbf{u}^\prime)\right)+p\left(F_2(\mathbf{v})+F_2^\prime(\mathbf{v}^\prime)\right)+p^2 \left( F_3(\mathbf{w})+ F_3^\prime(\mathbf{w}^\prime)\right)+p \left( G(\mathbf{u},\mathbf{v},\mathbf{w})+ G^\prime(\mathbf{u}^\prime,\mathbf{v}^\prime,\mathbf{w}^\prime)\right).$$
By Chevalley’s Theorem (see [@Ser p.5]) and the property (2), $D_1$ represents zero non-trivially unless $r_1,r_2,r_3\leq 3$. Similarly we can assume that $s_1,s_2,s_3\leq 3$. Since $m$ and $9-m$ do not take value $3$ or $6$, it can be deduced that there exists some $i_0\in\{1,2,3\}$ so that the value of the couple $(r_{i_0}, s_{i_0})$ belongs to $\{(2,1),(1,2),(3,1),(1,3),(2,3),(3,2),(2,2),(3,3)\}$. Without loss of generality, we suppose $(r_{i_0}, s_{i_0})=(2,1)$. (In other cases we can set the redundant variables 0 or change the order. And property (2) ensures that if appropriate variables are chosen, the induced form will not be identically zero.) Now $F_{i_0}+F^\prime_{i_0}$ is the sum of a $2$-form and a $1$-form, and the arguments of [@HB83 Proposition 2] leads to the conclusion that $F_{i_0}+F^\prime_{i_0}$ represents zero non-trivially in $\mathbb{Q}_p^9$. Let the variables that do not appear in $F_{i_0}$ and $F^\prime_{i_0}$ be zero. From the second statement of the property (1), it can be asserted that the terms $G$ and $G^\prime$ vanish. Hence $C$ also has non-trivial $\mathbb{Q}_p$- solutions.
Combing Theorem H, Theorem CS and Proposition \[PR1\], we obtain the following corollaries.
\[CO1\] Let $X\subseteq \mathbb{P}^{8}$ be a cubic hypersurface defined by a nonary cubic form that splits off an $m$-form, with $m\in [1,8]\setminus \{3,6\}$. Suppose that $X$ possesses at most isolated ordinary (double) points as singularities. Then $X(\mathbb{Q})\neq \emptyset$.
\[CO2\] Let $X\subseteq \mathbb{P}^{8}$ be a cubic hypersurface defined by a nonary cubic form that splits off an $m$-form, with $m\in [1,8]\setminus \{3,6\}$. Suppose that $X$ contains a set of three conjugate singular points. Then $X(\mathbb{Q})\neq \emptyset$.
Given a cubic extension $K$ of $\mathbb{Q}$, define the corresponding norm form $$N(x_1,x_2,x_3):=N_{K/\mathbb{Q}}(\omega_1x_1+\omega_2x_2+\omega_3x_3),$$ where $\{\omega_1,\omega_2,\omega_3\}$ is a basis of $K$ as a vector space over $\mathbb{Q}$.
For small $n$, the geometry of $X\subseteq \mathbb{P}^{n-1}$ defined by a cubic form is not hard to determine. The following lemma collects [@Browning1 Lemma 2-4].
\[LE1\] Suppose $X\subseteq \mathbb{P}^{n-1}$ is defined by $C=0$ with $C\in \mathbb{Z}[x_1,\ldots,x_n]$ a cubic form and $X(\mathbb{Q})=\emptyset$.
\(1) For $n=3$, either the curve $X$ is non-singular or $X$ contains precisely three conjugate singular points. In particular in the latter case, $C$ can be written as a norm form, i.e., $$C(\mathbf{x})=\text{N}_{K/\mathbb{Q}}(x_1\omega_1+x_2\omega_2+x_3\omega_3),$$ for some appropriate coefficients $\omega_1,\omega_2,\omega_3\in K$, where $K$ is the cubic number field obtained by adjoining one of the singularities.
\(2) For $n=4$, either the surface $X$ is non-singular or $X$ contains precisely three conjugate double points. In particular in the latter case we have the representation $$C(\mathbf{x})=\text{N}_{K/\mathbb{Q}}(x_1\omega_1+x_2\omega_2+x_3\omega_3)+ax_4^2\text{Tr}_{K/\mathbb{Q}}(x_1\omega_1+x_2\omega_2+x_3\omega_3)+bx_4^3,$$ for some appropriate coefficients $\omega_1,\omega_2,\omega_3\in K$ and $a,b\in \mathbb{Z}$.
\(3) For $n=5$, either the threefold $X$ is non-singular or $X$ is a geometrically integral cubic hypersurface whose singular locus contains precisely $\delta$ double points, with $\delta\in \{3,6,9\}$.
We first prove Theorem \[TH2\](2). Suppose that $C$ has the shape (\[split1\]). It is sufficient to handle the case $(m_1,m_2)=(4,5)$. By Lemma \[LE1\](2), either $X_1$ is non-singular or $X_1$ contains precisely three conjugate singular points. In the former case, $X_2$ has at most isolated double points (by Lemma \[LE1\](3)) and so does $X$. Since $C$ splits off a $4$-form. Corollary \[CO1\] ensures that $X(\mathbb{Q})\neq \emptyset$. In the latter case, $X$ contains a set of three conjugate singular points and $C$ splits off a $4$-form. Corollary \[CO2\] shows that $X(\mathbb{Q})\neq \emptyset$ also holds.
As for Theorem \[TH2\](1), the cases $m_1=4$ or $5$ can be implied from Theorem \[TH2\](2). The cases $m_1\geq 9$ can be deduced from [@Browning1 Theorem 2]. For $m_1=7$ or $8$. It is sufficient to handle $(m_1,9-m_1)$, i.e., we put $m_2=9-m_1$. Then $m_2=1$ or $2$. Since $C_1$ is non-singular, it follows that $C$ is a non-singular cubic form in 9 variables that splits off an $m_2$-form. By Corollary \[CO1\], it can be conclude that $X(\mathbb{Q})\neq \emptyset$. At last we suppose $m_1=6$ and $m_2\geq 4$. This case is dealt with in [@Bro12]. It can also be reduced to the case $(5,4)$, which is implied by Theorem \[TH2\](2).
By Lemma \[LE1\](1), either $X_2$ is non-singular or $X_2$ contains precisely three conjugate singular points. In the former case, $X_3$ has at most isolated double points (by Lemma \[LE1\](3)) and so does $X_0$. Since $C_0$ splits off a $1$-form. Corollary \[CO1\] ensures that $X_0(\mathbb{Q})\neq \emptyset$. In the latter case, $X_0$ contains a set of three conjugate singular points and $C_0$ splits off a $1$-form. Corollary \[CO2\] shows that $X_0(\mathbb{Q})\neq \emptyset$ also holds.
By Lemma \[LE1\](1,2), either both $X_2$ and $X_3$ are non-singular or at least one of them contains a set of three conjugate singular points. Note that $C_0$ splits a $4$-form. One obtains $X_0(\mathbb{Q})\neq \emptyset$ by Corollary \[CO1\] in the former case and by Corollary \[CO2\] in the latter case.
By lemma 2(3), $X_3$ has at most isolated double points and so does $X_0$. Corollary \[CO1\] shows that $X_0(\mathbb{Q})\neq \emptyset$.
The circle method
=================
For Theorem \[TH1\] it is suffice to handle the case $n_0=11$, since when $n_0>11$ we can simply force the redundant variables to be $0$. Most of the results outlined in §1 involve the circle method in the proof. We apply it to deal with $(1,1,9)$ and $(1,2,8)$-forms.
We use $C_0\in \mathbb{Z}[x_1,\ldots,x_{n_0}]$ to denote the cubic form stated in Theorem \[TH1\], which defines $X_0\subseteq \mathbb{P}^{n_0-1}$. Let $n_1,n_2,n_3\geq 1$ be integers such that $n_1+n_2+n_3=n_0$. Since $C_0$ splits off two forms, we write $$C_0(\mathbf{x})=C_1(\mathbf{x}_1)+C_2(\mathbf{x}_2)+C_3(\mathbf{x}_3), \label{split}$$ where $C_i\in \mathbb{Z}[\mathbf{x}_i]\,(1\leq i\leq 3)$ are cubic forms in $n_i$ variables and define $X_i\subseteq \mathbb{P}^{n_i-1}$, respectively. Without loss of generality, we assume that $n_1\leq n_2\leq n_3$.
If $C$ is degenerate, then $C=0$ has obvious non-zero integer solutions. If $C$ has no less than 3 variables and is not ‘good’, then $C=0$ also has non-zero integer solutions for ‘geometric reasons’ (see [@Dav]). Here a general cubic form $C$ is ‘good’ means that for any $H\geq 1$ and any $\varepsilon>0$, the upper bound $$\#\{\mathbf{x}\in \mathbb{Z}^n:|\mathbf{x}|\leq H, \text{rank}H(\mathbf{x})=r\}\ll H^{r+\varepsilon}$$ holds for each integer $0\leq r\leq n$, where $H(\mathbf{x})$ is the Hessian matrix of $C$. Any cubic form defining a hypersurface with at most isolated ordinary singularities is good, which is due to Hooley[@Hoo88]. Moreover, if $C_i=0$ for some $1\leq i\leq 3$ or $C_i+C_j=0$ for some $1\leq i<j\leq 3$ has non-trivial integer solutions, then we easily see that $C_0=0$ has non-trivial integer solutions. Now we can suppose that none of $C_j$ $(0\leq j\leq 3)$ and $C_i+C_j$ $(1\leq i<j\leq 3)$ has integer solutions, is degenerate, or is not ‘good’ whenever no less than 3 variables are possessed.
Write $e(x):=e^{2\pi i x}$. Define the cubic exponential sum $$S(\alpha)=S(\alpha;C,n,\rho,P):=\sum\limits_{\mathbf{x}\in\mathbb{Z}^n\atop |P^{-1}\mathbf{x}-\mathbf{z}|< \rho} e(\alpha C(x)),$$ where $\mathbf{z}$ is a fixed vector and $\rho>0$ is a fixed real number, both to be determined later. The precise value of $\mathbf{z}$ and $\rho$ are actually immaterial and the corresponding implied constants are allowed to depend on these quantities. Let $S_i(\alpha):=S(\alpha;C_i,n_i,\rho,P)$ for $0\leq i\leq 3$. From (\[split\]), one has $$S_0(\alpha)=S_1(\alpha)S_2(\alpha)S_3(\alpha).$$ Write $$\mathcal{N}(P):=\#\{\mathbf{x}\in \mathbb{Z}^{n_0}: |P^{-1}\mathbf{x}-\mathbf{z}|< \rho, \; C_0(\mathbf{x})=0\}.$$ On observing the simple equality $$\int_0^1 e(\alpha x)d\alpha =
\begin{cases}
1,\quad \text{if }x=0,\\
0,\quad \text{if }x\neq 0,
\end{cases}$$ the number of solutions of $C_0(\mathbf{x})=0$ counted by $\mathcal{N}(P)$ is exactly $$\int_0^1{S_0(\alpha)}d\alpha.$$ Next we divide the integral domain into two parts where different tools can be applied. Define the major arcs as $$\mathfrak{M}:=\bigcup\limits_{q\leq P^\Delta}\bigcup\limits_{(a,q)=1}\left[\frac{a}{q}-P^{-3+\Delta},\frac{a}{q}+P^{-3+\Delta}\right]$$ and the minor arcs as $\mathfrak{m}:=[0,1]\setminus \mathfrak{M}$, where $\Delta$ is a small positive integer to be specified later. The intervals in the major arcs are pairwise disjoint. The integral becomes $$\int_0^1{S(\alpha)}d\alpha=\int_{\mathfrak{M}}{S(\alpha)}d\alpha+\int_{\mathfrak{m}}{S(\alpha)}d\alpha.$$ If the first term on the right side (known as the main term) takes positive value and overwhelms the second term (known as the error term) for sufficiently large $P$, then we reach the declaration that $\mathcal{N}(P) \gg P^{\tau}$ ($\tau$ can be 8 according to Lemma \[LE2\] below) and then $C_0=0$ has non-trivial integer solutions. As a result, we have $X_0(\mathbb{Q})\neq \emptyset$ and Theorem \[TH1\] follows.
The following lemma ensures that the integral over major arcs are ‘large’.
\[LE2\] Let $n_0=11$. We have $$\int_\mathfrak{M} S_0(\alpha)d\alpha= \mathfrak{S}\mathfrak{J}P^{n_0-3}+\textit{o}\left(P^{n_0-3}\right),$$ where $$\mathfrak{S}:=\sum\limits_{q=1}^\infty \sum\limits_{(a,q)=1}q^{-n_0}S_{a,q}(C_0),\quad \mathfrak{J}:=\int_{-\infty}^\infty I(\beta;C_0) d\beta,$$ with $$S_{a,q}=S_{a,q}(C):=\sum\limits_{\mathbf{y} (\text{mod }q)} e_q(a C(\mathbf{y})),\quad I(\beta)=I(\beta;C):=\int_{|P^{-1}\mathbf{x}-\mathbf{z}|<\rho} e(\beta C(\mathbf{x}))d\mathbf{x}.$$
The proof of Lemma \[LE2\] uses standard arguments. One can see [@Dav Lemma 15.4, §16-18] or [@HB07 Lemma 2.1] for details. Since $C_0$ is good, Heath-Brown’s bound (see [@HB07 (7.1)]) $S_{a,q}(C_0)\ll q^{5n_0/6+\varepsilon}$ is effective and [@HB07 Theorem 4] ensures that $\mathfrak{S}$ is absolutely convergent for $n_0=11$. Standard argument (see [@Dav59 Lemma 7.3]) leads to $\mathfrak{S}>0$. Assuming that $C_3$ does not have a linear factor defined over $\mathbb{Q}$ (otherwise non-trivial integer solutions can be found easily), it is possible to choose appropriate $\mathbf{z}_3$ in the definition of $S_3$ so that $\mathbf{z}_3$ is a non-singular real solution to $C_3=0$. Now pick $\mathbf{z}_1=\mathbf{z}_2=\mathbf{0}$, then $\mathbf{z}=(\mathbf{z}_1,\mathbf{z}_2,\mathbf{z}_3)$ is a non-singular real solution to equation $C_0=0$. On selecting a sufficiently small value of $\rho>0$, we will have $\mathfrak{J}>0$. (See [@Dav §16] and [@HB83 §4] for details.)
Now we only need to show that the integral over the minor arcs is ‘small’ according to that over the major arcs.
\[PR2\] Suppose that $(n_1,n_2,n_3)\in \{(1,1,9),(1,2,8)\}$. Then either $X_i(\mathbb{Q})\neq \emptyset$ for some $1\leq i\leq 3$, or $$\int_\mathfrak{m}S_{n_0}(\alpha)d\alpha=\textit{o}\left(P^{n_0-3}\right).$$
In the latter case, we also have $X_0(\mathbb{Q})\neq \emptyset$ in view of Lemma \[LE2\]. Hence Theorem \[TH1\] follows from Proposition \[PR2\] immediately.
Note that the integral over the minor arcs is a kind of $L^1$ norm. The $L^\infty$ bound on the exponential sums is the essential content of Davenport’s result on cubic forms in 16 variables. Later Heath-Brown took use of Van der Corput’s method, with two additional techniques, resulted in a powerful $L^2$ bound and he successfully worked out the solubility of cubic forms in 14 variables with no restrictions. Combining these two kind of bounds, Browning showed an $L^v$ bound with $v\leq 2$. For $n=1,2$, there are good $L^v$ bounds with $v\geq 2$ (Hua’s inequality in dimension one and its analog in dimension two by Wooley). And we mention that for $n=3,4,5$, $L^2$ bounds resulted from the number of solutions of cubic forms are also available. We implant these analytic results in §4.
To apply such tools, we need to select the appropriate combination of the powers in Hölder’s inequality. For simplicity, we use the following notations: $$I_u(S;t,\mathfrak{a}):=P^{t+\varepsilon}\left(\int_{\mathfrak{a}}|S(\alpha)|^u d\alpha\right)^{1/u},$$ $$I_{u,v}(S_{n_1},S_{n_2};t,\mathfrak{a}):=P^{t+\varepsilon}\left(\int_{\mathfrak{a}}|S_{n_1}(\alpha)|^u d\alpha\right)^{1/u}\left(\int_{\mathfrak{a}}|S_{n_2}(\alpha)|^v d\alpha\right)^{1/v},$$ $$I_{u,v,w}(S_{n_1},S_{n_2},S_{n_3};t,\mathfrak{a}):=P^{t+\varepsilon}\left(\int_{\mathfrak{a}}|S_{n_1}(\alpha)|^u d\alpha\right)^{1/u}\left(\int_{\mathfrak{a}}|S_{n_2}(\alpha)|^v d\alpha\right)^{1/v}\left(\int_{\mathfrak{a}}|S_{n_3}(\alpha)|^w d\alpha\right)^{1/w}.$$
With these notations, Proposition \[PR2\] can be implied from $$I_{1}(S_1S_2S_3;0,\mathfrak{m})=\textit{o}\left(P^8\right).$$
Estimates on the cubic exponential sums
=======================================
\[LE3\] Let $\varepsilon>0$. Assume that $C\in \mathbb{Z}[x_1,\ldots,x_n]$ is a good cubic form. Let $\alpha \in [0,1]$ have the representation $$\alpha=a/q+\beta,\quad (a,q)=1,\quad 0\leq a<q\leq P^{3/2}$$ with $a,q\in \mathbb{Z}$. Then $$S(\alpha)\ll P^{n+\varepsilon}\left\{q|\beta|+\left(q|\beta|P^3\right)^{-1}\right\}^{n/8}.$$ If furthermore $|\beta|\leq q^{-1}P^{-3/2}$, then $$S(\alpha)\ll P^{n+\varepsilon}q^{-n/8}\min\left\{1,\left(|\beta|P^3\right)^{-n/8}\right\}.$$
\[LE4\] Let $\varepsilon>0$. Assume that $C\in \mathbb{Z}[x_1,\ldots,x_n]$ is a good cubic form. Let $1\leq R\leq P^{3/2}$ and $0<\phi\leq R^{-2}$. Define $$\mathcal{M}_v(R,\phi,\pm):=\sum\limits_{R\leq q< 2R} \sum\limits_{(a,q)=1} \int_\phi^{2\phi}\left|S\left(\frac{a}{q}\pm \beta\right)\right|^v d\beta.$$ Then $$\mathcal{M}_v(R\,\phi,\pm)\ll P^3+R^2\phi^{1-v/2}\left(\frac{\psi_H P^{2n-1+\varepsilon}}{H^{n-1}}F\right)^{v/2},$$ with $H$ any integer in $[1,P]$ and $$\psi_H:=\phi+\frac{1}{P^2 H},\quad F:=1+(RH^3\psi_H)^{n/2}+\frac{H^n}{R^{n/2}(P^2\psi_H)^{(n-2)/2}}.$$
\[LE5\] Let $\varepsilon>0$. The following bounds hold.
\(1) ([@Dav Lemma 3.2]) For $n=1$, one has $$\int_0^1|S(\alpha)|^{2^j}d\alpha \ll P^{2^j-j+\varepsilon},$$ for any $j\leq 3$.
\(2) ([@Woo99 Theorem 2]) For $n=2$, suppose that $C\in \mathbb{Z}[x_1,x_2]$ is a non-degenerate binary cubic form, then $$\int_0^1|S(\alpha)|^{2^{j-1}}d\alpha \ll P^{2^j-j+\varepsilon},$$ for any $j\leq 3$.
For $A,B,C\geq 0$, define $\mathcal{A}=\mathcal{A}(A,B,C)$ to be the set of $\alpha\in [0,1]$ for which there exists $a,q\in \mathbb{Z}$ such that $$\alpha=a/q+\beta,\quad (a,q)=1,\quad 1\leq a<q\leq P^A,\quad |\beta|\leq q^{-B}P^{-3+C}.$$
For $n=1$, define $$S^\ast (\alpha):= q^{-1} P\, S_{a,q} I(\beta P^3).$$ We try to approximate $S(\alpha)$ by $S^\ast (\alpha)$. Since $S^\ast (\alpha)$ has better $L^v$ $(v\geq 2)$ bounds, we can gain extra saving if their difference is small in some particular intervals. The following lemma can be derived from the book of Vaughan [@Vau97 §4] (or see [@Browning1 Lemma 10]).
\[LE6\] Let $\varepsilon>0$ and $n=1$. Suppose $A,B,C\geq 0$ and $A,B\leq 1$. Then for any $\alpha\in \mathcal{A}$, $$S(\alpha)=S^\ast(\alpha)+\textit{O}\left(P^{A/2+\varepsilon}+P^{(A+C-AB)/2+\varepsilon}\right).$$ Furthermore, if $k\geq 4$, then $$\int_\mathcal{A}|S^\ast(\alpha)|^k d\alpha \ll P^{k-3+\varepsilon}.$$
\[LE7\] Assume $\mathcal{A}$ is defined as above with $A,B,C\geq 0$. We have $$I_v(S;t,\mathcal{A})\ll
\begin{cases}
P^{n+t-(3-C)\left(\frac{1}{v}+\frac{n}{8}\right)+A\left(\frac{2}{v}+\frac{n}{8}-B\left(\frac{1}{v}+\frac{n}{8}\right)\right)+\varepsilon}+
P^{\frac{5}{8} n+t+\varepsilon+\min\left\{\frac{nA}{8} ,\;\; -(3-C)\left(\frac{1}{v}-\frac{n}{8}\right)+A\left(\frac{2}{v}-\frac{n}{8}-B\left(\frac{1}{v}-\frac{n}{8}\right)\right) \right\}},\quad \text{if }nv\leq 8.\\
P^{n+t-(3-C)\left(\frac{1}{v}+\frac{n}{8}\right)+A\left(\frac{2}{v}+\frac{n}{8}-B\left(\frac{1}{v}+\frac{n}{8}\right)\right)+\varepsilon}+
P^{n+t-\frac{3}{v}+A\left(\frac{2}{v}-\frac{n}{8}\right)+\varepsilon},\quad \text{if } 8\leq nv\leq 16.\\
P^{n+t-(3-C)\left(\frac{1}{v}+\frac{n}{8}\right)+A\left(\frac{2}{v}+\frac{n}{8}-B\left(\frac{1}{v}+\frac{n}{8}\right)\right)+\varepsilon}+
P^{n+t-\frac{3}{v}+\varepsilon},\quad \text{if }nv\geq 16.
\end{cases}$$
By dyadic summation, we have $$I_v(S;t,\mathcal{A})\ll P^{t+\varepsilon/2}(\log P)^2 \max\limits_{R,\phi,\pm} \mathcal{M}_v (R,\phi,\pm)^{1/v},$$ where the maximum runs over the possible sign changes and $R,\phi$ satisfy $R\leq P^A$, $\phi\leq R^{-B}P^{-(3-C)}$.
For the case $R\phi\geq P^{-3/2}$, Lemma \[LE3\] shows that $S(\alpha)\ll P^{n+\varepsilon}R^{n/8}\phi^{n/8}$. Then $$I_v(S;t,\mathcal{A})\ll P^{t+\varepsilon}(R^2\phi)^{\frac{1}{v}}\cdot P^{n+\varepsilon}R^{\frac{n}{8}}\phi^{\frac{n}{8}}
\ll P^{n+t+\varepsilon}R^{\frac{2}{v}+\frac{n}{8}}\phi^{\frac{1}{v}+\frac{n}{8}} \ll P^{n+t-(3-C)\left(\frac{1}{v}+\frac{n}{8}\right)+\varepsilon}R^{\frac{2}{v}+\frac{n}{8}-B\left(\frac{1}{v}+\frac{n}{8}\right)}.$$ For $R\phi\leq P^{-3/2}$ and $nv\leq 8$, one has $S(\alpha)\ll P^{5n/8+\varepsilon}R^{-n/8}\phi^{-n/8}$. Then $$I_v(S;t,\mathcal{A})\ll P^{t+\varepsilon}(R^2\phi)^{\frac{1}{v}}\cdot P^{\frac{5}{8} n+\varepsilon}R^{-\frac{n}{8}}\phi^{-\frac{n}{8}}
\ll P^{\frac{5}{8} n+t+\varepsilon}R^{\frac{2}{v}-\frac{n}{8}}\phi^{\frac{1}{v}-\frac{n}{8}} \ll P^{\frac{5}{8} n+t-(3-C)\left(\frac{1}{v}-\frac{n}{8}\right)+\varepsilon}R^{\frac{2}{v}-\frac{n}{8}-B\left(\frac{1}{v}-\frac{n}{8}\right)}.$$ Another estimate can be $$I_v(S;t,\mathcal{A})\ll P^{\frac{5}{8} n+t+\varepsilon}R^{\frac{2}{v}-\frac{n}{8}}\phi^{\frac{1}{v}-\frac{n}{8}} =P^{\frac{5}{8} n+t+\varepsilon}R^{\frac{n}{8}}(R^2\phi)^{\frac{1}{v}-\frac{n}{8}}\ll P^{\frac{5}{8} n+t+\varepsilon}R^{\frac{n}{8}}\ll
P^{\frac{5}{8} n+t+\frac{nA}{8}+\varepsilon}.$$ For $P^{-3}\leq \phi\leq R^{-1}P^{-3/2}$ and $nv\geq 8$, one has $$I_v(S;t,\mathcal{A})\ll P^{\frac{5}{8} n+ t+\varepsilon}R^{\frac{2}{v}-\frac{n}{8}}\phi^{\frac{1}{v}-\frac{n}{8}} \ll P^{\frac{5}{8} n+t-3\left(\frac{1}{v}-\frac{n}{8}\right)+\varepsilon}R^{\frac{2}{v}-\frac{n}{8}}=P^{n+t-\frac{3}{v}+\varepsilon}R^{\frac{2}{v}-\frac{n}{8}}. \label{lem7}$$ For $\phi\leq P^{-3}$ and $nv\geq 8$, one has $S(\alpha)\ll P^{n+\varepsilon}R^{-n/8}$. Then $$I_v(S;t,\mathcal{A})\ll P^{t+\varepsilon}(R^2\phi)^{\frac{1}{v}}\cdot P^{n+\varepsilon}R^{-n/8}
\ll P^{n+t+\varepsilon}R^{\frac{2}{v}-\frac{n}{8}}\phi^{\frac{1}{v}} \ll P^{n+t-\frac{3}{v}+\varepsilon}R^{\frac{2}{v}-\frac{n}{8}}. \label{lem7-2}$$ Now Lemma \[LE7\] follows.
When $nv> 16$, the exponent on $R$ is negative. And on $\mathfrak{m}$ we additionally have the fact that $R\leq P^{\Delta}$ and $\phi\leq P^{-3+\Delta}$ do not hold simultaneously. If $\phi\geq P^{-3+\Delta}$, the right side of (\[lem7\]) can be replaced by $P^{\frac{5}{8} n+t+\frac{nA}{8}-\Delta(\frac{n}{8}-\frac{1}{v})+\varepsilon}R^{\frac{2}{v}-\frac{n}{8}}$. Otherwise we will have $R\geq P^{\Delta}$ and the right side of (\[lem7\]), (\[lem7-2\]) can be bounded by $P^{n+t-\frac{3}{v}-\Delta(\frac{n}{8}-\frac{2}{v})+\varepsilon}$. Hence actually we can achieve the bound $$I_v(S;t,\mathcal{A}\cap \mathfrak{m})\ll P^{n+t-(3-C)\left(\frac{1}{v}+\frac{n}{8}\right)+A\left\{\frac{2}{v}+\frac{n}{8}-B\left(\frac{1}{v}+\frac{n}{8}\right)\right\}+\varepsilon}+
P^{n+t-\frac{3}{v}-\Delta(\frac{n}{8}-\frac{2}{v})+\varepsilon}, \label{remarkoflemma7}$$ provided that $nv>16$.
The next lemma is an extension of [@Browning1 Lemma 14]. A series of parameters and conditions are listed first. We are sorry that readers may be confused by these parameters and conditions at first glance. They do shorten the proof and make the lemma convenient to use. Actually when we try to apply it in definite cases, they can be computed and verified easily by computers.
Parameters: $$\begin{gathered}
\rho_0:=\frac{2}{n},\quad \pi_0:=\frac{-2\Lambda+2t+4n-3}{n},\\
\rho_1:=\frac{n(n-5)}{n^2 -5n+2},\quad \pi_1:=\frac{-2(n^2-2n(\Lambda-t-1)-2)}{n^2 -5n+2},\\
\rho_2:=\frac{n-8}{n-4},\quad \pi_2:=\frac{8\Lambda-5n-8t}{n-4},\\
\Upsilon:=\frac{-6\Lambda +6t+6n-3}{n-1},\quad Q:=P^{3\varepsilon}\left(1+P^{\Upsilon}\right),\\
\Xi:=\frac{-\pi_1+\pi_0}{\rho_1-\rho_0}=\frac{(3n-2)(-2\Lambda+2t+2n-3)}{n^2-6n+4},\\
\phi_0=R^{-\rho_0}P^{-\pi_0},\quad \phi_1=R^{-\rho_1}P^{-\pi_1},\quad \phi_2=R^{-\rho_2}P^{-\pi_2}\\\end{gathered}$$ Conditions: $$\begin{aligned}
&3/2-\Upsilon>0, \label{req0}\\
&\Lambda-t-3/2> 0, \label{req1}\\
&\Lambda-t-n/2> 0, \label{req2}\\
&2\Lambda-2t-n-2\Upsilon+3>0, \label{req3}\\
&10 \Lambda -10 t- 8 n+3 \geq 0, \label{req7}\\
&\Lambda-\left(\frac{2}{v}+\frac{n}{8}-\rho_1\left(\frac{1}{v}+\frac{n}{8}\right)\right)\cdot \Xi -n-t+\pi_1\left(\frac{1}{v}+\frac{n}{8}\right)>0, \label{req11}\\
&\pi_2-3\geq 0. \label{req12}\end{aligned}$$
The condition (\[req0\]) ensures that the $R$ we take into consideration always satisfies $R\leq P^{3/2}$ and then Lemma \[LE4\] can be applied. By Dirichlet’s approximation theorem, for any $\alpha\in [0,1]$ there exists integers $a$ and $q$ such that $$\alpha=a/q+\beta,\quad 1\leq a\leq q\leq Q,\quad (a,q)=1,\quad |\beta|\leq 1/(qQ). \label{apr}$$ By dyadic summation, we have $$I_v(S;t,\mathfrak{a})\ll P^{t+\varepsilon/2}(\log P)^2 \max\limits_{R,\phi,\pm} \mathcal{M}_v (R,\phi,\pm)^{1/v},$$ where the maximum runs over the possible sign changes and $R,\phi$ are in the range described by $\mathfrak{a}$ and satisfy $$1\leq R\leq Q,\quad 0< \phi\leq (RQ)^{-1}.$$ A direct deduction shows that $$1\leq R\leq Q,\quad R\phi \leq Q^{-1},\quad R^2\phi\leq 1. \label{apr_use}$$
\[LE8\] Let $n\geq 6$. Denote $\mathfrak{m}_0$ the set of $\alpha \in \mathfrak{m}$ with the representation (\[apr\]) with $$q\leq P^{\Xi+c\delta},\quad |\alpha-a/q|\leq \phi_2P^{\delta},$$ where $c$ is a positive constant depending only on $n$. Then
\(1) we have $I_2(S;t,\mathfrak{m}\setminus \mathfrak{m}_0)=\textit{o}\left(P^\Lambda\right)$, provided that (\[req0\])-(\[req11\]) holds.
\(2) we have $I_2(S;t,\mathfrak{m})=\textit{o}\left(P^\Lambda\right)$, provided that $\Xi \leq 0$ and (\[req0\])-(\[req12\]) holds.
We prove it through the following two lemmas. The constant in the expression $\textit{O}(\delta)$ occurring in the proof of this lemma only depends on $n$.
\[CL1\] Suppose that (\[req0\]), (\[req1\]), (\[req2\]) holds. We have $$I_2(S;t,\mathfrak{n}_1)=\textit{o}\left(P^\Lambda\right),$$ where $\mathfrak{n}_1$ is the set of $\alpha\in \mathfrak{m}$ with the representation (\[apr\]) with $$q\leq Q,\quad \max\{\phi_0,\phi_1 P^\delta\}\leq |\alpha-a/q|\leq (qQ)^{-1}.$$
It follows from Lemma \[LE4\] that $$I_2(S;t,\mathfrak{n}_1)\ll P^{t+\varepsilon}\left\{P^{3/2}+\max\limits_{1\leq R\leq Q \atop \max\{\phi_0,\phi_1P^\delta\}\leq \phi\leq (RQ)^{-1}}R\left(\frac{\psi_H P^{2n-1}}{H^{n-1}}F\right)^{1/2}\right\}.$$ The first term on the right side is $\textit{O}(P^{\Lambda})$ whenever (\[req1\]) holds. Now we estimate the second term $$E_1:=P^{t+\varepsilon}\max\limits_{R,\phi}R\left(\frac{\psi_H P^{2n-1}}{H^{n-1}}F\right)^{1/2}.$$
Take $H=\displaystyle \left\lfloor P^\varepsilon \left(1+R\phi^{1/2}P^{-\Lambda+t+n-1/2}\right)^{2/(n-1)}\right\rfloor$, then $\psi_H\asymp \phi$ when $\phi\geq \phi_0$. Combining (\[apr\]) and (\[req2\]), one has $$H\leq P^\varepsilon\left(1+ (R^2\phi)^{1/(n-1)}P^{(-2\Lambda+2t+2n-1)/(n-1)}\right)\leq P.$$ Hence the choice of $H$ is appropriate. Recall that $F=1+(RH^3\psi_H)^{n/2}+H^n R^{-n/2}P^{-(n-2)}\psi_H^{-(n-2)/2}$. Calculation reveals that $$RH^3\psi_H\ll R\phi P^{3\varepsilon}\left\{1+\left((R^2\phi)^{1/2}P^{-\Lambda+t+n-1/2}\right)^{6/(n-1)}\right\}\ll Q^{-1}P^{3\varepsilon}\left\{1+P^{6(-\Lambda+t+n-1/2)/(n-1)}\right\}\ll 1,$$ in view of (\[apr\_use\]) and the choice of $Q$. And $$H^n R^{-n/2}P^{-(n-2)}\psi_H^{-(n-2)/2}\ll R^{-\frac{n}{2}}\phi^{-\frac{n-2}{2}}P^{-(n-2)+n \varepsilon}+ R^{-\frac{n}{2}+\frac{2n}{n-1}}\phi^{-\frac{n-2}{2}+\frac{n}{n-1}}P^{-(n-2)+\frac{2n(-\Lambda+t+n-1/2)}{n-1}+n\varepsilon}$$ The exponent on $\phi$ in the second term is strictly negative for $n\geq 6$, hence the second term is $\textit{O}(1)$ provided that $\phi\geq \phi_1 P^\delta$. On assuming $\phi\geq \phi_1 P^\delta$, the first term is $$R^{-\frac{n}{2}}\phi^{-\frac{n-2}{2}}P^{-(n-2)+n \varepsilon}\ll R^{-\frac{n(n-4)}{ n^2 - 5n +2}} P^{-\frac{ n (n-2 )(-2\Lambda+2 t+2n-3)}{ n^2 - 5n +2}}\ll 1,$$ provided that $-2\Lambda+2t+2n-3\geq 0$ (noting that the exponent on $R$ is strictly negative when $n\geq 6$). When $-2\Lambda+2t+2n-3< 0$, one has $\pi_1< 2$. So $\phi\geq \phi_1 P^\delta \geq R^{-\rho_1}P^{-2+\delta}$ and $$R^{-\frac{n}{2}}\phi^{-\frac{n-2}{2}}P^{-(n-2)+n \varepsilon}\ll R^{-\frac{n}{2}}\left(R^{-\rho_1}P^{\delta}\right)^{-\frac{n-2}{2}}P^{n \varepsilon}\ll R^{-\frac{n(n-4)}{n^2-5n+2}}\ll 1.$$ Now we have $F\ll 1$ and $$E_1:=P^{t+\varepsilon}\max\limits_{R,\phi}R\left(\frac{\psi_H P^{2n-1}}{H^{n-1}}F\right)^{1/2}\ll \max\limits_{R,\phi} \frac{R\phi^{1/2}P^{(2n-1)/2+t+\varepsilon}}{H^{(n-1)/2}}\ll P^{\Lambda-\frac{n-3}{2}\varepsilon}.$$ Then Lemma \[CL1\] follows.
\[CL2\] Suppose that (\[req0\]), (\[req1\]), (\[req3\]), (\[req7\]) holds. Then $$I_2(S;t,\mathfrak{n}_2)=\textit{o}\left(P^\Lambda\right),$$ where $\mathfrak{n}_2$ collects $\alpha\in \mathfrak{m}$ with the representation (\[apr\]) with $$P^{\Xi+\delta} \leq q\leq Q,\quad |\alpha-a/q|\leq \min\{\phi_0,(qQ)^{-1}\}.$$
Similarly one has $$I_2(S;t,\mathfrak{n})\ll P^{t+\varepsilon}\left\{P^{3/2}+\max\limits_{1\leq R\leq Q \atop \phi \leq \min\{\phi_0,\phi_1 P^{\delta}, \phi_2P^{-\delta}\}}R\left(\frac{\psi_H P^{2n-1}}{H^{n-1}}F\right)^{1/2}\right\}:=P^{t+3/2+\varepsilon}+E_2.$$ And (\[req1\]) ensures the first term on the right is $\textit{O}(P^{\Lambda})$. Take $H=\left\lfloor P^{4\varepsilon} \left(1+RP^{-\Lambda+t+n-3/2}\right)^{2/n}\right\rfloor$, then $(P^2H)^{-1}\ll \psi_H\ll (P^2H)^{-1}P^{4\varepsilon}$ for $\phi\leq \phi_0$. Combining (\[apr\_use\]), (\[req3\]) and the choice of $Q$ gives $$H\ll P^{4\varepsilon}\left\{1+\left(QP^{-\Lambda+t+n-3/2}\right)^{2/n}\right\}\leq P,$$ i.e., the choice of $H$ is appropriate. On assuming (\[req0\]) and (\[req7\]), one reaches $$\begin{aligned}
RH^3\psi_H\ll RH^2P^{-2+4\varepsilon}\ll RP^{-2+12\varepsilon}\left\{1+\left(R P^{-\Lambda+t+n-3/2}\right)^{4/n}\right\} \nonumber\\
\ll QP^{-2+12\varepsilon}+Q^{1+\frac{4}{n}}P^{\frac{-4\Lambda+4t+2n-6}{n}+12\varepsilon} \label{reproof}\\
\ll 1+P^{\frac{(n+2) (-10 \Lambda + 10 t+ 8 n -3 )}{ n(n-1)}+18\varepsilon}\ll P^{18\varepsilon}. \nonumber\end{aligned}$$ Moreover, $$H^n R^{-n/2}P^{-(n-2)}\psi_H^{-(n-2)/2}\ll R^{-n/2}P^{(3n/2-1)4\varepsilon}+ R^{-n/2+3-2/n}P^{\left(-\Lambda+t+n-\frac{3}{2}\right)\left(3-\frac{2}{n}\right)+(3n/2-1)4\varepsilon}\ll 1,$$ whenever $R\geq P^{\frac{(3n-2) ( - 2 \Lambda + 2 t+ 2 n -3)}{ n^2 - 6 n +4}+\delta}=P^{\Xi+\delta}$. (Note that the exponent on $R$ in the second term is strictly negative when $n\geq 6$.) Now we have $F\ll P^{18\varepsilon}$ and $$E_2:=P^{t+\varepsilon}\max\limits_{R,\phi}R\left(\frac{\psi_H P^{2n-1}}{H^{n-1}}F\right)^{1/2}\ll \max\limits_{R,\phi} \frac{RP^{(2n-3)/2+t+12\varepsilon}}{H^{n/2}}\ll P^{\Lambda-(2n-10)\varepsilon}.$$ Then Lemma \[CL2\] follows.
First suppose that $R\geq P^{\Xi+c\delta}$, where $c=\max\left\{(\rho_1-\rho_0)^{-1},1\right\}$. Note that $$\rho_1-\rho_0=\frac{(n-1)(n^2-6n+4)}{n(n^2-5n+2)}>0$$ for $n\geq 6$, the positive constant $c$ depends only on $n$. One can check that $\phi_0\geq \phi_1 P^{\delta}$ under the condition $R\geq P^{\Xi+c\delta}$ and the choice of $c$. A combination of Lemma \[CL1\] and Lemma \[CL2\] shows that $$I_1(S;t,\mathfrak{m}_1)=\textit{o}\left(P^{\Lambda}\right),$$ where $\mathfrak{m}_1$ denotes the set of $\alpha\in\mathfrak{m}$ with the representation (\[apr\]) with $q\geq P^{\Xi+c\delta}$.
For $R\leq P^{\Xi+c\delta}$, Lemma \[CL1\] again shows that $I_1(S;t,\mathfrak{m}_2)=\textit{o}\left(P^{\Lambda}\right)$, where $\mathfrak{m}_2$ the $\alpha\in \mathfrak{m}$ with the representation (\[apr\]) with $q\leq P^{\Xi+c\delta}, \quad |\beta|\geq \phi_1P^{\delta}$.
As for the remaining range, an application of Lemma \[LE3\] yields $$\begin{aligned}
P^{t+\varepsilon}\mathcal{M}_2(R,\phi,\pm)^{1/2}\ll &P^{t+\varepsilon}(R^2\phi)^{1/2}P^{n+\varepsilon}\left\{R\phi+(R\phi P^3)^{-1}\right\}^{n/8}\ll P^{n+t+2\varepsilon}R^{1+\frac{n}{8}}(\phi_1P^{c\delta})^{\frac{1}{2}+\frac{n}{8}}+ P^{\frac{5n}{8}+t+2\varepsilon}R^{1-\frac{n}{8}}\phi^{\frac{1}{2}-\frac{n}{8}}\\
\ll &R^{1+\frac{n}{8}-\rho_1\left(\frac{1}{2}+\frac{n}{8}\right)}P^{n+t-\pi_1\left(\frac{1}{2}+\frac{n}{8}\right)+\textit{O}(\delta)}+R^{1-\frac{n}{8}}\phi^{\frac{1}{2}-\frac{n}{8}}P^{\frac{5n}{8}+t+2\varepsilon}.
\end{aligned}$$ The first term is $\textit{o}\left(P^\Lambda\right)$ when (\[req11\]) holds. The second term is $\textit{o}\left(P^\Lambda\right)$ when $\phi\geq \phi_2 P^{\delta}$. Then (1) follows.
Now if $\Xi\leq 0$, then $q\leq P^{\Xi+c \delta}$ and $|\beta|\leq \phi_2 P^{\delta}$ implies $q\leq P^\Delta$ and $|\beta|\leq P^{-\pi_2+\textit{O}(\delta)}\leq P^{-3+\Delta}$ on assuming that (\[req12\]) holds. Then $\alpha$ lies in the major arcs and $\mathfrak{m}_0=\emptyset$. Hence (2) follows.
The treatment of (1,1,9)-forms needs a certain fourth moment of a cubic exponential sum. The following lemma is a slight modification of Brüdern [@Bru91 Theorem 2], which involves an application of a Kloosterman refinement based on [@Hoo86].
Define the weight function $$w(x):=
\begin{cases}
e^{-\frac{1}{1-x^2}},\quad &\text{if }|x|<1,\\
0, &otherwise.
\end{cases}$$
Let $$T(\alpha)=\sum\limits_{x\in\mathbb{Z}}w(P^{-1}x)e\left(\alpha x^3\right).$$ Denote $$\mathfrak{N}=\mathfrak{N}(R,\phi):=\bigcup\limits_{R<q\leq 2R}\bigcup\limits_{a=1 \atop (a,q)=1}^q\left[\frac{a}{q}+\phi, \, \frac{a}{q}+2\phi\right].$$
\[LE9\] For $\phi\leq P^{-3}$, we have $$\int_{\mathfrak{N}(R,\phi)} |T(\alpha)|^4 d\alpha\ll P^{\varepsilon}\left(P^4 \phi +R^{7/2}\phi+R^2\phi P^2\right).$$ And for $\phi>P^{-3}$, we have $$\int_{\mathfrak{N}(R,\phi)} |T(\alpha)|^4 d\alpha\ll P^{\varepsilon}\left(\phi^{-1/3}+R^{7/2}\phi^3 P^6+R^2\phi P^2\right).$$
The weight function used here is slightly different from that in [@Bru91]. It is actually the weight in [@Hoo86]. However, the validity of the argument is not affected.
Bounding the minor arc estimates
================================
In this section, we bound the minor arc estimates and prove Proposition \[PR2\]. Recall the definition of $\mathfrak{m}_0$ and $\mathcal{A}$. They collect the $\alpha\in [0,1]$ with the representation (\[apr\]) with $$\begin{aligned}
&\mathfrak{m}_0\subseteq \mathfrak{m}: &&\quad q\leq P^{\Xi+\textit{O}(\delta)},\quad |\beta|\leq \phi_2P^{\delta}=R^{-\rho_2}P^{-\pi_2+\textit{O}(\delta)},\\
&\mathcal{A}(A,B,C): &&\quad q\leq P^A,\quad |\beta|\leq q^{-B}P^{-3+C}.
\end{aligned}$$
Recall $S_i(\alpha)=S(\alpha; C_i,n_i,\rho,P)$ $(0\leq i\leq 3)$. Lemma \[LE5\] shows that $$I_{4}(S_1;0,[0,1])\ll P^{1/2+\varepsilon}, \quad I_{4}(S_2;0,[0,1])\ll P^{5/4+\varepsilon}.$$ Taking $n=8,v=2,t=7/4,\Lambda=8$, Lemma \[LE8\](1) gives $\Xi =11/20, \rho_2=0,\pi_2=5/2$ and $$I_1(S_1S_2S_3;0,\mathfrak{m}\setminus \mathfrak{m}_0)\ll I_{4,4,2}(S_1,S_2,S_3;0,\mathfrak{m}\setminus \mathfrak{m}_0)\ll I_{2}(S_3;7/4,\mathfrak{m}\setminus \mathfrak{m}_0)=\textit{o}\left(P^8\right).$$ Noting that $\mathfrak{m}_0\subseteq \mathcal{A}\left(11/20+\textit{O}(\delta),0,1/2+\textit{O}(\delta)\right):=\mathcal{A}$. The $\textit{O}(\delta)$-term occurring here can be as small as we want, so substituting $\textit{O}(\delta)$ for $\varepsilon$ does not affect the validity of the lemmas. Lemma \[LE6\] gives $$S_{1}(\alpha)=S_{1}^\ast (\alpha)+\textit{O}\left(P^{21/40+\textit{O}(\delta)}\right), \quad I_{4}(S_1^\ast ;0,[0,1])\ll P^{1/4+\varepsilon}.$$ for $\alpha\in \mathcal{A}$. And $$I_1(S_1S_2S_3;0,\mathfrak{m}_0)\ll I_1(S_1^\ast S_2 S_3;0,\mathfrak{m})+I_1(S_2S_3;21/40,\mathcal{A}).$$ Taking $n=8,v=2,t=1/4+5/4=3/2, \Lambda=8$, Lemma \[LE8\](2) shows that $\Xi=0,\rho_2=0,\pi_2=3$ and $$I_1(S_1^\ast S_2 S_3;0,\mathfrak{m})\ll I_{4,4,2}(S_1^\ast,S_2,S_3;0,\mathfrak{m})\ll I_{2}(S_3;3/2; \mathfrak{m})=\textit{o}\left(P^8\right).$$ Denote $\widetilde{S}(\alpha):=S(\alpha;C_2+C_3,10,\rho,P)$. It follows from Lemma \[LE7\] that $$I_1(S_2S_3;21/40,\mathcal{A})=I_1(\widetilde{S};21/40,\mathcal{A})\ll P^{8-1/16+\textit{O}(\delta)}.$$ To conclude, we have $I_1(S_1S_2S_3;0,\mathfrak{m})=\textit{o}\left(P^8\right)$. Proposition \[PR2\] follows and $\mathcal{N}(P)\gg P^8$.
Define, for $i=1,2$, $$T_i(\alpha)=\sum\limits_{x\in\mathbb{Z}}w\left((\rho P)^{-1}x\right)e\left(\alpha C_{i}(x)\right).$$ The parameters $\rho\leq 1$ is determined in Lemma \[LE2\]. It is fixed and makes no difference to the validity of applying Lemma \[LE9\]. Let $\mathfrak{b}\subseteq \mathfrak{a}\subseteq \mathfrak{m}$ be the set of $\alpha \in \mathfrak{m}$ with the representation (\[apr\]) with $$\begin{aligned}
&\mathfrak{a}: &&\quad q\leq P^{25/31+\textit{O}(\delta)},\quad |\beta|\leq q^{-1/5}P^{-11/5+\textit{O}(\delta)}\\
&\mathfrak{b}: &&\quad q\leq P^{1145/1922+\textit{O}(\delta)},\quad |\beta|\leq q^{-1/5}P^{-1867/775+\textit{O}(\delta)}.
\end{aligned}$$
It is easy to see that, for $i=1,2$, one has $$I_{4}(T_i,0,[0,1])\ll P^{1/2+\varepsilon}$$ by regarding the left side as the weighted number of solutions to the equality $$C_i(x_1)+C_i(x_2)=C_i(x_3)+C_i(x_4).$$ Taking $n=9,v=2,t=1/2+1/2=1,\Lambda=8$, Lemma \[LE8\] yields $\Xi =25/31, \rho_2=1/5,\pi_2=11/5$ and $$I_1(T_1T_2S_3;0,\mathfrak{m}\setminus \mathfrak{a})\ll I_{4,4,2}(T_1,T_2,S_3;0,\mathfrak{m}\setminus \mathfrak{a})\ll I_{2}(S_3;1,\mathfrak{m}\setminus \mathfrak{a})=\textit{o}\left(P^8\right). \label{case1-1}$$ For $R,\phi$ in the range defined by $\mathfrak{a}$, Lemma 9 gives $$I_4(T_1;0,\mathfrak{a})\ll P^{\varepsilon} \max_{R,\phi} I_4(T_1;0,\mathfrak{N}(R,\phi))\ll P^{\frac{539}{310}\cdot \frac{1}{4}+\textit{O}(\delta)}= P^{\frac{539}{1240}+\textit{O}(\delta)}.$$ Taking $n=9,v=2,t=539/620,\Lambda=8$, Lemma \[LE8\] yields $\Xi =1145/1922, \rho_2=1/5,\pi_2=1867/775$ and $$I_1(T_1T_2S_3;0,\mathfrak{a}\setminus \mathfrak{b})\ll I_{4,4,2}(T_1,T_2,S_3;0,\mathfrak{a}\setminus \mathfrak{b}) \ll I_{2}(S_3;539/620,\mathfrak{m}\setminus \mathfrak{b})=\textit{o}\left(P^8\right). \label{case1-1}$$ For $R,\phi$ in the range defined by $\mathfrak{b}$, Lemma 9 again gives $$I_4(T_1;0,\mathfrak{b})\ll P^{\varepsilon} \max_{R,\phi} I_4(T_1;0,\mathfrak{N}(R,\phi))\ll P^{\frac{1}{4}+\textit{O}(\delta)}.$$ And noting that $\mathfrak{b}\subseteq \mathcal{A}(1145/1922+\textit{O}(\delta), 1/5, 458/775+\textit{O}(\delta)):=\mathcal{A}$. By (\[remarkoflemma7\]), it follows that $$I_1(T_1 T_2 S_3;0,\mathfrak{b})\ll I_{4,4,2}(T_1,T_2,S_3;0,\mathfrak{b})\ll I_{2}(S_3;1/2;\mathcal{A}\cap \mathfrak{m})\ll P^{8-\Delta/8+\textit{O}(\delta)}.$$ To sum up, we now have $$I_1(T_1T_2S_3;0,\mathfrak{m})=\textit{o}\left(P^8\right).$$
The weight function $w$ satisfies $w\geq 0$ in $\mathbb{R}$ and $w(x)\gg 1$ for $|x|<\rho P/2$, which ensures the argument in the treatment of singular integral in Lemma \[LE2\]. We conclude that $$\mathcal{N}(P)\gg P^8.$$
Further remarks
===============
The $(1,1,8)$ and $(1,2,7)$ cases are hard to solve. The estimates on exponential sums of the $1$-forms and $2$-forms are not small enough and Heath-Brown’s $L^2$ bound can not be used. Neither can it even in the case $(1,1,1,7)$. We say $C_0$ splits into four forms, and is a $(n_1,n_2,n_3,n_4)$-form, if $$C_0(\mathbf{x})=C_1(\mathbf{x}_1)+C_2(\mathbf{x}_2)+C_3(\mathbf{x}_3)+C_4(\mathbf{x}_4),$$ where $C_i\in \mathbb{Z}[\mathbf{x}_i]\,(1\leq i\leq 4)$ are cubic forms in $n_i$ variables and $n_1+n_2+n_3+n_4=n_0$. For $n_0=10$, according to (\[cases\]), the only case not solved is $(1,1,1,7)$. We need strong hypothesis, such as Hypothesis $HW_6$ (see [@HB98 p.10] for details), to solve the $(1,1,1,7)$ case. This hypothesis involves Riemann Hypothesis and standard analytic continuation of certain Hasse-Weil $L$-functions. We record the following proposition.
\[TH6\] Let $X\subseteq \mathbb{P}^{n-1}$ be a hypersurface defined by a cubic form that splits into four forms, with $n\geq 10$. Assuming Hypothesis $HW_6$, we have $X(\mathbb{Q})\neq \emptyset$.
The singular series $\mathfrak{S}$ is still absolutely convergent in the $(1,1,1,7)$ case (we have $S_{a,q}(C_i)\ll q^{2/3}$ for $1\leq i\leq 3$, which is better than $q^{5/6}$). The $(1,1,8)$ case remains unproved under this strong hypothesis. So it seems really hard to improve $n\geq 11$ to $n\geq 10$ in Theorem \[TH1\].
\[ackref\] We would like to thank Professor T. D. Browning for giving talks on this topic and suggesting this problem. He has also provided us with useful advices and great help, including pointing out the improved version of [@Browning1 Theorem 2], Theorem H, [@HB83 Proposition 2] and Brüdern’s fourth moment estimate. We also thank Yun Gao for her kind help. The first author is especially grateful to his supervisor, Professor Hongze Li, for his unending encouragement and his effort that provides the opportunities to communicate with foreign specialists.
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abstract: 'Equilibrium spatio-temporal correlation functions are central to understanding weak nonequilibrium physics. In certain local one-dimensional classical systems with three conservation laws they show universal features. Namely, fluctuations around ballistically propagating sound modes can be described by the celebrated Kardar-Parisi-Zhang (KPZ) universality class. Can such a universality class be found also in quantum systems? By unambiguously demonstrating that the KPZ scaling function describes magnetization dynamics in the SU(2) symmetric Heisenberg spin chain we show, for the first time, that this is so. We achieve that by introducing new theoretical and numerical tools, and make a puzzling observation that the conservation of energy does not seem to matter for the KPZ physics.'
author:
- 'Marko Ljubotina, Marko Žnidarič, and Tomaž Prosen'
title: 'Kardar-Parisi-Zhang physics in the quantum Heisenberg magnet'
---
**Introduction.–** Universality – where different systems can be described by the same underlying mathematical structure – is at the core of our understanding of nature. For instance, the properties of any thermalizing system can be described by the same equilibrium ensembles of statistical physics. Out of equilibrium less is known in general, in a way justifiably so, simply because the world of nonequilibrium is much richer. One of the more famous universality classes that can (among other) describe various nonequilibrium phenomena [@livibook] is that of the Kardar-Parisi-Zhang (KPZ) equation. The KPZ equation was originally introduced to describe stochastic growth of surfaces [@kardar86], and is a diffusion equation with the simplest possible nonlinearity (relevant at large scales) and an additional white noise term (equivalently, the surface’s slope is described by the stochastic Burgers equation). Besides describing surface dynamics it can be found in various contexts, ranging from exclusion processes to random matrix theory, for review see [@kpzrev1]. The KPZ equation itself harbors rich mathematical problems [@kpzrev2].
Nonequilibrium physics is one of the more propulsive areas of today’s theoretical physics. Close to equilibrium one can use Green-Kubo formulas and express nonequilibrium properties in terms of equilibrium correlation functions [@kubobook]. A downside to such an approach is that the calculation of spatio-temporal correlation functions is often very complicated. Any possible universality in their long-time behavior would therefore be highly appreciated. For classical fluids in one dimension such a picture has in fact been put forward [@beijeren12; @spohn14] in a form of nonlinear fluctuating hydrodynamics [@spohnrev] that describes (anomalous) fluctuations around sound peaks due to nonlinearity in one-dimensional systems that have 3 conservation laws (momentum, energy and mass), and are in general nonintegrable. That fluctuations are indeed described by the KPZ scaling function [@spohn04] has been verified in a number of classical systems [@kulkarni13; @mendl14; @das14; @das19; @unpub]. So-far there has been no observation of the KPZ universality class scaling function in quantum systems.
In this Letter we observe the KPZ scaling functions in an integrable quantum model that does not have any ballistic component. Namely, we show with an unprecedented accuracy (an order of magnitude larger than in simulations of classical systems) that an infinite temperature spin-spin correlation function in a paradigmatic SU(2) symmetric quantum Heisenberg chain has a KPZ form. Such accuracy is a result of two novelties: (i) using linear response formulation we show that one can calculate the equilibrium correlation function as an expectation value in a particular nonequilibrium state whose time evolution is easier to calculate, (ii) we directly treat an ensemble evolution, avoiding statistical averaging (as done in classical simulations), and which is, even more importantly, structurally stable. In addition, to discern the role played by conserved quantities, we show that in an integrable trotterized Floquet generalization [@vanicat18] of the model, that does not conserve the energy, the same KPZ scaling is observed. We note that the KPZ scaling exponents have been observed in various stochastic quantum settings, like random quantum circuits [@nahum17; @nahum18] or noisy evolution [@lamacraft18].
**The model.–** In classical systems the KPZ scaling function describes fluctuations around a sound mode, whose width scales as $\sim t^{1/z}$ with a dynamical exponent $z=\frac{3}{2}$. Therefore, to observe it one has to move to a ballistically moving reference frame, which, if the velocity is not known analytically, can introduce numerical inaccuracies. We are therefore going to look for KPZ physics at infinite temperature in the one-dimensional Heisenberg spin-$\frac{1}{2}$ chain at zero magnetization (half-filling) where the ballistic contribution is zero due to the spin-flip (particle-hole) symmetry and the spin transport shows a KPZ dynamical scaling exponent $z=\frac{3}{2}$. Namely, such superdiffusive magnetization transport has been observed in a nonequilibrium steady state where the current scales as $j \sim 1/L^{z-1}$ [@prl11] as well as in the spreading of an inhomogeneous initial state where the width scales as $\sigma \sim t^{1/z}$ [@ljubotina17]. The local Hamiltonian density is $${{h}}_{r,r+1}=J\left({{{s}}}^{{\rm x}}_{r}{{{s}}}^{{\rm x}}_{r+1}+{{{s}}}^{{\rm y}}_{r}{{{s}}}^{{\rm y}}_{r+1}+{{{s}}}^{{\rm z}}_{r}{{{s}}}^{{\rm z}}_{r+1}\right)\,,
\label{eq:localint}$$ where $s^{\alpha}_r=\frac{1}{2}\sigma^{\alpha}_r$, $\alpha\in\{{{\rm x}},{{\rm y}},{{\rm z}}\}$, are spin operators (Pauli matrices) at site $r\in\{-\frac{L}{2},\ldots,\frac{L}{2}-1\}$. Theoretical explanation of the scaling exponent $z=\frac{3}{2}$ is still lacking, but consistent derivations within assumptions of generalized hydrodynamics were recently given [@vasseur18]. In particular, it is possible to estimate the diffusion constant [@markomprl; @denardis18; @vasseur18] and prove its divergence, i.e. $z < 2$ [@enejprl].
Here, in order to observe precise spatio-temporal profiles of spin and current densities, we will consider two dynamical setings: [*continuous*]{} time evolution $U^t=e^{-i Ht}$ generated by $H=\sum_{r=-L/2}^{L/2-2} h_{r,r+1}$ (where we set $J=1$) or [*discrete*]{} time evolution with one step propagator ${{U}}={{U}}_{\rm e}{{U}}_{\rm o}$, with ${{U}}_{\rm o}=e^{-i\sum_{r}{{h}}_{2r-1,2r}}$ and ${{U}}_{\rm e}=e^{-i\sum_{r}{{h}}_{2r,2r+1}}$ (where we use $J=\frac{\pi}{2}$, and where one also observes the superdiffusive scaling $z=\frac{3}{2}$ [@ljubotina19]). Both settings are characterized by both a global $SU(2)$ symmetry and integrability.
In order to study transport we must derive the expressions for the local spin current density operators for both the continuous-time and discrete-time models. The former is the standard spin current in the Heisenberg model ${{j}}_r={{{s}}}_r^{{\rm x}}{{{s}}}_{r+1}^{{\rm y}}-{{{s}}}_r^{{\rm y}}{{{s}}}_{r+1}^{{\rm x}}$ which fulfills the continuity equation $\frac{{{\rm d}}{{{s}}}^{{\rm z}}_r}{{{\rm d}}t}={{j}}_{r-1}-{{j}}_{r}$. The current in the discrete-time model turns out to be slightly more complicated, with the operator being different on odd and even sites due to the staggered nature of the propagator $U$. The two currents densities satisfy a pair of continuity equations $$\begin{aligned}
&{{U}}^\dagger{{M}}_{2r}{{U}}-{{M}}_{2r}={{j}}_{2r-1}^{\rm o}-{{j}}_{2r+1}^{\rm o}\,,\\
&{{U}}^\dagger{{M}}_{2r-1}{{U}}-{{M}}_{2r-1}={{j}}_{2r-2}^{\rm e}-{{j}}_{2r}^{\rm e}\,,
\end{aligned}
\label{eq:disc_continuity}$$ where ${{M}}_r={{{s}}}_{r}^{{\rm z}}+{{{s}}}_{r+1}^{{\rm z}}$. The simpler odd current can then be seen to take the form $$\begin{aligned}
{{j}}_{2r-1}^{\rm o}=2\sin(J) j_{2r-1} -\frac{1}{2}\sin^2(J/2)({{{s}}}^{{\rm z}}_{2r}-{{{s}}}^{{\rm z}}_{2r-1})\,,
\end{aligned}
\label{eq:disc_curr}$$ whereas the even current is simply the odd current propagated by half a time step ${{j}}_{2r}^{\rm e}={{U}}^\dagger_{\rm e} {{j}}_{2r}^{\rm o}{{U}}_{\rm e}$ and acts on 4 adjacent sites.
![ Collapse of spin profiles for the continuous-time (top) and discrete-time (bottom) model in terms of the scaling parameter $\xi=x/t^{2/3}$ shown for several times. The continuous-time simulation was performed on a spin chain of length $L=400$ with bond dimension $\chi=400$ and polarization $\mu=0.0017$. The discrete-time simulation was performed with $L=7200$, $\chi=256$ and $\mu=0.0005$. The same parameters are used in other figures. In the discrete case there is an additional Floquet even-odd splitting whose size decays as $t^{-1/3}$ (the inset). []{data-label="fig:profiles"}](f11.pdf "fig:"){width="0.98\linewidth"} ![ Collapse of spin profiles for the continuous-time (top) and discrete-time (bottom) model in terms of the scaling parameter $\xi=x/t^{2/3}$ shown for several times. The continuous-time simulation was performed on a spin chain of length $L=400$ with bond dimension $\chi=400$ and polarization $\mu=0.0017$. The discrete-time simulation was performed with $L=7200$, $\chi=256$ and $\mu=0.0005$. The same parameters are used in other figures. In the discrete case there is an additional Floquet even-odd splitting whose size decays as $t^{-1/3}$ (the inset). []{data-label="fig:profiles"}](f12.pdf "fig:"){width="0.98\linewidth"}
We begin by preparing our system in a weakly polarized domain-wall mixed initial state [@ljubotina17] $$\begin{aligned}
\rho(t=0)&\propto \rho_\mu=\left(e^{\mu{{{s}}}^{{\rm z}}}\right)^{\otimes L/2}\otimes\left(e^{-\mu{{{s}}}^{{\rm z}}}\right)^{\otimes L/2}.
\end{aligned}
\label{eq:init_state}$$ An example of time evolution for both models is shown in Fig. \[fig:profiles\], using the scaling variable $\xi=\frac{r}{t^{1/z}}, z=\frac{3}{2}$. While this choice of the initial state provides a numerically stable and efficient way to study spin transport [@ljubotina17], we emphasize that for our purposes it provides us with an efficient way to study the infinite-temperature spin-spin correlation function $\langle{{{s}}}^{{\rm z}}_0{{{s}}}^{{\rm z}}_r(t)\rangle$, where $A(t)\equiv U^{-t}AU^t$ and $\langle\,\boldsymbol{\cdot}\,\rangle\equiv 2^{-L}{{\rm tr}}(\cdot)$ denotes the infinite-temperature expectation value. We explain that in the following section.
**Linear response.–** We start by expanding the initial state (\[eq:init\_state\]) to linear order in $\mu$, evolving it in time, and writing down the expectation value for a single spin, $$\langle{{{s}}}_r^{{\rm z}}(t)\rangle_\mu=-\mu\sum_{r'} \theta_{r'} \langle{{{s}}}_r^{{\rm z}}(t) {{{s}}}_{r'}^{{\rm z}}\rangle+\mathcal{O}(\mu^2)\,,
\label{eq:lin1}$$ where we introduced $\langle\,\boldsymbol{\cdot}\,\rangle_\mu = {{\rm tr}}[(\cdot)\rho_\mu]/{{\rm tr}}\rho_\mu$ as the expectation value in the weak domain-wall initial state (\[eq:init\_state\]) and $\theta_{r}\equiv 1 (-1)$ for $r\ge0 (<0)$. Accounting for the translational invariance of the infinite-temperature expectation value we obtain $$\begin{aligned}
&\langle{{{s}}}_{r-1}^{{\rm z}}(t)\rangle_\mu-\langle{{{s}}}_{r}^{{\rm z}}(t)\rangle_\mu\approx
\mu\langle{{{s}}}^{{\rm z}}_r(t)\sum_{r'}\theta_{r'}({{{s}}}^{{\rm z}}_{r'}-{{{s}}}^{{\rm z}}_{r'+1})\rangle\nonumber \\
&=2\mu \langle{{{s}}}^{{\rm z}}_r(t){{{s}}}^{{\rm z}}_0\rangle - 2\mu \langle{{{s}}}^{{\rm z}}_r(t){{{s}}}^{{\rm z}}_{-L/2}\rangle.
\label{eq:lin2}
\end{aligned}$$ In the thermodynamic limit $L \to \infty$ the second term vanishes as there are no correlations across infinite distances, and using the cyclic property of the trace we get $$\langle{{{s}}}_0^{{\rm z}}(0){{{s}}}_r^{{\rm z}}(t)\rangle=\lim_{\mu\to0}\frac{\langle{{{s}}}_{r-1}^{{\rm z}}(t)\rangle_\mu-\langle{{{s}}}_{r}^{{\rm z}}(t)\rangle_\mu}{2\mu}\,.
\label{eq:lin3}$$ This is our first main result.
{width="0.49\linewidth"} {width="0.49\linewidth"} {width="0.49\linewidth"} {width="0.49\linewidth"}
It shows that a weak domain wall initial state can be seen as a trick that allows us to calculate the infinite-temperature spin-spin correlation. We next recall [@spohnrev] why the LHS of Eq.(\[eq:lin3\]) is in certain classical systems described by the KPZ scaling function.
**Kardar-Parisi-Zhang equation.–** The KPZ stochastic partial differential equation was initially suggested to model the growth of surface $h(r,t)$ through random deposition [@kardar86] $$\partial_t h=\frac{1}{2}\lambda\left(\partial_rh\right)^2+\nu\partial_r^2h+\sqrt{\Gamma}\zeta\,,
\label{eq:kpz}$$ where $\zeta(r,t)$ is a space-time uncorrelated noise.
Of particular interest to us will be the correlation function $C(r,t)=\langle\left[h(r,t)-h(0,0)-t\langle\partial_th\rangle\right]^2\rangle$ – representing the fluctuations of the height around the expected value – and its second derivative $\frac{1}{2}\partial_r^2C(r,t)=\langle\partial_rh(0,0)\partial_rh(r,t)\rangle$ – describing the slope correlations (here brackets denote noise averaging). In terms of scaling functions $g(\varphi)$ and $f(\varphi)$ one has $$\begin{aligned}
g(\varphi)&=\lim_{t\to\infty}\frac{C\left((2\lambda^2t^2\Gamma\nu^{-1})^{-1/3}\varphi,t\right)}{\left(\frac{1}{2}\lambda t\Gamma^2\nu^{-2}\right)^{2/3}}\,,\\
f(\varphi)&=\frac{1}{4}g''(\varphi)\sim\partial_r^2C(r,t)\,.
\end{aligned}
\label{eq:kpz_scaling}$$ These can be obtained from the exact solution of the polynuclear growth model [@spohn04] (a model in the KPZ universality class), and have been tabulated with high precision in Ref. [@prahoferwww]. Nonlinear fluctuating hydrodynamics predicts that the correlation function of a conserved quantity, in our case $\langle{{{s}}}_0^{{\rm z}}(0){{{s}}}_r^{{\rm z}}(t)\rangle$, should be given by the so-called KPZ scaling function $f(\varphi)$.
Using Eq.(\[eq:lin3\]) this correlation function is equal to the magnetization difference on consecutive sites in the state $\rho(t) \propto U^t \rho_\mu U^{-t}$ (see Fig. \[fig:profiles\]). However, taking the discrete derivative increases numerical errors, so alternatively, one can also look at the scaling form of the current $j(r,t)=\langle{{j}}_r(t)\rangle_\mu$. In a diffusive process, the scaling forms of both the current as well as of the magnetization difference are Gaussian. Relation between the two in a general non-diffusive situation can be derived from the continuity equation.
Defining a shorthand notation $z(r,t)=\langle{{{s}}}_r^{{\rm z}}(t)\rangle_\mu$, and $\varphi=b\xi$, we write an ansatz $$\partial_rz(r,t)=\frac{a\mu}{t^{2/3}}f\left(\frac{br}{t^{2/3}}\right)\,,
\label{eq:scl1}$$ where we introduced two system-dependent parameters $a$ and $b$, and use continuum notation for the magnetization difference. Taking into account the continuity equation $\partial_tz=-\partial_rj$, one may obtain the shape of the spin current profile. Expressing everything in terms of $g(\varphi)$ (using per-partes integration and Eq. (\[eq:kpz\_scaling\])) we get $$\begin{aligned}
j(r,t)&=\frac{2a\mu}{3b^2t^{1/3}}h\left(\frac{br}{t^{2/3}}\right)\\
h(\varphi)&=\frac{g(\varphi)-\varphi g'(\varphi)}{4}\,.
\end{aligned}
\label{eq:scl3}$$ The form of $j(r,t)$, i.e. the function $h(\varphi)$, is therefore uniquely determined by the form of $\partial_r z(r,t)$, i.e., the KPZ function $f(\varphi)$.
![ Plotting the ratio between the gradient of spin density and spin current density in scaled units, we can observe that the numerical results for both models clearly do not obey Fick’s law. Instead, they are well described by the prediction from KPZ. Numerical data are plotted for maximum simulations times ($t=200$ for continuous and $t=3600$ for discrete time cases). The ratios are rescaled to $1$ at $\varphi=0$. []{data-label="fig:fick"}](f31.pdf){width="0.95\linewidth"}
We employ extensive numerical simulations [@foot3] using the time-evolving block decimation algorithm [@vidal03; @vidal04; @schollwock11] for matrix-product density operator in order to study the time evolution of a domain-wall like initial state in both the continuous and discrete time Heisenberg models. This allows us to compute the infinite-temperature spin-spin correlations (\[eq:lin3\]) in a numerically stable way with manageable bond dimensions $\chi$. Fig. \[fig:kpz\] shows the results and the best-fitting KPZ profile for both the spin and spin current. Due to higher numerical accuracy we only fit the data for the current, obtaining $a$ and $b$ (\[eq:scl3\]), which automatically fixes the spin difference profiles (\[eq:scl1\]). In order to avoid even-odd staggering in the discrete-time model we take the difference of two consecutive pairs of spins, rather than a difference of two spins, and appropriately scale the continuity equation. For comparison we also show best-fitting Gaussians. Because the KPZ scaling functions $f(\varphi)$ and $h(\varphi)$ are rather close to Gaussians for not too large arguments, one in fact needs at least two decades of accuracy to be able to distinguish the two. With our numerics we have accuracy over about three decades in the continuous model and about four in the discrete one. We can clearly confirm that the KPZ scaling functions emerge at sufficiently long times.
Free parameters $a$ and $b$ are found to be $a=b\approx0.67$, conjectured to be $\frac{2}{3 J^{2/3}}$, for the continuous-time model. Similarly, for the discrete-time model we find $a=b\approx0.43$, data for other values of $J$ are well described by the formula $a=b\approx\frac{2^{1/3}}{3|\tan(J/2)^{2/3}|}$.
Because the KPZ $f(\varphi)$ and $h(\varphi)$ are not Gaussian, their ratio $h/f\equiv w(br/t^{2/3})$ which appears in a relation $j(r,t)=[2t^{1/3}/(3b^2)]w(br/t^{2/3}) \,\partial_r z(r,t)$ is not a constant. Therefore, Fick’s law, even with a time-dependent diffusion constant [@foot2], is violated (Fig. \[fig:fick\]).
![ Dependence on bond dimension of the current profiles in a domain-wall state and for continuous ($t=200$) and discrete-time simulations ($t=3600$). Results are stable to increasing $\chi$ and converge to the KPZ scaling functions. We apply a moving average to the leftmost and rightmost $20\%$ of the data so that it is easier to see the decreasing truncation error in the tails. []{data-label="fig:bond"}](f41.pdf "fig:"){width="0.98\linewidth"} ![ Dependence on bond dimension of the current profiles in a domain-wall state and for continuous ($t=200$) and discrete-time simulations ($t=3600$). Results are stable to increasing $\chi$ and converge to the KPZ scaling functions. We apply a moving average to the leftmost and rightmost $20\%$ of the data so that it is easier to see the decreasing truncation error in the tails. []{data-label="fig:bond"}](f42.pdf "fig:"){width="0.98\linewidth"}
We also show the dependence of current profiles on the bond dimension $\chi$ used in simulations, Fig. \[fig:bond\]. In the discrete-time case we use slightly smaller $\chi$, however the acquired times are larger (Fig. \[fig:kpz\]), as well as the sizes ($L=7200$ vs. $L=400$). As a net result the wall-times of discrete model simulations are about half as long as for a continuous one despite about a decade better accuracy (Fig. \[fig:bond\]). We stress that in the best classical simulations (hard-point gas [@mendl14]) slightly less than two decades of agreement with KPZ are achieved. What distinguishes our quantum simulations is that we directly work with an ensemble, encoded in the many-body density matrix $\rho(t)$, so no averaging is needed. It is an interesting open problem how to do such efficient ensemble simulations for classical many-body models, in particular since for continuous variables the local function spaces are infinitely dimensional.
Lastly, we note that taking a slightly larger domain-wall step $\mu=0.02$ we are even able to observe (data not shown) second order $\mu^2$ corrections to the dynamics in the form of a small ballistically spreading front, traveling away from the site of the quench.
**Discussion.–** We have shown that the infinite temperature spin-spin correlation function in the isotropic Heisenberg spin-$\frac{1}{2}$ model obeys the Kardar-Parisi-Zhang scaling. This is the first such observation in a deterministic quantum model. We stress that in order to reliably show the KPZ physics one has to look at the full distribution function of fluctuations and not e.g. just the dynamical scaling exponent being $z=\frac{3}{2}$. For instance, a related spreading exponent $\frac{1}{3}$ generically appears in free or dilute models, see e.g. [@racz04; @vir18].
High accuracy of over four decades was achieved by using a trick where we simulate the melting of a slightly polarized domain wall by directly evolving the density operator, which is, through linear response, equivalent to studying the equilibrium spin-spin correlation function.
Besides providing a method to efficiently probe spatio-temporal correlation functions in quantum models, several new directions are opened. The most important is the question of universality. Namely, in nonlinear fluctuating hydrodynamics, so-far verified only in classical models, the KPZ universality is associated to the existence of 3 conservation laws. It is not clear which 3 conserved quantities (if at all) are responsible for the observed behavior. By studying a kicked Floquet generalization of the isotropic Heisenberg model which does not conserve the energy, but nevertheless shows the KPZ physics, we show that the energy is not one of them. It remains to be seen if the observed behavior is in any way related to integrability and the SU(2) symmetry of the model.
**Acknowledgements.–** We acknowledge useful related discussions with J. De Nardis, E. Ilievski, M. Medenjak, and H. Spohn. The authors acknowledge support by the European Research Council (ERC) through the advanced grant 694544 – OMNES and the grants P1-0402 and J1-7279 of the Slovenian Research Agency (ARRS).
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---
abstract: 'It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p-$space, the corresponding ergodic averages converge bilaterally almost uniformly for $1\leq p<2$ and almost uniformly for $2\leq p< {\infty}$. We show that these averages converge almost uniformly for all $1\leq p<{\infty}$.'
address: '76 University Drive, Pennsylvania State University, Hazleton 18202'
author:
- Semyon Litvinov
date: 'June 14, 2016'
title: 'On the Almost Uniform Convergence in Noncommutative Dunford-Schwartz Ergodic Theorem'
---
Preliminaries
=============
Let ${\mathcal}M$ be a semifinite von Neumann algebra equipped with a faithful normal semifinite trace $\tau$. Let ${\mathcal}P({\mathcal}M)$ stand for the set of projections in ${\mathcal}M$. If $\mathbf 1$ is the identity of ${\mathcal}M$ and $e\in {\mathcal}P({\mathcal}M)$, we write $e^{\perp}=\mathbf 1-e$. Denote by $L^0=L^0({\mathcal}M,\tau)$ the $*$-algebra of $\tau$-measurable operators affiliated with ${\mathcal}M$. Let $\| \cdot \|_{{\infty}}$ be the uniform norm in ${\mathcal}M$. Equipped with the [*measure topology*]{} given by the system $$V({\epsilon},{\delta})=\{ x\in L^0: \ \| xe\|_{{\infty}}\leq {\delta}\text{ \ for some \ } e\in {\mathcal}P({\mathcal}M) \text{ \ with \ } \tau(e^{\perp})\leq {\epsilon}\},$$ ${\epsilon}>0$, ${\delta}>0$, $L^0$ is a complete metrizable topological $*$-algebra [@ne].
Let $L^p=L^p({\mathcal}M.\tau)$, $1\leq p\leq {\infty}$, ($L^{{\infty}}={\mathcal}M$) be the noncommutative $L^p-$space associated with $({\mathcal}M,\tau)$.
For detailed accounts on the spaces $L^p({\mathcal}M,\tau)$, $p\in \{0\}\cup [1,{\infty})$, see [@se; @ye0; @px].
Denote by $\| \cdot \|_p$ the standard norm in the space $L^p$, $1\leq p< {\infty}$. A linear operator $T: L^1+{\mathcal}M\to L^1+{\mathcal}M$ is called a [*Dunford-Schwartz operator*]{} if $$\| T(x)\|_1\leq \| x\|_1 \ \ \forall \ x\in L^1 \text{ \ and \ } \| T(x)\|_{{\infty}}\leq \| x\|_{{\infty}} \ \ \forall \ x\in L^{{\infty}}.$$ If a Dunford-Schwartz operator $T$ is positive, that is, $T(x)\ge 0$ whenever $x\ge 0$, we will write $T\in DS^+$.
Given $T\in DS^+$ and $x\in L^1+L^{{\infty}}$, denote $$\label{e0}
A_n(x)=\frac 1n\sum_{k=0}^{n-1} T^k(x), \ \ n=1,2,\dots,$$ the corresponding ergodic averages of the operator $x$.
Note that, by [@jx Lemma 1.1], any $T\in DS^+$ can be uniquely extended to a positive linear contraction (also denoted by $T$) in $L^p$, $1\leq p<{\infty}$.
A sequence $\{ x_n\}{\subset}L^0$ is said to converge [*almost uniformly (a.u.) (bilaterally almost uniformly (b.a.u.))*]{} to $x\in L^0$ if for any given ${\epsilon}>0$ there is a projection $e\in {\mathcal}P({\mathcal}M)$ such that $\tau(e^{\perp})\leq {\epsilon}$ and $\| (x-x_n)e\|_{{\infty}}\to 0$ (respectively, $\| e(x-x_n)e\|_{{\infty}}\to 0$).
Note that a.u. convergence obviously implies b.a.u. convergence.
The following groundbreaking result was established in [@ye] as a corollary of a noncommutative maximal ergodic inequality [@ye Theorem 1]
\[ye\] Let $T\in DS^+$ and $x\in L^1$. Then the averages (\[e0\]) converge b.a.u. to some $\widehat x\in L^1$.
As it was noticed in [@cl Remark 1.2] (see also [@jx Lemma 1.1]), the class of iterating operators (kernels) that was considered in [@ye] coincides with the class of positive Dunford-Schwartz operators that was dealt with in [@jx].
In [@jx Corollary 6.2], Theorem \[ye\] was extended to noncommutative $L^p-$spaces, $1<p<{\infty}$, as follows (see also [@li Theorems 4.3, 4.4] and [@cl Proof of Theorem 1.5]).
\[jx\] Let $T\in DS^+$ and $x\in L^p$. Then the averages (\[e0\]) converge to some $\widehat x\in L^p$ b.a.u. for $1<p<2$ and a.u. for $p\ge 2$.
The aim of this article is to show that the averages (\[e0\]) converge almost uniformly for all $1\leq p<{\infty}$:
\[t2\] Let $T\in DS^+$ and $1\leq p<{\infty}$. Given $x\in L^p$, the averages (\[e0\]) converge a.u. to some $\widehat x\in L^p$.
Proof of Theorem \[t2\]
=======================
Let $\{ e_i\}_{i\in I}{\subset}{\mathcal}P({\mathcal}M)$. Denote by $\bigvee \limits_{i\in I} e_i$ the projection on the subspace ${\overline}{\sum \limits_{i\in I} e_i{\mathcal}H}$, and let $\bigwedge \limits_{i\in I} e_i$ stand for the projection on the subspace $\bigcap \limits_{i\in I} e_i{\mathcal}H$. It is well-known that ${\mathcal}P({\mathcal}M)$ is a complete lattice, since if $\{ e_i\}_{i\in I}{\subset}{\mathcal}P({\mathcal}M)$, then $\bigvee \limits_{i\in I} e_i\in {\mathcal}P({\mathcal}M)$ is the l.u.b. of $\{ e_i\}_{i\in I}$. Besides, a normal trace $\tau$ on ${\mathcal}M$ is countably subadditive, that is, given $\{ e_n\}_{n=1}^{\infty}{\subset}{\mathcal}P({\mathcal}M)$, we have $\tau \left (\bigvee \limits_{n=1}^{\infty}e_n\right )\leq \sum \limits_{n=1}^{\infty}\tau(e_n)$.
If $x\in L^0$, denote by $\mathbf l(x)$ ($\mathbf r(x)$) the left (respectively, right) support of the operator $x$.
The following lemma is contained in the proof of [@cls Theorem 2.2].
\[l1\] Assume that $x\in L^0$, ${\epsilon}>0$, and ${\delta}>0$ are such that $$\tau(h^\perp)\leq {\epsilon}\text{ \ \ and \ \ } \| hxh\|_{\infty}\leq {\delta}$$ for some $h\in {\mathcal}P({\mathcal}M)$. Then there exists $e\in {\mathcal}P({\mathcal}M)$ for which $$\tau(e^\perp)\leq 2{\epsilon}\text{ \ \ and \ \ } \| xe\|_{\infty}\leq {\delta}.$$
Let $f=\mathbf 1-\mathbf r(h^\perp x)$. Then, since for any $y\in L^0$ the projections $\mathbf l(y)\in {\mathcal}P({\mathcal}M)$ and $\mathbf r(y)\in {\mathcal}P({\mathcal}M)$ are equivalent [@sz 9.29], we have $$\tau(f^\perp)=\tau(\mathbf r(h^\perp x))=\tau(\mathbf l(h^\perp x))\leq \tau(h^\perp)\leq {\epsilon}.$$ Also, $$xf=hxf+h^\perp xf=hxf+h^\perp x-h^\perp x \cdot \mathbf r(h^\perp x)=hxf.$$ Therefore, letting $e=h\land f$, we have $\tau(e^\perp)\leq 2{\epsilon}$ and $$xe=xfe=hxfe=hxhe,$$ implying $\| xe\|_{\infty}\leq \| hxh\|_{\infty}\leq {\delta}$.
For what follows it is convenient to employ the notion of bilateral uniform equicontinuity at zero of a sequence of maps from a normed space into $L^0$:
Let $(X, \| \cdot \|)$ be a normed space. A sequence of maps $M_n: X\to L^0$ is called [*bilaterally uniformly equicontinuous in measure (b.u.e.m.) at zero*]{} if for every ${\epsilon}>0$ and ${\delta}>0$ there exists ${\gamma}>0$ such that, given $x\in X$ with $\| x\|<{\gamma}$, there is a projection $e\in {\mathcal}P({\mathcal}M)$ satisfying conditions $$\tau(e^{\perp})\leq {\epsilon}\text{ \ \ and \ \ } \sup_n\| eM_n(x)e\|_{{\infty}}\leq {\delta}.$$
It is easy to see that, in the commutative case, the notion of bilateral uniform equicontinuity in measure at zero of a sequence $M_n: X\to L^0$ is precisely the continuity in measure at zero of the maximal operator $M^*(f)=\sup \limits_n |M_n(f)|, \ f\in X$.
The next property was noticed in [@li Corollary 2.1 and Proposition 4.2].
\[p2\] The sequence $\{A_n\}$ given by (\[e0\]) is b.u.e.m. at zero on $L^p$, $1\leq p<{\infty}$.
\[l2\] Given $1\leq p<{\infty}$, let a sequence $\{ y_m\}{\subset}L^p$ be such that $\| y_m\|_p \to 0$. Then for any ${\epsilon}>0$ and ${\delta}>0$ there exist a projection $e\in {\mathcal}P({\mathcal}M)$ and a sequence $\{ x_k \}$ with $x_k\in \{y_m\}$ for every $k$ such that $$\tau(e^\perp)\leq {\epsilon}\text{ \ \ and \ \ } \sup_n\| A_n(x_k)e\|_{\infty}\leq {\delta}\text{ \ \ for all \ } k.$$ In particular, given ${\epsilon}>0$, ${\delta}>0$, there exist $e\in {\mathcal}P({\mathcal}M)$ and $x\in \{y_m\}$ such that $$\tau(e^\perp)\leq {\epsilon}\text{ \ \ and \ \ } \sup_n\| A_n(x)e\|_{\infty}\leq {\delta}.$$
Let $ k$ be a positive integer. Due to Proposition \[p2\], there exist $x_k\in \{ y_m\}$ and $g_k\in {\mathcal}P({\mathcal}M)$ such that $$\tau(g_k^\perp)\leq \frac {\epsilon}{k\cdot 2^{k+1}} \text{ \ \ and \ \ } \sup_n\| g_kA_n(x_k)g_k\|_{\infty}\leq {\delta}.$$ In particular, $$\sup_{n\leq k}\| g_kA_n(x_k)g_k\|_{\infty}\leq {\delta}\text{ \ \ for every\ } k.$$
Given $n\leq k$, let $$h_{k,n}=g_k \text{\ \ whenever \ }n\leq k.$$ Then we have $$\| h_{k,n}A_n(x_k)h_{k,n}\|_{\infty}\leq {\delta}\text{ \ \ for all \ } k \text{\ and\ } n\leq k.$$ By Lemma \[l1\], for any $k$ and $n\leq k$ there is $e_{k,n}\in {\mathcal}P({\mathcal}M)$ such that $$\tau(e_{k,n}^\perp)\leq 2\cdot \tau(h_{k,n}^\perp)\leq \frac {\epsilon}{k\cdot 2^k}
\text{ \ \ \ and \ \ \ } \| A_n(x_k)e_{k,n}\|_{\infty}\leq {\delta}.$$ If $e_k=\bigwedge\limits_{n\leq k}e_{k,n}$, then $$\tau(e_k^\perp)\leq \frac {\epsilon}{2^k} \text{ \ \ and \ \ } \sup_{n\leq k} \| A_n(x_k)e_k\|_{\infty}\leq {\delta}\text{ \ \ for each \ } k.$$ Letting $e=\bigwedge\limits_ke_k$, we obtain the required inequalities: $$\tau(e^\perp)\leq {\epsilon}\text{ \ \ and \ \ } \sup_n\| A_n(x_k)e\|_{\infty}\leq {\delta}\text{ \ \ for all \ } k.$$
A proof of the following technical lemma can be found in [@cl0 Lemma 1.6].
\[l3\] Let $(X,+)$ be a semigroup, and let $M_n:X\to L^0$ be a sequence of additive maps. Assume that $x\in X$ is such that for every ${\epsilon}>0$ there exist a sequence $\{ x_k\}{\subset}X$ and a projection $e\in {\mathcal}P({\mathcal}M)$ satisfying conditions
\(i) $\{ M_n(x+x_k)\}$ converges a.u. as $n\to {\infty}$ for each $k$;
\(ii) $\tau(e^\perp)\leq {\epsilon}$;
\(iii) $\sup_n\| M_n(x_k)e\|_{\infty}\to 0$ as $k\to {\infty}$.
Then the sequence $\{M_n(x)\}$ converges a.u.
\[c1\] Let $1\leq p<{\infty}$. Then the set $${\mathcal}C= \{ x\in L^p: \ \{ A_n(x)\} \text{\ converges \ } a.u. \}$$ is closed in $L^p$.
Let a sequence $\{ z_m\}{\subset}{\mathcal}C$ and $x\in L^p$ be such that $\| z_m-x\|_p\to 0$. Denote $y_m=z_m-x$. Fix ${\epsilon}>0$. As $\| y_m\|_p\to 0$, it follows from Lemma \[l2\] that, given $k$, there exist $e_k\in {\mathcal}P({\mathcal}M)$ and $x_k=y_{m_k}$ such that $$\tau(e_k^\perp)\leq \frac {\epsilon}{2^k} \text{ \ \ and \ \ } \sup_n\|A_n(x_k)e_k\|_{\infty}\leq \frac 1k.$$ We have $x+x_k=z_{m_k}\in {\mathcal}C$, hence the sequence $\{ A_n(x+x_k)\}$ converges a.u. for each $k$. In addition, if $e=\bigwedge\limits_k e_k$, then $\tau(e^\perp)\leq {\epsilon}$ and $\sup_n\| A_n(x_k)e\|_{\infty}\leq \frac 1k\to 0$. This, by Lemma \[l3\], implies that $x\in {\mathcal}C$, so ${\mathcal}C$ is closed in $L^p$.
Now we can finish proof of Theorem \[t2\]:
It is well known (see, for example, [@cl Proof of Theorem 1.5]) that the averages $A_n(x)$ converge a.u. whenever $x\in L^2$. Therefore, since the set $L^p\cap L^2$ is dense in $L^p$, Corollary \[c1\] entails that the averages $A_n(x)$ converge a.u. for each $x\in L^p$ (to some $\widehat x\in L^0$).
Clearly $A_n(x)\to \widehat x$ in measure. As each $A_n$ is a contraction in $L^p$, we conclude, by [@cls Theorem 1.2], that $\widehat x\in L^p$.
[**Acknowledgement**]{}: The author is deeply grateful to Professor Vladimir Chilin for fruitful discussions and careful examination of the manuscript.
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|
---
abstract: 'In [@actorobserver] we introduced a dataset linking the first and third-person video understanding domains, the Charades-Ego Dataset. In this paper we describe the egocentric aspect of the dataset and present annotations for Charades-Ego with 68[,]{}536 activity instances in 68.8 hours of first and third-person video, making it one of the largest and most diverse egocentric datasets available. Charades-Ego furthermore shares activity classes, scripts, and methodology with the Charades dataset [@charades], that consist of additional 82.3 hours of third-person video with 66[,]{}500 activity instances. Charades-Ego has temporal annotations and textual descriptions, making it suitable for egocentric video classification, localization, captioning, and new tasks utilizing the cross-modal nature of the data.'
author:
- |
Gunnar A. Sigurdsson $^1\footnotemark[1]$ Abhinav Gupta $^{1}$ Cordelia Schmid $^{2}$ Ali Farhadi $^{3}$ Karteek Alahari $^{2}$\
$^1$Carnegie Mellon University $^2$Inria$\footnotemark[2]$ $^3$Allen Institute for Artificial Intelligence\
[[allenai.org/plato/charades](allenai.org/plato/charades)]{}
bibliography:
- 'egocentric.bib'
title: 'Charades-Ego: A Large-Scale Dataset of Paired Third and First Person Videos'
---
Introduction
============
In recent years, the field of egocentric action understanding [@lee2012discovering; @Li_2015_CVPR; @pirsiavash2012detecting; @ryoo2013first; @jayaraman2015learning; @Rhinehart_2017_ICCV] has exploded due to a variety of applications, including augmented and virtual reality. If we can create a link between third and first person video understanding, then we can use billions of easily available third-person videos to improve egocentric video understanding.
In the *Charades-Ego dataset* we followed the “Hollywood in Homes” approach [@charades], originally used to collect the Charades dataset [@charades; @challenge]. That is, we recruited crowd workers over the internet to record themselves in both first and third-person performing a given script of activities.
Whereas [@actorobserver] focused on the multi-domain aspects of the dataset. In this paper we describe the egocentric side of the dataset, and present an egocentric dataset with 68[,]{}536 activity instances in 68.8 hours of first and third-person video. Charades-Ego furthermore shares activity classes, scripts, and methodology with the Charades dataset [@charades], that consist of 82.3 hours of third-person video with 66[,]{}500 activity instances. Example egocentric frames are presented in Figure \[fig:examples\], and video visualizations are available on the website.
![Example frames from the Charades-Ego dataset.[]{data-label="fig:examples"}](examples.png){width="1.0\linewidth"}
The Charades-Ego Dataset
========================
In this section we discuss the egocentric aspects of the Charades-Ego dataset, for more information about the dataset, particularly the multi-modal nature of the dataset, please refer to [@actorobserver]. We asked users on Amazon Mechanical Turk to record two videos: (1) one with them acting out the script from the third-person; and (2) another one with them acting out the same script in the same way, with a camera fixed to their forehead. This offers a compromise between a synchronized lab setting to record both views simultaneously, and scalability. In fact, our dataset is one of the largest first-person datasets available [@fathi2011learning; @lee2012discovering; @pirsiavash2012detecting; @firstthird2017cvpr], has significantly more diversity (112 actors in many rooms), and most importantly, is the only large-scale dataset to offer pairs of first and third-person views that we can learn from.
Most of the scripts in Charades-Ego come from the Charades dataset. In addition, to ensure sufficient diversity in Charades-Ego, we collected an additional 1000 scripts using the method outlined in the Charades dataset [@charades]. Therefore, $78.7\%$ of the scripts in Charades-Ego are shared with the training set of the Charades dataset. The users are “given the choice to hold the camera to their foreheads, and do the activities with one hand, or create their own head mount and use two hands. We encouraged the latter option by incentivizing the users with an additional bonus for doing so. We compensated AMT workers \$1.5 for each video pair, and \$0.5 in additional bonus. This strategy worked well, with over 60% of the submitted videos containing activities featuring both hands, courtesy of a home-made head mount holding the camera.” [@actorobserver].
The activities are the same 157 activities as in the Charades dataset. In total there are 34.4 hours of third person video and another 34.4 hours of corresponding first person videos consisting of 68[,]{}536 activity instances, or $8.72$ activities per video on average. For comparison, the largest concurrent egocentric dataset EGTEA Gaze+, has 28 hours (10[,]{}325 instances) [@Li_2015_CVPR]. The videos come from various rooms in 112 homes across the world.
We split the videos into 80/20 training set (6167 videos) and test set (1693 videos), ensuring that no subject occurs in both sets. The smallest number of examples per category in training and test sets is 52 and 24 examples, respectively.
Video annotation
================
To get a pool of activities that are likely to be present in the videos, we both copy over the activities from videos in the original Charades dataset that share the same script, as well as run the third person videos through video-level annotation following the methodology for the Charades dataset [@sigurdsson2016much].
The videos were temporally annotated by presenting both third and first person videos of from the same subject at the same time, and having the workers annotate both. The third-person video is used to obtain better annotations for the first-person video. For example, the consensus among human annotators for egocentric videos is only 4.2% lower than for the fully-observable third-person videos (77.0% versus 72.8%). Each annotation task consisted of a single pair of videos and 5 activities for annotation on average. The annotation interface is presented in Figure \[fig:interface\].
![The interface used to collect annotations for the joint third and first person dataset. The videos are annotated simultaneously. Moving the sliders seeks in the corresponding video to help with annotation.[]{data-label="fig:interface"}](interface.pdf){width="1.0\linewidth"}
Baselines on Charades-Ego v1.0
==============================
To encourage research on both first and third person videos we present baselines trained on both first and third person videos and tested on either first or third person videos. These experiments are presented in Table \[tbl:thirdfirst\]. All models start with the same ResNet-152 models pretrained on the original Charades third-person dataset[^1]. *Third-Person Training* refers to using that original model, and *First-Person Training* or *First/Third Training* refers to using the labelled data in Charades-Ego for fine-tuning. We see that the different kinds of training do not improve third-person accuracy when we already have a good third-person model. However, we clearly obtain a significant improvement when using the first-person labelled data and testing on the egocentric test set.
To establish an egocentric benchmark on the Charades-Ego dataset we present results of multiple baselines on v1.0 of the Charades-Ego Egocentric test set. This is presented in Table \[tbl:egocentric\]. Following [@actorobserver], we also present results for baselines that have not been trained with first-person labelled data, that is, zero-shot egocentric recognition. We include the baseline ResNet-152 Transfer, which uses the Charades model to predict the activities in the third person video, and then uses those labels as supervision for the first-person video.
Discussion
==========
We hope this type of data is a step in bringing the fields of third-person and first-person activity recognition together.
#### Acknowledgements
The authors would like to thank Yin Li, Nick Rhinehart, and Kris Kitani for suggestions and advice on the dataset.
[^1]: From [[github.com/gsig/charades-algorithms](github.com/gsig/charades-algorithms)]{}.
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---
abstract: 'One of the main challenges in the vision-based grasping is the selection of feasible grasp regions while interacting with novel objects. Recent approaches exploit the power of the convolutional neural network (CNN) to achieve accurate grasping at the cost of high computational power and time. In this paper, we present a novel unsupervised learning based algorithm for the selection of feasible grasp regions. Unsupervised learning infers the pattern in data-set without any external labels. We apply k-means clustering on the image plane to identify the grasp regions, followed by an axis assignment method. We define a novel concept of Grasp Decide Index (GDI) to select the best grasp pose in image plane. We have conducted several experiments in clutter or isolated environment on standard objects of Amazon Robotics Challenge 2017 and Amazon Picking Challenge 2016. We compare the results with prior learning based approaches to validate the robustness and adaptive nature of our algorithm for a variety of novel objects in different domains.'
author:
- 'Siddhartha Vibhu Pharswan$^{1,2}$, Mohit Vohra$^{1}$, Ashish Kumar$^{1}$ and Laxmidhar Behera$^{1}$ [^1][^2]'
title: '**Domain Independent Unsupervised Learning to grasp the Novel Objects** '
---
INTRODUCTION
============
A robot with grasping capability has tremendous applications in warehouse industries, construction industries, or in medical sector. Various solutions have been proposed to enhance the grasping capability of manipulators. Authors [@pas15], proposed geometric methods to predict the grasp points for unknown objects present in a clutter. In [@lev17] & [@pin16], authors collected large scale data set to train a giant convolutional neural network (CNN) to predict the grasp regions. Although CNN-based solutions have shown some good results, but their performance depends on the data set. Thus making these solutions data biased and domain (background) dependent. The effect of domain specific nature can easily be examined by taking, an example of a network trained on one particular background [@lev17], but can take equal or less training time when transferred to the different background. Selecting proper hyper-parameters can reduce the training time but still the difference is not that significant. Consequently, robot grasping is dealing with generalization at the cost of high computation power and time.
![Our robotic hardware grasping object in dense clutter. The robot is grasping the object after the final selection is done by GDI in one of the experiments.[]{data-label="fig1"}](fig1){width="8.15cm" height="12cm"}
When dealing with novel objects unsupervised nature like human beings is needed in algorithm to identify the grasp regions. This behaviour involves the single shot screening of work-space with all the grasp regions along with final grasp. While the current approaches rely on creating or accumulating data, our method is not data specific. We structured the algorithm using an unsupervised learning baseline to find out the grasp regions in a clutter. Collision phenomenon [@ulr17] is used to avoid the gripper collision with surrounding objects. We represent the two-finger gripper as a rectangle [@jia11] in the image plane as shown in Fig. 2. In this representation, the opening of the gripper is represented by the length of the red line. Green line and centroid point represent the two fingers and palm of the gripper. The advantage of representing gripper in the image plane allows us to limit the search space also, it helps in identifying whether a grasp pose is feasible or not. Using the point cloud data corresponding to the sampled points inside gripper rectangles, collision with other objects is avoided or completely eliminated. For finding the feasible grasp regions, we divide our algorithm into 5 stages where each stage is hierarchically related to its previous state. In the first stage, poses are sampled uniformly at random orientations in the image plane. After sampling, poses which belong to no object regions are removed. These stages are followed by cluster and axes assignment and finally, the most feasible grasps are selected by GDI.
One of the significant contributions of our work is, neither our work requires any pre-processing step, nor does it need any prior information about the objects. We are thus providing a generalized solution. Besides this, our framework is light enough to run on a single computer making it an economical solution.
RELATED WORK
============
To solve the robot grasping problem, numerous solutions have been proposed for the last decades. Most of them are focused on known object grasping, while only a few are more concerned about the novel object grasping [@pin16], [@lev17]. In [@sax08], authors uses machine learning approaches to predict the grasp points on two images of the object, and the points are triangulated to predict the 3D location of the grasp point. The results in [@sax08] are quite specific and not even shown in real practical clutter. In [@katz14], authors estimate the surface of unknown objects in a clutter using depth discontinuities in depth image and normal vector at each point, and train a network to predict the grasp regions for unknown objects in a pile.[@bou15] uses Voxel Cloud Connectivity Segmentation method to detect surface of the object and proposes a reinforcement learning based system which can learn to manipulate the objects by trial and error. [@chit12] have done modelling of robot grasping environment to perform the manipulation task. While this technique completely relies on the combination of multiple sensors with in the grasping system. Key advantage of our method is, neither we require any segmentation techniques nor we need any normal vector for surface estimation, thus making our approach less complex.
Because of high learning capability and generalize nature of CNN, various CNN-based methods [@red15], [@sul17] have been proposed which predict the grasp region directly using RGB images. Key requirement of CNN-based methods is the large scale data-set. In [@lev17], to show the generalization of the grasp framework, authors use 800000 grasp attempts and then transfer the same strategy to other robots. [@pin16] shows the generalization of the framework by 700 hours of robot training. Authors [@her14] have proposed a template based learning approach, which depends upon the creation of template data for set of objects and make it generalise on other similar or dissimilar objects. While this approach also affects the same problems that CNN faces like lightening conditions, data creation strategies etc.
In [@mah16], authors use a large data-set of 10000 3D models, with 2.5 million gripper poses for training. The grasp metrics and data-set for grasping objects is proposed by [@boh15]. In [@len15], [@red15], authors uses CNN-based architectures to predict the rectangular grasp regions in the image plane. In [@keh13], authors present a system architecture for a cloud-based robot grasping which uses massive parallel computation power and real-time sharing of vast data resource. They used google object recognition system followed by creating 3D models in offline mode for analyzing the object poses and grasp regions. During testing, image of object is sent to object recognition server for pose estimation and grasp region selection. A more related review of previous work in the field of grasping is given in [@boh16].
While all these works are creating a bunch of data to solve the problem, our approach focuses on finding the top feasible grasp poses.
![Gripper representation as rectangle and lines of varying lengths on the object to be grasped. The length of red line (gripper opening) is maximum opening of gripper mapped to image plane. Green points on every line gripper are the gripper fingers in image plane.[]{data-label="fig2"}](fig2){width="6.75cm" height="6cm"}
We have shown in the experiments that just by single Kinect sensor how grasping can be solved robustly at less cost. The way we have posed the problem, I think is the first time unsupervised learning in the whole pipeline is used without any labelled data-set and training.
PROBLEM STATEMENT
=================
Given an image ***I*** and corresponding point cloud ***P*** $\mathbf{\subset \ R^3}$ of the robot task-space from any RGB-D sensor (in our case Kinect V1 of resolution 640 $\times$ 480), how to decide the feasible grasp regions without any prior knowledge of objects. The main step in our problem is to find the object and no-object regions in the image plane. We used a sampling based strategy to find the object and no-object regions which exempt us to use any standard segmentation techniques.
Gripper Representation in Image Plane
-------------------------------------
Depending on the stage of our algorithm, we represent the gripper either by a rectangle or by a line. For sampling stage, gripper is represented by a line having parameters ($x_c,\ y_c,\ l_v,\ \theta$) and for GDI calculation, it is treated as a rectangle with three parameters ($x_c,\ y_c,\ \theta$). Here $x_c$, $y_c$ are the palm co-ordinates of gripper in image plane, $l_v$ is the length of line segment which represents the opening of gripper in image plane, $\theta$ is angle of rectangle or line with horizontal axis. Fig. \[fig2\] shows gripper representation. To decide the optimum opening of gripper in image plane for known objects, an object of maximum width is selected for mapping along with additional clearance. On the other side, to pick up the novel objects gripper opening is decided on the basis of task category. For small household objects, opening of gripper is kept half of its’ maximum opening. Width of gripper also depends on the geometric constraints of gripper design as well. We used the gripper with maximum opening of . To avoid collision of gripper fingers with near by objects, clearance of 2cm (mapped to image plane) is provided at each side of gripper finger.
Sampling and Filtering
----------------------
To grasp an object, either robot has prior object information \[16\], \[17\] or it has learnt by trials \[6\], \[7\]. In real time situations, we do not have any prior information of the objects. To identify the object and no-object regions, previous methods [@pin16], [@lev17] either trained models rigorously or prefer object modelling [@qui09]. In our method, we used the depth information of the work-space from the 3D sensor (Kinect in our case). Given a work-space, we perform uniform sampling of the line configurations of the gripper in the image plane within the task-space of robot. These configurations are at random angles with fixed $l_v$ (depends on maximum width of object) for a particular clutter. This sampling stage is the $1^{st}$ stage of the algorithm. Large number of line configurations are sampled to cover the entire task-space in image plane.
As all the sampled configurations will not be over object regions, to remove the false samples we perform filtering operation in two levels. In first level, samples belong to no-object regions are removed. Here, we filtered the false poses using the Z-values of center point of line configurations and task-space surface. As the center point of every line pose represents the palm of the gripper, so to grasp an object firmly, the z-value of center point must be greater than the z-value of background (domain where objects are placed). So we will select those poses whose Z-value of centre points are greater than the background z-value. In the second level, we remove the samples which have high probability of object gripper collision. In the gripper representation, corner points of line represent the finger of the gripper. If difference between the z-values of corner points is large, it means one of the finger is in collision with the objects, hence we will reject these poses as well. Above conditions can remove all the false poses when objects are placed separately, but in a clutter environment still there will be some false poses left which will be removed in further stages. Output of filtering stage helps in segmenting the object and no-object regions using point cloud information only, which makes the algorithm to work in any lightening conditions as well. This nature of domain independence is inherited from the nature of 3D sensor used.
K-Means Clustering
------------------
Once the configurations in object regions are obtained, next step is to group the particular set of configurations and localise it in image plane. For this task, we apply k-means clustering on the remaining configurations. The cluster centroids are taken from the center points of gripper configurations. Center points are taken as input because it forcibly allows the k-centers to lie on the objects surface. If corner points were taken as input to the k-means clustering algorithm, then the sampled poses present at the corner sides of objects would be grouped together by k-means clustering. Thus the chance of local minima is more. Local minima can be seen in the following common situations *i)* sampled poses on two objects are identified by a same k-center *ii)* more than two k-centers on the same pose family. Later situation is more effective because in robot grasping, that case helps to grasp the object in multiple ways. So selecting centre points as input can minimize the chance of local minima but it can not be avoided in case of dense clutter. In experiment section we demonstrate that performance of our algorithm is not effected by local minima due to the ranked selection of grasps.
![Axis assignment for one of the point families cluster. Contour line represents the region of poses belongs to the index of green cluster centroid.[]{data-label="fig3"}](pose_cal){width="8cm" height="6cm"}
Axis Assignment
---------------
K-means clustering divides the object region into multiple segments, and every segment has a set of line posses. To assign axes to a single segment, the corner points of line poses which are present inside that segment, will be used. For each segment, the corner points of all the line poses are treated as points family.
So if $k^{th}$ segment has S poses, then segment will have 2S (T) corner points and $k^{th}$ cluster will have a point family of 2S points. For each point family, axis assignment (major axis angle $\phi$) is done using the formula\[5\] given below. Let $\mu_x,\ \mu_y$ represents the centroid of cluster having T corner points. Let us assume that $i^{th}$ corner point can be represented as ${P^i}_{kx},\ {P^i}_{ky}$ and $\phi$ is the angle of major axis with horizontal axis. Relation between centroid, corner points and major axis angle is given by $$tan(2\phi) = 2* (\frac{\sum_{i=1}^{T} ({P^i}_{kx} - \mu_x)*({P^i}_{ky} - \mu_y)}{\sum_{i=1}^{T} [({P^i}_{kx} - \mu_x)^2 + ({P^i}_{ky} - \mu_y)^2]}) {\stepcounter{equation}\ (\theequation)}$$ When objects are placed separately, major axis will lie along the major axis of the objects. Axis assignment is described on Fig. \[fig3\] using one point family and its associated cluster centroid.
![Sampled pixels inside the rectangle representing finger of gripper in image plane over point regions. A,B,C etc are the point family clusters for corresponding cluster centroid (black dot) in image plane.[]{data-label="fig4"}](gdi){width="8.5cm" height="4cm"}
![Top and side views of extreme GDI cases in practical conditions. Top view shows the sampled points inside rectangle in image plane and side view represents z-values corresponding to them. Intermediate cases gets cancelled from the above formula.[]{data-label="fig5"}](mash){width="8.75cm" height="4cm"}
Grasp Decide Index
------------------
In axes assignment stage, we assign a major axis to each segment. But output of above stage leads to k-axes groups assigned to every point family. It is because the direction of major axis is decided by the point family of a segment, and in a clutter environment it could be possible that more than one object could be the part of that segment. To filter out the false assignments and select the final graspable assignment, we propose an index. This index takes into account the collision of gripper fingers with surrounding objects. For GDI calculation, we represent every cluster by a rectangle with centre at centroid of cluster, and orientation of rectangle is $\theta$ as $\phi + 90$. To represent the fingers of gripper in image plane, we have sampled points near rectangle periphery but inside it as shown in Fig. 4. Let the z-values of $i^{th}$ sampled pixel and rectangle center (palm center) is $Z_i$ and $Z_c$ respectively. The number of positive deviations ($\Delta Z = Z_i - Z_c$) are different for every rectangle in image plane. The more the number of positive deviations, a rectangle has less the chance of collision. The Grasp Decide Index (GDI) is formulated as follows: $$GDI = \max_{N} (Z-Z_c),\;where\;Z \in R\,^N {\stepcounter{equation}\ (\theequation)}$$ ***N*** is the number of sampled pixels in each rectangle, ***max*** denotes the maximum positive deviation for a rectangle over N-pixels. Final rectangle out of those is selected on the basis of one with highest GDI. Avoiding the random grasp selection strategy, index ensures the safety of grasp which can be seen in Fig. 5.
EXPERIMENTS AND RESULTS
=======================
Objects Used for Grasping
-------------------------
We have used total 50 objects out of which 35 were standard objects from Amazon Robotics Challenge 2017 and Amazon Picking Challenge 2016. Remaining objects are household and office stationary objects. These objects are placed randomly in clutter or separately on different domains for all the experiments. We have used two different backgrounds i.e. red bin and wooden table. Fig. 6 shows a major part of object set used for the experiments.
{width="8cm" height="6cm"} \[fig6\]
Parameters Selection and Positive-Negative Effect
-------------------------------------------------
In second stage, to remove the false poses we perform two level of filtering using depth value. Since we placed the Kinect at a height of 1.3m above the workspace and because of noisy point cloud, we add a depth margin value of 0.025m for filtering. In our experiments we found that, this margin is sufficient to grasp the thin objects (around 2.5m thick) and also we can decrease this margin value by placing Kinect nearer to workspace.
Environment NoT OP MT $\alpha$($\%$) $\beta$($\%$)
--------------- ----- ---- ---- ---------------- ---------------
Clutter(Red) 11 10 12 90.90 91.67
Clutter(Wood) 20 18 22 90 90.9
Clutter(Wood) 15 13 17 86.67 88.24
Clutter(Wood) 15 14 16 93.33 93.75
Clutter(Red) 10 8 13 80 76.92
: Results of cluttered environment[]{data-label="tablec1"}
Environment NoT OP MT $\alpha$(%) $\beta$(%)
----------------- ----- ---- ---- ------------- ------------
Seperated(Red) 15 14 16 93.33 93.75
Seperated(Wood) 15 15 15 100 100
Seperated(Red) 15 14 16 93.33 93.75
: Results of no clutter[]{data-label="tablec2"}
{width="17.5cm" height="13cm"} \[fig7\]
In our experiments, we found that k-means clustering methods stuck in local minima. This situation occurs when we try to assign a single centroid to more than one family points, or when we assign multiple centroids to a single point family. But the performance of our system is not affected because in the first case, the grasp pose will be removed by the GDI stage, while in second case multiple centroids to a single family provide various ways to grasp the object. In all our experiments value of K is taken as 6 to 12 without any consideration of how many objects are there in bin because as the grasping proceeds clutter will decrease and objects can be grasped easily (latter case of local minima). While for the case of isolation, K is equal to number of objects present in the task-space. All stages start from the sampling to final pose selection are shown in Fig. 7 for two different domains. We have tabulated the results of our experiments in the Table I and Table II for clutter and no clutter respectively where number of trials (NoT) robot take is equal to number of objects present.
{width="17.5cm" height="11cm"} \[fig8\]
Total 8 experiments are conducted. For five experiments, objects were placed in clutter inside red bin and on wooden table. Remaining three experiments are performed for uncluttered domains(red bin or wooden table). For each uncluttered experiment, objects were placed separately in random pose, even sometimes in overlapping situations, while for cluttered experiments objects are placed randomly in the bin or table. For the visualization, we select the top 5 poses with rank 1 to rank 5 based on their decreasing GDI values respectively. Final selected pose is the rank 1 pose having highest GDI among the selected poses and has minimum chance of collision. To measure the performance of our algorithm, we have used three main factors i) maximum robot trials to clear the task-space (MT), ii) object picked (OP), iii) number of trials (NoT) which are equal to the number of objects present in the task-space. Two parameters $\alpha = OP/NoT$ and $\beta=NoT/MT$, measure the grasp success rate on the basis of robot trials. $\alpha$ strictly focuses on slipping failure and algorithm bad prediction cases. Percentages of objects picked over number of trials (NoT) and objects picked over maximum trials (MT), when average over all clutter experiments show the accuracy of 88.18% and 88.292% respectively. Table II shows the result of no-clutter cases where the $\alpha$ and $\beta$, averaged over all experiments are 95.55% and 95.83% respectively. Our accuracy measure of experiments is considered taking NoT as the reference because domain will obviously be cleared for some extra trials which is the case with $\beta$. We have not tried rigorous trials to empty the work-space like [@pin16].
Comparisons with past grasping techniques
-----------------------------------------
We have compared our results with random action strategy, heuristics strategies like i) grasping near the centroid, ii) grasping along minimum eigen axes (which is reflected just to fit the object in gripper) and iii) grasping the top in clutter \[13\]. In learning based strategies, we have compared the results of our algorithm with that of [@pin16]. In experiments [@pin16], the algorithm has high accuracy on train set but less at test set, so to make valid comparison we have taken average of accuracy of train set and test set.
Random Strategy[@katz14] 15%
------------------------------- --------------------------------------
Heuristic Strategies[@katz14] 92% (No clutter), 40% (Clutter)
Learning Strategy[@pin16] (40K trials) 86.3%
Proposed Strategy 95.5% (No Clutter), 88.18% (Clutter)
: Comparative study of grasping[]{data-label="table5"}
Their results are published on 150 variety of objects and we had also tested our algorithm on standard and similar type of objects like brush, headphones, hand- drill etc. We have compared the $\alpha$ value of our experiments with these strategies as it reflects the real-effect of algorithm prediction nature. We have not focused on maximum number of trials percentage as it is more specific to empty the bin instead of algorithm success.
{width="8.5cm" height="4.8cm"} \[fig9\]
Causes of success and failure
-----------------------------
It has been found that our method is biased on sampling. More dense sampling will lead to better results. During our initial experiments we sampled very less poses with $l_v$ taken as 80 and got very poor results in clutter. Improvement in results is achieved by reducing the value of $l_v$ to 30 (in pixel units) with dense sampling. If objects are placed separately these values does not matter too much. Another noticeable fact is, GDI includes the safety measure, it does not consider the success of grasp. GDI, mostly selects the collision free grasps without considering its’ success for a particular shape of the object. In some cases, if k-center gets assigned to two clusters or vice-versa, then the axes assignment step will be the cause of failure. As, the axes assignment is done blindly, it will only consider the effect of high deviation within the point families, which may result in a some axes assignment in no object regions or along the minor axes of object itself as shown in Fig. 8.
CONCLUSIONS
===========
We have proposed a novel real time grasp pose estimation technique using unsupervised learning baseline. To demonstrate the performance of our grasp framework, we have conducted several experiments presented in this paper. Starting from the sampled poses, our framework finds the best poses in clutter with high percentage of success. Our method does not rely on any standard segmentation techniques which allows it to deal with any background. If we consider only the removal of clutter in our experiments then our algorithm will outperform in comparison to previous methods as shown in all the experiments.
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[^1]: $^{1}$Intelligent Systems and Control Lab (ISL) Indian Institute of Technology Kanpur, Uttar Pradesh 208016, India. [vibhu@iitk.ac.in, mvohra@iitk.ac.in, krashish@iitk.ac.in, lbehera@iitk.ac.in ]{}
[^2]: $^{2}$Department of Mechanical Engineering Indian Institute of Technology Kanpur, Uttar Pradesh 208016, India.
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---
abstract: 'We study the Hartree ground state of a dipolar condensate of atoms or molecules in an three-dimensional anisotropic geometry and at $T=0$. We determine the stability of the condensate as a function of the aspect ratios of the trap frequencies and of the dipolar strength. We find numerically a rich phase space structure characterized by various structures of the ground-state density profile.'
author:
- 'O. Dutta[^1] and P. Meystre'
title: 'Ground-state structure and stability of dipolar condensates in anisotropic traps '
---
Introduction
============
The realization of Bose-Einstein condensation (BEC) in low-density atomic vapors [@bec1; @bec2] has led to an explosion of experimental and theoretical research on the physics of quantum-degenerate atomic and molecular systems. While much of the work so far has concentrated on systems characterized by $s$-wave two-body interactions, the recent demonstration of a condensate of chromium atoms [@chbec] opens up the study of gases that interact via long-range, anisotropic magnetic dipole interactions. In a parallel development, it can be expected that quantum degenerate samples of heteronuclear polar molecules will soon be available through the use of Feshbach resonances [@Kett; @Jin1], photoassociation [@DeM; @HetDeM1], or a combination of the two. When in their vibrational ground state, these molecules interact primarily via the electric dipole potential, and they are expected to provide a fascinating new type dipole-dominated condensates in the near future.
As a result of the anisotropy and long-range nature of the dipole-dipole interaction, a number of novel phenomena have been predicted to occur in low-density quantum-degenerate dipolar atomic and molecular systems, both in conventional traps and in optical lattices. An early study of the ground state of polar condensates was presented in Ref. [@stabdp], which determined its stability diagram as a function of the number of atoms and $s$-wave scattering length. It identified a stable structured ground state for a specific range of parameters. At about the same time, the effect of trap geometry on the stability of the condensate was considered in Ref. [@stabdp1] for a system dominated by the dipole interaction. This was followed by the prediction [@oplat] of the existence of a number of quantum phases for dipolar bosons in optical lattices. Recent work [@dvor1; @dvor2] considers the structural phases of vortex lattices in rotating dipolar Bose gases.
A novel feature of dipolar condensates, as compared to their scalar cousins, is the appearance of a roton minimum in their Bogoliubov spectrum. This feature was discussed in the context of atomic condensates in Ref. [@rotmax], which considered the impact of the roton-maxon feature in the excitation spectrum and the stability of pancake-shaped dipolar condensates. For this particular geometry it was found that the excitation spectrum can touch the zero-energy axis for a non-zero wave vector [@2dd], which points to the instability of homogeneous condensates and the onset of density modulations [@sup]. A roton minimum was also found [@odell1] for the case of laser-induced dipolar interactions in self-bound BECs with cylindrical symmetry. Quasi-2D dipolar bosons with a density-modulated order parameter were determined to be unstable within the mean-field theory [@coop], and cigar-shaped quasi-one dimensional condensates were likewise found [@odell] to be dynamically unstable for dipoles polarized along the axis of the cylindrical trap. The stability of dipolar condensates in pancake traps was also recently discussed in Ref. [@stabdp2], which found the appearance of biconcave condensates for certain values of the trap aspect ratio and strength of the dipole interaction. From the Bogoliubov excitation spectrum it was possible to attribute the instability of the condensate under a broad range of conditions to its azimuthal component.
Further building on these studies, the present note reports the results of a detailed numerical analysis of the stability and structure of the Hartree ground state of dipolar condensates confined in anisotropic harmonic trap. We proceed by introducing the trap frequencies $\omega_x, \omega_y$ and $\omega_z$, respectively, in the $x$, $y$ and $z$ directions, and the corresponding trap aspect ratios $\lambda_y = \omega_y/\omega_x$ and $\lambda_z=\omega_z/\omega_x$. Thus $\lambda_z =1$ corresponds to a pancake trap, whereas $\lambda_z=0$ corresponds to a cylindrical trap with free motion in $z$ direction. We further assume for concreteness that an external field polarizes the dipoles along the $y$ axis. The stability of the condensate is then determined as a function of the trap aspect ratios and of an effective dipolar interaction strength that is proportional to the number of atoms or molecules in the condensate. Various ground state structures of the condensate are identified in the stable region of parameter space.
The remainder of this paper is organized as follows: Section II introduces our model and comments on important aspects of our numerical approach. Section III summarizes our results, identifying up to five different types of possible ground states, depending on the tightness of the trap and the particle number. Finally, Section IV is a summary and conclusion.
Formal development
==================
The dipole-dipole interaction between two particles separated by a distance $r$ is $$V_{\rm dd}(r) = g_{\rm dd} \frac{1 - 3 y^2/r^2}{r^3},$$ where $g_{\rm dd}$ is the dipole-dipole interaction strength and $\hat{y}$ is the polarization direction. For atoms with a permanent magnetic dipole moment we have $g_{\rm dd} = \mu_0
\mu^2_m/4 \pi$ while for dipolar molecules $g_{\rm dd} = \mu^2_e/4
\pi \epsilon_0$, $\mu_m$ and $\mu_e$ being the magnetic moment of the atoms and the electric dipole moment of the molecules, respectively.
Within the mean-field approximation, the condensate order parameter $\phi(r)$ satisfies the Gross-Pitaevskii (GP) equation $$\begin{aligned}
\label{gpeq}
E \phi(r) & = & \left [ H_0 + g \left | \phi^2(r)
\right | \right . \nonumber\\
& + & \left . N \int V_{\rm dd}(r - r')
\left | \phi(r') \right |^2 d^3r' \right ] \phi(r) ,
\end{aligned}$$ where $$H_0 = -\frac{\hbar^2}{2 m} \nabla^2 + \frac{1}{2} m \omega_x^2 \left
( x^2 + \lambda^2_y y^2 + \lambda^2_z z^2 \right )$$ is the sum of the kinetic energy and the trapping potential and $N$ denotes the number of particles in the condensate. The second term on the right-hand side of Eq. (\[gpeq\]) is the contact interaction, $g=4\pi\hbar^2 a/m$ being proportional to the $s$-wave scattering strength $a$, and the third term describes the effects of the nonlocal dipole-dipole interaction. For dipole interaction dominated systems, $g$ is small compared to $V_{\rm
dd}$. This is the case that we consider here, and in the following we neglect the $s$-wave scattering term altogether.
For convenience we introduce the dimensionless parameter $$D=N g_{\rm dd} m/(\ell_x \hbar^2)$$ that measures the effective strength of the dipole-dipole interaction, where the oscillator length $\ell_x=\sqrt{\hbar/(m
\omega_x)}$. The condensate ground state is then determined numerically by solving the Gross-Pitaevskii equation (\[gpeq\]) for imaginary times. The term involving the dipole interaction energy is calculated using the convolution theorem, $$\begin{aligned}
&&\int V_{\rm dd}(r - r') | \phi(r') |^2 d^3r'=
\nonumber \\
&&{\cal F}^{-1}
\left \{{\cal F} \left [ V_{\rm dd}(r) \right ] * {\cal F} \left
[ | \phi(r) |^2 \right ] \right \}, \nonumber\end{aligned}$$ where ${\cal F}$ and ${\cal F}^{-1}$ stand for Fourier transform and inverse Fourier transform, respectively. The dipole-dipole interaction is calculated analytically in momentum space as [@stabdp], $${\cal F}\left [ V_{\rm dd}(r) \right ]= \frac{4\pi}{3} \left [ 3
\frac{k^2_y}{k^2_x+k^2_y+k^2_z} - 1 \right ],$$ $k_x, k_y, k_z$ are the momentum components in $x, y, z$ direction.
The initial order parameter was chosen randomly, and the stability diagram was generated for each pair of parameters $(\lambda_y,
\lambda_z)$ by increasing the effective dipolar strength $D$ until a critical value $D_{\rm cr}$ above which the condensate collapses. Because of the random initial condition this value varies slightly from run to run. The plotted results show the average over 100 realizations of the initial wave function, the error bars indicating the maximum deviation from of $D_{\rm cr}$ from its mean $\bar{D}_{\rm cr}$. This approach typically resulted in numerical uncertainties similar to those of Ref. [@dip].
results
=======
A good starting point for the discussion of our results is the observation that in the case of a cylindrical trap, $\lambda_z=0$, we found no stable structured condensate ground state. (By structured profiles, we mean profiles that are not simple gaussians.) In particular, solutions exhibiting density modulations along the $z$-axis were found to be unstable. Moving then to the case of a pancake trap by keeping $\lambda_y$ fixed but increasing $\lambda_z$ from $0$ to $1$, we found for $\lambda_y \gtrsim 4$ the appearance of a small parameter region where the stable ground state is characterized by a structured density profile, the domain of stability of this structured solution increasing with $\lambda_y$. A $\{\lambda_z - D\}$ phase space stability diagram typical of this regime is shown in Fig. \[fig1\] for $\lambda_y=5$. In this figure, region $I$ is characterized by the existence of a usual condensate with its familiar, structureless gaussian-like density profile. As $\lambda_z$ is increased, the condensate becomes unstable for decreasing values of the effective dipole interaction strength $D$, or alternatively of the particle number $N$. For $.525 <
\lambda_z < .7$, though, the ground state changes from a gaussian-shaped to a double-peaked density profile (domain II in the figure), before the system becomes unstable.
Figures \[fig4\] and \[fig5\] show surface plots and corresponding 3-D renditions of density profiles typical of the various situations encountered in our study. Figures \[fig4\]a and \[fig5\]a are illustrative of the present case. The appearance of two density peaks away from the center of the trap results from the interplay between the repulsive nature of the dipoles in a plane transverse to its polarization direction, the $(x-z)$ plane, and the confining potential.
Increasing the tightness of the trap along the polarization direction $y$, i.e, increasing $\lambda_y$, results in the emergence of additional types of structured ground states. One such case is illustrated in Fig. \[fig2\], which is for $\lambda_z=5.5$. For small values of $\lambda_z$, i.e., a weak trapping potential along the $z$-direction, we observe the appearance of a domain (region III in the figure) characterized by a double-peaked ground state with the maximum density along the $z$ direction and a gaussian-like density in $x$ direction. This type of double-peaked structure along the weak trapping axis was first predicted in Ref. [@dpeak]. Typical density profiles in this region resemble those in Figs. 4a and 5a, but with a rotation by 90 degrees in the $(x,z)$-plane. The regions II and III are separated by a small additional domain IV characterized by a ground-state distribution with a quadruple-peaked structure as illustrated in Figs. 4b and 5b, as might be expected. In general, these characteristics of the ground state density profile persist until $\lambda_y \approx 6.5$.
Figure \[fig3\] shows a stability diagram typical of higher values of the aspect ratio $\lambda_y$, in this case $\lambda_y =
7$, for $0.4 < \lambda_z<1$. As $D$ is increased, the ground-state density of the condensate first undergoes a transition from a gaussian-like to a double-peaked profile of the type illustrated in Figs. 4b and 5b (region III). As $D$ is further increased, this domain is followed for $\lambda_z$ close enough to unity by a second transition to a domain (region V) with the appearance of a density minimum near trap center. Initially, this minimum is surrounded by a region with a radial density modulation, see Fig. 4c, but for larger values of $\lambda_z$ this modulation is reduced, see Figs. 4d and 5d. In that region, the density profile resembles the solution previously reported in Ref. [@stabdp2] for a similar parameter range.
.
In the case of atoms a typical magnetic moment of $6 \mu_B$, and we find that the range of critical dipole strengths $D$ corresponding to structured ground states can be achieved for $10^4-10^5$ atoms for trapping frequencies $\omega_x \approx
1kHz$. For molecules with a typical electric dipole moment of $1
Debye$ the corresponding number is $ 10^3-10^4$ molecules. While these are relatively high particle numbers, especially for the atomic case, they do not seem out of reach of experimental realization.
conclusion
==========
In conclusion, we have performed a detailed numerical study of the ground state structure and stability of ultracold dipolar bosons in an anisotropic trap for dipoles polarized along the $y$-direction. The trap aspect ratios along $y$ and $z$ direction, $\lambda_y$ and $\lambda_z$ were used as control parameters, and the mean-field stability diagram has established as function of these parameters and a dimensionless interaction strength $D$. For small $\lambda_y$ the system was found to exhibit a standard density profile, but for larger values, and depending on $\lambda_z$, various structured ground state were found to appear before reaching the unstable regime where the condensate collapses. These include a four-peak structured solution in the $x-z$ plane, a ring-like ground state with a modulated radial density profile. For $\lambda_y \sim 7$ and $\lambda_z=1$, we found a biconcave condensate profile, as already reported in [@stabdp2].
For strong confining potentials along the dipole polarization direction, i.e for large $\lambda_y$, increasing $\lambda_z$ can be viewed as resulting in a change from a quasi-one dimensional to a quasi-two-dimensional geometry. As such we can think of the various ground-state structures as a result of dimensional crossover in a trapping geometry. To gain a deeper understanding of these structures as we approach the instability region, future work study the Bogoliubov spectrum of the trapped system.
We thank Drs. D. O’Dell, D. Meiser and R. Kanamoto for numerous useful discussions and comments. This work is supported in part by the US Office of Naval Research, by the National Science Foundation, by the US Army Research Office, by the Joint Services Optics Program, and by the National Aeronautics and Space Administration.
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[^1]: Corresponding author. Email address: dutta@physics.arizona.edu
|
[ **Some properties of Kaehler submanifolds\
with recurrent tensor fields** ]{}
[**Irina I. Bodrenko**]{} [^1]
[**Abstract**]{}\
The properties of Kaehler submanifolds with recurrent the second fundamental form in spaces of constant holomorphic sectional curvature are being studied in this article.
Introduction {#introduction .unnumbered}
============
Let $M^{2m+2l}$ be a Kaehler manifold of complex dimension $m+l$ $(m\geq 1, l\geq 1)$ with almost complex structure $J$ and a Riemannian metric $\widetilde g$, $\widetilde\nabla$ be the Riemannian connection coordinated with $\widetilde g$, $\widetilde
R$ be the Riemannian curvature tensor of manifold $M^{2m+2l}$. Let $F^{2m}$ be a Kaehler submanifold of complex dimension $m$ in $M^{2m+2l}$ with induced Riemannian metric $g$. The restriction $J$ to $F^{2m}$ defines induced almost complex structure on $F^{2m},$ which we will denote by the same symbol $J$. Let $\nabla$ be the Riemannian connection coordinated with $g$, $D$ be the normal connection, $b$ be the second fundamental form, $R^\bot$ be the tensor of normal curvature of submanifold $F^{2m}$, $\overline\nabla=
\nabla\oplus D$ be the connection of van der Waerden — Bortolotti. $b$ is called [*parallel*]{} if $\overline\nabla
b\equiv 0$. A tensor of normal curvature $R^\bot$ is called [*parallel*]{} if $\overline\nabla R^\bot \equiv 0$.
According to the definition of recurrent tensor field (see \[1\] , note 8), nonzero form $b\ne 0$ is called [*recurrent*]{} if there exists 1-form $\mu$ on $F^{2m}$ such that $\overline\nabla b = \mu\otimes b$.
\[theorem 1\]. Let $F^{2m}$ be a Kaehler submanifold of complex dimension $m$ in a Kaehler manifold $M^{2m+2l}(c)$ of complex dimension $m+l$ and constant holomorphic sectional curvature $c$. If $F^{2m}$ has recurrent the second fundamental form $b$ then the tensor of normal curvature $R^\bot\ne 0$ is parallel.
It is known (see \[1\] , note 8, theorem 3), that for a Riemannian manifold $M$ with recurrent tensor of Riemannian curvature $\widetilde R$ and irreducible narrowed linear group of holonomy, it is necessary that the tensor of Riemannian curvature $\widetilde R$ be parallel (i.e. $\widetilde\nabla\widetilde R
\equiv 0$) with the condition $\dim M \geq 3$. A Riemannian manifold $M$ is called [*locally symmetric*]{} if $\widetilde\nabla \widetilde R\equiv 0$.
\[theorem 2\]. Let $F^{2m}$ be a Kaehler submanifold of complex dimension $m$ in a Kaehler manifold $M^{2m+2l}(c)$ of complex dimension $m+l$ and constant holomorphic sectional curvature $c$. If $F^{2m}$ has recurrent the second fundamental form $b$ then $F^{2m}$ is locally symmetric\
submanifold.
Main notations and formulas.
============================
Let $M^{n+p}$ be $(n+p)$-dimensional $(n\geq 2, p\geq 2)$ smooth Riemannian manifold, $\widetilde g$ be a Riemannian metric on $M^{n+p}$, $\widetilde\nabla$ be the Riemannian connection coordinated with $\widetilde g$, $F^n$ be $n$-dimensional smooth submanifold in $M^{n+p}$, $g$ be the induced Riemannian metric on $F^n$, $\nabla$ be the Riemannian connection on $F^n$ coordinated with $g$, $TF^n$ and $T^\bot F^n$ be tangent and normal bundles on $F^n,$ respectively, $R$ and $R_1$ be the tensors of Riemannian and Ricci curvature of connection $\nabla,$ respectively, $b$ be the second fundamental form $F^n$, $D$ be the normal connection, $R^\bot$ be the tensor of normal curvature, $\overline\nabla$ be the connection of Van der Waerden — Bortolotti.
The formulas of Gauss and Weingarten have, respectively, the following form \[2\] : $$\widetilde\nabla_X Y = \nabla_X Y + b(X, Y),
\eqno (1.1)$$ $$\widetilde\nabla_X\xi = -A_\xi X + D_X\xi,
\eqno (1.2)$$ for any vector fields $X, Y,$ tangent to $F^n$, and vector field $\xi$ normal to $F^n$.
The equations of Gauss, Peterson — Codacci and Ricci have, respectively, the following form \[2\] : $$\widetilde R(X, Y, Z, W) = R (X, Y, Z, W) +
\widetilde g(b(X, Z), b(Y, W)) -\widetilde g(b(X, W), b(Y, Z)),
\eqno (1.3)$$ $$(\widetilde R (X, Y)Z)^\bot
=(\overline\nabla_X b)(Y, Z) - (\overline\nabla_Y b)(X, Z),
\eqno (1.4)$$ $$\widetilde R (X, Y, \xi,\eta)
= R^\bot(X, Y, \xi, \eta) - g([A_\xi, A_\eta]X, Y),
\eqno (1.5)$$ for any vector fields $X, Y, Z, W$, tangent to $F^n$, and vector fields $\xi, \eta$, normal to $F^n$.
For any vector field $\xi$ normal to $F^n$, we denote as $A_\xi$ the second fundamental tensor relatively to $\xi$. For $A_\xi$ the relation holds $$\widetilde g(b(X, Y), \xi) = \widetilde g(A_\xi X, Y),
\eqno (1.6)$$ for any vector fields $X,Y$, tangent to $F^n$.
A normal vector field $\xi$ is called [*nondegenerate*]{} if $\det
A_\xi \ne 0$.
Covariant derivatives $\overline\nabla b$, $(\overline\nabla
A)_\xi$ and $\overline\nabla R^\bot,$ are defined by the following equalities,\
respectively ( \[2\] ): $$(\overline\nabla_X b)(Y, Z) = D_X (b(Y, Z)) - b(\nabla_X Y,Z)
- b(Y, \nabla_X Z),
\eqno (1.7)$$ $$(\overline\nabla_X A)_\xi Y= \nabla_X (A_\xi Y) - A_\xi(\nabla_X Y)
- A_{D_X \xi} Y,
\eqno (1.8)$$ $$(\overline\nabla_X R^\bot)(Y, Z)\xi = D_X (R^\bot(Y, Z)\xi)
- R^\bot(\nabla_X Y,Z)\xi - R^\bot(Y, \nabla_X Z)\xi -
R^\bot (Y,Z) D_X\xi,
\eqno (1.9)$$ for any vector fields $X, Y, Z$, tangent to $F^n$, and vector field $\xi$ normal to $F^n$.
Let indices, in this article, take the following values: $i, j, k,
s, t = 1, \dots, n$, $\alpha, \beta,\gamma = 1, \dots, p.$ We will use the Einstein rule.
Let $x$ be an arbitrary point $F^n$, $T_xF^n$ and $T^\bot_x F^n$ be the tangent and normal spaces $F^n$ at point $x,$ respectively, $U(x)$ be some neighborhood of point $x$, $(u^1,\dots, u^n)$ be local coordinates on $F^n$ in $U(x)$, $\{\partial/\partial u^i\}$ be a local basis in $TF^n$, $\{n_{\alpha |}\}$ be a field of bases of normal vectors in $T^\bot F^n$ in $U(x)$. We may always choose the basis $\{n_{\alpha |}\}$ orthonormalized and assume that $\widetilde g (n_{\alpha |}, n_{\beta|}) = \delta_{\alpha \beta}$, where $\delta_{\alpha \beta}$ is the Kronecker symbol. We introduce the following designations: $$g_{ij} =
g
\left(
\frac{\partial}{\partial u^i},\frac{\partial}{\partial u^j}
\right),
\quad
b_{ij}
=
b
\left(
\frac{\partial}{\partial u^i},\frac{\partial}{\partial u^j}
\right)
=
b^{\alpha}_{ij} n_{\alpha |},
\quad
\Gamma_{ij, k}
=
g
\left(
\nabla_{\frac{\partial}{\partial u^i}}\frac{\partial}{\partial u^j},
\frac{\partial}{\partial u^k}
\right),$$ $$\Gamma_{ij}^k = g^{kt}\Gamma_{ij, t},
\quad
\nabla_ib^{\alpha}_{jk}=
\frac{\partial b^{\alpha}_{jk}}{\partial u^i}
- \Gamma_{ij}^t b^\alpha_{tk} -\Gamma_{ik}^t b^\alpha_{jt},
\quad
\Gamma^\bot_{\alpha\beta |i}=
\widetilde g\left(n_{\alpha |},
\widetilde\nabla_{\frac{\partial}{\partial u^i}} n_{\beta |}\right),$$ $$\Gamma^{\bot\alpha}_{\beta |i}
=\delta^{\alpha\gamma} \Gamma^\bot_{\gamma\beta |i},
\quad
\Gamma^{\bot}_{\alpha\beta |i}
=\delta_{\alpha\gamma} \Gamma^{\bot\gamma}_{\beta |i},
\quad
\Gamma^{\bot}_{\alpha\beta |i} + \Gamma^{\bot}_{\beta \alpha|i} = 0,
\quad
\overline\nabla_i b^\alpha_{jk}
=\left(
\nabla_i b^\alpha_{jk}
+
\Gamma^{\bot\alpha}_{\beta |i}
b^\beta_{jk}
\right),$$ $$\left(
\overline\nabla_{\frac{\partial}{\partial u^i}} b
\right)
\left(
\frac{\partial}{\partial u^j},
\frac{\partial}{\partial u^k}
\right)
=
\overline\nabla_i b^\alpha_{jk} n_{\alpha |},
\qquad
b_{\alpha | ik} = \delta_{\alpha\beta} b^\beta_{ik},
\qquad
a_{\alpha| i}^j = b_{\alpha | ik} g^{kj},$$ $$\nabla_i a_{\alpha| j}^k
=
\frac{\partial a_{\alpha| j}^k}{\partial u^i}
- \Gamma_{ij}^t a_{\alpha| t}^k +\Gamma_{it}^k a_{\alpha| j}^t,
\qquad
a_{\alpha| i}^j \frac{\partial}{\partial u^j}
= A_{n_{\alpha |}}
\left(
\frac{\partial}{\partial u^i}
\right),$$ $$\overline\nabla_i a_{\alpha| j}^k
=
\left(
\nabla_i a_{\alpha | j}^k
-
\Gamma^{\bot\beta}_{\alpha |i}
a_{\beta | j}^k
\right),
\qquad
\left(
\overline\nabla_{\frac{\partial}{\partial u^i}} A
\right)_{n_{\alpha |}}
\left(
\frac{\partial}{\partial u^j}
\right)
=
\overline\nabla_i a_{\alpha| j}^k \frac{\partial}{\partial u^k},$$ where $\|g^{kt}\|$ and $\|\delta^{\alpha\beta}\|$ are inverse matrixes to $\|g_{kt}\|$ and $\|\delta_{\alpha\beta}\|,$ respectively.
We assume that a Riemannian manifold $M^{n+p}$ is almost Hermitian manifold with almost complex structure $J$ (see \[3\] , chapter 6, section 6.1). Then $M^{n+p}$ has even dimension: $n+p = 2(m+l)$, where a number $m+l$ is called [*complex dimension*]{} of $M^{n+p}$; the Riemannian metric $\widetilde g$ is almost Hermitian, i.e. for any vector fields $\widetilde X, \widetilde
Y$, tangent to $M^{n+p}$, the following condition holds: $$\widetilde g (J\widetilde X, J\widetilde Y) =
\widetilde g (\widetilde X, \widetilde Y).
\eqno (1.10)$$
Almost Hermitian manifold $M^{n+p}$ is called [*Kaehler manifold*]{} ( \[3\] ) if almost complex structure $J$ is parallel, i.e. for any vector fields $\widetilde X, \widetilde Y$, tangent to $M^{n+p}$, the following condition holds: $$\widetilde\nabla_{\widetilde X} J\widetilde Y =
J \widetilde\nabla_{\widetilde X}\widetilde Y.
\eqno (1.11)$$
A submanifold $F^n$ of a Kaehler manifold $M^{n+p}$ is called [*Kaehler submanifold*]{} if for any vector field $X\in TF^n,$ vector field $JX\in TF^n$. $F^n$ is Kaehler manifold relative to induced almost complex structure $J$ and induced almost Hermitian metric $g$ (see \[3\] , chapter 6, par. 6.7). Kaehler submanifold $F^n$ in Kaehler manifold $M^{n+p},$ has even dimension $n = 2m$ and codimension $p= 2l.$ Number $m$ is called [*complex dimension,*]{} and number $l$ is called [*complex codimension*]{} of Kaehler submanifold $F^n$.
We denote by $M^{2m+2l}(c),$ a Kaehler manifold of complex dimension $m+l$ of constant holomorphic sectional curvature $c$. The tensor of Riemannian curvature $\widetilde R$ of space $M^{2m+2l}(c)$ complies with the formula \[1\] : $$\widetilde R (\widetilde X, \widetilde Y)\widetilde Z
=
\frac{c}{4}
\left(
\widetilde g(\widetilde Y, \widetilde Z)\widetilde X
-\widetilde g(\widetilde X, \widetilde Z)\widetilde Y
+
\widetilde g(J\widetilde Y, \widetilde Z)J\widetilde X
-\widetilde g(J\widetilde X, \widetilde Z)J\widetilde Y
+ 2\widetilde g(\widetilde X, J\widetilde Y)J\widetilde Z
\right),
\eqno (1.12)$$ for any vector fields $\widetilde X, \widetilde Y, \widetilde Z$, tangent to $M^{2m+2l}(c)$.
The properties of covariant derivative $\overline\nabla$.
=========================================================
\[lemma 1\]. Let $F^n$ be a submanifold in a Riemannian manifold $M^{n+p}$. Then the following equality holds: $$\widetilde g((\overline\nabla_Z A)_\xi X, Y)
=
\widetilde g((\overline\nabla_Z b)(X, Y), \xi)
\quad
\forall X, Y, Z \in TF^n,
\quad
\forall \xi \in T^\bot F^n.
\eqno (2.1)$$
[**Proof.**]{} We will find the expressions of the left and the right parts of the equality (2.1), in local coordinates. We assume $$Z =Z^i \frac{\partial}{\partial u^i},
\quad
X = X^j \frac{\partial}{\partial u^j},
\quad
Y =Y^k \frac{\partial}{\partial u^k},
\quad
\xi = \xi^\alpha n_{\alpha |}.
\eqno (2.2)$$ We have: $$\widetilde g((\overline\nabla_Z A)_\xi X, Y)
=
Z^i X^j Y^k
\xi^\alpha
g_{sk}\overline\nabla_i a_{\alpha| j}^s
=
Z^i X^j Y^k
\left(
\xi^\alpha
g_{sk}
\nabla_i a_{\alpha | j}^s
-
\xi^\alpha
g_{sk}
\Gamma^{\bot\beta}_{\alpha |i}
a_{\beta | j}^s
\right)
=$$ $$=
Z^i X^j Y^k
\left(
\xi^\alpha
\nabla_i (g_{sk} a_{\alpha | j}^s)
-
\xi^\alpha
\Gamma^{\bot\beta}_{\alpha |i}
g_{sk} a_{\beta | j}^s
\right)
=
Z^i X^j Y^k
\left(
\xi^\alpha
\nabla_i b_{\alpha | jk}
-
\xi^\alpha
\Gamma^{\bot\beta}_{\alpha |i}
b_{\beta | jk}
\right)
=$$ $$=
Z^i X^j Y^k
\left(
\xi^\alpha
\nabla_i (\delta_{\alpha\beta}b^\beta_{jk})
-
\xi^\alpha
\Gamma^{\bot\beta}_{\alpha |i}
\delta_{\beta\gamma}
b^\gamma_{jk}
\right)
=
Z^i X^j Y^k
\left(
\xi^\alpha
\delta_{\alpha\beta}
\nabla_i b^\beta_{jk}
-
\xi^\alpha
\Gamma^{\bot\beta}_{\alpha |i}
\delta_{\beta\gamma}
b^\gamma_{jk}
\right)
=$$ $$=
Z^i X^j Y^k
\left(
\xi^\alpha
\delta_{\alpha\beta}
\nabla_i b^\beta_{jk}
-
\xi^\alpha
\Gamma^{\bot}_{\gamma\alpha |i}
b^\gamma_{jk}
\right)
=$$ $$=
Z^i X^j Y^k
\left(
\xi^\alpha
\delta_{\alpha\beta}
\left(
\overline\nabla_i b^\beta_{jk}
- \Gamma^{\bot\beta}_{\gamma |i}
b^\gamma_{jk}
\right)
-
\xi^\alpha
\Gamma^{\bot}_{\gamma\alpha |i}
b^\gamma_{jk}
\right)
=$$ $$=
Z^i X^j Y^k
\left(
\xi^\alpha
\delta_{\alpha\beta}
\overline\nabla_i b^\beta_{jk}
-
\xi^\alpha
\delta_{\alpha\beta}
\Gamma^{\bot\beta}_{\gamma |i}
b^\gamma_{jk}
-
\xi^\alpha
\Gamma^{\bot}_{\gamma\alpha |i}
b^\gamma_{jk}
\right)
=$$ $$=
Z^i X^j Y^k
\left(
\xi^\alpha
\delta_{\alpha\beta}
\overline\nabla_i b^\beta_{jk}
-
\xi^\alpha
\Gamma^{\bot}_{\alpha\gamma |i}
b^\gamma_{jk}
-
\xi^\alpha
\Gamma^{\bot}_{\gamma\alpha |i}
b^\gamma_{jk}
\right)
=
Z^i X^j Y^k
\delta_{\alpha\beta}
\xi^\alpha
\overline\nabla_i b^\beta_{jk}
=$$ $$=
\widetilde g((\overline\nabla_Z b)(X, Y), \xi).$$ Lemma is proved.
\[lemma 2\]. Let $F^{2m}$ be a Kaehler submanifold in a Kaehler manifold $M^{2m+ 2l}.$ Then for any $X\in TF^{2m}$ and for any $\xi \in T^\bot F^{2m}$ the following equality holds: $$\left(
\overline\nabla_X
A
\right)_{J\xi}
=
J
\left(
\overline\nabla_X
A
\right)_\xi
\eqno (2.3)$$
[**Proof.**]{} From (1.1), because of (1.11), we obtain the following equalities (see , for example, \[3\] , chapter 6, section 6.1, lemma 6. 26): $$\nabla_X JY = J \nabla_X Y,
\quad
J b(X, Y) = b(X, JY),
\qquad
\forall X, Y\in TF^{2m}.
\eqno (2.4)$$ From (1.2) we have: $$\widetilde\nabla_X J\xi = -A_{J\xi} X + D_X J\xi,
\quad
J \widetilde\nabla_X\xi = J(-A_\xi X + D_X\xi).$$ Hence, because of (1.11), we obtain: $$-A_{J\xi} X + D_X J\xi = J(-A_\xi X + D_X\xi).$$ Therefore, $$-A_{J\xi} X + J A_\xi X = J D_X\xi - D_X J\xi.$$ Since $F^{2m}$ is a Kaehler submanifold, then, from here, we have $$A_{J\xi}X = J A_\xi X,
\quad
D_X(J\xi)= J D_X\xi,
\qquad
\forall X, Y\in TF^{2m}.
\eqno (2.5)$$ From (1.7) we have $$(\overline\nabla_X A)_{J\xi} Y
= \nabla_X (A_{J\xi} Y) - A_{J\xi}(\nabla_X Y)
- A_{D_X (J\xi)} Y.$$ Hence, using (2.4) and (2.5), we have: $$(\overline\nabla_X A)_{J\xi} Y
= \nabla_X J(A_\xi Y) - J A_\xi(\nabla_X Y)
- A_{J(D_X \xi)} Y
=$$ $$=
J \nabla_X (A_\xi Y) - J A_\xi(\nabla_X Y)
- J A_{D_X \xi} Y
= J (\overline\nabla_X A)_\xi Y.$$ Lemma is proved.
\[lemma 3\]. Let $F^{2m}$ be a Kaehler submanifold in a Kaehler manifold $M^{2m+2l}(c)$ of constant holomorphic sectional curvature $c$. Then for any $X, Y, Z \in TF^{2m}$ and for any $\xi \in T^\bot F^{2m}$ the following equalities hold: $$(\overline\nabla_{JZ} b)(X,Y)
=
J
\left(
(\overline\nabla_Z b)(X,Y)
\right),
\eqno (2.6)$$ $$(\overline\nabla_{JZ} A)_\xi
= - J (\overline\nabla_Z A)_\xi,
\eqno (2.7)$$ $$J A_\xi = - A_\xi J,
\eqno (2.8)$$ $$J(\overline\nabla_Z A)_\xi = -(\overline\nabla_Z A)_\xi J.
\eqno (2.9)$$
[**Proof.**]{} 1. Because of (1.12), the equation (1.4) takes the following form: $$(\overline\nabla_X b)(Y, Z) = (\overline\nabla_Y b)(X, Z),
\quad
\forall X, Y, Z \in TF^{2m}.
\eqno (2.10)$$ Using (2.10), from (1.7) we obtain: $$(\overline\nabla_{JZ} b)(X,Y)
=
(\overline\nabla_X b)(JZ,Y)
= D_X(b(JZ,Y)) - b(\nabla_X(JZ),Y) -b(JZ, \nabla_X Y).$$ Hence, using (2.4) and (2.5), we have: $$(\overline\nabla_{JZ} b)(X,Y)
= D_X(J(b(Z,Y))) -b(J\nabla_X Z,Y) -b(JZ, \nabla_X Y)
=$$ $$=
J(D_X (b(Z,Y))) - J(b(\nabla_X Z,Y)) - J(b(Z, \nabla_X Y))
=$$ $$=
J(D_X (b(Z,Y)) - b(\nabla_X Z,Y) - b(Z, \nabla_X Y))
= J
\left(
(\overline\nabla_Z b)(X,Y)
\right).$$ The equality (2.6) is proved.
2\. Using (2.6), from (2.1) we obtain: $$\widetilde g((\overline\nabla_{JZ} A)_\xi X, Y)
=
\widetilde g((\overline\nabla_{JZ} b)(X, Y), \xi)
=
\widetilde g(J((\overline\nabla_Z b)(X, Y)), \xi).$$ Hence, because of (1.10) and equality $J^2 = -I$, we have: $$\widetilde g((\overline\nabla_{JZ} A)_\xi X, Y)
=
- \widetilde g((\overline\nabla_Z b)(X, Y), J\xi)
=
-\widetilde g((\overline\nabla_Z A)_{J\xi} X, Y)
=
-\widetilde g(J((\overline\nabla_Z A)_\xi X), Y).$$ From here we get (2.7).
3\. From (1.6), using (2.4), we obtain: $$\widetilde g(J A_\xi X, Y) = -\widetilde g(A_\xi X, JY)
= -\widetilde g(b(X, JY), \xi)
= -\widetilde g(b(JX, Y), \xi)
= -\widetilde g(A_\xi JX, Y).$$ Thus, $$\widetilde g(J A_\xi X, Y)
= -\widetilde g(A_\xi JX, Y)
\quad
\forall X,Y\in TF^{2m},
\quad
\forall \xi \in T^\bot F^{2m}.$$ The derived equality is equivalent to (2.8).
4\. From (1.8), using (2.4) and (2.8), for any $X,Y\in TF^{2m}$ and for any $\xi\in T^\bot F^{2m},$ we have: $$J
\left(
(\overline\nabla_X A)_\xi Y
\right)
=
J
\left(
\nabla_X (A_\xi Y) - A_\xi(\nabla_X Y) - A_{D_X \xi} Y
\right)
=$$ $$=
\nabla_X J(A_\xi Y) + A_\xi J(\nabla_X Y) + A_{D_X \xi} (JY)
=$$ $$=
-\nabla_X (A_\xi JY) + A_\xi (\nabla_X JY) + A_{D_X \xi} (JY)
=
-
(\overline\nabla_X A)_\xi (JY).$$ Thus, $$J
\left(
(\overline\nabla_X A)_\xi Y
\right)
=
-
(\overline\nabla_X A)_\xi (JY),
\quad
\forall X,Y\in TF^{2m},
\quad
\forall \xi \in T^\bot F^{2m}.$$ The obtained equality is equivalent to (2.9). Lemma is proved.
\[lemma 4\]. Let $F^{2m}$ be a Kaehler submanifold in a Kaehler manifold $M^{2m+2l}.$\
Then the following equality holds $$\overline\nabla_Z
\left(
\widetilde g(X, JY)J\xi
\right) = 0
\qquad
\forall X, Y, Z \in TF^{2m},
\quad
\forall \xi \in T^\bot F^{2m}.
\eqno (2.11)$$
[**Proof.**]{} By the definition of covariant derivative $\overline\nabla,$ we have: $$\overline\nabla_Z
\left(
\widetilde g(X, JY)J\xi
\right)
=$$ $$=
D_Z
\left(
\widetilde g(X, JY)J\xi
\right)
-
\widetilde g(\nabla_Z X, JY)J\xi
-
\widetilde g(X,\nabla_Z(JY))J\xi
-
\widetilde g(X, JY)D_Z(J\xi).$$ We transform the right part of the last equality, writing it in local coordinates and using the designations (2.2): $$\left(
\frac{\partial
\left(
g_{kl}X^k(JY)^l(J\xi)^\tau
\right)
}{\partial u^i}
+
\Gamma^{\bot\tau}_{\sigma| i}
g_{kl}X^k(JY)^l(J\xi)^\sigma
\right)
Z^i n_\tau
-$$ $$-
g_{kl}
\left(
\frac{\partial X^k}{\partial u^i} + \Gamma^k_{im} X^m
\right)
(JY)^l(J\xi)^\tau
Z^i n_\tau
-
g_{lk}
\left(
\frac{\partial (JY)^l}{\partial u^i} + \Gamma^l_{im} (JY)^m
\right)
X^k(J\xi)^\tau
Z^i n_\tau
-$$ $$-
g_{kl} X^k (JY)^l
\left(
\frac{\partial (J\xi)^\tau}{\partial u^i}
+ \Gamma^{\bot \tau}_{\sigma| i}(J\xi)^\sigma
\right)
Z^i n_\tau
=$$ $$=
\left(
\frac{\partial g_{kl}}{\partial u^i} X^k (JY)^l(J\xi)^\tau
-
g_{kl}\Gamma^k_{im} X^m (JY)^l(J\xi)^\tau
-
g_{lk}\Gamma^l_{im} X^k (JY)^m(J\xi)^\tau
\right)
Z^i n_\tau
=$$ $$=
\left(
\frac{\partial g_{kl}}{\partial u^i}
-
g_{ml}\Gamma^m_{ik}
-
g_{mk}\Gamma^m_{il}
\right)
X^k (JY)^l
(J\xi)^\tau
Z^i n_\tau
=0.$$ Lemma is proved.
\[lemma 5\]. Let $F^{2m}$ be a Kaehler submanifold in a Kaehler manifold $M^{2m+2l}(c)$ of constant holomorphic sectional curvature $c$. Then the following equality holds $$R^\bot (X, Y)\xi =
\frac{c}{2}\widetilde g(X, JY)J\xi
+ b(X, A_\xi Y) - b(Y, A_\xi X),$$ $$\forall X, Y \in TF^{2m},
\quad
\forall \xi \in T^\bot F^{2m}.
\eqno (2.12)$$
[**Proof.**]{} Because of (1.12), we have: $$\widetilde R (X,Y,\xi, \eta)=
\widetilde g (\widetilde R (X,Y)\xi, \eta)=
\frac{c}{2}\widetilde g(X, JY)\widetilde g(J\xi,\eta).$$ Then the equation (1.5) takes the form: $$R^\bot(X, Y, \xi, \eta)
= \frac{c}{2}\widetilde g(X, JY)\widetilde g(J\xi,\eta)
+ \widetilde g([A_\xi, A_\eta]X, Y).$$ We transform the second term in the right part of the obtained equality, using self-adjointness of operator $A_\xi$: $$\widetilde g([A_\xi, A_\eta]X, Y)
=
\widetilde g((A_\xi A_\eta - A_\eta A_\xi)X, Y)
=
\widetilde g(A_\xi (A_\eta X), Y)
- \widetilde g(A_\eta (A_\xi X), Y)
=$$ $$= \widetilde g(A_\eta X, A_\xi Y) - \widetilde g(A_\xi X, A_\eta Y)
=
\widetilde g(b(A_\xi Y, X),\eta) - \widetilde g(b(A_\xi X, Y),\eta).$$ Then for any $\eta\in T^\bot F^{2m}$ we have: $$R^\bot (X, Y, \xi, \eta)
\equiv
\widetilde g(R^\bot (X, Y)\xi, \eta)
=$$ $$=
\widetilde g(\frac{c}{2}\widetilde g(X, JY)J\xi, \eta)
+
\widetilde g(b(A_\xi Y, X),\eta) - \widetilde g(b(A_\xi X, Y),\eta).$$ From here we obtain the equality (2.12). Lemma is proved.
\[lemma 6\]. Let $F^{2m}$ be a Kaehler submanifold in a Kaehler manifold $M^{2m+2l}(c)$ of constant holomorphic sectional curvature $c$. Then the following equality holds $$(\overline\nabla_Z R^\bot)(X,Y) \xi
=$$ $$=
(\overline\nabla _Z b)(X, A_\xi Y)
+
b(X, (\overline\nabla_Z A)_\xi Y)
-
(\overline\nabla _Z b)(Y, A_\xi X)
-
b(Y, (\overline\nabla_Z A)_\xi X),$$ $$\forall X, Y, Z \in TF^{2m},
\quad
\forall \xi \in T^\bot F^{2m}.
\eqno (2.13)$$
[**Proof.**]{} From formula (1.9), using (2.12), we obtain: $$(\overline\nabla_Z R^\bot)(X,Y) \xi
= D_Z
\left(
\frac{c}{2}\widetilde g(X, JY)J\xi
+ b(X, A_\xi Y) - b(Y, A_\xi X)
\right)
-$$ $$-
\left(
\frac{c}{2}\widetilde g(\nabla_Z X, JY)J\xi
+ b(\nabla_Z X, A_\xi Y) - b(Y, A_\xi (\nabla_Z X))
\right)
-$$ $$-
\left(
\frac{c}{2}\widetilde g(X, J(\nabla_Z Y))J\xi
+ b(X, A_\xi (\nabla_Z Y)) - b(\nabla_Z Y, A_\xi X)
\right)
-$$ $$-
\left(
\frac{c}{2}\widetilde g(X, JY))J(D_Z \xi)
+ b(X, A_{D_Z\xi} Y) - b(Y, A_{D_Z\xi} X)
\right)
=$$ $$=
\frac{c}{2}
\Biggl(
D_Z(\widetilde g(X, JY)J\xi)
- \widetilde g(\nabla_Z X, JY)J\xi
- \widetilde g(X, J(\nabla_Z Y))J\xi
- \widetilde g(X, JY) J(D_Z \xi)
\Biggr)
+$$ $$+
D_Z(b(X, A_\xi Y)) - D_Z(b(Y, A_\xi X))
- b(\nabla_Z X, A_\xi Y) + b(Y, A_\xi (\nabla_Z X))
-$$ $$-
b(X, A_\xi (\nabla_Z Y)) + b(\nabla_Z Y, A_\xi X)
- b(X, A_{D_Z\xi} Y) + b(Y, A_{D_Z\xi} X).$$ Hence, using (2.3) and (2.4), we have $$(\overline\nabla_Z R^\bot)(X,Y) \xi
=$$ $$=
\frac{c}{2} \overline\nabla_Z(\widetilde g(X, JY)J\xi)
+
D_Z(b(X, A_\xi Y)) - D_Z(b(Y, A_\xi X))
- b(\nabla_Z X, A_\xi Y)
+$$ $$+ b(Y, A_\xi (\nabla_Z X))
-
b(X, A_\xi (\nabla_Z Y)) + b(\nabla_Z Y, A_\xi X)
- b(X, A_{D_Z\xi} Y) + b(Y, A_{D_Z\xi} X).$$ Therefore, because of (2.11), we obtain: $$(\overline\nabla_Z R^\bot)(X,Y) \xi
=
\Biggl(
D_Z(b(X, A_\xi Y))
- b(\nabla_Z X, A_\xi Y)
- b(X, A_\xi (\nabla_Z Y))
- b(X, A_{D_Z\xi} Y)
\Biggr)
-$$ $$-
\Biggl(
D_Z(b(Y, A_\xi X))
- b(\nabla_Z Y, A_\xi X)
- b(Y, A_\xi (\nabla_Z X))
- b(Y, A_{D_Z\xi} X)
\Biggr).$$ Hence, using (1.7), we obtain: $$(\overline\nabla_Z R^\bot)(X,Y) \xi
=
\Biggl(
(\overline\nabla_Z b)(X, A_\xi Y)
+ b(X, \nabla_Z (A_\xi Y))
- b(X, A_\xi (\nabla_Z Y))
- b(X, A_{D_Z\xi} Y)
\Biggr)
-$$ $$-
\Biggl(
(\overline\nabla_Z b)(Y, A_\xi X)
+ b(Y, \nabla_Z (A_\xi X))
- b(Y, A_\xi (\nabla_Z X))
- b(Y, A_{D_Z\xi} X)
\Biggr).$$ Now, using (1.8), we obtain: $$(\overline\nabla_Z R^\bot)(X,Y) \xi
=$$ $$=
\Biggl(
(\overline\nabla_Z b)(X, A_\xi Y)
+ b(X, (\overline\nabla_Z A)_\xi Y)
\Biggr)
-
\Biggl(
(\overline\nabla_Z b)(Y, A_\xi X)
+ b(Y, (\overline\nabla_Z A)_\xi X)
\Biggr).$$ Lemma is proved.
\[lemma 7\]. Let $F^{2m}$ be a Kaehler submanifold in a Kaehler manifold $M^{2m+2l}(c)$ of constant holomorphic sectional curvature $c$. Then the following equality holds $$(\overline\nabla_Z R^\bot)(X,Y, \xi, \eta)
=
\widetilde g([(\overline\nabla _Z A)_\xi, A_\eta]X, Y)
+
\widetilde g([A_\xi, (\overline\nabla_Z A)_\eta] X, Y),$$ $$\forall X, Y, Z \in TF^{2m},
\quad
\forall \xi,\eta \in T^\bot F^{2m}.
\eqno (2.14)$$
[**Proof.**]{} Because of (2.13), we have: $$(\overline\nabla_Z R^\bot)(X,Y, \xi, \eta)
\equiv
\widetilde g
\left(
(\overline\nabla_Z R^\bot)(X,Y)\xi, \eta
\right)
=
\widetilde g
\left(
(\overline\nabla_Z b)(X, A_\xi Y), \eta
\right)
-$$ $$-
\widetilde g
\left(
(\overline\nabla_Z b)(Y, A_\xi X), \eta
\right)
+
\widetilde g
\left(
b(X, (\overline\nabla_Z A)_\xi Y), \eta
\right)
-
\widetilde g
\left(
b(Y, (\overline\nabla_Z A)_\xi X), \eta
\right).$$ In the derived equality, we transform the first and the second terms using (2.1), the third and the fourth using (1.6): $$(\overline\nabla_Z R^\bot)(X,Y, \xi, \eta)
=
\widetilde g
\left(
(\overline\nabla_Z A)_ \eta X, A_\xi Y
\right)
-
\widetilde g
\left(
(\overline\nabla_Z A)_ \eta Y, A_\xi X
\right)
+$$ $$+
\widetilde g
\left(
A_\eta X, (\overline\nabla_Z A)_\xi Y)
\right)
-
\widetilde g
\left(
A_\eta Y, (\overline\nabla_Z A)_\xi X)
\right).$$ Hence, because of self-adjointness of operators $A_\xi$ and $(\overline\nabla A)_\xi,$ we obtain: $$(\overline\nabla_Z R^\bot)(X,Y, \xi, \eta)
=
\widetilde g
\left(
A_\xi(\overline\nabla_Z A)_ \eta X, Y
\right)
-
\widetilde g
\left(
Y, (\overline\nabla_Z A)_ \eta A_\xi X
\right)
+$$ $$+
\widetilde g
\left(
(\overline\nabla_Z A)_\xi A_\eta X, Y)
\right)
-
\widetilde g
\left(
Y, A_\eta (\overline\nabla_Z A)_\xi X)
\right)
=$$ $$=
\widetilde g
\left(
[A_\xi, (\overline\nabla_Z A)_ \eta] X, Y
\right)
+
\widetilde g
\left(
[(\overline\nabla_Z A)_\xi, A_\eta] X, Y)
\right)
.$$ Lemma is proved.
\[lemma 8\]. Let $F^{2m}$ be a Kaehler submanifold in a Kaehler manifold $M^{2m+2l}(c)$ of constant holomorphic sectional curvature $c$. Then the following equality holds $$(\overline\nabla_{JZ} R^\bot)(X,Y, \xi, \eta)
=
(\overline\nabla_Z R^\bot)(X,Y, J\xi, \eta)
-
2\widetilde g([(\overline\nabla _Z A)_{J\xi}, A_\eta]X, Y),$$ $$\forall X, Y, Z \in TF^{2m},
\quad
\forall \xi,\eta \in T^\bot F^{2m}.
\eqno (2.15)$$
[**Proof.**]{} From (2.14) we obtain: $$(\overline\nabla_{JZ} R^\bot)(X,Y, \xi, \eta)
=
\widetilde g([(\overline\nabla _{JZ} A)_\xi, A_\eta]X, Y)
+
\widetilde g([A_\xi, (\overline\nabla_{JZ} A)_\eta] X, Y).$$ Hence, using (2.7), we have: $$(\overline\nabla_{JZ} R^\bot)(X,Y, \xi, \eta)
=
\widetilde g([-J(\overline\nabla _Z A)_\xi, A_\eta]X, Y)
+
\widetilde g([A_\xi, -J(\overline\nabla_Z A)_\eta] X, Y).$$ In the derived equality, we transform the second term using (2.8) and (2.9): $$[A_\xi, J(\overline\nabla_Z A)_\eta]
=
A_\xi J (\overline\nabla_Z A)_\eta - J(\overline\nabla_Z A)_\eta A_\xi
=$$ $$=
-J A_\xi (\overline\nabla_Z A)_\eta + (\overline\nabla_Z A)_\eta J A_\xi
=
- [J A_\xi , (\overline\nabla_Z A)_\eta].$$ Therefore, $$(\overline\nabla_{JZ} R^\bot)(X,Y, \xi, \eta)
=
-\widetilde g([J(\overline\nabla _Z A)_\xi, A_\eta]X, Y)
+
\widetilde g([J A_\xi, (\overline\nabla_Z A)_\eta] X, Y).$$ Hence, because of (2.3) and (2.5), and using (2.14), we obtain: $$(\overline\nabla_{JZ} R^\bot)(X,Y, \xi, \eta)
=
-\widetilde g([(\overline\nabla _Z A)_{J\xi}, A_\eta]X, Y)
+
\widetilde g([A_{J\xi}, (\overline\nabla_Z A)_\eta] X, Y)
=$$ $$=
(\overline\nabla_Z R^\bot)(X,Y, J\xi, \eta)
-
2\widetilde g([(\overline\nabla _Z A)_{J\xi}, A_\eta]X, Y).$$ Lemma is proved.
Proofs of theorems 1, 2.
========================
[**Proof theorem 1.**]{}
Let for some 1-form $\mu$ on $F^{2m},$ the following condition holds $$(\overline\nabla_X b)(Y,Z) = \mu (X) b(Y,Z) \quad \forall X, Y,
Z\in TF^{2m}. \eqno (3.1)$$ Then for any vector field $\xi \in T^\bot F^{2m},$ we have: $$\widetilde g
(
\left(
\overline\nabla_X b
\right)
(Y,Z), \xi )
=
\widetilde g(\mu (X) b(Y,Z), \xi).$$ Hence, using (2.1) and (1.6), we obtain: $$\widetilde g
(
\left(
\overline\nabla_X A
\right)_\xi Y, Z)
=
\widetilde g (\mu (X) A_\xi Y, Z)
\quad
\forall X, Y, Z\in TF^{2m},
\quad
\forall \xi \in T^\bot F^{2m}.$$ Thus, the condition (3.1) is equivalent to the condition $$\left(
\overline\nabla_X A
\right)_\xi
=
\mu (X) A_\xi,
\quad
\forall X \in TF^{2m},
\quad
\forall \xi \in T^\bot F^{2m}.
\eqno (3.2)$$ From (3.2) we obtain the equality: $$\left(
\overline\nabla_{JX} A
\right)_\xi
=
\mu(JX) A_\xi
\quad
\forall X \in TF^{2m},
\quad
\forall \xi \in T^\bot F^{2m}.
\eqno (3.3)$$ On the other hand, from (3.2), because of (2.7), we have: $$\left(
\overline\nabla_{JX} A
\right)_\xi
= -J
\left(
\mu(X) A_\xi
\right),
\quad
\forall X \in TF^{2m},
\quad
\forall \xi \in T^\bot F^{2m}.
\eqno (3.4)$$ From (3.3) and (3.4), we obtain: $$\mu(JX) A_\xi
= -J
\left(
\mu(X) A_\xi
\right),
\quad
\forall X \in TF^{2m},
\quad
\forall \xi \in T^\bot F^{2m}.$$ Hence, for any $Y\in TF^{2m},$ we have: $$\mu(JX) A_\xi Y
= - \mu(X)
J
\left(
A_\xi Y
\right),
\quad
\forall X \in TF^{2m},
\quad
\forall \xi \in T^\bot F^{2m}.
\eqno (3.5)$$ Using (3.5), we obtain: $$\mu(JX)
\widetilde g( A_\xi Y , A_\xi Y )
= - \mu(X)
\widetilde g(
J
\left(
A_\xi Y
\right),
A_\xi Y) = 0,$$ $$\forall X, Y \in TF^{2m},
\quad
\forall \xi \in T^\bot F^{2m}.
\eqno (3.6)$$ Since $b\ne 0$ then there exists nondegenerate vector field $\xi\in T^\bot F^{2m}$, and from (3.6) we come to the equality: $$\mu (X) = 0
\quad
\forall X\in TF^{2m}.$$ Then 1-form $\mu\equiv 0$ and, therefore, $$\left(
\overline\nabla_X A
\right)_\xi
= 0,
\quad
\forall X \in TF^{2m},
\quad
\forall \xi \in T^\bot F^{2m}.
\eqno(3.7)$$ Hence, because of (2.14), we obtain the conclusion of the theorem.
[**Proof of theorem 2.**]{}
Form (1.3) we obtain: $$\nabla_W R(X,Y,Z,V) =
\widetilde g((\overline\nabla_Wb)(X, V), b(Y, Z))+
\widetilde g(b(X, V), (\overline\nabla_Wb)(Y, Z)) -$$ $$-
\widetilde g((\overline\nabla_Wb)(X, Z), b(Y, V)) -
\widetilde g(b(X, Z), (\overline\nabla_Wb)(Y, V))
\quad
\forall X,Y,Z,V,W\in TF^{2m}.$$ Therefore, because of (3.7), $\nabla R\equiv 0$. Theorem is proved.
\
\
$[1]$ Kobayashi S., Nomizu K. Foundations of differential geometry. Vol. 2. M.: Nauka. 1981.\
$[2]$ Chen B.-Y. Geometry of submanifolds. N.-Y.: M. Dekker. 1973.\
$[3]$ Gray A. Tubes. M.: Mir. 1993.
[^1]: ©Irina I. Bodrenko, associate professor, Department of Mathematics,\
Volgograd State University, University Prospekt 100, Volgograd, 400062, RUSSIA.\
E.-mail: bodrenko@mail.ru http://www.bodrenko.com http://www.bodrenko.org
|
---
abstract: |
The Standard Model (SM) of the electroweak has proven to be successful in describing all the available precision experimental data. However, the Higgs mechanism, responsible for the electroweak symmetry breaking in the SM, still remains one of the most important open questions of the theory. The effect of new operators that give rise to anomalous Higgs boson coupling to two photons is examined in the two-photon processes $\gamma\gamma \to H \to b\bar{b}, \gamma\gamma,
W^+W^-,ZZ$ at a high energy linear $e^+ e^-$ collider (NLC).
address: |
Theory Group, Lawrence Berkeley National Laboratory,\
Berkeley, CA 94720, USA.
author:
- 'S. M. Lietti'
title: 'Anomalous Higgs-Photon Interactions in Photon Fusion Processes at NLC'
---
Introduction
============
The Standard Model (SM) of the electroweak interactions based on the gauge group $SU(2)_L \times U(1)_Y$ has proven to be successful in describing all the available precision experimental data [@sm_data]. This applies particularly to the predictions for the couplings of the gauge bosons to the matter fermions. The recent measurements of the gauge-boson self couplings at LEPII [@lep2] and Tevatron [@tevatron] collider also shed some light on the correctness of the SM predictions for these interactions.
On the other hand, the precise mechanism of the electroweak symmetry breaking still remains one of the most important open questions of the theory. In the SM, the breaking is realized via the Higgs mechanism in which a scalar $SU(2)$-doublet, the Higgs boson, is introduced [*ad hoc*]{} and the symmetry is spontaneously broken by the vacuum expectation value (VEV) of the Higgs field. However, in this simple realization, the theory presents problems since the running Higgs mass shows a quadratic divergence at some high scale. This may imply the existence of a cut-off scale $\Lambda$ above which new physics must appear.
The experiments which will take place at the Next Linear electron–positron Collider (NLC) will be able to explore the nature of the Higgs boson and its couplings to other particles [@review_higgs_nlc]. Deviations from the SM predictions for these couplings would indicate the existence of new physics effects.
In general, such deviations can be parametrized in terms of effective Lagrangians by adding to the SM Lagrangian higher dimensional operators that describe the new phenomena [@effective]. This model–independent approach accounts for new physics that shows up at an energy scale $\Lambda$, larger than the electroweak scale. The effective Lagrangians are constructed with the light particle spectrum that exists at low energies, while the heavy degrees of freedom are integrated out. They are invariant under the $SU(2)_L \times U(1)_Y$ and, in the linearly realized version, they involve, in addition to the usual gauge–boson fields, also the light Higgs particle. The most general dimension–6 effective Lagrangian, containing all SM bosonic fields, that is $C$ and $P$ even, was constructed in Ref. [@hisz].
Out of the eleven independent operators constructed in Ref.[@hisz], three of them describe new interaction between the Higgs particle and the photon, $${\cal L}_{\text{eff}} = \frac{f_{WW}}{\Lambda^2} \;
\Phi^{\dagger} \hat{W}_{\mu \nu} \hat{W}^{\mu \nu} \Phi +
\frac{f_{BB}}{\Lambda^2} \; \Phi^{\dagger} \hat{B}_{\mu \nu}
\hat{B}^{\mu \nu} \Phi +
\frac{f_{BW}}{\Lambda^2} \; \Phi^{\dagger} \hat{B}_{\mu \nu}
\hat{W}^{\mu \nu} \Phi\; ,
\label{lagrangian}$$ where, in the unitary gauge, the Higgs doublet becomes $\Phi =
(1/\sqrt{2}) [\, 0 \, , \, (v + H) \, ]^T$, $\hat{B}_{\mu
\nu} = i (g'/2) B_{\mu \nu}$, and $\hat{W}_{\mu \nu} = i (g/2)
\sigma^a W^a_{\mu \nu}$, with $B_{\mu \nu}$ and $ W^a_{\mu \nu}$ being the field strength tensors of the $U(1)$ and $SU(2)$ gauge fields respectively, and $\Lambda$ represents the energy scale for new physics.
The operators of Eq. (\[lagrangian\]) describe the effect, at one–loop level [@arz], of new heavy states predicted by the underlying theory that should be valid at very high energies. The possible existence of heavy fermions and/or bosons, that couple to the (light) bosonic sector of the SM, should indirectly manifest itself in the Higgs boson couplings via equation (\[lagrangian\]), after all the heavy degrees of freedom are integrated out. Anomalous Higgs boson couplings have already been studied in Higgs and Z boson decays [@hagiwara2], in $e^+ e^-$ [@ee; @our; @our2], $\gamma\gamma$ [@gamma], and $p\bar{p}$ colliders [@fer].
In this paper, we explore the consequence of new operators that give rise to an anomalous Higgs boson coupling to photons ($H\gamma\gamma$). In particular, we study the anomalous production of the Higgs boson via two-photon processes in a electron-positron collider, with the subsequent decay of the Higgs boson into two particles. It is important to notice that we have also taken into account the SM one–loop Higgs contributions [@hgg; @hgz] to this vertex in our analyses.
The lagrangian in Eq. (\[lagrangian\]) induces, besides the $H\gamma\gamma$ coupling, other anomalous Higgs couplings like $HZ\gamma$, $HZZ$, and $HW^+W^-$. In the unitary gauge, Eq. (\[lagrangian\]) can be written for the anomalous Higgs couplings as, $${\cal L}_{\text{eff}}^{\text{H}} =
g_{H\gamma\gamma} H A_{\mu\nu} A^{\mu\nu} +
g_{HZ\gamma} H A_{\mu\nu} Z^{\mu\nu} +
g_{HZZ} H Z_{\mu\nu} Z^{\mu\nu} +
g_{HWW} H W^+_{\mu\nu} W_-^{\mu\nu}\;\; ,
\label{lagrangian_ug}$$ where $A_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ and the same for $Z_{\mu\nu}$. The effective couplings $g_{H\gamma\gamma}$, $g_{HZ\gamma}$, $g_{HZZ}$, and $g_{HWW}$ are related to the coefficients of the operators appearing in Eq. (\[lagrangian\]). In particular, for the $H\gamma\gamma$ coupling one has, $$g_{H\gamma\gamma} = - \frac{g m_W \sin^2\theta_W }{2}
\left(\frac{f_{BB}+f_{WW}-f_{BW}}{\Lambda^2}\right) \;\; .
\label{coupling_haa}$$
Considering only the effect of one operator at a time and combining information from precision measurements at LEPI and at low energy, one has the following constraints at 95% CL (in units of TeV$^{-2}$) [@concha], for $m_H=200$ GeV and $m_{\text{top}}=175$ GeV, $$\begin{aligned}
-1. \leq \frac{f_{BW}}{\Lambda^2} \leq 8.6 \;\;,\;\;
-79 \leq \frac{f_{BB}}{\Lambda^2} \leq 47 \;\;,\;\;
-24 \leq \frac{f_{WW}}{\Lambda^2} \leq 14 \;\;.
\label{limits}\end{aligned}$$
Anomalous Higgs boson production via vector boson fusion can be also generated by the anomalous couplings $HZ\gamma$ – $HW^+W^-$ and $HZZ$ of Eq. (\[lagrangian\_ug\]) are not considered here since their effect is much less important than the anomalous Higgs boson production via two-photon process due to phase space reduction.
In order to impose limits on the dimension–6 operators of Eq. (\[lagrangian\]) which generate the new $H\gamma\gamma$ interaction, we examine the Higgs boson production via two-photon process at the NLC with the subsequent decay into $\gamma\gamma$, $b \bar{b}$, $W^+ W^-$, $ZZ$. The SM background considered for these reactions can be divided in two groups:
- [Set I: all the SM direct contribution for $e^+ e^- \to \gamma\gamma$, $b \bar{b}$, $W^+ W^-$, and $ZZ$; ]{}
- [Set II: the vector boson fusion contributions $e^+ e^- \to W^+ W^-(\nu \bar{\nu}) \to \gamma\gamma(\nu \bar{\nu})$, $b \bar{b}(\nu \bar{\nu})$, $W^+ W^-(\nu \bar{\nu})$, $ZZ(\nu \bar{\nu})$, and $e^+ e^- \to Z Z(e^+ e^-) \to \gamma\gamma(e^+ e^-)$, $b \bar{b}(e^+ e^-)$, $W^+ W^-(e^+ e^-)$, $ZZ(e^+ e^-)$.]{}
Vector Boson Fusion at NLC
==========================
In the case of a high energy electron-positron collider, virtual gauge bosons can be produced nearly on–shell and collinear with the initial particles. If one uses the effective boson approximation [@eba; @jot] one may regard the fermion beams as sources of gauge bosons and at leading log ignore the virtuality of these bosons in calculating the cross section. The process $f_a+f_b \to f_{a'}+f_{b'}+X$, where both $f_a$ and $f_b$ serve as source of a vector boson, can be evaluated by the effective boson approximation formula [@jot] $$\begin{aligned}
\sigma_{f_a+f_b \to f_{a'}+f_{b'}+X}(s_0)=
\int dx_1 \int dx_2 f_n(x_1)f_m(x_2)
\hat{\sigma}^{nm}_{V_1+V_2 \to f_{a'}+f_{b'}+X}
(\hat{s}_0)
\label{evb}\end{aligned}$$ where $\hat{s}_0=x_1 x_2 s_0$, $V_{1,2}=W^\pm$ or $Z^0$, and $m,n=-1,0,1$ are the vector boson helicities. If one writes the elementary coupling between the fermions and the vector boson as $\bar{\Psi} \Gamma_\mu \Psi V^\mu$ with $$\begin{aligned}
\Gamma_\mu= g_R \frac{\gamma_\mu(1+\gamma_5)}{2} +
g_L \frac{\gamma_\mu(1-\gamma_5)}{2},
\nonumber \end{aligned}$$ one obtains the following distribution function: $$\begin{aligned}
f_{-1}&=&g_R^2 h_1 + g_L^2 h_2,
\nonumber \\
f_{0}&=&(g_L^2 + g_R^2) h_0,
\nonumber \\
f_{1}&=&g_L^2 h_1 + g_R^2 h_2,
\nonumber \end{aligned}$$ where $$\begin{aligned}
h_0 &=& \left[ \frac{x}{16 \pi^2} \right]
\left[\frac{2(1-x)\xi}{w^2 x} - \frac{2\Delta(2-w)}{w^3} \log
\left(\frac{x}{\Delta'}\right)\right],
\nonumber \\
h_1 &=& \left[ \frac{x}{16 \pi^2} \right]
\left[\frac{-(1-x)(2-w)}{w^2}+\frac{(1-w)(\xi-w^2)}{w^3}\log
\left(\frac{1}{\Delta'}\right)
-\frac{\xi-2xw}{w^3}\log\left(\frac{1}{x}\right)\right],
\nonumber \\
h_2 &=& \left[ \frac{x}{16 \pi^2} \right]
\left[\frac{-(1-x)(2-w)}{w^2(1-w)}+\frac{\xi}{w^3}\log
\left(\frac{x}{\Delta'}\right)\right],
\nonumber \end{aligned}$$ where $w=x-\Delta$, $\xi=x+\Delta$, $\Delta=M_V^2/s^0$, and $\Delta'=\Delta/(1-w)$.
This approach will be used to evaluate the vector boson fusion background (Set II) described in Section \[introduction\].
Anomalous Higgs Boson Production via Two-Photon Process at NLC
==============================================================
For a high energy electron-positron collider, two-photon processes can be calculated using an effective photon approximation [@photon-aprox] so that if a cross section $\sigma_{\gamma \gamma \to X}$ is known, the cross section for $e^+e^- \to e^+e^- X$ via the two photon mechanism is given by: $$\sigma_{e^+e^- \to e^+e^- X}(s_0) = \left[\frac{\alpha}{2 \pi}
\log\left( \frac{s_0}{4 \hat{m}_e^2}\right) \right]^2
\int_0^1 f(\tau) \sigma_{\gamma\gamma \to X}(\tau s_0) d \tau,
\label{ph-apr}$$ where $s_0$ is the square of the center of mass energy of the initial $e^+e^-$ and $$f(\tau)=\frac{1}{\tau}[(2+\tau)^2 \log \frac{1}{\tau} - 2(1-\tau)(3+\tau)].
\label{ftau}$$ In this expression the total cross section for $e^+e^- \to e^+e^- X$ is given if one takes $\hat{m}_e = m_e = 0.5$ MeV as the mass of the electron. However, if one wishes to observe the $e^+e^-$ in the final state, experimental constrains require that a minimum cut on the transverse momentum of the final state electron $P_{T_{min}}$ be used. In this case the result is given by taking $\hat{m}_e = P_{T_{min}}$.
In order to study the anomalous Higgs boson production via two-photon process at NLC we do not necessarily need to observe the final $e^+ e^-$ pair because it is not a product of the Higgs boson decay. Besides, for a NLC with $\sqrt{s_0}=500(1000)$ GeV and requiring $P_{T_{min}}=20(40)$ GeV one finds that the cross section of Eq. (\[ph-apr\]) is only 3.7(3.3)% of the total cross section for $\hat{m}_e = m_e = 0.5$ MeV. Therefore, more than 96 % of the anomalous Higgs boson production via photon fusion happens when the final $e^+ e^-$ pair is undetected.
The anomalous contributions for the $H \gamma \gamma$ interaction are significant only when the Higgs boson is produced on–mass–shell, as we will see in section \[results\]. For this reason, we will use the production of $b\bar{b}$ and $\gamma \gamma$ pairs to study a light Higgs boson mass range of $100 \leq m_H \leq150$ GeV at a NLC with energy $\sqrt{s_0}=500$ GeV and integrated luminosity ${\cal L}=50$ fb$^{-1}$. The production of $W^+ W^-$ and $ZZ$ pairs will be used to study a heavier Higgs boson mass range of $200\leq m_H\leq350$ GeV at a NLC with energy $\sqrt{s_0}=1$ TeV and integrated luminosity ${\cal L}=100$ fb$^{-1}$. We have considered in our analyses a 80% detection efficiency for each photon and quark bottom, and a 80% overall detection efficiency for the $W^+ W^-$ and $Z Z$ final state.
Results
=======
In order to compute the contributions for the signal of anomalous Higgs boson production via photon fusion with subsequent decay into pairs of bottom quarks, photons, and massive gauge bosons $W$ and $Z$, as well as for the background for these final state pairs of particles via direct, photon, and vector boson fusion production, we have have incorporated all anomalous couplings in Helas–type [@helas] Fortran subroutines. These new subroutines were used to adapt a Madgraph [@madgraph] output to include all the anomalous contributions. We have checked that our code passed the non–trivial test of electromagnetic gauge invariance. We employed Vegas [@vegas] to perform the Monte Carlo phase space integration to obtain the differential and total cross sections for the signal and the background (Sets I and II).
To estimate the impact of the anomalous coefficients $f_{BB}$, $f_{WW}$, and $f_{BW}$ in the Higgs boson production via photon fusion, we have evaluated the total cross section for signal and background for all processes described in Section (\[introduction\]). The signal was obtained considering that all anomalous operators coefficients have the same value $f_{all} = f_{BB}=
f_{BW}= f_{WW}= 30$ TeV$^{-2}$, which is in agreement with the limits of Eq.(\[limits\]). In order to avoid infrared divergences in the two photons final state, we have required these photons to have a transverse momentum of $p_{T_\gamma}\geq 25$ GeV. An analysis of the significance of the signal (Significance = Signal/$\sqrt{\text{Background}}$) shows that the $b \bar{b}$ production is a better option compared to the $\gamma\gamma$ production in order to impose limits on the anomalous coefficients for a light Higgs mass ($100\leq M_H(GeV) \leq 150$) as one can see in Tables \[table1\] and \[table2\]. For higher masses ($200\leq M_H(GeV) \leq 350$), Tables \[table3\] and \[table4\] show that the $W^+W^-$ production is a better option compared to the $ZZ$ production.
In order to improve the sensitivity of NLC to the anomalous Higgs boson production, we have investigated different distributions of the final state particles for both signal ($f_{all} = 30$ TeV$^{-2}$) and background. One of most promising variables is the transverse momentum of the final particles whose distribution is presented in Fig. \[fig:1\] (a) for the $b\bar{b}$ final state and in Fig. \[fig:2\] (a) for the $W^+ W^-$ final state. In both cases, we observe that the contribution of the anomalous Higgs production reaches its maximum contribution in $$p_{T_{ano}}=\frac{1}{2}\sqrt{M_H^2-M_{pair}^2},$$ where $M_H$ is the mass of the Higgs boson and $M_{pair}$ is the sum of the masses of the final particles that should be a product of the Higgs boson decay. Similar behaviour is observed for the $\gamma \gamma$ and $ZZ$ final states. Therefore, we require the transverse momentum of these final state particles to be in the range $$25 < p_T(\text{GeV}) < (p_{T_{ano}}+5) \;\;.
\label{cut_pt}$$ In this way, the significance of the signal is enhanced by a factor of at least 1.75, compared to the previous analyses without any cut, as we can see in Tables \[table1\]-\[table4\]. The 95% CL allowed values for the coefficients $f_{BB}$, $f_{WW}$, $f_{BW}$, and $f_{all}$ using the cut (\[cut\_pt\]) are shown in Figures \[fig:3\] and \[fig:4\] (dashed lines) for all the final pair productions.
Another promising variable is the invariant mass of the particles produced in the Higgs decay, presented in Fig. \[fig:1\] (b) for the $b\bar{b}$ final state and Fig. \[fig:2\] (b) for the $W^+ W^-$ final state. Since the contribution of the anomalous couplings is dominated by on–mass–shell Higgs production with the subsequent $H \to b \bar{b},
\gamma\gamma, W^+W^-,ZZ$ decays, as can be clearly seen in the Figures \[fig:1\] (b) and \[fig:2\] (b), a more drastic cut would be to require $$(M_H-5) < M_{pair}^{inv}(\text{GeV}) < (M_H+5) \;\;,
\label{cut_minv}$$ where $M_{pair}^{inv}$ is the invariant mass of the final $b \bar{b}$, $\gamma\gamma$, $W^+ W^-$, or $ZZ$ pairs. The best constraints are obtained at NLC when this cut is applied, as we can be seen through the enhancement of the significance in Tables \[table1\]-\[table4\]. The 95%CL results obtained using the cut (\[cut\_minv\]) are also shown in Figures \[fig:3\] and \[fig:4\]. These results are more restrictive than the constraints obtained at LEPI and at low energy \[Eq. (\[limits\])\] and for $ZZ\gamma$ and $Z\gamma\gamma$ production at LEPII and NLC [@our2], especially for $f_{BB}$ and $f_{WW}$.
Conclusions
===========
The search for the effect of higher dimensional operators that give rise to anomalous Higgs boson couplings may provide important information on physics beyond the SM and should be pursued in all possible reactions. In this paper, we have studied the $b \bar{b}$, $\gamma\gamma$, $W^+ W^-$,and $ZZ$ production in high energy $e^+ e^-$ colliders (NLC) via photon fusion, focusing on the operators that generate anomalous $H\gamma\gamma$ coupling.
We established the limits that can be imposed at NLC through the analysis of the impact of the anomalous coupling over the total cross section of processes involving two final bottons, photons, W’s, and Z’s bosons. In order to improve the sensitivity of NLC to this anomalous Higgs boson production, the limits were evaluated for the cases where a convenient cut on the transverse momentum spectrum and on the invariant mass spectrum of the final state particles is used.
Typical values of a few TeV$^{-2}$ are reached in our analyses. Our results are more restrictive than the constraints obtained at low energy data, at LEPI, and for $ZZ\gamma$ and $Z\gamma\gamma$ production at LEPII and NLC [@our2]. Therefore, the NLC should provide important hints about the existence of new physics beyond the Standard Model.
I would like to thank S.F. Novaes and J.K. Mizukoshi for very useful discussions. This work was supported in part by the Director, Office of Science, Office of Science, Office of Basic Energy Services, of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP).
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-2 cm
-2 cm
Cross Section (fb) Without Cuts Transverse Momentum Cut Invariant Mass Cut
---------------------------------------------- -------------- ------------------------- --------------------
Background ($e^+ e^- \to b \bar{b}$) 381 10.9 0
Background ($\gamma \gamma \to b \bar{b}$) 783 6.8 1.7
Background ($W^+ W^- \to b \bar{b}$) 9.0 6.2 3.1
Background ($Z Z \to b \bar{b}$) 0.36 0.16 0.13
Signal ($\gamma \gamma \to H \to b \bar{b}$) 66.4 57.6 66.4
Significance 11 66 168
: Background and Signal ($f_{all}=30$ TeV$^{-2}$ and $M_H=100$ GeV) cross sections (in fb) for the $b \bar{b}$ final state at NLC with a center of mass energy of 500 GeV. The Significance is given by the fraction $\left(\frac{\text{Number of Signal events}}
{\sqrt{\text{Number of Background events}}}\right)$ for ${\cal L}=50$ fb$^{-1}$ and $e_f$=64% overall detection efficiency. The Transverse Momentum Cut is $25<P_{T_{b,\bar{b}}}(GeV)<(P_{T_{ano}}+5)$ while the Invariant Mass Cut is $(M_H-5)< M^{inv}_{b \bar{b}}(GeV)<(M_H+5)$.[]{data-label="table1"}
Cross Section (fb) $P_{T_{\gamma_1,\gamma_2}}>25$ GeV Transverse Momentum Cut Invariant Mass Cut
-------------------------------------------------- ------------------------------------ ------------------------- ----------------------
Background ($e^+ e^- \to \gamma \gamma$) 2981 942 0
Background ($\gamma \gamma \to \gamma \gamma$) $< 8 \times 10^{-5}$ $< 8 \times 10^{-5}$ $< 8 \times 10^{-5}$
Background ($W^+ W^- \to \gamma \gamma$) $< 3 \times 10^{-2}$ $< 2 \times 10^{-2}$ $< 5 \times 10^{-3}$
Background ($Z Z \to \gamma \gamma$) $<9 \times 10^{-5}$ $<9 \times 10^{-5}$ $< 9 \times 10^{-5}$
Signal ($\gamma \gamma \to H \to \gamma \gamma$) 14.7 12.7 14.7
Significance 1.5 2.3 $>$1200
: Background and Signal ($f_{all}=30$ TeV$^{-2}$ and $M_H=100$ GeV) cross sections (in fb) for the $\gamma\gamma$ final state at NLC with a center of mass energy of 500 GeV. The Significance is given by the fraction $\left(\frac{\text{Number of Signal events}}
{\sqrt{\text{Number of Background events}}}\right)$ for ${\cal L}=50$ fb$^{-1}$ and $e_f$=64% overall detection efficiency. The Transverse Momentum Cut is $25<P_{T_{\gamma_1,\gamma_2}}(GeV)<(P_{T_{ano}}+5)$ while the Invariant Mass Cut is $(M_H-5)< M^{inv}_{\gamma\gamma}(GeV)<(M_H+5)$.[]{data-label="table2"}
Cross Section (fb) Without Cuts Transverse Momentum Cut Invariant Mass Cut
-------------------------------------------- -------------- ------------------------- --------------------
Background ($e^+ e^- \to W W$) 2655 633 0
Background ($\gamma \gamma \to W W$) 196 87 9.8
Background ($W^+ W^- \to W W$) 5.4 3.4 2.5
Background ($Z Z \to W W$) 0.42 0.32 0.27
Signal ($\gamma \gamma \to H \to W^+ W^-$) 36.3 33.8 31.9
Significance 6 11 81
: Background and Signal ($f_{all}=30$ TeV$^{-2}$ and $M_H=200$ GeV) cross sections (in fb) for the $W^+ W^-$ final state at NLC with a center of mass energy of 1 TeV. The Significance is given by the fraction $\left(\frac{\text{Number of Signal events}}
{\sqrt{\text{Number of Background events}}}\right)$ for ${\cal L}=100$ fb$^{-1}$ and $e_f$=80% overall detection efficiency. The Transverse Momentum Cut is $25<P_{T_{W^+, W^-}}(GeV)<(P_{T_{ano}}+5)$ while the Invariant Mass Cut is $(M_H-5)< M^{inv}_{W^+ W^-}(GeV)<(M_H+5)$.[]{data-label="table3"}
Cross Section (fb) Without Cuts Transverse Momentum Cut Invariant Mass Cut
---------------------------------- -------------- ------------------------- --------------------
B($e^+ e^- \to Z Z$) 147 21.2 0
B($\gamma \gamma \to Z Z$) 0.06 0.05 0.05
B($W^+ W^- \to Z Z$) 1.58 0.88 0.92
B($Z Z \to Z Z$) 0.10 0.08 0.09
S($\gamma \gamma \to H \to Z Z$) 6.9 4.6 5.2
Significance ($S/\sqrt{B}$) 5 9 45
: Background and Signal ($f_{all}=30$ TeV$^{-2}$ and $M_H=200$ GeV) cross sections (in fb) for the $ZZ$ final state at NLC with a center of mass energy of 1 TeV. The Significance is given by the fraction $\left(\frac{\text{Number of Signal events}}
{\sqrt{\text{Number of Background events}}}\right)$ for ${\cal L}=100$ fb$^{-1}$ and $e_f$=80% overall detection efficiency. The Transverse Momentum Cut is $25<P_{T_{Z_1, Z_2}}(GeV)<(P_{T_{ano}}+5)$ while the Invariant Mass Cut is $(M_H-5)< M^{inv}_{ZZ}(GeV)<(M_H+5)$.[]{data-label="table4"}
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abstract: |
Any singular level of a completely integrable system (c.i.s.) with non-degenerate singularities has a singular affine structure. We shall show how to construct a simple c.i.s. around the level, having the above affine structure. The cotangent bundle of the desingularised level is used to perform the construction, and the c.i.s. obtained looks like the simplest one associated to the affine structure.
This method of construction is used to provide several examples of c.i.s. with different kinds of non-degenerate singularities.
author:
- 'Carlos Currás-Bosch [^1]'
title: '**Singular cotangent model**'
---
\[section\]
\[section\] \[section\] \[section\]
\[section\] \[section\] \[section\] \[section\] \[section\]
[: 53D05, 37J35,37G05 : Symplectic manifold; Singular Lagrangian foliation; Integrable Hamiltonian system; Affine structure on a level; Normal forms]{}
Introduction
=============
Let $(M^{2n},\omega, F)$ be a non-degenerate integrable system. This means that $(M^{2n},\omega)$ is a symplectic manifold and $F=(f_{1},...,f_{n})$ is a proper moment map, which is non-singular almost everywhere, and its singularities are of Morse-Bott type.
The $\Bbb R^{n}$-action generated by the Hamiltonian vector fields $H_{f_{1}},..., H_{f_{n}}$ gives a singular Lagrangian foliation on $(M^{2n},\omega)$. Any leaf is an orbit of this actions. At the same time, any connected component of $F^{-1}(c)$ is a level. We know that the regular levels are $n$-dimensional tori and the singular levels are finite union of several leaves.
A semilocal classification of such integrable systems is still open. It consists in finding a complete system of invariants describing symplectically a neighborhood of a level. Some approaches to solve this question has been currently made, see [@BM], [@Mir], [@MiZ], [@Mo]. As the local description of non-degenerate singularities is given in terms of products of elliptic, hyperbolic and focus-focus components, the number of elliptic, hyperbolic and focus-focus components at each point of the level will play an essential role in the classification problem.
As in the regular case, the Hamiltonian vector fields $H_{f_{1}},...,H_{f_{n}}$ endow any leaf and any level with an affine structure with singularities. In studying the semi-local classification, we have seen that this affine structure gives strong conditions on the set of invariants found. Our proposal in this paper is to prove that this affine structure allows to the construction of a completely integrable system around a given level $L_{0}$ of $(M^{2n},\omega, F)$, such that the affine structure on $L_{0}$ is the given one. This construction looks like the simplest one with the given affine structure on $L_{0}$. As the 1-jet of the former completely integrable system (c.i.s. from now on) and the one constructed in this way coincide, it takes sense to denote this c.i.s. as the linearized c.i.s. of the initial one.
The process of construction of the linearized c.i.s. indicates us a way to construct c.i.s. (in fact thew will be linearized completely integrable systems) with prescribed non-degenerate singularities along a given singular level. Some constructions are given.
I would like to express my gratitude to Pierre Molino for interesting and fruitful conversations on this subject.
Definition and basic properties.
================================
Local expressions {#subsection2.1}
-----------------
Let $(M^{2n},\omega,f_{1},...,f_{n})$ be an integrable system, i.e., $(M^{2n},\omega)$ is a symplectic manifold. The functions $f_{1},...,f_{n}$ are Poisson commuting (first integrals of a given Hamiltonian system), such that $df_{1}\wedge \cdot \cdot \cdot \wedge df_{n}\ne 0$ on a dense subset of $M^{2n}$, and the moment map $F:M^{2n}
\longrightarrow \Bbb R^{n}$, $F=(f_{1},...,f_{n})$, is proper. Such an integrable system is said to be non-degenerate if, in a neighborhood of each point $p_{0}\in M^{2n}$, there exist canonical coordinates $(x_{1},y_{1},...,x_{n},y_{n})$, and $n$ local functions $h_{1},...,h_{n}$, which have one of the following expressions: $$h_{i}=y_{i} \qquad \text{(regular terms)}$$ $$h_{i}=(x_{i})^{2}+(y_{i})^{2} \qquad \text{(elliptic
terms)}$$ $$h_{i}=x_{i}y_{i} \qquad \text{(hyperbolic terms)}$$ $$\begin{cases} h_{i}=x_{i}y_{i}+x_{i+1}y_{i+1} \\
h_{i+1}=x_{i}y_{i+1}-y_{i}x_{i+1} \qquad \text{(focus-focus
terms)} \end{cases}$$ such that:
1. $f_{1},...,f_{n}$ Poisson commute with $h_{1},...,h_{n}$.
2. $\{j_{p_{0}}^{2}f_{1},...,j_{p_{0}}^{2}f_{n} \} $ and $\{j_{p_{0}}^{2}h_{1},...,j_{p_{0}}^{2}h_{n} \}$ generate the same space of $2$-jets at $p_{0}$.
This notion of non-degeneracy implies obvious conditions on the space of $2$-jets generated at each point by $f_{1},...,f_{n}$. Conversely, assuming these infinitesimal conditions, the existence of such adapted coordinates is an important result, due to H.Eliasson [@E]. The demonstration has been completed by E.Miranda [@Mir] and E.Miranda & V.N.San [@MirSan] . Adapted coordinates will be referred to as Eliasson coordinates, or simply E-coordinates.
Let $\cal {L}$ be the Lagrangian singular foliation associated to this c.i.s., i.e. the leaves of $\cal{L}$ are the orbits of the $\Bbb R^{n}$-action generated by $H_{f_{1}},...,H_{f_{n}}$.
From a practical point of view, one can locally regard at $(M^{2n},\omega, \cal {L})$ as $(\Bbb
R^{2n},\omega_{0},{\cal{L}}_{0})$, where $\omega_{0}$ is the standard symplectic two-form on $\Bbb R^{2n}$ and ${\cal {L}}_{0}$ is given by $dh_{i}=0$, $i=1,...,n.$
Let $L_{0}$ be a singular level, and $p_{0}$ a point of $L_{0}$. By taking E-coordinates around $p_{0}$ we have the following characteristic numbers of the point $p_{0}$: the numbers $k_{e}$, $k_{h}$ and $k_{f}$ correspond to the number of elliptic, hyperbolic and focus-focus terms of the set $(h_{1},...,h_{n})$. The leaf through $p_{0}$ is $\Bbb T^{c}\times \Bbb R^{o}$, and the numbers $c$ and $o$ are called the degrees of closedness and openness of the leaf, respectively.
Following Zung (see [@Z2]), the $5$-tuple $(k_{e},k_{h},k_{f},c,o)$ is called the leaf-type and $(k_{e},k_{h},k_{f})$ the Williamson type of $p_{0}$. In general one has $k_{e}+k_{h}+2k_{f}+c+o=n$. In [@Z2] it is proved that the three numbers $k_{e},
k_{f}+c, k_{h}+k_{k}+o$, are invariants of each level. These numbers are known as the degrees of ellipticity, closedness and openness of the level.
Summarizing, one can say that a singular level $L_{0}$ is a compact $(n-k_{e})$-manifold, with self-intersections provided with a manifold structure of dimension less than $(n-k_{e})$. This level is endowed with a non-degenerate $\Bbb R^{n-k_{e}}$-action. Non-degenerate action means that the isotropy of this action at each point is linearizable.
A we have proved (see [@Cu]) that a completely integrable system, wit $k_{e}\ne 0$, is equivalent to a product of a standard elliptic model (with degree of ellipticity equal to $k_{e}$) with a c.i.s. with zero ellipticity degree, we will restrict our attention along this paper to the case $k_{e}=0$.
Desingularized level
--------------------
We assume that on the singular level $L_{0}$, with singular affine structure $\nabla _{0}$, the degree of ellipticity vanishes. In order to give a construction of a standard c.i.s., such that $L_{0}$ is a singular level, and the induced singular affine structure on it coincides with $\nabla _{0}$, we start by giving the construction of the so called desingularized level: As $k_{e}=0$ on $L_{0}$, $\text{dim.} L_{0}=n$. Let $\psi$ be a differentiable embedding of $M^{2n}$ in an euclidean space $\Bbb R^{l}$. Let $L_{0}^{r}\subset L_{0}$ be the subset of regular points of $L_{0}$, i.e., a point $p\in L_{0}$ lies in $L_{0}^{r}$ if and only if the rank of $dF$ is equal to $n$ at $p$. Note that $L_{0}^{r}$ is not, in general, a connected submanifold. We consider the following embedding of $L_{0}^{r}$ $$\varphi :L_{0}^{r} \hookrightarrow M^{2n}\times G_{n}(\Bbb R^{l})
\quad , \quad p \mapsto (p,[T_{\psi (p)}(\psi (L_{0}))]),$$ where $G_{n}(\Bbb R^{l})$ is the $n$ dimensional Grassmann manifold of $\Bbb R^{l}$.
We define the desingularized level, ${\hat{L}}_{0}$ as the closure of $\varphi (L_{0}^{r})$.
\[Claim 2.1\]$ {\hat{ L}}_{0}$ is a $n$-dimensional submanifold of $M^{2n}\times G_{n}(\Bbb R^{l})$
Let $q$ be a point of $\varphi (L_{0}^{r})$. As $\varphi$ is an embedding, we have a $n$-dimensional natural chart defined around $q$. Let $q=(p,[V])$ be a point in $\text{Cl}(\varphi(L_{0}^{r})) \setminus \varphi(L_{0}^{r})$. Obviously $p\in L_{0}\setminus L_{0}^{r}$. We know that there is a chart $U$ in $M^{2n}$, with coordinates $(x_{1},y_{1},...,x_{n},y_{n})$, centered at $p$, and $L_{0}\cap
U$ is given by $h_{i}=0$, $i=1,...,n$, where the functions $h_{i}$ are in the form of section \[subsection2.1\]. Let us have a look at the tangent space to $L_{0}$ at a point near to $p$: from the expressions of $h_{i}$, we see $$L_{0}\cap U=\prod _{j=1}^{s} (L^{j}_{0} \cap U^{j}),$$ where each $ L_{0}^{j} \cap U^{j}$ is of the form a) $U^{j}=D_{2}$, with coordinates $(x_{j},y_{j})$, $h_{j}=x_{j}$, and $L_{0}^{j}\cap U^{j}$ is given by $h_{j}=0$ (regular term). b) $U^{j}=D_{2}$, with coordinates $(x_{j},y_{j})$, $h_{j}=x_{j}y_{j}$, and $L_{0}^{j}\cap U^{j}$ is given by $h_{j}=0$ (hyperbolic term). c) $U_{j}=D_{2}\times D_{2}$, with coordinates $ (x_{j},y_{j},x_{j+1},y_{j+1})$, $h_{j}=x_{j}y_{j}+x_{j+1}y_{j+1}$, $h_{j+1}=x_{j}y_{j+1}-x_{j+1}y_{j}$, and $L_{0}^{j}\cap U^{j}$ is given by $h_{j}=h_{j+1}=0$ (focus-focus term). So, the tangent space at a point near to $p$ will be: In the case a) $$[T_{(0,y_{j})}(L_{0}^{j}\cap
U^{j})]=[<\frac{\partial}{\partial y_{j}}>],$$ so $$[T_{(0,0)}(L_{0}^{j}\cap U^{j})]=[<\frac{\partial}{\partial
y_{j}}>]$$ in the case b) $$[T_{(0,y_{j})}(L_{0}^{j}\cap U^{j})]=[<\frac{\partial}{\partial
y_{j}}
>]$$ or $$[T_{(x_{j},0)}(L_{0}^{j}\cap U^{j})]=[<\frac{\partial }{\partial x_{j}}>],$$ so $[T_{(0,0)}(L_{0}^{j}\cap U^{j})]$ is either $$[<\frac{\partial}{\partial y_{j}}>] \quad \text{or } \quad [<\frac{\partial
}{\partial x_{j}}>]$$.
and in the case c) $$[T_{(0,y_{j},0,y_{j+1})}(L_{0}^{j}\cap U^{j})]=[<\frac{\partial}{\partial y_{j}},\frac{\partial}{\partial y_{j+1}}>] ,$$ $$\text{or} \quad
[T_{(x_{j},0,x_{j+1},0)}(L_{0}^{j}\cap
U^{j})]=[<\frac{\partial}{\partial x_{j}},\frac{\partial}{\partial
x_{j+1}}
>],$$ so $$[T_{(0,0,0,0)}(L_{0}^{j}\cap U^{j})]=[<\frac{\partial}{\partial y_{j}},\frac{\partial}{\partial y_{j+1}}>]
\quad \text {or}$$ $$[T_{(0,0,0,0)}(L_{0}^{j}\cap U^{j})]=[<\frac{\partial}{\partial
x_{j}}, \frac{\partial}{\partial x_{j+1}}
>].$$
So, in fact,there are $2^{h+f}$ possibilities for the class of the grassmannian $[V]$ at $p \in L_{0} \setminus L_{0}^{r}$. As the structure of the chart around any one of these points will be a product structure, it will be sufficient to show the differentiability in the hyperbolic and in the focus-focus case. Let us do it, for instance, in the focus-focus cases: we can consider that $p=(0,0,0,0)$ and $[V]=[<\frac{\partial}{\partial x_{j}},\frac{\partial}{\partial x_{j+1}}>]$, then the coordinates we take around $(p,[V])$ are $(x_{j},x_{j+1})$. One can check easily that with these charts, and the previous charts around the regular points, we have a structure of $C^{\infty}$-manifold on ${\hat {L_{0}}}$.
The map $j:{\hat{L}}_{0} \longrightarrow M^{2n} \quad , \quad
(p,[V])\longmapsto p $ is differentiable, $j({\hat{L}}_{0})=L_{0}$, and the preimage of any singular point on $L_{0}$ consists of $2^{h+f}$ points on $ {\hat{L}}_{0}$. This map is a Lagrangian immersion in $(M^{2n},\omega)$
The singular cotangent model
============================
Affine structure on the desingularized level
--------------------------------------------
The Poisson action of the local $F$-basic functions defines a singular affine structure $\nabla _{0}$ on $L_{0}$. On each leaf of the level, the affine structure is defined by considering the infinitesimal generators of the $\Bbb R^{n}$-action as parallel vector fields. This $\Bbb R^{n}$-action can be lifted in a natural way to ${\hat{L}}_{0}$. Note that this lift is possible because the map $j:{\hat{L}}_{0} \longrightarrow M^{2n}$ is an immersion. As the affine structure on $L_{0}$ is provided by the Hamiltonian vector fields $H_{f_{1}}\vert_{L_{0}}=X_{1},...,H_{f_{n}}\vert
_{L_{0}}=X_{n}$ it seems natural to write as ${\hat{X_{i}}}$, $i=1,...,n$, the vector fields on ${\hat {L}}_{0}$ induced through the immersion $j$, and ${\hat{\nabla}}_{0}$ this affine structure.
By using the affine structure on ${\hat {L}}_{0}$ we are going to construct a completely integrable system on $(T^{*}{\hat {L}}_{0}, {\hat {\omega}}_{0})$, where ${\hat
{\omega}}_{0}$ is the standard symplectic two-form on $T^{*}{\hat
{L}}_{0}$, and such that the affine structure on ${\hat
{L}}_{0}$, induced by this c.i.s. is ${\hat {\nabla}} _{0}$. As a last step, by a standard process of gluing, by using natural local identifications, we will obtain a c.i.s., such that $L_{0}$ is one level, and the affine structure on $L_{0}$ will be $\nabla _{0}$.
A completely integrable system on $T^{*}{\hat{L}}_{0}$ {#subsection 3.2}
------------------------------------------------------
We define $n$ differentiable functions $g_{1}$,...,$g_{n}$ on $T^{*}{\hat{L}}_{0}$ by $$g_{i}(
(p,[V]),w):=<{\hat{X}}_{i}(p,[V]),w>$$ Our proposal is to see that $(T^{*}{\hat{L}}_{0},{\hat{\omega}}_{0},(g_{1},...,g_{n}))$ is completely integrable. To do it, we need only to prove that $\{
g_{i},g_{j}\} \vert _{0} =0$. By a continuity argument, it will be sufficient to prove it at the points $((p,[V]),-)\in {\hat
{L}}_{0}$, where $p$ is a regular point. We can take Eliasson coordinates $(x_{1},y_{1},...,x_{n},y_{n})$ around the point $p$, and as any basic function only depends on $(y_{1},...,y_{n})$ in a neighborhood of $p$, the expression of $H_{f_{i}}$ in this neighborhood will be of the form $\sum_{j=1}^{n} {\frac {\partial
f_{i}}{\partial y_{j}} }(0,...,0)\frac {\partial}{\partial
x_{j}}$, i.e. is a vector fields with constant coefficients (which is obvious because the vector field is affine parallel). Let $\alpha$ be the Liouville form on $T^{*}{\hat
{L}}_{0}$, in this neighborhood $\alpha =\sum _{i=1}^{n}
y_{i}dx_{i}$, $g_{i}=\sum _{k=1}^{n} y_{k} {\frac {\partial
f_{i}}{\partial y_{k}}}(0,...,0)$, so $dg_{i}=\sum_{k=1}^{n}{\frac
{\partial f_{i}}{\partial y_{k}}}(0,....,0) dy_{k}$, and obviously $$\Lambda ^{0} (dg_{i},dg_{j})=0.$$
Once we know that $(T^{*}{\hat{L}}_{0},{\hat{\omega}}_{0},(g_{1},...,g_{n}))$ is completely integrable, we remark that, in general, is not proper: let us assume, for instance that $\text{dim.}L_{0}=1$, and let $p$ be an hyperbolic point, we can write $\omega_{0}=dx\wedge dy$, and $h=xy$, then $x=0$ is a leaf. In order to have a proper completely integrable system around $L_{0}$we have to define a gluing between points on ${\hat {L}}_{0}$ which are projected, via $j$, to the same singular point on $L_{0}$. As the singular composition of each point is, in fact, a product of hyperbolic and focus-focus components, we need show how this gluing is done in hyperbolic and focus-focus cases.
Let $(p,[V_{1}]), (p,[V_{2}])\in {\hat {L}}_{0}$, where $p \in L_{0}$ is a purely hyperbolic point (degree of hyperbolicity equal to one). Then in Eliasson coordinates $(x,y)$ around the point $p$, we may assume $[V_{1}]=[<\frac {\partial }{\partial x}>]$ and $[V_{2}]=[<\frac{\partial}{\partial y}>]$. We take canonical coordinates in $T^{*}{\hat {L}}_{0}$ in two neighborhoods $U_{1}$ of $(p,[V_{1}])$, $U_{2}$ of $(p,[V_{2}])$, and we denote them by $(x,y)$ and $(X,Y)$ respectively. Any basic function in these neighborhoods depends on $xy$ and $XY$ respectively, and the canonical symplectic two form is $dx\wedge dy$ and $dX\wedge dY$ respectively. The symplectomorphism from $U_{1}$ onto $U_{2}$, we are searching for, is expressed in these coordinates by $X=-y, Y=x$.
We can proceed in a similar way for two points $(p,[V_{1}])$, $(p,[V_{2}])$, in the preimage of a focus-focus point $p\in
L_{0}$. Regarding to Claim \[Claim 2.1\] one can consider $$[V_{1}]=[<\frac{\partial }{\partial x_{1}},\frac{\partial }{\partial
x_{2}}>] \quad , \quad [V_{2}]=[<\frac{\partial }{\partial
y_{1}},\frac{\partial }{\partial y_{2}}>].$$ We take canonical coordinates in $T^{*}{\hat {L}}_{0}$ in two neighborhoods $U_{1}$ of $(p,[V_{1}])$ and $U_{2}$ of $(p,[V_{2}])$, and we denote them by $(x_{1},x_{2},y_{1},y_{2})$ and $(X_{1},X_{2},Y_{1},Y_{2})$ respectively. Any basic function in these neighborhoods depends on $x_{1}y_{1}+x_{2}y_{2} ,
-y_{1}x_{2}+x_{1}y_{2}$ and $X_{1}Y_{1}+X_{2}Y_{2},
-Y_{1}X_{2}+X_{1}Y_{2}$, respectively. The symplectomorphism from $U_{1}$ to $U_{2}$ we need is given by: $X_1=-y_{1}$, $X_2=-Y_2$, $Y_1
=X_1$, $Y_2=-X_2$.
The singular cotangent model
----------------------------
Once we have shown how to identify the neighborhoods of the points in the same preimage we get a germ of $2n$-dimensional symplectic manifold $(N, \omega _{0})$ containing $L_{0}$ as a singular Lagrangian submanifold. Now we check that the functions $g_{i}$, $i=1,...,n$, can be projected to $N$. To do it, we must see that the functions $g_{i}$ remain invariant under the above defined identifications in the hyperbolic and the focus-focus cases. In the hyperbolic case, we see that in the above defined neighborhood $U_{1}$, any function $g_{i}$ is of the form $y\frac {\partial h_{i}}{\partial y}(0)$, and a similar expression in $U_{2}$. Having in mind the symplectomorphism $Y=-x, X=y$, one sees that $h_{i}$ is preserved. For the focus-focus identification the proof is similar.
It is quite obvious from the given construction of $(N, \omega
_{0},(g_{1},...,g_{n}))$that the singular affine structure on $L_{0}$ is the previous one.
Construction of c.i.s. with prescribed singularities
----------------------------------------------------
As a sort of application of the above considerations, let us point out how to give some c.i.s. with prescribed singularities, around a singular level. The intrinsic geometry of the level, i.e. the number and kind of singular points, is obviously related with the kind of singularities along its singular points.
We show how to obtain a c.i.s. around a $2$-dimensional singular level with a focus-focus point and one circle of hyperbolic points: Let us consider the $2$-sphere $S^{2}$. This manifold $S^{2}$ will play the role of $\hat{L}_{0}$ of the above sections. We shall consider $T^{*}S^{2}$ and two vector fields, with singularities, which will be used to define two functions on $T^{*}S^{2}$. These functions have singularities at several points, and by furnishing the gluings at the corresponding singular points, we will get the c.i.s. around the singular level. Let $\theta$ (longitude) and $\varphi$ (latitude) be polar coordinates on $S^{2}$.The vector fields to consider are: $X=\frac{\partial}{\partial \theta}$ and $Y=h(\varphi)\frac
{\partial}{\partial \varphi}$. We have to give a goodexpression for $h(\varphi)$, obviously we take $h(-\frac{\pi}{2})=0$, $h(\frac{\pi}{2})=0$, and $h(-\frac{\pi}{4})=h(\frac{\pi}{4})=0$. We consider four open subsets of $S^{2}$ $$U_{1}=\{(\theta,\varphi)\quad \vert \quad -\frac{\pi}{2}\le \varphi
<-\frac{\pi}{2}+\varepsilon \}$$ $$V_{1}=\{(\theta,\varphi)\quad \vert \quad
-\frac{\pi}{4}-\varepsilon <\varphi < \frac{\pi}{4}+\varepsilon \}$$ $$V_{2}=\{(\theta,\varphi)\quad \vert \quad \frac{\pi}{4}-\varepsilon <\varphi
<\frac{\pi}{4}+\varepsilon \}$$ $$U_{2}=\{ (\theta,\varphi)\quad \vert \quad
\frac{\pi}{2}-\varepsilon <\varphi \le \frac{\pi}{2} \}$$ where $\varepsilon <\frac{\pi}{8}$.
On $U_{1}$ and $U_{2}$we can take as coordinates the first two cartesian coordinates $(x_{1},x_{2})$, and consider the vector fields: on $U_{1}$, $x_{1}\frac
{\partial}{\partial x_{1}}+x_{2}\frac{\partial}{\partial
x_{2}}=-\frac{\cos \varphi}{\sin \varphi}\cdot
\frac{\partial}{\partial \varphi}$ and $-x_{2}\frac{\partial}{\partial
x_{1}}+x_{1}\frac{\partial}{\partial
x_{2}}=\frac{\partial}{\partial \theta}$.
on $U_{2}$, $-x_{1}\frac {\partial}{\partial
x_{1}}-x_{2}\frac{\partial}{\partial x_{2}}=\frac{\cos
\varphi}{\sin \varphi}\cdot \frac{\partial}{\partial \varphi}$ and $-x_{2}\frac{\partial}{\partial
x_{1}}+x_{1}\frac{\partial}{\partial
x_{2}}=\frac{\partial}{\partial \theta}$.
On $V_{1}$ and $V_{2}$ we can take $(\theta , \varphi)$ as coordinates and the following vector fields: on $V_{1}$, $-(\varphi
+\frac{\pi}{4})\frac{\partial}{\partial \varphi}$ and $\frac{\partial}{\partial \theta}$. on $V_{2}$, $(\varphi-\frac{\pi}{4})\frac{\partial}{\partial
\varphi}$ and $\frac{\partial}{\partial \theta}$. Now we see that the function $h(\varphi)$ we are searching for can be a differentiable function of $\varphi$, such that its values in $U_{1},V_{1},U_{2},V_{2}$ are the above prescribed and not vanishing on $S^{2}\setminus (U_{1}\cup V_{1}\cup U_{2} \cup
V_{2})$.
Finally, we consider on $T^{*}S^{2}$ the pair of functions $f, g$ associated to the vector fields $X,Y$, defined as follows: for any point $(z,w)\in T^{*}S^{2}$, $f(z,w):=<X(z),w>$, $g(z,w):=<Y(z),w>$. The c.i.s. we are searching for is obtained by a process of gluing from $(T^{*}S^{2},\omega _{0},(f,g))$. This gluing can be easily established by defining two symplectomorphisms: one of them is a local symplectomorphism between $(T^{*}U_{1}, (x_{1}=0,x_{2}=0,0,0))$ and $(T^{*}U_{2},(X_{1}=0,X_{2}=0,0,0))$. We recall (see hyperbolic gluing in section\[subsection 3.2\]) that the mapping is given by $$(x_{1},x_{2},y_{1},y_{2})\longmapsto
(X_{1}=-y_{1},X_{2}=-y_{2},Y_{1}=x_{1},Y_{2}=x_{2}).$$
The other symplectomorphism we need to conclude the gluing is a semi-local symplectomorphism from $(T^{*}V_{1},(\theta,\varphi=-\frac{\pi}{4},0,0))$ in $T^{*}V_{2},(\theta,\varphi=\frac{\pi}{4},0,0))$, described as follows: on $T^{*}V_{1}$ we take $(\theta,
\varphi +\frac{\pi}{4}, \Theta,\Phi)$ as canonical coordinates. In the same form, we take $({\bar{\theta}}=\theta,{\bar{\varphi}}
-\frac{\pi}{4}=\varphi-\frac{\pi}{4},{\bar{\Theta}}=\Theta,{\bar{\Phi}})$ as canonical coordinates in $T^{*}V_{2}$. The symplectomorphism we need to define the gluing is $$(\theta,\varphi +\frac{\pi}{4},\Theta,\Phi) \longmapsto
({\bar{\theta}}=\theta,{\bar{\varphi}}
-\frac{\pi}{4}=\Phi,{\bar{\Theta}}=\Theta,{\bar{\Phi}}=-(\varphi +
\frac{\pi}{4})).$$
Thus, the quotient of a germ of neighborhood of $S^{2}$ in $T^{*}S^{2}$, by using these identifications provides us a germ of completely integrable system around a level having one singular point of focus-focus type and one circle of hyperbolic points.
One sees from this construction that the same arguments can serve to give c.i.s. with several points of focus-focus type; it should be necessary to use different copies of $S^{2}$ and define a gluing by using the poles alternatively. The above construction suggests different ways of having circles of hyperbolic points in the level.
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Departament d’Àlgebra i Geometria, Universitat de Barcelona (Spain)
E-mail address:carloscurrasbosch@ub.edu
[^1]: partially supported by DGICYT: MTM 2006-04353.
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abstract: 'The beam energy dependence of correlation lengths (the Hanbury-Brown-Twiss radii) is calculated by using a blast-wave model and the results are comparable with those from RHIC-STAR beam energy scan data as well as the LHC-ALICE measurements. A set of parameter for the blast-wave model as a function of beam energy under study are obtained by fit to the HBT radii at each energy point. The transverse momentum dependence of HBT radii is presented with the extracted parameters for Au + Au collision at $\sqrt{s_{NN}} = $ 200 GeV and for Pb+Pb collisions at 2.76 TeV. From our study one can learn that particle emission duration can not be ignored while calculating the HBT radii with the same parameters. And tuning kinetic freeze-out temperature in a range will result in system lifetime changing in the reverse direction as it is found in RHIC-STAR experiment measurements.'
author:
- 'S. Zhang'
- 'Y. G. Ma[^1]'
- 'J. H. Chen'
- 'C. Zhong'
title: 'Beam energy dependence of Hanbury-Brown-Twiss radii from a blast-wave model'
---
Introduction
============
The Quark-Gluon-Plasma (QGP) predicted by quantum chromodynamics (QCD) [@QCD-QGP] can be formed in relativistic heavy-ion collisions. It is believed that this kind of new state of matter is produced in the early stage of central Au + Au collisions at the top energy in the Relativistic Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory [@RHICWithePaper]. It was concluded that the hot-dense matter is a strongly interacting partonic matter named as sQGP under extreme temperature and energy density with sufficient experimental evidences [@RHIC-SQGP; @RHIC-SQGP2; @RHIC-SQGP3; @RHIC-SQGP4; @RHIC-SQGP5]. Recently, many results in Pb + Pb and p+Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV in the Large Hadron Collider (LHC) were also reported for exploring properties of the hot-dense quark-gluon matter [@ALICE-chDen; @ALICE-chCDepen; @ALICE-chRAA; @ALICE-pionSpectra]. Mapping the QCD phase diagram and locating the phase boundary and possible critical end point becomes hot topic in the field [@Itoch-QaurkStars; @QCD-QGP; @Alford; @Costa; @Zong]. The properties inherited from QGP will imprint signal on observables which can reflect phase transition information. The geometry of the system shall undergo phase space evolution from QGP stage to hadron kinetic freeze-out stage, which can be considered as an observable that is sensitive to the equation of state [@HBT-BES-Th; @HBT-BES-EXP-STAR]. Hanbury-Brown-Twiss (HBT) technique invented for measuring sizes of nearby stars [@HBT-STARSize] was extended to particle physics [@HBT-Particle] and heavy-ion collisions [@Koonin; @Pratt; @HBT-HeavyIon; @Boal; @HBT-UHeinz; @Ma; @Hu; @ZhangZQ]. The HBT technique can also be applied to extract the precise space-time properties from particle emission region at kinetic freeze-out stage in heavy-ion collisions. Furthermore this technique has been evolved to search for new particles and to measure particle interactions [@STAR-Lambda-Lambda; @STAR-pLambda; @STAR-antiProtonInter].
Experimental results on HBT study in high energy nuclear reaction were reported by STAR [@HBT-STAR-200GeV] and PHENIX [@HBT-PHENIX-200GeV] at RHIC top energy in Au+Au collisions, as well as by ALICE [@HBT-ALICE] at $\sqrt{s_{NN}} = $ 2.76 TeV in Pb+Pb collisions. Recently STAR and PHENIX Collaborations have also presented beam energy dependence of HBT radii [@HBT-BES-EXP-STAR; @HBT-BES-EXP-PHENIX] and a non-monotonic changing behaviour for the square difference between outward radius and sideward radius ($R_{out}^2-R_{side}^2$) with increase of beam energy was found. This behaviour could be sensitive to equation of state and was considered as a probe related to the critical end point of QGP phase transition [@HBT-BES-Th]. A finite-size scaling (FSS) analysis of experimental data was performed in Ref. [@HBT-BES-Th] and the analysis suggested that a second order phase transition was taken place with a critical end point located at a chemical freeze-out temperature of $\sim$ 165 MeV and a baryon chemical potential of $\sim$ 95 MeV.
In this paper we present beam energy dependence of HBT radii calculated from a blast-wave model. Firstly, experimental data of HBT radii from RHIC-STAR and LHC-ALICE are fitted and parameters for the blast-wave model are configured as a function of beam energy. The transverse momentum dependence of HBT radii are calculated at RHIC top energy and LHC energy with these parameters. From the results, it was found that particle emission duration is important for calculating transverse momentum dependence of HBT radii and changing of kinetic freeze-out temperature will result in system lifetime changing in reverse direction as that in the RHIC-STAR experimental analysis [@HBT-BES-EXP-STAR].
The paper is organised as following. In Sec. II, blast-wave model and HBT correlation function are briefly introduced. Some kinetic parameters are presented as a function of beam energy. Section III presents energy dependence of extracted HBT radii with various kinetic temperature, system lifetime and particle emission duration etc. Transverse momentum dependence of HBT radii is discussed in Section IV. Finally Section V gives the summary.
Blast-wave model and HBT correlation function
=============================================
The particle emission function $S(x,p)$ in heavy-ion collisions used in this study is similar as in reference [@BLWave-Fabrice],
$$\begin{aligned}
S(x,p) & = & m_T\cosh(\eta-Y)\Omega(r,\phi_s)e^{-(\tau-\tau_0)^2/2\Delta\tau^{2}}\frac{1}{e^{K\cdot u/T_{kin}}\pm1}\nonumber\\
& = & m_T\cosh(\eta-Y)\Omega(r,\phi_s)e^{-(\tau-\tau_0)^2/2\Delta\tau^{2}}\sum^{\infty}_{n=1}(\mp)^{n+1}e^{-K\cdot u/T_{kin}}\nonumber\\
& \simeq & m_T\cosh(\eta-Y)\Omega(r,\phi_s)e^{-(\tau-\tau_0)^2/2\Delta\tau^{2}}e^{-K\cdot u/T_{kin}}.
\label{eq:EmissionFun}\end{aligned}$$
In cylindrical coordinates, source moving four-velocity and momentum can be written respectively as, $$\begin{aligned}
u_{\mu}(x) = & (\cosh\eta\cosh\rho(r,\phi_s),\sinh\rho(r,\phi_s)\cos\phi_b,\nonumber\\
& \sinh\rho(r,\phi_s)\sin\phi_b,\sinh\eta\cosh\phi_b),
\label{eq:Velocity}\end{aligned}$$ and $$\begin{aligned}
K_{\mu} = & (m_T\cosh Y, p_T\cos\phi_p, p_T\sin\phi_p, m_T\sinh Y).
\label{eq:Momentum}\end{aligned}$$ And the flow rapidity is given by, $$\rho(r,\phi_s) = \tilde{r}[\rho_0+\rho_2\cos(2\phi_b)],
\label{eq:FlowRap}$$ here the normalized elliptical radius, $$\tilde{r} \equiv \sqrt{\frac{[r\cos\phi_s]^2}{R_x^2}+\frac{[r\sin\phi_s]^2}{R_y^2}},
\label{eq:Tilde_r}$$ with $$\tan\phi_s = \left(\frac{R_y}{R_x}\right)^2\tan\phi_b.
\label{eq:phis_phib}$$
In equation Eq. (\[eq:EmissionFun\]), spatial weighting of source elements is selected as a simple pattern [@BLWave-Fabrice], $$\begin{aligned}
\Omega(r,\phi_s) = \left\{ \begin{array}{ll}
1 &, \tilde{r}<1\\
0 &, \tilde{r}>1
\end{array}
\right ..
\label{eq:Omega}\end{aligned}$$
![\[fig:Tkin\] (Color online) Kinetic freeze-out temperature as a function of centre-of-mass energy $\sqrt{s_{NN}}$. Data are from [@STARSYS-Spectra; @STAR-Kumar-BES]. ](fig1-Tkin-sNN){width="8.2cm"}
Here are the main parameters in this model, the kinetic freeze-out temperature $T_{kin}$, the radial flow parameter $\rho_0$, the “elliptic flow parameter” $\rho_2$ which controls second-order oscillation of transverse rapidity by the relation as in Eq. (\[eq:phis\_phib\]), the system lifetime $\tau_0$ and the particle emission duration $\Delta\tau$, $R_x$ and $R_y$ related to system size and space asymmetry. In this calculation we assume that the system is in most central heavy-ion collisions and thus set the $R_x = R_y = R_0$, $\rho_2 = 0$. In experimental measurement, hadron spectra can be fitted by the blast-wave model with integrating the emission function except the $p_T$ and $Y$. The kinetic freeze-out temperature $T_{kin}$ and the averaged radial flow $\langle\beta\rangle$ was extracted from the fit. For detail technique information, one may refer to [@STARSYS-Spectra]. The averaged radial flow is related to the flow rapidity $\rho = \tanh^{-1}\beta$, from which the radial flow parameter $\rho_0$ is calculated. Figure \[fig:Tkin\] and \[fig:beta\] present the measured $T_{kin}$ and $\langle\beta\rangle$ at a wide beam energy range respectively. The data come from [@STARSYS-Spectra; @STAR-Kumar-BES]. The kinetic freeze-out temperature $T_{kin}$ and the averaged radial flow $\langle\beta\rangle$ can be parametrised as a function of $\sqrt{s_{NN}}$ by empirical formula,
$$\begin{aligned}
T_{kin} & = & T_{lim}\frac{1}{(1+\exp(8.559-\ln(\sqrt{s_{NN}})/0.093))/(\sqrt{s_{NN}}^{0.057}/0.846)}\nonumber\\
\langle\beta\rangle & = & \beta_{lim}\frac{1}{(1+\exp(5.666-\ln(\sqrt{s_{NN}})/0.124))\sqrt{s_{NN}}^{0.065})},
\label{eq:TBeta_sNN}\end{aligned}$$
where $T_{lim}$ = 169.171 MeV and $\beta_{lim}$ = 0.399. And then free parameters in the blast-wave model will be the $R_0$, the $\tau_0$ and the $\Delta\tau$, which are all related to expanding characters of the collision system. And it will be determined by the HBT correlation calculation which will be discussed below in detail.
![\[fig:beta\] (Color online) The averaged radial flow $\langle\beta\rangle$ as a function of center-of-mass energy $\sqrt{s_{NN}}$. Data are from [@STARSYS-Spectra; @STAR-Kumar-BES]. ](fig2-beta-sNN){width="8.2cm"}
In our previous works, the blast-wave model was coupled with thermal equilibrium model to describe the hadron production and its spectra with a range of thermal parameters [@SZhang-LHC], and with coalescence mechanism to calculate the light nuclei production and to predict the di-baryons production rate [@BW-XueLiang; @BW-Neha]. In addition, the DRAGON model [@DRAGON] and the THERMINATOR2 [@THERMINATOR] model have also been developed as event generator to study the phase-space distribution of hadrons at freeze-out stage. It is also successfully applied in experimental data analysis [@STARSYS-Spectra; @STAR-Kumar-BES] to extract the kinetic freeze-out properties and to provide the phase-space distribution to calculate the HBT correlation in theory [@BLWave-Fabrice; @HBT-UAW].
The identical two particle HBT correlation function can be written as [@HBT-UAW; @HBT-UHeinz], $$\begin{aligned}
C(\vec{K},\vec{q}) = 1+%\left|\left<e^{i\vec{q}\cdot(\vec{x}-\vec{\beta}t)}\right>\right|
\left|\frac{\int d^4xe^{i\vec{q}\cdot(\vec{x}-\vec{\beta}t)}S(x,K)}{\int d^4xS(x,K)}\right|,
\label{eq:HBTCorrFun}\end{aligned}$$ here $K$ is average momentum for the two particles, $K=\frac{1}{2}(p_1+p_2)$, $q$ denotes relative momentum between two particles, $q=p_1-p_2$, and $\vec{\beta}=\vec{K}/\vec{K_0}$. From Refs. [@HBT-UAW; @BLWave-Fabrice; @HBT-oslSys], the “out-side-long" coordinates system is used in this calculation, in which the long direction $R_{long}$ is parallel to the beam, the sideward direction $R_{side}$ is perpendicular to the beam and total pair momentum, and the outward direction $R_{out}$ is perpendicular to the long and sideward directions. After expanding angular dependence of $C(K,q)$ in a harmonic series with the “out-side-long" coordinates system, the HBT radii can be written as [@HBT-UAW; @BLWave-Fabrice],
$$\begin{aligned}
R_{side}^2&=&\frac{1}{2}(\langle\tilde{x}^2\rangle+\langle\tilde{y}^2\rangle)-\frac{1}{2}(\langle\tilde{x}^2\rangle-\langle\tilde{y}^2\rangle)\cos(2\phi_p)-\langle\tilde{x}\tilde{y}\rangle\sin(2\phi_p),\nonumber\\
%
R_{out}^2&=&\frac{1}{2}(\langle\tilde{x}^2\rangle+\langle\tilde{y}^2\rangle)+\frac{1}{2}(\langle\tilde{x}^2\rangle-\langle\tilde{y}^2\rangle)\cos(2\phi_p)+\langle\tilde{x}\tilde{y}\rangle\sin(2\phi_p)-2\beta_T(\langle\tilde{t}\tilde{x}\rangle\cos\phi_p+\langle\tilde{t}\tilde{y}\rangle\sin\phi_p)+\beta_T^2\langle\tilde{t}^2\rangle,\nonumber\\
%
R_{long}^2&=&\langle\tilde{z}^2\rangle-2\beta_l\langle\tilde{t}\tilde{z}\rangle+\beta_l^2\langle{\tilde{t}}^2\rangle,
\label{eq:HBT-Radii}\end{aligned}$$
where $$\begin{aligned}
\langle f(x)\rangle(K)&\equiv&\frac{\int d^4xf(x)S(x,K)}{\int d^4xS(x,K)},\nonumber\\
\tilde{x}^{\mu}&\equiv&x^{\mu}-\langle\tilde{x}^{\mu}\rangle(K).\end{aligned}$$
In the calculation, observables are related to integrals of emission function (\[eq:EmissionFun\]) over phase space $d^4x=dxdydzdt=\tau d\tau d\eta rdrd\phi_s$, weighted with some quantities $B(x,K)$. If $B(x,K)=B'(r,\phi_s,K)\tau^i\sinh^j\eta\cosh^k\eta$, then the integrals can be written as in [@BLWave-Fabrice], $$\begin{aligned}
\int^{2\pi}_0d\phi_s\int^{\infty}_0rdr\int^{\infty}_{-\infty}d\eta\int^{\infty}_{-\infty}\tau d\tau S(x,K)B(xK)\nonumber\\
=m_TH_i\{B'\}_{j,k}(K),\end{aligned}$$ and some useful integrals, $$\begin{aligned}
H_i&\equiv&\int^{\infty}_{-\infty}d\tau\tau^{i+1}e^{-(\tau-\tau_0)^2/2\Delta\tau^2},\nonumber\\
G_{j,k}(x,K)&\equiv&\int^{\infty}_{-\infty}d\eta e{-\beta\cosh\eta}\sinh^j\eta\cosh^{k+1}\eta,\nonumber\\
\{B'\}_{j,k}(K)&\equiv&\int^{2\pi}_0d\phi_s\int^{\infty}_0rdrG_{j,K}(x,K)B'(x,K)\nonumber\\
&\times&e^{\alpha\cos(\phi_b-\phi_p)}\Omega(r,\phi_s),\end{aligned}$$ where we define, $$\begin{aligned}
\alpha&\equiv&\frac{p_T}{T}\sinh\rho(r,\phi_s),\nonumber\\
\beta&\equiv&\frac{m_T}{T}\cosh\rho(r,\phi_s).\end{aligned}$$
Retieère and Lisa [@BLWave-Fabrice] have provided a systematic analysis of parameter range for the blast-wave model and investigated the $p_T$ spectra, the collective flow, and the HBT correlation of hadrons produced in heavy-ion collisions. In this calculation we will use the algorithm developed in Ref. [@BLWave-Fabrice; @HBT-UAW] to study the energy and transverse momentum dependence of pion HBT correlation radii. Based on the discussion above, the free parameters will be $R_0$, $\tau_0$ and $\Delta\tau$ which can be determined by fitting experimental data by Eq. (\[eq:HBT-Radii\]). Before the study of energy dependence on HBT radii, we calculated pion’s spectra by using this algorithm in the blast-wave model, $$\begin{aligned}
\frac{dN}{p_Tdp_T} = \int d\phi_p\int d^4xS(x,K).\end{aligned}$$
Figure \[fig:pionSpectra\] presents pion’s spectra which are comparable with experimental data from STAR at $\sqrt{s_{NN}}$ = 200 GeV in central Au+Au collisions [@STAR-pionSpectra] and ALICE at $\sqrt{s_{NN}}$ = 2.76 TeV in central Pb+Pb collisions [@ALICE-pionSpectra], respectively.
![\[fig:pionSpectra\] Comparison of pion’s spectra from blast-wave model (lines) and the data in central Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV [@STAR-pionSpectra] and the data in central Pb+Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV [@ALICE-pionSpectra]. ](fig3-pionSpectra){width="8.2cm"}
energy dependence of HBT radii
==============================
The parameters are configured as following. The kinetic freeze-out temperature $T_{kin}$ and the averaged radial flow $\langle\beta\rangle$ are from Eq. (\[eq:TBeta\_sNN\]) as a function of $\sqrt{s_{NN}}$, but in some case $T_{kin}$ are fixed to 90, 100 and 120 MeV for comparison. In numerical calculation, the particle emission duration $\Delta\tau$ is set to zero and in another case the energy dependence of $\Delta\tau$ will be extracted by fit on the data. The $R_0$ will also be given by fit the data at each energy point. The experimental results of HBT radii are taken from the STAR and the ALICE collaborations [@HBT-BES-EXP-STAR; @HBT-ALICE] at centre-of-mass energy $\sqrt{s_{NN}}$ points, 7.7, 11.5, 19.6, 27, 39, 62.4, 200 and 2760 GeV. The difference between calculated radii results and the experimental data should reach a minimum value ($\delta_s, \delta_o, \delta_l$) for each energy point, $$\begin{aligned}
R_{side}(\text{th})-R_{side}(\text{exp})&=&\delta_s,\nonumber\\
R_{out}(\text{th})-R_{out}(\text{exp})&=&\delta_o,\nonumber\\
R_{long}(\text{th})-R_{long}(\text{exp})&=&\delta_l.\nonumber\end{aligned}$$ Actually from Eq.( \[eq:HBT-Radii\]) and the algorithm in [@BLWave-Fabrice; @HBT-UAW], one can find the HBT radii parameter dependence as following: $$\begin{aligned}
R_{side}^2&=&R_{side}^2(T_{kin},\rho_0,R_0),\nonumber\\
R_{out}^2&=&R_{out}^2(T_{kin},\rho_0,R_0,\tau_0,\Delta\tau),\nonumber\\
R_{long}^2&=&R_{long}^2(T_{kin},\rho_0,R_0,\tau_0,\Delta\tau).\end{aligned}$$ So $R_0$ can be determined directly by fit on $R_{side}^2$. And the $\tau_0$, $\Delta\tau$ can be extract by fit on $R_{out}^2$ and $R_{long}^2$ simultaneously. We then learnt that the difference of $R_{out}^2-R_{side}^2$ not only depends on the system lifetime $\tau_0$ but also on the particle emission duration $\Delta\tau$.
{width="15cm"}
![\[fig:R0\] (Color online) $R_0$ as a function of center-of-mass energy $\sqrt{s_{NN}}$ with different kinetic temperature parametrisation. ](fig5-R0){width="8.2cm"}
Figure \[fig:Rosl\] presents our calculation on HBT radii for identical charged pion-pion correlation with the configured parameters. The HBT radii show an increasing trend with the increasing of centre-of-mass energy $\sqrt{s_{NN}}$. In the case of $\Delta\tau\neq 0$, the results can describe experimental data successfully. However, for $\Delta\tau$ = 0.0, the $R_{out}$ cannot be fitted despite $R_{long}$ can be well matched by the calculation. Since $T_{kin}$ and $\rho_0$ are taken from experimental results, $R_{side}$ will only depend on parameter $R_0$, which reflects the system size where particles are emitted. Figure \[fig:R0\] displays the extracted $R_0$ as a function of $\sqrt{s_{NN}}$. It demonstrates a similar trend of energy dependence as $R_{side}$. With fixed temperature of $T_{kin}$ (90, 100, 120 MeV), it is found that a large $R_0$ is needed to fit the data while $T_{kin}$ sets to small value. This is consistent with evolution of the fireball created in heavy-ion collisions, where temperature becomes lower while system size increases.
$R_{out}$ and $R_{long}$ not only depend on $\tau_0$ but also on $\Delta\tau$. Figure \[fig:tau0Deltatau\] shows $\tau_0$ and $\Delta\tau$ as a function of centre-of-mass energy $\sqrt{s_{NN}}$ from fit to the data. The $\Delta\tau$ slightly depends on the $\sqrt{s_{NN}}$. From figure \[fig:tau0Deltatau\] one can see that $\tau_0$ generally increases with the increasing of $\sqrt{s_{NN}}$ in trends but there exists a minimum value at $\sqrt{s_{NN}}\sim$ 39 GeV. It may imply that the system in higher energy (such as at LHC) will undergo a longer time evolution than in lower energy before hadron rescattering ceases (the kinetic freeze-out status). With fixed temperature of $T_{kin}$ (90, 100, 120 MeV), the system lifetime $\tau_0$ and the particle emission duration $\Delta\tau$ are all in reverse order to the temperature $T_{kin}$. This suggests that a system expanding with a long lifetime and a broad duration will result in a lower temperature, which is consistent with the behaviour of $R_0$ as discussed above. We learnt that our results are comparable with the experimental results with $\Delta\tau\neq 0$. With the system lifetime and HBT radii calculation all taken into account, it can be concluded that the particle emission duration can not be ignored while fitting the HBT radii ($R_{side}$, $R_{out}$ and $R_{long}$) at the same time.
{width="15cm"}
After $R_{out}$ and $R_{side}$ are all calculated, difference of $R_{out}^2-R_{side}^2$ as a function of centre-of-mass energy $\sqrt{s_{NN}}$ can be obtained as shown in Figure \[fig:RoRsDiff\]. In the case of $\Delta\tau\neq 0$, the calculated results can describe the data very well. However, it is unsuccessful to fit the data with $\Delta\tau$=0 for the current parameter configuration. Energy dependence of the difference of $R_{out}^2-R_{side}^2$ demonstrates a non-monotonic increasing trend with the increasing of $\sqrt{s_{NN}}$. The peak of experimental results locates at $\sqrt{s_{NN}}\sim$17.3 GeV [@HBT-BES-EXP-STAR] and the calculated results give a very similar behaviour for the peak emerging. And in reference [@HBT-BES-Th], the theoretical work proposes the critical end point (CEP) for deconfinement phase transition at $\sqrt{s_{NN}}$ = 47.5 GeV by applying FSS. Anyway other observables, such as elliptic flow and fluctuations, should be considered together and other basic theoretical calculations are awaiting for comparison, which contribute to locate the CEP and understand underlying physics around this energy region.
{width="15cm"}
transverse momentum dependence of HBT radii
===========================================
With the above parameter configuration, we also calculated the transverse momentum dependence of HBT radii at $\sqrt{s_{NN}}$=200 GeV and 2760 GeV in central heavy-ion collisions. Figure \[fig:RsolpTSTAR\] and \[fig:RsolpTALICE\] show the HBT radii as a function of transverse momentum in central Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV and in central Pb+Pb collisions at $\sqrt{s_{NN}}$=2760 GeV, respectively. The experimental data is from Ref. [@HBT-BES-EXP-STAR; @HBT-ALICE]. $R_{side}$, $R_{out}$ and $R_{long}$ decrease with the increasing of transverse momentum $p_T$ as shown in figure \[fig:RsolpTSTAR\], which indicates that high $p_T$ particles are emitted from near the centre of the fireball. It is found that the calculated results fit the STAR data in the case of $\Delta\tau\neq 0$ but fails to describe the $R_{out}$ with $\Delta\tau$ = 0. The similar $p_T$ dependence trend is found in central Pb+Pb collisions at $\sqrt{s_{NN}}$ = 2760 GeV as shown in figure \[fig:RsolpTALICE\]. In the $\Delta\tau\neq 0$ case, the calculated results reproduce the $R_{side}$ and $R_{out}$ exactly but slightly underestimate the value of $R_{long}$. Again, a reasonable parameter configuration can not be found for fitting ALICE data in the case of $\Delta\tau$ = 0. These results suggest that the system lifetime and particle emission duration should be taken into account at the same time while describing $R_{side}$, $R_{out}$ and $R_{long}$ with the same parameter configuration in the blast-wave model.
{width="15cm"}
{width="15cm"}
Summary
=======
The HBT radii ($R_{side}$, $R_{out}$ and $R_{long}$) are calculated from the blast-wave model in the “out-side-long" ($o s l$) coordinates system. In comparison with the experimental data [@HBT-BES-EXP-STAR; @HBT-ALICE], we found that: in the case of $\Delta\tau\neq 0$, the parameter configuration for blast-wave model can successfully describe the experimental results of collision energy and transverse momentum dependence of $R_{side}$, $R_{out}$ and $R_{long}$. Since the collision system has different temperature at each centre-of-mass energy point, the configured parameters can be considered as the preferred values with a case of $T_{kin}$ as a function of $\sqrt{s_{NN}}$ and $\Delta\tau\neq 0$ as shown in the Figure \[fig:R0\] and \[fig:tau0Deltatau\]. However, it can not be configured for the blast-wave parameter to fit the experimental data while setting the $\Delta\tau$ to zero. This may imply that the particle emission duration plays an important role to describe the system expanding and can not be ignored while calculating the $R_{side}$, $R_{out}$ and $R_{long}$ to fit the data at the same time. And the difference of $R_{out}^2-R_{side}^2$ presents a non-monotonic increasing trend with the increasing of $\sqrt{s_{NN}}$ as seen in the experimental analysis [@HBT-BES-EXP-STAR], which is sensitive to the equation of state and might be related to the critical end point with other observables taken into account.
This work was supported in part by the Major State Basic Research Development Program in China under Contract No. 2014CB845400, the National Natural Science Foundation of China under contract Nos. 11421505, 11220101005, 11105207, 11275250, 11322547 and U1232206, and the CAS Project Grant No. QYZDJ-SSW-SLH002.
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[^1]: Author to whom all correspondence should be addressed: ygma@sinap.ac.cn
|
---
abstract: 'Multi-species reaction-diffusion systems, with more-than-two-site interaction on a one-dimensional lattice are considered. Necessary and sufficient constraints on the interaction rates are obtained, that guarantee the closedness of the time evolution equation for $E^{\mathbf a}_n(t)$’s, the expectation value of the product of certain linear combination of the number operators on $n$ consecutive sites at time $t$.'
---
[**Exactly solvable models through the generalized empty interval method: multi-species and more-than-two-site interactions**]{} 2 cm
[Amir Aghamohammadi[[^1]]{} & Mohammad Khorrami[[^2]]{}]{} 5 mm
[*[Department of Physics, Alzahra University, Tehran 19938-91167, Iran. ]{}*]{}
[**PACS numbers:**]{} 05.40.-a, 02.50.Ga
[**Keywords:**]{} reaction-diffusion, generalized empty-interval method, next-nearest-neighbor interaction, multi-species
Introduction
============
The study of the reaction-diffusion systems, has been an attractive area. A reaction-diffusion system consists of a collection of particles (of one or several species) moving and interacting with each other with specific probabilities (or rates in the case of continuous time variable). In the so called exclusion processes, any site of the lattice the particles move on, is either vacant or occupied by one particle. The aim of studying such systems, is of course to calculate the time evolution of such systems. But to find the complete time evolution of a reaction-diffusion system, is generally a very difficult (if not impossible) task.
Reaction-diffusion systems have been studied using various methods: analytical techniques, approximation methods, and simulation. The success of the approximation methods, may be different in different dimensions, as for example the mean field techniques, working good for high dimensions, generally do not give correct results for low dimensional systems. A large fraction of analytical studies, belong to low-dimensional (specially one-dimensional) systems, as solving low-dimensional systems should in principle be easier [@ScR; @ADHR; @KPWH; @HS1; @PCG; @HOS1; @HOS2; @AL; @AKK; @RK; @RK2; @AKK2; @AAMS; @AM1].
Various classes of reaction-diffusion systems are called exactly-solvable, in different senses. In [@AA; @RK3; @RK4], integrability means that the $N$-particle conditional probabilities’ S-matrix is factorized into a product of 2-particle S-matrices. This is related to the fact that for systems solvable in this sense, there are a large number of conserved quantities. In [@BDb; @BDb1; @BDb2; @BDb3; @Mb; @HH; @AKA; @KAA; @MB; @AAK], solvability means closedness of the evolution equation of the empty intervals (or their generalization).
The empty interval method (EIM) has been used to analyze the one dimensional dynamics of diffusion-limited coalescence [@BDb; @BDb1; @BDb2; @BDb3]. Using this method, the probability that $n$ consecutive sites are empty has been calculated. For the cases of finite reaction-rates, some approximate solutions have been obtained. EIM has been also generalized to study the kinetics of the $q$-state one-dimensional Potts model in the zero-temperature limit [@Mb].
In [@AKA], all the one dimensional reaction-diffusion models with nearest neighbor interactions which can be exactly solved by EIM have been studied. EIM has also been used to study a model with next nearest neighbor interaction [@HH]. In [@KAA], exactly solvable models through the empty-interval method, for more-than-two-site interactions were studied. There, conditions were obtained which are sufficient for a one-species system to be solvable through the EIM. In [@MB], the conventional EIM has been extended to a more generalized form. Using this extended version, a model has been studied, which can not be solved by conventional EIM.
In a recent article [@AAK], we considered nearest-neighbor multi-species models on a one-dimensional lattice. we obtained necessary and sufficient conditions on the reaction rates, so that the time evolution equation for $E^{\mathbf a}_{k,n}(t)$ is closed. Here $E^{\mathbf a}_{k,n}(t)$ is the expectation of the product of a specific linear combination of the number operators (corresponding to different species) at $n$ consecutive sites beginning from the $k$-th site. All single-species left-right symmetric reaction-diffusion systems solvable through the generalized empty-interval method (GEIM), were classified. In this article, the method introduced in [@AAK] is generalized to more than two-site interactions.
In section 2, multi-species systems with three-site interactions are investigated, which are solvable through the GEIM. Conditions necessary and sufficient for closedness of the evolution equation of $E^{\mathbf a}_{k,n}(t)$ are obtained, and the evolution equation is explicitly given.
In section 3, the more general case of multi-species systems with $k$-site interactions, solvable through the GEIM, is investigated. Again, conditions necessary and sufficient for closedness of the evolution equation of $E^{\mathbf a}_{k,n}(t)$ as well as the evolution equation itself, are obtained.
Finally, in section 4, as an example a specific model with a three-site interaction is introduced and exactly solved.
Three-site interactions
=======================
Consider a periodic lattice with $L+1$ sites. Each site is either empty, or occupied with a particle of one of $p$ possible species. Denote by $N_i^\alpha$, the number operator of the particles of type $\alpha$ at the site $i$. $\alpha=p+1$ is regarded as a vacancy. $N^\alpha_i$ is equal to one, if the site $i$ is occupied by a particle of type $\alpha$. Otherwise, $N^\alpha_i$ is zero. We also have a constraint $$\label{1}
s_\alpha N^\alpha_i=1,$$ where ${\mathbf s}$ is a covector the components of which ($s_\alpha$’s) are all equal to one. The constraint , simply says that every site, either is occupied by a particle of one type, or is empty. A representation for these observables is $$\label{2}
N_i^\alpha:=\underbrace{1\otimes\cdots\otimes 1}_{i-1}\otimes
N^\alpha\otimes\underbrace{1\otimes\cdots\otimes 1}_{L+1-i},$$ where $N^\alpha$ is a diagonal $(p+1)\times(p+1)$ matrix the only nonzero element of which is the $\alpha$’th diagonal element, and the operators 1 in the above expression are also $(p+1)\times(p+1)$ matrices. It is seen that the constraint can be written as $$\label{3}
{\mathbf s}\cdot{\mathbf N}=1,$$ where ${\mathbf N}$ is a vector the components of which are $N^\alpha$’s. The state of the system is characterized by a vector $$\label{4}
{\mathbf P}\in\underbrace{{\mathbb V}\otimes\cdots\otimes{\mathbb
V}}_{L+1},$$ where ${\mathbb V}$ is a $(p+1)$-dimensional vector space. All the elements of the vector ${\mathbf P}$ are nonnegative, and $$\label{5}
{\mathbf S}\cdot{\mathbf P}=1.$$ Here ${\mathbf S}$ is the tensor-product of $L+1$ covectors ${\mathbf s}$.
As the number operators $N^\alpha_i$ are zero or one (and hence idempotent), the most general observable of such a system is the product of some of these number operators, or a sum of such terms.
The evolution of the state of the system is given by $$\label{6}
\dot{\mathbf P}={\mathcal H}\;{\mathbf P},$$ where the Hamiltonian ${\mathcal H}$ is stochastic, by which it is meant that its non-diagonal elements are nonnegative and $$\label{7}
{\mathbf S}\; {\mathcal H}=0.$$ The interaction is a next-nearest-neighbor interaction: $$\label{8}
{\mathcal H}=\sum_{i=1}^{L+1}H_{i,i+1,i+2},$$ where $$\label{9}
H_{i,i+1,i+2}:=\underbrace{1\otimes\cdots\otimes 1}_{i-1}\otimes H
\otimes\underbrace{1\otimes\cdots\otimes 1}_{L-i-1}.$$ (It has been assumed that the sites of the system are identical, that is, the system is translation-invariant. Otherwise $H$ in the right-hand side of would depend on $i$.) The three-site Hamiltonian $H$ is stochastic, that is, its non-diagonal elements are nonnegative, and the sum of the elements of each of its columns vanishes: $$\label{10}
({\mathbf s}\otimes{\mathbf s}\otimes{\mathbf s})H=0.$$
Now consider a certain class of observables, namely $$\label{11}
{\mathcal E}^{\mathbf a}_{k,n}:=\prod_{l=k}^{k+n-1}({\mathbf
a}\cdot{\mathbf N}_l),$$ where ${\mathbf a}$ is a specific $(p+1)$-dimensional covector, and ${\mathbf N}_i$ is a vector the components of which are the operators $N_i^\alpha$. We want to find criteria for $H$, so that the evolutions of the expectations of ${\mathcal E}^{\mathbf
a}_{k,n}$’s are closed, that is, the time-derivative of their expectation is expressible in terms of the expectations of ${\mathcal E}^{\mathbf a}_{k,n}$’s themselves. Denoting the expectations of these observables by $E^{\mathbf a}_{k,n}$, $$\label{12}
E^{\mathbf a}_{k,n}:={\mathbf S}\;{\mathcal E}^{\mathbf
a}_{k,n}{\mathbf P}.$$ For $1<n<L$, we have $$\begin{aligned}
\label{13}
\dot E^{\mathbf a}_{k,n}=&{\mathbf S}\;{\mathcal E}^{\mathbf
a}_{k,n}
{\mathcal H}\;{\mathbf P},\nonumber\\
=&\sum_{l=-1}^{n}{\mathbf S}\;{\mathcal E}^{\mathbf
a}_{k,n}H_{k+l-1,k+l,k+l+1}
\;{\mathbf P}\nonumber\\
=&{\mathbf S}\;{\mathcal E}^{\mathbf a}_{k,n}H_{k-2,k-1,k}\;{\mathbf P}\nonumber\\
&+{\mathbf S}\;{\mathcal E}^{\mathbf a}_{k,n}H_{k-1,k,k+1}\;{\mathbf P}\nonumber\\
&+\sum_{l=1}^{n-2}{\mathbf S}\;{\mathcal E}^{\mathbf
a}_{k,n}H_{k-1+l,k+l,k+l+1}
\;{\mathbf P}\nonumber\\
&+{\mathbf S}\;{\mathcal E}^{\mathbf a}_{k,n}H_{k+n-2,k+n-1,k+n}\;{\mathbf P}\nonumber\\
&+{\mathbf S}\;{\mathcal E}^{\mathbf a}_{k,n}H_{k+n-1,k+n,k+n+1}\;{\mathbf P}.\end{aligned}$$ In the right-hand side of the last equality, the first two and the last two terms are contributions of the interactions at the boundaries of the block, while the summation term is the contribution of the sites in the bulk.
To proceed, one may use the following identity $$\label{14}
{\mathbf s}({\mathbf b}\cdot{\mathbf N})={\mathbf b},$$ using which, one arrives at $$\label{15}
({\mathbf s}\otimes{\mathbf s }\otimes{\mathbf s})[({\mathbf
a}\cdot{\mathbf N})\otimes ({\mathbf a}\cdot{\mathbf N})\otimes
({\mathbf a}\cdot{\mathbf N})H]=({\mathbf a}\otimes{\mathbf
a}\otimes{\mathbf a})H.$$ Demanding that the bulk terms in (\[13\]) be expressible in terms of the expectations of ${\mathcal E}^{\mathbf a}_{k,n}$’s themselves, one arrives at the following condition for the bulk terms.
$$\label{17}
({\mathbf a}\otimes{\mathbf a}\otimes{\mathbf a})H=\lambda
({\mathbf a}\otimes{\mathbf a}\otimes{\mathbf a}),$$
for some constant $\lambda$. By similar arguments, the condition coming from the left boundary terms of (\[13\]) can be obtained: $$\begin{aligned}
\label{18}
({\mathbf s}\otimes{\mathbf s}\otimes{\mathbf a}\otimes{\mathbf a})[(1\otimes H)
+(H\otimes 1)]=
&\nonumber \\
\mu_{\mathrm L}{\mathbf s}\otimes{\mathbf s}\otimes{\mathbf s}\otimes{\mathbf s}&+
\nu_{\mathrm L}{\mathbf s}\otimes{\mathbf s}\otimes{\mathbf s}\otimes{\mathbf a}+
\gamma_{\mathrm L}{\mathbf s}\otimes{\mathbf s}\otimes{\mathbf a}\otimes{\mathbf a}
\nonumber \\
&+ \delta_{\mathrm L}{\mathbf s}\otimes{\mathbf a}\otimes{\mathbf a}\otimes{\mathbf a}+
\phi_{\mathrm L}{\mathbf a}\otimes{\mathbf a}\otimes{\mathbf a}\otimes{\mathbf a},\end{aligned}$$ and finally, the condition coming from the right boundary terms of (\[13\]) is $$\begin{aligned}
\label{19}
({\mathbf a}\otimes{\mathbf a}\otimes{\mathbf s}\otimes{\mathbf s})[(1\otimes H)
+(H\otimes 1)]=
&\nonumber \\
\mu_{\mathrm R}{\mathbf s}\otimes{\mathbf s}\otimes{\mathbf s}\otimes{\mathbf s}&+
\nu_{\mathrm R}{\mathbf a}\otimes{\mathbf s}\otimes{\mathbf s}\otimes{\mathbf s}+
\gamma_{\mathrm R}{\mathbf a}\otimes{\mathbf a}\otimes{\mathbf s}\otimes{\mathbf s}
\nonumber \\
&+ \delta_{\mathrm R}{\mathbf a}\otimes{\mathbf a}\otimes{\mathbf
a}\otimes{\mathbf s}+ \phi_{\mathrm R}{\mathbf a}\otimes{\mathbf
a}\otimes{\mathbf a}\otimes{\mathbf a},\end{aligned}$$ where the multipliers in the right-hand sides of (\[18\]) and (\[19\]) are constants. Arranging all these together, the evolution equation for $E^{\mathbf a}_{k,n}$, for $1<n<L$, becomes $$\begin{aligned}
\label{19-19}
\dot E^{\mathbf a}_{k,n}=&\mu_{\mathrm L}E^{\mathbf a}_{k+2,n-2}+
\mu_{\mathrm R}E^{\mathbf a}_{k,n-2}+\nu_{\mathrm L}E^{\mathbf
a}_{k+1,n-1}+\nu_{\mathrm R}E^{\mathbf a}_{k,n-1}\nonumber\\
&+[\gamma_{\mathrm L}+(n-2)\lambda+\gamma_{\mathrm R}]E^{\mathbf
a}_{k,n}\nonumber\\
&+\delta_{\mathrm L}E^{\mathbf a}_{k-1,n+1}+\delta_{\mathrm
R}E^{\mathbf a}_{k,n+1}+\phi_{\mathrm L}E^{\mathbf
a}_{k-2,n+2}+\phi_{\mathrm R}E^{\mathbf a}_{k,n+2}.\end{aligned}$$ In general, even if the initial conditions are not translationally invariant, one can solve the above equation. However, assuming that the initial conditions are translationally invariant, simplifies the calculations. Assuming that the initial conditions are translationally invariant, as the dynamics is also translationally invariant, $E^{\mathbf a}_{k,n}$ is independent of $k$, and so one arrives at $$\begin{aligned}
\label{19-2}
\dot E^{\mathbf a}_{n}=&(\mu_{\mathrm L}+ \mu_{\mathrm
R})E^{\mathbf a}_{n-2}+(\nu_{\mathrm L}+\nu_{\mathrm R})E^{\mathbf
a}_{n-1}+[\gamma_{\mathrm L}+(n-2)\lambda+\gamma_{\mathrm R}]
E^{\mathbf a}_{n}\nonumber\\
&+(\delta_{\mathrm L}+\delta_{\mathrm
R})E^{\mathbf a}_{n+1}+(\phi_{\mathrm L}+\phi_{\mathrm
R})E^{\mathbf a}_{n+2}.\end{aligned}$$
The cases of zero-, one-, $L$-, and $(L+1)$-point functions should be considered separately. The zero-point function is equal to one. In the case of the one-point function, one arrives at an additional condition $$\begin{aligned}
\label{18-2}
({\mathbf s}\otimes{\mathbf s}\otimes{\mathbf a}\otimes{\mathbf
s}\otimes{\mathbf s})[(1\otimes 1\otimes H) +(1\otimes H\otimes
1)+( H\otimes 1\otimes 1)]=
\nonumber \\
\mu_{0}({\mathbf s}\otimes{\mathbf s}\otimes{\mathbf
s}\otimes{\mathbf s}\otimes{\mathbf s})+ \mu_{4}({\mathbf
s}\otimes{\mathbf s}\otimes{\mathbf a}\otimes{\mathbf
s}\otimes{\mathbf s})\nonumber
\\ + \mu_{8}({\mathbf s}\otimes{\mathbf a}\otimes{\mathbf
s}\otimes{\mathbf s}\otimes{\mathbf s})+ \mu_{16}({\mathbf
a}\otimes{\mathbf s}\otimes{\mathbf s}\otimes{\mathbf
s}\otimes{\mathbf s})&\nonumber \\ + \mu_{12}({\mathbf
s}\otimes{\mathbf a}\otimes{\mathbf a}\otimes{\mathbf
s}\otimes{\mathbf s})+ \mu_{24}({\mathbf a}\otimes{\mathbf
a}\otimes{\mathbf s}\otimes{\mathbf s}\otimes{\mathbf s})
&\nonumber \\ + \mu_{28}({\mathbf a}\otimes{\mathbf
a}\otimes{\mathbf a}\otimes{\mathbf s}\otimes{\mathbf s}) +
\mu_{2}({\mathbf s}\otimes{\mathbf s}\otimes{\mathbf
s}\otimes{\mathbf a}\otimes{\mathbf s}) &\nonumber \\ +
\mu_{6}({\mathbf s}\otimes{\mathbf s}\otimes{\mathbf
a}\otimes{\mathbf a}\otimes{\mathbf s}) + \mu_{14}({\mathbf
s}\otimes{\mathbf a}\otimes{\mathbf a}\otimes{\mathbf
a}\otimes{\mathbf s}) &\nonumber \\ + \mu_{7}({\mathbf
s}\otimes{\mathbf s}\otimes{\mathbf a}\otimes{\mathbf
a}\otimes{\mathbf a}) + \mu_{3}({\mathbf s}\otimes{\mathbf
s}\otimes{\mathbf s}\otimes{\mathbf a}\otimes{\mathbf a})
&\nonumber \\ + \mu_{1}({\mathbf s}\otimes{\mathbf
s}\otimes{\mathbf s}\otimes{\mathbf s}\otimes{\mathbf a}).\end{aligned}$$ Then the evolution equation for $E^{\mathbf a}_{1}$ is $$\begin{aligned}
\dot E^{\mathbf a}_{1}=&(\mu_{7}+\mu_{14}+\mu_{28})E^{\mathbf
a}_{3} +(\mu_{3}+\mu_{6}+\mu_{12}+\mu_{24})E^{\mathbf
a}_{2}\nonumber
\\ &+(\mu_{1}+\mu_{2}+\mu_{4}+\mu_{8}+\mu_{16})E^{\mathbf a}_{1}+\mu_{0}.\end{aligned}$$
In the case $n=L$, one arrives at the following additional condition $$\begin{aligned}
\label{18-3}
({\mathbf a}\otimes{\mathbf a}\otimes{\mathbf s}\otimes{\mathbf
a}\otimes{\mathbf a})[(1\otimes 1\otimes H) +(1\otimes H\otimes
1)+( H\otimes 1\otimes 1)]=
\nonumber \\
\nu_{31}({\mathbf a}\otimes{\mathbf a}\otimes{\mathbf
a}\otimes{\mathbf a}\otimes{\mathbf a})+ \nu_{27}({\mathbf
a}\otimes{\mathbf a}\otimes{\mathbf s}\otimes{\mathbf
a}\otimes{\mathbf a})\nonumber
\\ + \nu_{23}({\mathbf a}\otimes{\mathbf s}\otimes{\mathbf
a}\otimes{\mathbf a}\otimes{\mathbf a})+ \nu_{15}({\mathbf
s}\otimes{\mathbf a}\otimes{\mathbf a}\otimes{\mathbf
a}\otimes{\mathbf a})&\nonumber \\ + \nu_{19}({\mathbf
a}\otimes{\mathbf s}\otimes{\mathbf s}\otimes{\mathbf
a}\otimes{\mathbf a})+ \nu_{7}({\mathbf s}\otimes{\mathbf
s}\otimes{\mathbf a}\otimes{\mathbf a}\otimes{\mathbf a})
&\nonumber \\ + \nu_{3}({\mathbf s}\otimes{\mathbf
s}\otimes{\mathbf s}\otimes{\mathbf a}\otimes{\mathbf a}) +
\nu_{29}({\mathbf a}\otimes{\mathbf a}\otimes{\mathbf
a}\otimes{\mathbf s}\otimes{\mathbf a}) &\nonumber \\ +
\nu_{25}({\mathbf a}\otimes{\mathbf a}\otimes{\mathbf
s}\otimes{\mathbf s}\otimes{\mathbf a}) + \nu_{17}({\mathbf
a}\otimes{\mathbf s}\otimes{\mathbf s}\otimes{\mathbf
s}\otimes{\mathbf a}) &\nonumber \\ + \nu_{24}({\mathbf
a}\otimes{\mathbf a}\otimes{\mathbf s}\otimes{\mathbf
s}\otimes{\mathbf s}) + \nu_{28}({\mathbf a}\otimes{\mathbf
a}\otimes{\mathbf a}\otimes{\mathbf s}\otimes{\mathbf s})
&\nonumber \\ + \nu_{30}({\mathbf a}\otimes{\mathbf
a}\otimes{\mathbf a}\otimes{\mathbf a}\otimes{\mathbf s}).\end{aligned}$$ Then the evolution equation for $E^{\mathbf a}_{L}$ is $$\begin{aligned}
\dot E^{\mathbf a}_{L}=&(\nu_{3}+\nu_{17}+\nu_{24})E^{\mathbf
a}_{L-2}+(\nu_{7}+\nu_{19}+\nu_{25}+\nu_{28})E^{\mathbf a}_{L-1}
\nonumber \\
&+(\nu_{15}+\nu_{23}+\nu_{27}+\nu_{29}+\nu_{30})E^{\mathbf
a}_{L}+\nu_{31}E^{\mathbf a}_{L+1}.\end{aligned}$$
The evolution equation for the case $n=L+1$ is simply $$\label{19-3}
\dot E^{\mathbf a}_{L+1}=\lambda (L+1)E^{\mathbf a}_{L+1}.$$
$k$-site interactions
=====================
Now, let’s go further and consider the more general case of $k$-site interactions. The problem of single-species $k$-site interactions solvable in the framework of empty interval method was addressed in [@KAA]. In that article necessary conditions for a system to be solvable through the EIM was obtained. Here we obtain the necessary and sufficient conditions for a system to be solvable through the GEIM. To fix notation, let’s introduce $$\begin{aligned}
\label{26}
{\cal A}^{ijl}:&={\mathbf s}^{\otimes i}\otimes
{\mathbf a}^{\otimes j}\otimes {\mathbf s}^{\otimes l},\cr
{\cal B}^{ijl}:&={\mathbf a}^{\otimes i}\otimes
{\mathbf s}^{\otimes j}\otimes {\mathbf a}^{\otimes l},\end{aligned}$$ where $$\label{27}
{\mathbf b}^{\otimes i}:=\underbrace{{\mathbf b}\otimes\cdots
\otimes {\mathbf b}}_i.$$ First consider the bulk terms in the time derivative of ${\mathcal
E}^{\mathbf a}_{n}$’s. Demanding that the bulk terms can be expressed in terms of the expectations of ${\mathcal E}^{\mathbf
a}_{n}$’s, one arrives at the following condition for the bulk terms. $$\label{28}
{\cal A}^{0k0}H=\lambda {\cal A}^{0k0}.$$ The condition coming from the left boundary of the block is $$\label{29}
{\cal A}^{k-1,k-1,0}[{\mathbf 1}^{\otimes k-2}\otimes H +
+\cdots + H\otimes {\mathbf 1}^{\otimes
k-2}]= \sum _{{{m,l}\atop{m+l=2k-2}}}C^L_{ml0}{\cal A}^{ml0},$$ while the condition coming from the right boundary of the block $$\label{30}
{\cal A}^{0,k-1,k-1}[{\mathbf 1}^{\otimes k-2}\otimes H +
\cdots + H\otimes {\mathbf 1}^{\otimes
k-2}]= \sum _{{{m,l}\atop{m+l=2k-2}}}C^R_{0lm}{\cal A}^{0lm}.$$
For the blocks of the length $j$ with $j<k-1$, one needs the condition $$\label{31}
{\cal A}^{k-1,j,k-1}[{\mathbf 1}^{\otimes j+k-2}\otimes H +
\cdots + H\otimes {\mathbf 1}^{\otimes
j+k-2}]= \sum _{{{m,p,l}\atop{m+p+l=j+2k-2}}}C_{mpl}{\cal A}^{mpl}.$$ Finally for the blocks of the length $(L+1-j)$ with $j<k-1$, one needs the condition $$\label{32}
{\cal B}^{k-1,j,k-1}[{\mathbf 1}^{\otimes j+k-2}\otimes H +
\cdots + H\otimes {\mathbf 1}^{\otimes
j+k-2}]= \sum _{{{m,p,l}\atop{m+p+l=j+2k-2}}}D_{mpl}{\cal B}^{mpl}.$$ Now it is easy to write the evolution equation of ${ E}^{\mathbf
a}_n$. For $k-1\leq n\leq L+2-k$, $$\begin{aligned}
\label{33}
\dot { E}^{\mathbf
a}_n =(n-k+1)\lambda { E}^{\mathbf a}_n
&+\sum_{l=0}^{2k-2}{ E}^{\mathbf a}_{l-k+1+n}C^L_{2k-2-l,l,0}\nonumber\\
&+\sum_{l=0}^{2k-2}{ E}^{\mathbf a}_{l-k+1+n}C^R_{0,l,2k-2-l}, \qquad k-1\leq n\leq L+2-k.\end{aligned}$$ For $0<n<k-1$, $$\label{33-1}
\dot { E}^{\mathbf a}_n =\sum_{p=0}^{n+2k-2}
\sum_{m=0}^{n+2k-2-p}C_{m,p,n+2k-2-p-m}{ E}^{\mathbf a}_{p}, \qquad 0<n<k-1.$$ Finally, for $L+2-k<n<L+1$, $$\label{34}
\dot { E}^{\mathbf a}_n =\sum_{p=0}^{n+2k-2}
\sum_{m=0}^{n+2k-2-p}D_{m,p,n+2k-2-p-m}{ E}^{\mathbf a}_{L+1-p}, \qquad L+2-k<n<L+1.$$ One also has, $$\begin{aligned}
\label{36-1}
{ E}^{\mathbf a}_0&=1,\nonumber\\
\dot { E}^{\mathbf a}_{L+1}&=(L+1)\lambda { E}^{\mathbf a}_{L+1}.\end{aligned}$$
Equations (\[33\]) to (\[36-1\]) can be solved to obtain a certain set of correlation functions ($E^{\mathbf a}_n$’s).
A three-site model with annihilation, as an example
===================================================
Here, a model with three-site interactions is considered. Each site may be occupied or be a vacancy. A particle at each site may be annihilated, and the annihilation rate depends on its neighboring sites. The interactions are $$\begin{aligned}
\label{35}
AAA&\to A\emptyset A& \qquad\hbox{with the rate }\Lambda_1,\cr
AA\emptyset &\to A\emptyset \emptyset & \qquad\hbox{with the rate }\Lambda_2,\cr
\emptyset AA&\to \emptyset \emptyset A &\qquad\hbox{with the rate }\Lambda_3,\cr
\emptyset A\emptyset & \to \emptyset \emptyset \emptyset &
\qquad\hbox{with the rate }\Lambda_4.\end{aligned}$$ ($A$ and $\emptyset$ denote an occupied site and an empty site, respectively.) Choosing ${\mathbf a}=(1\quad 0)$, it is seen that this model fulfills the conditions needed for solvability. This means that the evolution equations for $E_n$’s with $E_n=\langle
N_1 \cdots N_n\rangle$ are closed. Using the method introduced in previous sections, one can obtain $E_n$’s. The evolution equation for $E_n$’s are $$\begin{aligned}
\label{37}
&\dot E_n=-[(n-2)\Lambda_1+\Lambda_2+\Lambda_3)]E_n+(\Lambda_2+\Lambda_3-2\Lambda_1)E_{n+1},\quad
2\leq n\leq L-1\nonumber\\
&\dot
E_1=-\Lambda_4E_1-(\Lambda_2+\Lambda_3-2\Lambda_4)E_2+(\Lambda_2+\Lambda_3-\Lambda_1-\Lambda_4)E_3,\nonumber\\
&\dot
E_L=-(\Lambda_2+\Lambda_3)E_L+(\Lambda_2+\Lambda_3-2\Lambda_1)E_{L+1}\nonumber\\
&\dot E_{L+1}=-(L+1)\Lambda_1E_L\end{aligned}$$ To simplify the model, assume $\Lambda_1=0$. This means that a block of particles, can only be altered at the boundaries. Defining $$\begin{aligned}
\label{38-0}
\alpha:&=\Lambda_2+\Lambda_3,\cr
\beta:&=\Lambda_4,\end{aligned}$$ (\[37\]) is rewritten as $$\begin{aligned}
\label{38}
&\dot E_n=\alpha(E_{n+1}-E_n),\quad
2\leq n\leq L\cr
&\dot E_1=\beta(E_2-E_1)+(\alpha-\beta)(E_3-E_2),\cr &\dot
E_{L+1}=0.\end{aligned}$$ There are two cases.
**i**) $\alpha=0$. In this case, all the $E_n$’s for $n\ne 1$ are constant and $$\label{39}
E_1(t)=e^{-\beta t}[E_1(0)-2E_2(0)+E_3(0)]+2E_2(0)-E_3(0).$$
**ii**) $\alpha\ne 0$. In this case, $E_{L+1}$ is constant. Using this, one can obtain $E_L$, and then $E_{L-1}$, ..., and $E_2$. The result is
$$\label{42}
E_{L-k}=E_{L+1}+\left[\sum_{j=0}^k\frac{\gamma_{L-k+j}}{j!}\,
(\alpha\,t)^j\right] e^{-\alpha\, t},$$
where $\gamma_k$’s are arbitrary constants to be determined from the initial conditions. $E_1$ is also easily determined from the second equation of (\[38\]).\
\
[**Acknowledgement**]{} The authors would like to thank M. Kaboutari, for useful discussions.
[99]{} G. M. Schütz; “Exactly solvable models for many-body systems far from equilibrium” in “Phase transitions and critical phenomena, vol. **19**”, C. Domb & J. Lebowitz (eds.), (Academic Press, London, 2000). F. C. Alcaraz, M. Droz, M. Henkel, & V. Rittenberg; Ann. Phys. **230** (1994) 250. K. Krebs, M. P. Pfannmuller, B. Wehefritz, & H. Hinrichsen; J. Stat. Phys. **78**\[FS\] (1995) 1429. H. Simon; J. Phys. **A28** (1995) 6585. V. Privman, A. M. R. Cadilhe, & M. L. Glasser; J. Stat. Phys. **81** (1995) 881. M. Henkel, E. Orlandini, & G. M. Schütz; J. Phys. **A28** (1995) 6335. M. Henkel, E. Orlandini, & J. Santos; Ann. of Phys. **259** (1997) 163. A. A. Lushnikov; Sov. Phys. JETP **64** (1986) 811 \[Zh. Eksp. Teor. Fiz. **91** (1986) 1376\]. M. Alimohammadi, V. Karimipour, & M. Khorrami; Phys. Rev. **E57** (1998) 6370. F. Roshani & M. Khorrami; Phys. Rev. **E60** (1999) 3393. F. Roshani & M. Khorrami; J. Math. Phys. **43** (2002) 2627. M. Alimohammadi, V. Karimipour, & M. Khorrami; J. Stat. Phys. **97** (1999) 373. A. Aghamohammadi, A. H. Fatollahi, M. Khorrami, & A. Shariati; Phys. Rev. **E62** (2000) 4642. A. Aghamohammadi & M. Khorrami; J. Phys. **A33** (2000) 7843. M. Alimohammadi, & N. Ahmadi; Phys. Rev. **62** (2000) 1674. F. Roshani & M. Khorrami; Phys. Rev. **E64** (2001) 011101. F. Roshani & M. Khorrami; Eur. Phys. J. **B36** (2003) 99. M. A. Burschka, C. R. Doering, & D. ben-Avraham; Phys. Rev. Lett. **63** (1989) 700. D. ben-Avraham; Mod. Phys. Lett. **B9** (1995) 895. D. ben-Avraham; in “Nonequilibrium Statistical Mechanics in One Dimension”, V. Privman (ed.), pp 29-50 (Cambridge University press,1997). D. ben-Avraham; Phys. Rev. Lett. **81** (1998) 4756. T. Masser, D. ben-Avraham; Phys. Lett. **A275** (2000) 382. M. Alimohammadi, M. Khorrami, & A. Aghamohammadi; Phys. Rev. **E64** (2001) 056116. M. Henkel & H. Hinrichsen; J. Phys. **A34**, 1561 (2001). M. Khorrami, A. Aghamohammadi, & M. Alimohammadi; J. Phys. **A36** (2003) 345. M. Mobilia & P. A. Bares; Phys. Rev. **E64** (2001) 066123. A. Aghamohammadi, M. Alimohammadi, & M. Khorrami; Eur. Phys. J. **B31** (2003) 371.
[^1]: mohamadi@azzahra.ac.ir
[^2]: mamwad@mailaps.org
|
---
abstract: |
We provide further evidence that a massless cosmological scalar field with a non-minimal coupling to the Ricci curvature of the type $M^2_{\rm pl}(1+\xi \sigma^n/M_{\rm pl}^n)
$ alleviates the existing tension between local measurements of the Hubble constant and its inference from CMB anisotropies and baryonic acoustic oscillations data in presence of a cosmological constant. In these models, the expansion history is modified compared to $\Lambda$CDM at early time, mimicking a change in the effective number of relativistic species, and gravity weakens after matter-radiation equality. Compared to $\Lambda$CDM, a quadratic ($n=2$) coupling increases the Hubble constant when [*Planck*]{} 2018 (alone or in combination with BAO and SH0ES) measurements data are used in the analysis. Negative values of the coupling, for which the scalar field decreases, seem favored and consistency with Solar System can be naturally achieved for a large portion of the parameter space without the need of any screening mechanism. We show that our results are robust to the choice of $n$, also presenting the analysis for $n=4$.
author:
- Matteo Braglia
- Mario Ballardini
- 'William T. Emond'
- Fabio Finelli
- 'A. Emir Gümrükçüoğlu'
- Kazuya Koyama
- Daniela Paoletti
title: 'A larger value for $H_0$ by an evolving gravitational constant'
---
Introduction
============
Despite its simplicity, the six parameters $\Lambda$ Cold Dark Matter (CDM) concordance model has been extremely successful in explaining cosmic microwave background (CMB) anisotropies, baryon acoustic oscillations (BAO), the abundance of primordial light element by Big Bang Nucleosynthesis (BBN), luminosity distance of type Ia supernovae (SNe Ia) and several other cosmological observations. However, the unknown nature of the Dark Energy (DE) and Cold Dark Matter (CDM) permeating our Universe justifies the search for other alternatives. These doubts have been recently corroborated by growing discrepancies between the present rate of the expansion of the Universe $H_0$ inferred from CMB anisotropies measurements and the one estimated by low-redshift distance-ladder measurements [@Verde:2019ivm].
The value of the Hubble constant inferred from [*Planck*]{} 2018 data, $H_0 = (67.36 \pm 0.54)$ km s$^{-1}$Mpc$^{-1}$ [@Aghanim:2018eyx], is in a 4.4$\sigma$ tension with the most recent distance-ladder measurement from the SH0ES team [@Riess:2019cxk], $H_0 = (74.03 \pm 1.42)$ km s$^{-1}$Mpc$^{-1}$, determined by using Cepheid-calibrated SNe Ia with new parallax measurements from HST spatial scanning [@Riess:2018byc]. This is a recent snapshot of a long-standing tension of distance-ladder measurements of $H_0$ with a much wider set of cosmological data rather than Planck data only [@Lemos:2018smw], whose magnitude is possibly affected by unaccounted effects such as uncertainties in calibration [@Efstathiou:2013via; @Freedman:2019jwv; @Freedman:2020dne; @Yuan:2019npk] or in the luminosity functions of SNIa [@Efstathiou:2013via; @Rigault:2014kaa; @Rigault:2018ffm; @Freedman:2019jwv; @Freedman:2020dne; @Yuan:2019npk]. Other determinations of $H_0$ at low-redshift, such as from strong-lensing time delay [@Wong:2019kwg], also point to a higher $H_0$ than the one inferred by Planck data.
Assuming that this $H_0$ *tension* is not due to unknown systematics or unaccounted effects as those mentioned above, some new physics is therefore needed to solve it. One way to address the tension is to modify early time (for redshifts around matter-radiation equality) physics in order to reduce the inferred value of the comoving sound horizon at baryon drag $r_s$. Indeed, a smaller value of the comoving sound horizon at baryon drag $r_s$ can provide a higher value of $H_0$ without spoiling the fit to CMB anisotropies data and changing the BAO observables [@Bernal:2016gxb; @Aylor:2018drw; @Knox:2019rjx]. A prototypical example of such an early-time modification is an excess in the number $N_\mathrm{eff}$ of relativistic degrees of freedom, eventually interacting with hidden dark sectors [@Riess:2011yx; @Wyman:2013lza; @Cyr-Racine:2013jua; @Lancaster:2017ksf; @Buen-Abad:2017gxg; @DiValentino:2017oaw; @DEramo:2018vss; @Poulin:2018zxs; @Kreisch:2019yzn; @Blinov:2019gcj].
An alternative solution to $N_\mathrm{eff}$, which can substantially alleviate the tension, consists in Early Dark Energy (EDE) models [@Poulin:2018cxd; @Agrawal:2019lmo; @Alexander:2019rsc; @Lin:2019qug; @Smith:2019ihp; @Braglia:2020bym]. In these models a scalar field minimally coupled to gravity is subdominant and frozen by the Hubble friction at early times and starts to move around the matter-radiation equality when its effective mass becomes comparable to the Hubble flow and quickly rolls to the minimum of its potential, injecting an amount of energy in the cosmic fluid sharply to sizeably reduce $r_s$. The parameters of the potential and the initial value of the scalar field, which can be remapped in the critical redshift at which the scalar field moves $z_c$ and the maximum value of the energy injection $\Omega_\phi(z_c)$, have to be fine tuned to successfully ease the Hubble tension[^1].
In this paper we study the capability of a massless scalar field $\sigma$ with a non-minimal coupling of the form $F(\sigma)=M_\textup{pl}^2[1+\xi(\sigma/M_\textup{pl})^n]$, where $M_\textup{pl}=1/\sqrt{8\pi G}=2.435 \times 10^{18}$ GeV is the reduced Planck mass, and $n$ is taken as an even and positive integer, to reduce the $H_0$ tension. This simple model relies on the degeneracy between a non-minimal coupling to the Ricci curvature and the Hubble parameter which has been studied in previous works on the constraints on scalar-tensor theories of gravity[^2] [@Umilta:2015cta; @Ballardini:2016cvy; @Rossi:2019lgt; @Sola:2019jek]. In general, scalar-tensor models modify both the early (in a way that resembles a contribution of an extra *dark* radiation component) and late time expansion of the Universe [@Rossi:2019lgt]. By our embedding of a massless $\sigma$ in $\Lambda$CDM, we focus on the early-type of modification in this paper. In the case of a negative coupling $\xi<0$, the scalar field decreases because of the coupling to matter, leading to cosmological post-Newtonian parameters which can be naturally consistent with Solar System constraints $\gamma_{\rm PN}-1 = (2.1 \pm 2.3) \times 10^{-5}$ at 68% CL [@Bertotti:2003rm] and $\beta_{\rm PN}-1 = (4.1\pm7.8) \times 10^{-5}$ at 68% CL [@Will:2014kxa], extending what already emphasized for a conformal coupling (CC, i.e. $\xi=-1/6$) in [@Rossi:2019lgt]. We also investigate to the case where $N_\textup{eff}$, which describes the effective number of relativistic species, is included in the analysis.
This paper is organized as follows. In Sec. \[sec:background\], we describe the background evolution of the model and compare it to other existing solutions to the $H_0$ tension. We describe the datasets and the details of our MCMC analysis in Sec. \[sec:datasets\] and present our results in Sec. \[sec:results\]. We end by discussing our results in the conclusions \[sec:conclusions\].
Background evolution {#sec:background}
====================
The model that we consider is described by the action $$\label{eq:action}
S = \int {\mathrm{d}}^{4}x \sqrt{-g} \left[ \frac{F(\sigma)}{2} R + \frac{(\partial\sigma)^2}{2} - \Lambda + {\cal L}_m \right] \,,$$ where $F(\sigma)\coloneqq M_\textup{pl}^2[1+\xi(\sigma/M_\textup{pl})^n]$ is the non-minimal coupling (NMC) of the scalar field to the Ricci scalar $R$, $(\partial\sigma)^2\coloneqq g^{\mu\nu}\partial_\mu\sigma\partial_\nu\sigma$, ${\cal L}_m$ is the Lagrangian density describing the matter sector, and $M_\textup{pl}$, $\Lambda$ are the reduced Planck mass and bare cosmological constant, respectively. The $n=2$ case has been studied in Refs. [@Rossi:2019lgt; @Ballardini:2020iws] with a potential $V \propto F^2$, which is, however, close to a flat potential for the range of $\xi$ allowed by observations [^3].
The Friedmann and the Klein-Gordon (KG) equations in the spatially flat FLRW background are given by:
\[eq:eoms\] $$\begin{aligned}
3 F H^2 \ =& \ \rho \: + \: \frac{\dot{\sigma}^2}{2} \: + \: \Lambda \: - \: 3\dot{F}H \\ \coloneqq& \ \rho \: + \: \rho_\sigma \;, \nonumber
\end{aligned}$$ $$\begin{aligned}
\label{eq:KG}
\ddot{\sigma} \: + \: 3H\dot{\sigma} \ =& \ \frac{F_{,\sigma}}{2F + 3F^2_{,\sigma}}\Big[\rho \: - \: 3p \: + \: 4\Lambda \: \nonumber\\&- \: \big(1 \: + \: 3F_{,\sigma\sigma}\big)\dot{\sigma}^2 \: \Big] \;,
\end{aligned}$$
where $\rho\,\, (p)$ collectively denotes the total matter energy density (pressure), with $\rho_\sigma\,\,(p_\sigma)$ denoting the energy density of the scalar field, and a subscript $\sigma$ denotes the derivative with respect to the scalar field. Because of the NMC, the Newton constant in the Friedmann equations is replaced by $G_N\coloneqq (8\pi F)^{-1}$ that now varies with time. This has not to be confused with the effective *gravitational constant* that regulates the attraction between two test masses and is measured in laboratory experiments, which is instead given by [@Boisseau:2000pr]: $$\label{eq:Geff}
G_{\mathrm{eff}}=\frac{1}{8\pi F}\left(\frac{2F+4F_{,\sigma}^{2}}{2F+3F_{,\sigma}^{2}}\right) .$$
The deviations from general relativity (GR) can also be parameterized by means of the so-called Post-Newtonian (PN) parameters [@Will:2014kxa], which are given within NMC by the following equations [@Boisseau:2000pr]: $$\begin{aligned}
\label{eqn:gammaPN}
\gamma_{\rm PN}&=1-\frac{F_{,\sigma}^{2}}{F+2F_{,\sigma}^{2}},\\
\label{eqn:betaPN}
\beta_{\rm PN}&=1+\frac{FF_{,\sigma}}{8F+12F_{,\sigma}^{2}}\frac{{\mathrm{d}}\gamma_{\rm PN}}{{\mathrm{d}}\sigma},\end{aligned}$$ where the prediction from GR, i.e. $\gamma_{\rm PN}=\beta_{\rm PN}=1$, is tightly constrained from Solar System experiments. Note that $\gamma_{\rm PN}<1$ in our models.
![We plot the evolution of the energy injection $\Omega_i\coloneqq\rho_i/\rho_c$ \[Top\], the scalar field \[Center\] and the deviation from 1 of the effective (solid lines) and cosmological (dot-dashed lines) Newton constant \[Bottom\] for the models with $n=2,\,\xi<0$ (purple lines), $n=4,\,\xi<0$ (magenta lines), $n=2,\,\xi>0$ (red lines) and $n=4,\,\xi>0$ (brown lines), together with the EDE model of Ref. [@Agrawal:2019lmo] (orange lines) and the $\Lambda$CDM+$N_\textup{eff}$ model (cyan lines). In order to compare the evolution of our model to the aforementioned ones, we set the cosmological parameters to the bestfit values in Table 3 of Ref. [@Agrawal:2019lmo] and set $\xi=-1/6$. In the cases with $\xi>0$, we change the values of the initial conditions on the scalar field and the coupling $\xi$ as in the plot legends.[]{data-label="fig:Background"}](Omega_88mm.pdf "fig:"){width=".85\columnwidth"} ![We plot the evolution of the energy injection $\Omega_i\coloneqq\rho_i/\rho_c$ \[Top\], the scalar field \[Center\] and the deviation from 1 of the effective (solid lines) and cosmological (dot-dashed lines) Newton constant \[Bottom\] for the models with $n=2,\,\xi<0$ (purple lines), $n=4,\,\xi<0$ (magenta lines), $n=2,\,\xi>0$ (red lines) and $n=4,\,\xi>0$ (brown lines), together with the EDE model of Ref. [@Agrawal:2019lmo] (orange lines) and the $\Lambda$CDM+$N_\textup{eff}$ model (cyan lines). In order to compare the evolution of our model to the aforementioned ones, we set the cosmological parameters to the bestfit values in Table 3 of Ref. [@Agrawal:2019lmo] and set $\xi=-1/6$. In the cases with $\xi>0$, we change the values of the initial conditions on the scalar field and the coupling $\xi$ as in the plot legends.[]{data-label="fig:Background"}](sigma_88mm.pdf "fig:"){width=".85\columnwidth"} ![We plot the evolution of the energy injection $\Omega_i\coloneqq\rho_i/\rho_c$ \[Top\], the scalar field \[Center\] and the deviation from 1 of the effective (solid lines) and cosmological (dot-dashed lines) Newton constant \[Bottom\] for the models with $n=2,\,\xi<0$ (purple lines), $n=4,\,\xi<0$ (magenta lines), $n=2,\,\xi>0$ (red lines) and $n=4,\,\xi>0$ (brown lines), together with the EDE model of Ref. [@Agrawal:2019lmo] (orange lines) and the $\Lambda$CDM+$N_\textup{eff}$ model (cyan lines). In order to compare the evolution of our model to the aforementioned ones, we set the cosmological parameters to the bestfit values in Table 3 of Ref. [@Agrawal:2019lmo] and set $\xi=-1/6$. In the cases with $\xi>0$, we change the values of the initial conditions on the scalar field and the coupling $\xi$ as in the plot legends.[]{data-label="fig:Background"}](G_88mm.pdf "fig:"){width=".85\columnwidth"}
Before analyzing the background evolution of our model, note that the NMC to the gravity sector induces some conditions that the theory needs to satisfy in order to have a stable FLRW evolution. For the action , we find that there are in total three physical degrees of freedom associated with the gravity sector (that is, the metric and the $\sigma$ field) [@Gannouji:2006jm]. In order to avoid negative kinetic energy states in the tensor sector, we need $$\label{eq:condition1}
F > 0 \,,$$ and the positivity of the kinetic term in the reduced quadratic action of the scalar field perturbations leads to the second condition $$\label{eq:condition2}
F\, (2\,F+3\,F_\sigma^2)>0\,.$$ For the matter sector, any fluid that satisfies the null energy condition and has real sound speed will be stable. Note that the conditions and also ensure the positivity of the effective gravitational and cosmological Newton constants.
The evolution of relevant background quantities is shown in Fig. \[fig:Background\] for the case of $n=2$ and $n=4$ (see caption for the parameters used in the plots). As can be seen from the central panel in Fig. \[fig:Background\], the scalar field is nearly frozen deep in the radiation era, and is driven by the coupling to non-relativistic matter around the radiation-matter equality era $z\sim\mathcal{O}(10^3-10^4)$, as evident from the Klein-Gordon equation , decreasing (growing) for $\xi<0$ ($\xi>0$).
Since the goal of our paper is to ease the $H_0$ tension, we also plot the relevant quantities for two other reference models, i.e. the case of a varying number of relativistic degrees of freedom in addition to $\Lambda$CDM, and the Rock’n Roll model introduced Ref. [@Agrawal:2019lmo]. This second model is a representative case of EDE models in Einstein gravity [@Poulin:2018cxd; @Agrawal:2019lmo; @Alexander:2019rsc; @Lin:2019qug; @Smith:2019ihp], where a non-negligible energy density is injected around recombination, leading to a larger value of $H_0$.
Let us now stress the important differences between the model studied here, and the two other reference cases. By considering our model as Einstein gravity [@Boisseau:2000pr; @Gannouji:2006jm], the resulting effective DE has an equation of state $w_\mathrm{DE}\equiv p_\mathrm{DE}/\rho_\mathrm{DE} \sim 1/3$ during radiation era (see e.g. Fig. 2 of Ref. [@Rossi:2019lgt] and their Eqs. (13) and (14) for the definitions of $\rho_\mathrm{DE}$ and $p_\mathrm{DE}$) and the contribution of the scalar field[^4] to the total expansion rate $H(z)$ thus resembles the one from an extra *dark* radiation component. This is confirmed by the top panel in Fig. \[fig:Background\], where we plot the energy fraction of the scalar field, parameterized by $\Omega_\sigma = \rho_\mathrm{DE}/3 H^2 F_0$ - where the subscript $0$ denotes quantities evaluated at $z=0$ - and compare it to the $\Lambda$CDM+$N_{\textup{eff}}$ model. As can be seen, when $\xi<0$, the scalar field contributes to the total energy density in a way that is very similar to the $\Lambda$CDM +$N_{\textup{eff}}$ model. Having started with the same $\xi<0$ and initial condition $\sigma_i/M_{\rm pl}<1$ in both the $n=2$ and $n=4$ case, the term multiplying the square bracket in Eq. is smaller in the latter case and the rolling of the scalar field towards smaller values is less efficient. The equation of state $w_\mathrm{DE}$ is not $1/3$ anymore in general when the scalar field is subsequently driven by matter.
Our model is therefore different from EDE models recently proposed in the literature (see e.g. Refs. [@Agrawal:2019lmo; @Poulin:2018cxd; @Alexander:2019rsc; @Lin:2019qug; @Smith:2019ihp]) for which the equation of state is close to $-1$ at early times. Note also that, in our model, the scalar field moves in a natural way after radiation-matter equality, being driven by non-relativistic matter, and is not important just around recombination.
In general, a distinct feature of our model is the modification to gravity induced by $\sigma$ which is plotted in the bottom panel of Fig. \[fig:Background\]. For $\xi <0$, since the scalar field contribution becomes negligible at late times, both $G_N$ and $G_\textup{eff}$ are very close to $G$ today. For this reason our model is consistent with laboratory and Solar System experiments for a large volume of the parameter space, as we will show in this paper. We do not show the evolution of the PN parameters defined in Eqs. and as they behave similarly.
Methodology and data sets {#sec:datasets}
=========================
We run a Markov-chain Monte Carlo (MCMC) using the publicly available code [MontePython-v3]{}[^5] [@Audren:2012wb; @Brinckmann:2018cvx] wrapped either with [CLASSig]{} [@Umilta:2015cta], a modified version of the [CLASS]{}[^6] [@Lesgourgues:2011re; @Blas:2011rf] for scalar-tensor theory of gravity, or with a modified version of [hiCLASS]{} [@Zumalacarregui:2016pph; @Bellini:2019syt] which allows to study consistently oscillating scalar fields. The agreement of [CLASSig]{} and [hiCLASS]{} for the precision of current and future experiments has been demonstrated in [@Bellini:2017avd]. Mean values and uncertainties on the parameters reported, as well as the contours plotted, have been obtained using [GetDist]{}[^7] [@Lewis:2019xzd]. For all our runs we set the scalar field in slow-roll and use adiabatic initial conditions for the scalar field perturbations [@Rossi:2019lgt; @Paoletti:2018xet].
We study cosmological models in Eq. with $n=2 \,, 4$ and free $\xi$, and devote particular attention to the value of $\xi=-1/6$, which is obviously nested in the previous class with $n=2$. We sample the cosmological parameters $\{\omega_b,\,\omega_{cdm}, \,\theta_s,\,\ln 10^{10}A_s,\,n_s,\,\tau_\textup{reio},\,\xi,\,\sigma_i\}$ fixing $n=2,\,4$ and using Metropolis-Hastings algorithm. We consider flat priors consistent with the stability conditions in Sec. \[sec:background\] on the extra parameters we consider $\xi\in[-0.9,\,0.9]$ and $\sigma_i/M_\textup{pl}\in[0,\,0.9]$, for $n=2$ case with free $\xi$ and $\sigma_i/M_\textup{pl}\in[0,\,0.9]$ in the CC case. For the case with $n=4$, we change our prior to $\xi\in[-0.9,\,0.2]$ as larger positive values for the coupling $\xi$ lead to a deviation of order $10^{-1}$ from GR as can be seen from Fig. \[fig:Background\]. As in [@Ballardini:2016cvy], we take into account the different value of the effective gravitational constant in the modified Big Bang Nucleosynthesis (BBN) condition for the helium, and the baryon density tabulated in the public code PArthENoPE [@Pisanti:2007hk]. We consider the chains to be converged using the Gelman-Rubin criterion $R-1<0.01$.
We constrain the cosmological parameters using several combination of data sets. We use the CMB measurements from the [*Planck*]{} 2018 release (hereafter P18) on temperature, polarization, and weak lensing CMB angular power spectra [@Aghanim:2019ame; @Akrami:2018vks]. We use the following likelihood combination, the so-called [*Planck*]{} baseline: on high-multipoles, $\ell \geq 30$, we use the [Plik]{} likelihood, on the lower multipoles we use the [Commander]{} likelihood for temperature and [SimAll]{} for the E-mode polarization [@Aghanim:2019ame], for the lensing likelihood we the conservative multipoles range, i.e. $8 \leq \ell \leq 400$ [@Akrami:2018vks].
Baryon acoustic oscillation (BAO) measurements from galaxy redshift surveys are used as primary astrophysical data set to constrain these class of theories providing a complementary late-time information to the CMB anisotropies. We use the Baryon Spectroscopic Survey (BOSS) DR12 [@Alam:2016hwk] “consensus” in three redshift slices with effective redshifts $z_{\rm eff} = 0.38,\,0.51,\,0.61$ [@Ross:2016gvb; @Vargas-Magana:2016imr; @Beutler:2016ixs] in combination with measurements from 6dF [@Beutler:2011hx] at $z_{\rm eff} = 0.106$ and the one from SDSS DR7 [@Ross:2014qpa] at $z_{\rm eff} = 0.15$. We consider a Gaussian likelihood based on the latest determination of $H_0$ from SH0ES, i.e. $H_0 = 74.03 \pm 1.42$ km s$^{-1}$Mpc$^{-1}$ [@Riess:2019cxk], which we will denote as R19 in the following. We also consider a tighter Gaussian likelihood, i.e. $H_0 = 73.3 \pm 0.8$ km s$^{-1}$Mpc$^{-1}$ [@Verde:2019ivm], obtained from a combination of $H_0$ measurements from SH0ES [@Riess:2019cxk], MIRAS [@Huang:2019yhh], CCHP [@Freedman:2019jwv], H0LiCOW [@Wong:2019kwg], MCP [@Reid:2008nm] and SBF which we will denote as V19 in the following. We should warn the reader that the V19 value is obtained by neglecting covariances between the aforementioned observations, as stressed in Ref. [@Verde:2019ivm]. Nevertheless, V19 can give an idea of how our model can respond to a possible future worsening of the $H_0$ tension.
Note that our analysis differs from [@Rossi:2019lgt] not only in the updated data, but also in theoretical priors: in this paper we consider flat priors on $(\xi, \sigma_i)$, whereas in [@Rossi:2019lgt] flat priors were assumed on $(\xi, M_{\rm pl})$, with $\xi>0$ and $\xi <0$ considered separately, and $M_{\rm pl}$ was also allowed to vary, with a boundary condition on $\sigma_0$ (the value of the scalar field today) to fix consistency between $G_\mathrm{eff}$ and $G$. We have however verified that these different priors have a very small effect on the resulting posterior distributions of the parameters, at least for $\xi=-1/6$.
Results {#sec:results}
=======
The results of our cosmological analysis for the CC ($n=2$ with free $\xi$) model are summarized in Fig. \[fig:Resultscc\] (Fig. \[fig:Resultsn2\]), where we plot the reconstructed two-dimensional posterior distributions of main and derived parameters, and in Table \[tab:cc\] (Table \[tab:n2\]), where we report the reconstructed mean values and the 68% and 95% CL. We also report our results for the $n=4$ case in Table \[tab:n4\].
We find similar values for $H_0$ in all the models, but larger than in $\Lambda$CDM. We find $H_0=68.47^{+0.58}_{-0.86}$ ($H_0=68.40^{+0.59}_{-0.80}$) km s$^{-1}$Mpc$^{-1}$ at 68% CL for CC (for free $\xi$) with P18 data only. As in other similar models, we find larger values for $n_s \,, \omega_c \,, \sigma_8$ and smaller values for $\omega_{b}$ compared to the baseline $\Lambda$CDM model. When BAO and SH0ES data are combined, i.e. P18+BAO+R19, we obtain $H_0=69.29^{+0.59}_{-0.72}$ ($H_0=69.10^{+0.49}_{-0.66}$) km s$^{-1}$Mpc$^{-1}$ for CC (for free $\xi$). Higher values for $H_0$ can be obtained by substituting the combination of measurements V19 to R19, as can be seen from Tables \[tab:cc\] and \[tab:n2\]. Note that similar results are also obtained in the $n=4$ case, for which we find a slightly smaller value of $H_0=68.05\pm 0.56 $ ($H_0=69.09^{+0.52}_{-0.69} $) km s$^{-1}$Mpc$^{-1}$ with P18 (P18+BAO+R19) data. For this reason, we focus our discussion on the $n=2$ case in the following, commenting only when results for $n=4$ substantially differ.
In Tables I, II, III, we also report the difference in the best-fit of the model with respect to $\Lambda$CDM, i.e. $\Delta \chi^2 = \chi^2 - \chi^2 (\Lambda\mathrm{CDM})$, where negative values indicate an improvement in the fit of the given model with respect to the $\Lambda$CDM for the same dataset [^8]. Although our models provide a similar or slightly worst fit to P18 data compared to $\Lambda$CDM, we find $\Delta \chi^2 \sim -5$ ($-6.8$) for CC (free $\xi$) when BAO+R19 are combined. Higher values of $\Delta \chi^2$ are obviously obtained by substituting V19 to R19. We also compute values of the Aikike (Bayes) information criteria $\Delta {\rm AIC}$ ($\Delta {\rm BIC}$) defined as $\Delta {\rm AIC}=\Delta\chi^2+2\Delta p$ ($\Delta {\rm BIC}=\Delta\chi^2+\Delta p\ln N$), where $\Delta p$ is the number of extra parameters with respect to $\Lambda$CDM model and $N$ is the number of data points considered in our MCMC analysis [^9] [@Liddle:2007fy]. According to both criteria, all our models are penalized compared to $\Lambda$CDM for P18 data only due to the addition of parameters. Only for AIC we find that our model with $n=2$ is (strongly) favoured for (CC) free $\xi$ compared to $\Lambda$CDM when BAO and R19 are combined. Substituting V19 to R19 makes the statistical preference of our model stronger in general.
[**Constraints on modified gravity parameters:**]{} The constraints on the modified gravity parameter are very different in the CC and $n=2$ case, which are a one- and two-parameter extension of the $\Lambda$CDM model. Although the mean values are very similar, constraints are very much looser in the latter case. This is because, when $\xi$ is large and negative, the decreasing of the scalar field is very efficient and thus its effect redshifts away even before matter-radiation equality, leaving smaller imprints on the CMB. Note that positive values of $\xi$, for which the scalar field increases after matter-radiation equality contributing to the late-time background evolution, seem disfavoured by the data for our priors. In particular for P18, we find an upper bound $\xi<0.052$ ($\xi<0.02$) at the 2$\sigma$ level for $n=2$ ($n=4$). The upper bound is even more stringent when we add to the analysis BAO+R19 data for which we find $\xi<0.047$ ($\xi<-0.026$) at the 2$\sigma$ level for $n=2$ ($n=4$).
[**Comparison with BBN constraints:**]{} With our priors, the departure of $\sqrt{F}$ from $M_\mathrm{pl}$ can also be constrained by BBN [@Copi:2003xd; @Bambi:2005fi; @Coc:2006rt]. Since the scalar field is frozen at very early times, the BBN constraints reported in [@Copi:2003xd; @Bambi:2005fi] would imply $\xi\sigma_i^n=0.01^{+0.20}_{-0.16}$ at 68% CL, which are consistent, but less stringent, than the constraints reported in Tables \[tab:cc\], \[tab:n2\] and \[tab:n4\], as already mentioned in previous works on scalar-tensor [@Ballardini:2016cvy]. We find $ -0.014^{+0.026}_{-0.052}$ ($ >-0.0150$) for the $n=2$ (CC[^10]) and $-0.0010^{+0.0029}_{-0.0076}$ for the $n=4$ case at 95% CL using P18 data only. When adding BAO+R19 we obtain a higher $\xi\sigma_i^n$ and the constraints change to $-0.025^{+0.037}_{-0.070}$ ($>-0.0234 $) for the $n=2$ (CC) and $-0.013^{+0.021}_{-0.038} $ for the $n=4$ case at 95% CL. Note that $\xi\sigma_i^n$ is more constrained in the CC case compared to $n=2$ and $n=4$, as the coupling is fixed to $\xi=-1/6$.
[**Comparison with PN:**]{} The derived cosmological PN parameters are well consistent with GR and their uncertainties are comparable with bounds from Solar System experiments [@Bertotti:2003rm; @Will:2014kxa]. Again, because of the large errors on $\xi$, the bounds in the $n=2$ model are somewhat looser than in the CC model. Therefore, the CC ($n=2$) model potentially offers a simple one (two) modified gravity parameter extension to the baseline $\Lambda$CDM that naturally eases the $H_0$ tension and can be consistent at 2$\sigma$ with Solar System constraints on the deviation from GR. We have checked that the inclusion of Solar System constraints in our analysis by means of a Gaussian prior based on the Cassini constraint $\gamma_{\rm PN} - 1 = 2.1\pm2.3\times10^{-5}$ [@Bertotti:2003rm] has a very small impact in our constraints on the six standard cosmological parameters.
For the representative example of $n=2$ with free $\xi$ the constraint on $H_0$ obtained from P18+BAO+R19 changes to $H_0=69.00^{+0.47}_{-0.57} $ km s$^{-1}$Mpc$^{-1}$. The constraints on the modified gravity parameters instead change substantially. Thanks to the constraining power of the prior we find $\sigma_i=0.19^{+0.13}_{-0.08} \,M_\mathrm{pl}$ at 68% CL and $\gamma_{\rm PN}-1> -2.2 \cdot 10^{-6} $ and a bound on $\xi< -0.15$ at 95% CL. Although $\xi$ remains unconstrained, we note that the upper limit is tighter than the the one obtained without the prior information on $\gamma_{\rm PN}$. Negative values of $\xi$ are more favored as they lead to a more efficient rolling of the scalar field toward smaller values, and therefore a smaller $\gamma_{\rm PN}-1$.
![Constraints on main and derived parameters of the CC model with $n=2$ and $\xi=-1/6$ from Planck 2018 data (P18), P18 in combination with BAO and SH0ES measurements and P18 in combination with BAO and a combined prior which takes into account all the late time measurements. Parameters on the bottom axis are our sampled MCMC parameters with flat priors, and parameters on the left axis are derived parameters (with $H_0$ in \[km s$^{-1}$Mpc$^{-1}$\]). Constraints for the $\Lambda$CDM model obtained with P18 data are also shown for a comparison. Contours contain 68% and 95% of the probability.[]{data-label="fig:Resultscc"}](cc.pdf){width=".85\columnwidth"}
[**Robustness and caveats of the inclusion of SNe data:**]{} So far we did not use the SNe Ia luminosity distance because the time evolution of gravitational constant changes the peak luminosity of SNe and this needs to be properly accounted in the analysis [@GarciaBerro:1999bq; @Riazuelo:2001mg; @Nesseris:2006jc; @Wright:2017rsu]. However, for the bestfit value obtained from P18 + BAO + R19 with the priors on $\gamma_{\rm PN}$, the relative change of $G_\textup{eff}$ from $G$ today is at most $10^{-5}$ in the relevant range of redshifts for SNe Ia. Under the assumption that we can ignore the effect of time evolution of $G_\textup{eff}$ on the magnitude-redshift relation of SNe Ia, we use the Pantheon Sample of SNe to check the robustness of our constraint on $H_0$ [@Scolnic:2017caz]. We obtain $H_0=69.28^{+0.58}_{-0.74} $ ($H_0= 68.98^{+0.46}_{-0.54} $) km s$^{-1}$Mpc$^{-1}$ for CC (for free $\xi$) using P18+BAO+R19+Pantheon with the prior on $\gamma_{\rm PN}$. This shows that the inclusion of SNe Ia data does not change the constraint on $H_0$. Note also that the modification of the gravitational constant can also change the low-redshift distance ladder measurements of the Hubble constant [@Desmond:2019ygn; @Desmond:2020wep]. However, again due to the smallness of the relative change of $G_\textup{eff}$ from $G$ today, this effect can be ignored safely in our models.
[**Comparison with other EDE models:**]{} Models based on a sharp energy injection around the time of matter-radiation equality lead to a value of $H_0$ which can be higher than the ones we found within our model for any choice of $n$ and $\xi$ although this is model dependent (see e.g. Refs. [@Poulin:2018cxd; @Agrawal:2019lmo; @Alexander:2019rsc; @Lin:2019qug; @Smith:2019ihp]). However, the radiation-like behavior of the scalar field in theories described by the action , is completely generic and, provided that the coupling $\xi$ is negative, the scalar field contribution quickly becomes negligible thanks to the coupling to non-relativistic matter and modifies essentially only the early time dynamics. For this reason, a higher $H_0$ than in $\Lambda$CDM is a natural outcome of the NMC for a large portion of the parameter space compared to EDE models, which have more extra parameters to tune.
[**Addition of $N_\mathrm{eff}$:**]{}
As already mentioned in the introduction, the archetypal way to reduce the sound horizon at baryon drag is to allow the number of relativistic species $N_\textup{eff}$ to vary [@Riess:2011yx; @Wyman:2013lza]. By varying $N_\textup{eff}$, we find for P18+BAO+R19 $\Delta \chi^2 \sim -2.8$ with $H_0=70.01\pm 0.89$ km s$^{-1}$Mpc$^{-1}$ and $N_\textup{eff}=3.30\pm 0.14$. Despite the higher mean value for $H_0$, the improvement in the fit is smaller than what we obtain for CC case, and even smaller for NMC with $n=2$. We then investigate to which extent the addition of extra relativistic species ($N_\textup{eff}$) to our model with $n=2$ can further ease the tension.
We allow $N_\textup{eff}$ to vary with a flat prior $N_\textup{eff}\in[0,6]$ and we restrict to the combination of P18, BAO and V19. The results of our analysis are shown in Fig. \[fig:ResultsNur\], where we plot for the CC and $n=2$ case the 2D posterior distributions of the main parameters $\sigma_i$ and $N_\textup{eff}$ and the derived $H_0$, $\gamma_{\rm PN}$ and $\xi\sigma_i^2$. To provide the reader with a comparison, we also plot the constraints on the $\Lambda$CDM+$N_\textup{eff}$ model for the same dataset. As in the case where $N_\textup{eff}$ is fixed, constraints on the other cosmological parameters are nearly the same in both the models. Again, we find very similar results, i.e. $N_\textup{eff} = 3.43^{+0.16}_{-0.13}$, $H_0 = 71.45\pm 0.68$ km s$^{-1}$Mpc$^{-1}$ in the CC model and $N_\textup{eff} = 3.44^{+0.15}_{-0.12}$, $H_0=71.44\pm 0.67$ km s$^{-1}$Mpc$^{-1}$ in the $n=2$ model at 68% CL.
![Constraints on main and derived parameters of the model with $n=2$ and $\xi$ as a main parameter from [*Planck*]{} 2018 data (P18), P18 in combination with BAO and SH0ES measurements and P18 in combination with BAO and a combined prior which takes into account all the late time measurements. Parameters on the bottom axis are our sampled MCMC parameters with flat priors, and parameters on the left axis are derived parameters (with $H_0$ in \[km s$^{-1}$Mpc$^{-1}$\]). Constraints for the $\Lambda$CDM model obtained with P18 data are also shown for a comparison. Contours contain 68% and 95% of the probability.[]{data-label="fig:Resultsn2"}](n2.pdf){width=".85\columnwidth"}
CC P18 P18 + BAO + R19 P18 + BAO + V19
----------------------------------------- ------------------------------------------------------ ------------------------------------------------------- ------------------------------------------------------
$10^{2}\omega_{\rm b}$ $2.242\pm 0.015$ $2.248\pm 0.014$ $2.252\pm0.013$
$\omega_{\rm c}$ $0.1197\pm 0.0012 $ $0.11910\pm 0.00099$ $0.1188\pm 0.0010$
$100*\theta_{s }$ $1.04194\pm 0.00030 $ $1.04205\pm 0.00028 $ $1.042\pm 0.00028$
$\tau_\textup{reio }$ $0.0547\pm 0.0077$ $0.0570\pm 0.0071$ $0.05803\pm 0.0075$
$\ln \left( 10^{10} A_{\rm s} \right)$ $3.046\pm 0.015$ $3.049\pm 0.014$ $3.053\pm 0.015$
$n_{\rm s}$ $0.9675\pm 0.0046$ $0.9695\pm 0.0038 $ $0.9734\pm 0.0037$
$\sigma_i$ \[M$_\mathrm{pl}$\] $0.1312_{-0.13}^{+0.039}$ $0.224^{+0.13}_{-0.081}$ $0.3585^{+0.078}_{-0.047}$
$H_0$ \[km s$^{-1}$Mpc$^{-1}$\] $68.47^{+0.58}_{-0.86}$ $69.29^{+0.59}_{-0.72}$ $70.56\pm0.6$
$\sigma_8$ $0.8272^{+0.0063}_{-0.0081}$ $0.8313^{+0.0079}_{-0.011}$ $0.841\pm0.010$
$r_s$ \[Mpc\] $146.97^{+0.33}_{-0.29}$ $146.83^{+0.48}_{-0.34}$ $146.4\pm 0.45$
$\xi\sigma^2_i$ \[$M_\mathrm{pl}^2]$ $>-0.0150$ $>-0.0234$ $-0.022^{+0.016}_{-0.015}$
$\sigma_0$ \[M$_\mathrm{pl}$\] $0.004017_{-0.004}^{+0.0012}$ $0.006841_{-0.0025}^{+0.004}$ $0.01102_{-0.0015}^{+0.0024}$
$\gamma_{\rm PN}-1$ $> -0.95\cdot 10^{-5}$ $> -1.5\cdot 10^{-5}$ $\left(\,-1.4^{+1.0}_{-0.9}\,\right)\cdot 10^{-5}$
$\beta_{\rm PN}-1$ $\left(\,0.23^{+0.61}_{-0.34}\,\right)\cdot 10^{-6}$ $ \left(\,0.53^{+0.75}_{-0.61}\,\right)\cdot 10^{-6}$ $\left(\,1.16^{+0.78}_{-0.84}\,\right)\cdot 10^{-6}$
$\Delta \chi^2$ $+0.42$ $-5.0$ $-13.64$
$n=2$ P18 P18 + BAO + R19 P18 + BAO + V19
----------------------------------------- ----------------------------------------------------- ---------------------------------------------------- ----------------------------------------------------
$10^{2}\omega_{\rm b}$ $2.241\pm 0.015$ $2.249\pm 0.014$ $2.253\pm 0.014$
$\omega_{\rm c}$ $0.1198\pm0.0012$ $0.11903^{+0.00095}_{-0.0011}$ $ 0.1190\pm 0.0012$
$100*\theta_{s }$ $1.04193\pm0.00030$ $1.04205\pm 0.00031$ $1.04210\pm 0.00029$
$\tau_\textup{reio }$ $0.0544\pm0.0076$ $ 0.0564\pm 0.0076$ $ 0.0578\pm0.0072$
$\ln \left( 10^{10} A_{\rm s} \right)$ $3.045\pm0.0014$ $3.048\pm 0.015 $ $3.052\pm 0.014$
$n_{\rm s}$ $0.9673\pm 0.0046$ $ 0.9699\pm 0.0046$ $0.9724\pm 0.0041$
$\sigma_i$ \[M$_\mathrm{pl}$\] $<0.224$ $0.260^{+0.088}_{-0.19}$ $> 0.46$
$\xi$ $ < 0.052$ (95% CL) $< 0.047$ (95% CL) $<-0.0283$(95% CL)
$H_0$ \[km s$^{-1}$Mpc$^{-1}$\] $68.40^{+0.59}_{-0.80}$ $69.10^{+0.49}_{-0.66} $ $70.64\pm 0.71$
$\sigma_8$ $0.8456_{-0.018}^{+0.013}$ $0.8370^{+0.0072}_{-0.020}$ $ 0.8450^{+0.0088}_{-0.014}$
$r_s$ \[Mpc\] $147.01\pm 0.36 $ $146.95^{+0.48}_{-0.30}$ $146.08^{+0.77}_{-0.89}$
$\xi\sigma^2_i$ \[$M_\mathrm{pl}^2$\] $ -0.014^{+0.026}_{-0.052}$ $-0.025^{+0.037}_{-0.070}$ $-0.030^{+0.030}_{-0.074} $
$\sigma_0$ \[M$_\mathrm{pl}$\] $ 0.1046^{+0.40}_{-0.18}$ $ 0.09^{+0.46}_{-0.19}$ $0.20^{+0.33}_{-0.26}$
$\gamma_{\rm PN}-1$ $> -1.73\cdot 10^{-3}$ $> -1.56\cdot 10^{-3}$ $> -1.26\cdot 10^{-3}$
$\beta_{\rm PN}-1$ $-\left(\,3.0^{+1.8}_{-1.6}\,\right)\cdot 10^{-5} $ $-\left(\,3.0^{+1.7}_{-1.4}\,\right)\cdot 10^{-5}$ $-\left(\,1.5^{+2.9}_{-2.5}\,\right)\cdot 10^{-5}$
$\Delta \chi^2$ $+0.52$ $-6.8$ $-18.44$
$n=4$ P18 P18 + BAO + R19 P18 + BAO + V19
----------------------------------------- ------------------------------------------------------ --------------------------------------------------- ----------------------------------------------------
$10^{2}\omega_{\rm b}$ $2.240\pm 0.015$ $2.250\pm 0.013$ $2.258\pm 0.013$
$\omega_{\rm c}$ $0.1198\pm 0.0012$ $ 0.11892\pm 0.00093$ $ 0.11830\pm 0.00097$
$100*\theta_{s }$ $1.04190\pm 0.00028$ $1.04205\pm 0.00028$ $1.04217\pm 0.00028 $
$\tau_\textup{reio }$ $0.0545\pm 0.0074$ $ 0.0564\pm 0.0076$ $0.0596^{+0.0070}_{-0.0078} $
$\ln \left( 10^{10} A_{\rm s} \right)$ $3.045\pm 0.014$ $3.049\pm 0.015 $ $3.055\pm 0.015 $
$n_{\rm s}$ $0.9662\pm0.0043$ $ 0.9706^{+0.0037}_{-0.0042} $ $0.9757^{+0.0039}_{-0.0044}$
$\sigma_i$ \[M$_\mathrm{pl}$\] $< 0.257$ $0.37^{+0.20}_{-0.17}$ $0.55^{+0.13}_{-0.11}$
$\xi$ $ <0.02$ (95% CL) $< -0.026$ (95% CL) $< -0.031$(95% CL)
$H_0$ \[km s$^{-1}$Mpc$^{-1}$\] $68.05\pm 0.56 $ $69.09^{+0.52}_{-0.69} $ $70.23\pm 0.54$
$\sigma_8$ $0.8247\pm 0.0061$ $0.8370^{+0.0072}_{-0.020}$ $0.845^{+0.010}_{-0.018} $
$r_s$ \[Mpc\] $147.06\pm 0.28 $ $146.96^{+0.39}_{-0.33}$ $ 146.69^{+0.38}_{-0.43}$
$\xi\sigma^4_i$ \[$M_\mathrm{pl}^4$\] $ -0.0010^{+0.0029}_{-0.0076}$ $-0.013^{+0.021}_{-0.038}$ $-0.035^{+0.038}_{-0.057} $
$\sigma_0$ \[M$_\mathrm{pl}$\] $0.18^{+0.39}_{-0.22}$ $0.18^{+0.25}_{-0.17}$ $ 0.20^{+0.21}_{-0.13}$
$\gamma_{\rm PN}-1$ $>-1.72\cdot10^{-4}$ $>-1.65\cdot10^{-4}$ $>-2.34\cdot10^{-4}$
$\beta_{\rm PN}-1$ $\left(\,-0.8^{+11.0}_{-9.4}\,\right)\cdot 10^{-6} $ $\left(\,0.4^{+6.1}_{-3.8}\,\right)\cdot 10^{-6}$ $\left(\,2.5^{+7.4}_{-6.6}\,\right)\cdot 10^{-6} $
$\Delta \chi^2$ $-0.58$ $-1.14$ $-9.42$
![Constraints on some of the main and derived parameters of the CC and $n=2$ model with the addition of $N_\textup{eff}$ from P18 in combination with BAO and a combined prior which takes into account all the late time measurements. Parameters on the bottom axis are our sampled MCMC parameters with flat priors, and parameters on the left axis are derived parameters (with $H_0$ in \[km s$^{-1}$Mpc$^{-1}$\]). Contours contain 68% and 95% of the probability.[]{data-label="fig:ResultsNur"}](ccNur.pdf){width=".85\columnwidth"}
It is interesting to note that the value we find $\Delta N_\textup{eff} \sim 0.39$ is similar to the case of an additional thermalized massless boson which decouples at a temperature $T>100 $ MeV [@Weinberg:2013kea]. We note that the relevant parameter that regulates the scalar field modification to $H(z)$, that is $\xi\sigma_i^2$, is now much smaller than in the correspondent case with $N_\textup{eff}$ fixed (see Table \[tab:cc\] and \[tab:n2\]), that is $\xi\sigma_i^2> -0.0193$ in the CC model and $\xi\sigma_i^2= -0.012^{+0.018}_{-0.003}$ in the $n=2$ model: this means that the higher value of $H_0$ is now driven by a combination of a higher $N_\textup{eff}$ with the non-minimally coupled scalar field $\sigma$. In fact, in the case of the $\Lambda$CDM+$N_\textup{eff}$ model we find a larger $N_\textup{eff} = 3.50\pm 0.12$ at 68% CL consistently with the scalar field effectively contributing as an extra dark radiation component in the CC and $n=2$ case.
Discussion and conclusions {#sec:conclusions}
==========================
In this paper we have studied the addition of a cosmological massless scalar field $\sigma$ to $\Lambda$CDM with a coupling to the Ricci scalar of the form $F(\sigma)=M_\textup{pl}^2[1+\xi(\sigma/M_\textup{pl})^n]$, in the case of $n=2,\,4$. This class of models has one (as for CC) or two extra parameters with respect to $\Lambda$CDM. The scalar field $\sigma$ is frozen deep in the radiation era, essentially contributing to the expansion history of the Universe as an effective relativistic degree of freedom, and the coupling to non-relativistic matter acts as a driving force for the scalar field around radiation-matter equality [@Umilta:2015cta; @Ballardini:2016cvy; @Rossi:2019lgt]. The basic assumption of a cosmological constant $\Lambda$ minimizes the deviations from $\Lambda$CDM at late time which are present in scalar-tensor theories and allows to focus on the early time dynamics.
We have used the most recent [*Planck*]{}, BAO and SH0ES data to perform a MCMC analysis and constrain the parameters of our model. We find that Planck 18 (+BAO+R19) constrain the expansion rate of the Universe from $H_0=68.40^{+0.59}_{-0.80}$ ($H_0=69.10^{+0.49}_{-0.66} $) km s$^{-1}$Mpc$^{-1}$ for $n=2$. Similar results for the cosmological parameters can also be obtained in the CC case.
Compared to other attempts to alleviate the $H_0$ tension such as EDE models, we obtain a lower expansion rate. However, we stress that EDE models require two or three extra parameters with respect to $\Lambda$CDM, which have to be fine tuned to inject the precise amount of energy to the cosmic fluid in a very narrow range of redshift. The models considered here have only one or two extra parameters and can be easily embedded in a consistent theoretical framework of scalar-tensor theories of gravity.
We find that our constraints on $\xi\sigma^n$, the deviation from GR, are consistent with those obtained from BBN [@Copi:2003xd; @Bambi:2005fi] and the constraints on the PN parameters from the Solar System measurements [@Bertotti:2003rm; @Will:2014kxa]. Higher values for $H_0$ can be obtained by further allowing $N_\mathrm{eff}$ to vary or by using the tighter prior V19 on $H_0$ rather than R19. In the former case, we find tighter constraints on $\xi \sigma_i^n$ that regulates the scalar field contribution to the expansion history during the radiation era and the larger value of $H_0$ is driven by a cooperation with the extra relativistic species described by $N_\mathrm{eff}$.
MBr acknowledges the Marco Polo program of the University of Bologna for supporting a visit to the Institute of Cosmology and Gravitation at the University of Portsmouth, where this work started. MBa, FF, DP acknowledge financial contribution from the contract ASI/INAF for the Euclid mission n.2018-23-HH.0. AEG and KK received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 646702 “CosTesGrav”). KK is also supported by the UK STFC ST/S000550/1. WTE is supported by an STFC consolidated grant, under grant no. ST/P000703/1. FF and DP acknowledge financial support by ASI Grant 2016-24-H.0. Numerical computations for this research were done on the Sciama High Performance Compute cluster, which is supported by the ICG, SEPNet, and the University of Portsmouth. MBr thanks Gui Brando and Jascha A. Schewtschenko for help in the use of Sciama.
**Note added:** While this project was near to completion, a related paper [@Ballesteros:2020sik], also studying how a massless non-minimally coupled scalar field with $n=2$ with a flat potential could ease the tension, appeared on the arXiv. Where a comparison is possible, we find consistency in the estimate of cosmological parameters, but our findings for $\Delta \chi^2$ are at odds with [@Ballesteros:2020sik]. Not only NMC with $n=2$ leads to a larger improvement in the fit than the addition of $N_\mathrm{eff}$ for P18+BAO+R19, but also the CC does.
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[^1]: See Refs. [@Berghaus:2019cls; @Sakstein:2019fmf] for recent proposals that reduce the degree of fine-tuning in EDE models.
[^2]: See also Ref. [@Zumalacarregui:2020cjh] for a related mechanism in the framework of an exponentially coupled cubic Galileon model.
[^3]: Note that the choice of $V \propto F^2$ corresponds to a cosmological constant in the corresponding Einstein frame ($\hat g_{\mu \nu} \propto F g_{\mu \nu}$) in which the canonically rescaled scalar field is universally coupled to the trace of the matter energy-momentum tensor.
[^4]: Note that $\Omega_\sigma$ becomes slightly negative in Fig. \[fig:Background\]. This is not a physical problem as $\Omega_\sigma$ only parameterizes the contribution of the scalar field to the total expansion rate $H(z)$ when the Einstein equations are written in the Einstein gravity form, see e.g. Ref. [@Boisseau:2000pr; @Gannouji:2006jm].
[^5]: <https://github.com/brinckmann/montepython_public>
[^6]: <https://github.com/lesgourg/class_public>
[^7]: <https://getdist.readthedocs.io/en/latest>
[^8]: Note that the $\Lambda$CDM reference cosmology in our case has massless neutrinos, differently from the assumption adopted by the Planck collaboration of one massive neutrino with $m_\nu=0.06$ eV consistent with a normal hierarchy with minimum mass allowed by particle physics. The differences with respect to the baseline $Planck$ results in the estimate of the cosmological parameters due the choice $N_\mathrm{eff}=3.046$ and $m_\nu=0$ is small, except for a shift towards higher values for $H_0$, as $H_0 = 67.98 \pm 0.54$ ($H_0 = 68.60 \pm 0.43$) km s$^{-1}$Mpc$^{-1}$ for P18 (P18+BAO+R19).
[^9]: We consider 2352 points for P18, 8 for BAO and 1 (6) for R19 (V19).
[^10]: Note that, in the CC case, $\xi\sigma_i^2<0$ by construction.
|
---
bibliography:
- 'Bibli.bib'
---
[**Trapped modes and reflectionless modes as\
eigenfunctions of the same spectral problem**]{}
<span style="font-variant:small-caps;">Anne-Sophie Bonnet-Ben Dhia</span>$^1$, <span style="font-variant:small-caps;">Lucas Chesnel</span>$^2$, <span style="font-variant:small-caps;">Vincent Pagneux</span>$^3$\
$^1$ Laboratoire Poems, CNRS/ENSTA/INRIA, Ensta ParisTech, Université Paris-Saclay, 828, Boulevard des Maréchaux, 91762 Palaiseau, France;\
$^2$ INRIA/Centre de mathématiques appliquées, École Polytechnique, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau, France;\
$^3$ Laboratoire d’Acoustique de l’Université du Maine, Av. Olivier Messiaen, 72085 Le Mans, France.\
E-mails: `bonnet@ensta.fr`, `lucas.chesnel@inria.fr`, `vincent.pagneux@univ-lemans.fr`\
– –
**Abstract.** We consider the reflection-transmission problem in a waveguide with obstacle. At certain frequencies, for some incident waves, intensity is perfectly transmitted and the reflected field decays exponentially at infinity. In this work, we show that such reflectionless modes can be characterized as eigenfunctions of an original non-selfadjoint spectral problem. In order to select ingoing waves on one side of the obstacle and outgoing waves on the other side, we use complex scalings (or Perfectly Matched Layers) with imaginary parts of different signs. We prove that the real eigenvalues of the obtained spectrum correspond either to trapped modes (or bound states in the continuum) or to reflectionless modes. Interestingly, complex eigenvalues also contain useful information on weak reflection cases. When the geometry has certain symmetries, the new spectral problem enters the class of $\mathcal{PT}$-symmetric problems.\
**Key words.** Waveguides, transmission spectrum, backscattering, trapped modes, complex scaling, $\mathcal{P}\mathcal{T}$ symmetry.
Introduction
============
In bounded domains, eigenvalues and corresponding eigenmodes completely determine the solutions to wave equations. In open systems, it is necessary to take into account also *complex resonances*, sometimes known as *leaky modes* or *quasi normal modes* in literature [@aguilar1971class; @balslev1971spectral; @simon1972quadratic; @moiseyev1998quantum; @AsPV00; @sjostrand1991complex; @zworski1999resonances; @zworski2011lectures; @moiseyev2011non]. Complex resonances provide the backbone of phenomena of wave propagation. For example, they indicate if there is a rapid variation of the scattering coefficients. We emphasize that real eigenvalues and complex resonances are intrinsic and universal objects that can be computed using different approaches. Among them, let us mention methods based on complex scaling or *Perfectly Matched Layers* (PMLs) [@Bera94] or techniques relying on Hardy space infinite elements [@HoNa09; @HoNa15]. However complex resonances fail to provide useful information on recurrent questions such as “how large is the backscattered energy?” or “is there good transmission in the system?”. Such questions are of tremendous importance in many topics of intense study in wave physics: extraordinary optical transmission [@ebbesen1998extraordinary], topological states immune to backscattering [@wang2009observation; @lu2014topological], perfect transmission resonances [@zhukovsky2010perfect], transmission eigenchannels through disordered media [@kim2012maximal; @sebbah2017scattering], reflectionless metamaterials [@rahm2008optical] or metasurfaces [@yu2014flat; @asadchy2015broadband].\
In this article, we define a new complex spectrum to quantify these aspects. In particular, our spectrum allows one to identify wavenumbers for which there is an incident field such that the backscattered field is evanescent (Figure \[figIntro\]). For construction of waveguides where such phenomenon occurs at a prescribed frequency, we refer the reader to [@BoNa13; @ChNa16; @BoCN17; @ChNPSu]. Our spectrum also contains usual trapped modes [@Urse51; @Evan92; @EvLV94; @DaPa98; @LiMc07; @Naza10c] called *Bound States in the Continuum* (BSCs or BICs) in quantum mechanics [@SaBR06; @Mois09; @GPRO10; @HZSJS16]. To compute this new spectrum, we use complex scaling/PMLs techniques in an original way. Our approach is based on the following basic observation: if for an incident wave, the backscattered field is evanescent, then the total field is ingoing in the input lead and outgoing in the output lead. Therefore, changing the sign of the imaginary part of the usual complex scaling/PML in the input lead should allow one to compute *reflectionless modes* (see the exact definition after (\[defEquationDilatation\])). We show that this simple idea, inspired by the results obtained in [@HeKS11] on a 1D $\mathcal{PT}$-symmetric problem, indeed works. Interestingly, we will observe numerically that complex eigenvalues in this spectrum seem to contain useful information as well, indicating settings where reflection is weak.
(-2.7,0.3) rectangle (2.7,1.7); plot(,[1.7+0.1\*(+1)\^4\*(-1)\^4\*(+3)]{});
plot \[smooth cycle, tension=1\] coordinates [(-0.6,0.9) (0,0.5) (0.7,1) (0.5,1.5) (-0.2,1.4)]{};
(-2.7,0.3)–(2.7,0.3); (-2.7,1.7)–(-1,1.7); (1,1.7)–(2.7,1.7); (-2.7,0.3)–(-3,0.3); (2.7,0.3)–(3,0.3); (-2.7,1.7)–(-3,1.7); (2.7,1.7)–(3,1.7); at (-2.85,1.3)[ $u_i$]{}; at (-2.85,0.7)[$u_s$]{}; at (2.85,0.7)[$u_s$]{};
plot\[very thick,domain=0:pi/4-0.2,samples=100\] (,[-0.8+0.02\*exp(-(-4))]{});
at (0.85,0)[$\boldsymbol{+}$]{};
plot\[domain=0:pi/4,samples=100\] (,[0.25\*sin(20\*r)]{});
plot\[domain=0:pi/4,samples=100\] (0.2+,[0.25\*sin(20\*r)]{}); at (1.3,0)[$\boldsymbol{+}$]{}; (0.2,-0.3)–(1,0.3); (1,-0.3)–(0.2,0.3);
plot\[very thick,domain=8-pi/4+0.2:8,samples=100\] (-8.3,[0.02\*exp((-4))]{});
plot\[domain=0:pi/4,samples=100\] (,[0.25\*sin(20\*r)]{});
To make the presentation as simple as possible, we stick to a simple 2D wave problem in a waveguide with Neumann boundary conditions. We assume that this waveguide contains a penetrable obstacle (Figure \[figsetting\]). Everything presented here can be generalized to other types of obstacles, to other kinds of boundary conditions (Dirichlet, ...), to higher dimension (${\mathrm{3D}}$) and to more complex geometries (one input lead/several output leads).\
The paper is organized as follows. In section \[SectionSetting\], the governing equations of the scattering problem are presented and the reflectionless case is defined. Section \[SectionClassical\] is a reminder of the technique of complex scaling allowing one to compute classical complex resonances. In section \[SectionUnusual\], we explain how to use conjugated complex scalings to identify reflectionless frequencies. We provide examples of spectra in section \[SectionNumerics\] dedicated to numerics and we give some proofs of the theoretical results in section \[SectionProofs\]. Eventually, section \[SectionConclusion\] contains some concluding remarks.
Setting {#SectionSetting}
=======
(-1.5,0) rectangle (1.5,1); (-1,0.25) rectangle (1,0.75); (-1.5,0)–(1.5,0); (-1.5,1)–(1.5,1); (-1.5,0)–(-1.8,0); (1.5,0)–(1.8,0); (-1.5,1)–(-1.8,1); (1.5,1)–(1.8,1); (-1,-0.1)–(-1,0.1); (1,-0.1)–(1,0.1); at (-1,-0.3)[$-L$]{}; at (1,-0.3)[$L$]{}; at (0,0.5)[$\gamma=5$]{}; at (-1.5,0.5)[$\gamma=1$]{}; at (-2.2,-0.3)[(a)]{};
(-1.5,0) rectangle (1.5,1); (-1,0.25) rectangle (1,0.75); (-1,0.5) rectangle (0,0.75); (-1.5,0)–(1.5,0); (-1.5,1)–(1.5,1); (-1.5,0)–(-1.8,0); (1.5,0)–(1.8,0); (-1.5,1)–(-1.8,1); (1.5,1)–(1.8,1); at (0.5,0.5)[$\gamma=5$]{}; at (-1.5,0.5)[$\gamma=1$]{};
(3,1.2)–(3.5,1.2); (3.1,1.1)–(3.1,1.6); at (3.65,1.3)[$x$]{}; at (3.25,1.6)[$y$]{};
(-1,-0.1)–(-1,0.1); (1,-0.1)–(1,0.1); at (-1,-0.3)[$-L$]{}; at (1,-0.3)[$L$]{}; at (-2.2,-0.3)[(b)]{};
\
We are interested in the propagation of waves in a waveguide with a bounded penetrable obstacle (Figure \[figsetting\]). We assume that the waveguide coincides with the region ${\Omega}:=\{(x,y)\in{\mathbb{R}}^2\,|\,0<y<1\}$ and that the propagation of waves is governed by the Helmholtz equation with Neumann boundary conditions $$\label{PbInitial}
\begin{array}{|rcll}
\Delta u + k^2\gamma u & = & 0 & \mbox{ in }{\Omega}\\[3pt]
\partial_y u & = & 0 & \mbox{ on }y=0\mbox{ and }y=1.
\end{array}$$ Here $u$ can represent for instance the acoustic pressure in a compressible fluid, or the transverse component of the displacement field in an elastic solid, or one component of the electromagnetic field in a dielectric medium. $\Delta$ denotes the $\mbox{2D}$ Laplace operator. Moreover $k$ is the wavenumber such that $k={\omega}/c$ where $c$ denotes the waves velocity and ${\omega}$ is the angular frequency, corresponding to a time harmonic dependence in $e^{-i{\omega}t}$. We assume that $\gamma$ is a positive and bounded function such that $\gamma=1$ for $|x|\ge L$ where $L>0$ is given. In other words, the obstacle is located in the region ${\Omega}_L:=\{(x,y)\in{\Omega}\,|\,|x|<L\}$.\
Pick $k\in(N\pi;(N+1)\pi)$, with $N\in{\mathbb{N}}:=\{0,1,\dots\}$. For $n\in{\mathbb{N}}$, set $\beta_n=\sqrt{k^2-n^2\pi^2}$. In this work, the square root is such that if $z=|z|\,e^{i{\mathrm{arg}}(z)}$ with ${\mathrm{arg}}(z)\in[0;2\pi)$, then $\sqrt{z}=\sqrt{|z|}\,e^{i{\mathrm{arg}}(z)/2}$. With this choice, we have $\Im m\,\sqrt{z}\ge0$ for all $z\in{\mathbb{C}}$. The function $$\label{defModes}
w_n^{\pm}(x,y)=(2|\beta_n|)^{-1/2}e^{\pm i\beta_n x}\varphi_n(y),$$ with $\varphi_0(y)=1$, $\varphi_n(y)=\sqrt{2}\cos(n\pi y)$ for $n\ge1$, solves Problem (\[PbInitial\]) without obstacle ($\gamma\equiv1$). For $n=0,\dots,N$, the wave $w_n^{\pm}$ propagates along the $(Ox)$ axis from $\mp\infty$ to $\pm\infty$. On the other hand, for $n> N$, $w_n^{\pm}$ is exponentially growing at $\mp\infty$ and exponentially decaying at $\pm\infty$. For $n=0,\dots,N$, we consider the scattering of the wave $w_n^{+}$ by the obstacle located in ${\Omega}$. It is known that Problem (\[PbInitial\]) admits a solution $u_n=w_n^{+}+u^s_n$ with the *outgoing* *scattered field* $u^s_n$ written as $$\label{RadiationCondition}
u^s_n=\sum_{p=0}^{+\infty} s^{\pm}_{np}w_p^{\pm}\qquad\mbox{ for }\pm x\ge L$$ with $(s^{\pm}_{np})\in{\mathbb{C}}^{{\mathbb{N}}}$. The solution $u_n$ is uniquely defined if and only if *Trapped Modes* (TMs) do not exist at the wavenumber $k$. We remind the reader that trapped modes are non zero functions $u\in {{\mathrm{L}}}^2(\Omega)$ satisfying (\[PbInitial\]), where ${{\mathrm{L}}}^2(\Omega)$ is the space of square-integrable functions in $\Omega$. We denote by $\mathscr{K}_{{\mathrm{t}}}$ the set of $k^2$ such that TMs exist at the wavenumber $k$. On the other hand, the scattering coefficients $s^{\pm}_{np}$ in (\[RadiationCondition\]) are always uniquely defined, including for $k^2\in\mathscr{K}_{{\mathrm{t}}}$. In the following, we will be particularly interested in the features of the *reflection matrix* (whose size, determined by the number of propagative modes, depends on $k$) $$\label{reflection matrix}
R(k):=(s^{-}_{np})_{0\le n,p\le N}\in{\mathbb{C}}^{N+1\times N+1}.$$
Let $k\in(0;+\infty)\setminus\pi{\mathbb{N}}$. We say that the wavenumber $k$ is reflectionless if ${{\mathrm{ker}}\,}R(k)\ne\{0\}$.
Let us explain this definition. In general, by linearity, for an incident field $$\label{DefIncidentField}
u_i=\sum_{n=0}^Na_n w_n^{+},\qquad (a_n)_{n=0}^N\in{\mathbb{C}}^{N+1},$$ Problem (\[PbInitial\]) admits a solution $u$ such that $u=u_i+u_s$ with $$\label{DefScatteredField}
u_s=\sum_{p=0}^{+\infty}b_p^{\pm} w_p^{\pm}\mbox{ for }\pm x\ge L\ \mbox{ and }\ b_p^{\pm}=\sum_{n=0}^Na_{n}s^{\pm}_{np}\in{\mathbb{C}}.$$ The above definition says that, if $k$ is reflectionless, then there are $(a_n)_{n=0}^N\in{\mathbb{C}}^{N+1}\setminus\{0\}$ such that the $b_p^{-}$ in (\[DefScatteredField\]) satisfy $b_p^{-}=0$, $p=0,\dots,N$. In other words, the scattered field is exponentially decaying for $x\le -L$. Let us notice finally that the corresponding total field $u=u_i+u_s$ decomposes as $$\label{DefReflectionlessMode}
\begin{array}{ll}
{\displaystyle}u=\sum_{n=0}^{N}a_n w_n^{+}+\tilde{u}&\mbox{ for } x\le -L\\[13pt]
{\displaystyle}u=\sum_{n=0}^{N}t_n w_n^{+}+\tilde{u}&\mbox{ for } x\ge L\\
\end{array}$$ where $t_n=a_n+b_n^+$ and where $\tilde{u}$ decays exponentially for $\pm x\ge L$. In other words, the total field is *ingoing* for $x\le-L$ and *outgoing* for $x\ge L$.\
In the following, we call *Reflectionless Modes* (RMs) the functions $u$ admitting expansion (\[DefReflectionlessMode\]) and we denote by $\mathscr{K}_{{\mathrm{r}}}$ the set of $k^2$ such that the wavenumber $k$ is reflectionless. The main objective of this article is to explain how to determine directly the set $\mathscr{K}_{{\mathrm{r}}}$ and the corresponding RMs by solving an eigenvalue problem, instead of computing the reflection matrix for all values of $k$.
Classical complex scaling {#SectionClassical}
=========================
As a first step, in this section we remind briefly how to use a complex scaling to compute trapped modes. Define the unbounded operator $A$ of ${{\mathrm{L}}}^2({\Omega})$ such that $$Au=-\cfrac{1}{\gamma}\,\Delta u$$ with Neumann boundary conditions $\partial_{y}u=0\mbox{ on }y=0\mbox{ and }y=1$. It is known that $A$ is a selfadjoint operator (${{\mathrm{L}}}^2({\Omega})$ is endowed with the inner product $(\gamma\,\cdot,\cdot)_{{{\mathrm{L}}}^2({\Omega})}$) whose spectrum $\sigma(A)$ coincides with $[0;+\infty)$. More precisely, we have $\sigma_{{\mathrm{ess}}}(A)=[0;+\infty)$ where $\sigma_{{\mathrm{ess}}}(A)$ denotes the essential spectrum of $A$. By definition, $\sigma_{{\mathrm{ess}}}(A)$ corresponds to the set of $\lambda \in{\mathbb{C}}$ for which there exists a so-called singular sequence $(u^{(m)})$, that is an orthonormal sequence $(u^{(m)})\in{{\mathrm{L}}}^2(\Omega)^{{\mathbb{N}}}$ such that $((A-\lambda)u^{(m)})$ converges to 0 in ${{\mathrm{L}}}^2(\Omega)$. Besides, $\sigma(A)$ may contain eigenvalues (at most a sequence accumulating at $+\infty$) corresponding to TMs. In order to reveal these eigenvalues which are embedded in $\sigma_{{\mathrm{ess}}}(A)$, one can use a complex change of variables. For $0<\theta<\pi/2$, set $\eta=e^{i\theta}$ and define the function $\mathcal{I}_{\theta}:{\mathbb{R}}\to{\mathbb{C}}$ such that $$\label{defDilatationx}
\mathcal{I}_{\theta}(x)=\begin{array}{|ll}
-L+(x+L)\,\eta&\mbox{ for }x\le- L\\
x&\mbox{ for }|x|< L\\
+ L+(x-L)\,\eta&\mbox{ for } x\ge L.
\end{array}$$ For the sake of simplicity, we will use abusively the same notation $\mathcal{I}_{\theta}$ for the following map: $\{{\Omega}\to {\mathbb{C}}\times(0;1), \;
(x,y) \mapsto (\mathcal{I}_{\theta}(x),y)\}$. Note that with this definition, the left inverse $\mathcal{I}_{\theta}^{-1}$ of $\mathcal{I}_{\theta}$, acting from $\mathcal{I}_{\theta}({\Omega})$ to ${\Omega}$, is equal to $\mathcal{I}_{-\theta}$. One can easily check that for all $n\geq 0$, $w_n^+\circ \mathcal{I}_{\theta}$ is exponentially decaying for $x\ge L$, while $w_n^-\circ \mathcal{I}_{\theta}$ is exponentially decaying for $x\le-L$. As a consequence, defining from expansion (\[DefScatteredField\]) the function $v_{\theta}=u_s\circ \mathcal{I}_{\theta}$, one has $v_{\theta}=u_s$ for $|x|< L$ and $v_{\theta}\in {{\mathrm{L}}}^2(\Omega)$ (which is in general not true for $u_s$). Moreover $v_{\theta}$ satisfies the following equation in ${\Omega}$: $$\label{defEquationDilatation}
\alpha_{\theta}\frac{\partial}{\partial x}\Big(\alpha_{\theta}\frac{\partial v_{\theta}}{\partial x}\Big)+\frac{\partial^2 v_{\theta}}{\partial y^2}+k^2\gamma v_{\theta}=k^2(1-\gamma)u_i$$ with $\alpha_{\theta}(x)=1$ for $|x|<L$ and $\alpha_{\theta}(x)=\eta^{-1}=\overline{\eta}$ for $\pm x\ge L$. In particular, for a TM, $v_{\theta}$ solves (\[defEquationDilatation\]) with $u_i=0$. This leads us to consider the unbounded operator $A_{\theta}$ of ${{\mathrm{L}}}^2({\Omega})$ such that $$\label{defOpClassicalPMLs}
A_{\theta}v_{\theta}=-\cfrac{1}{\gamma}\left(\alpha_{\theta}\frac{\partial}{\partial x}\Big(\alpha_{\theta}\frac{\partial v_{\theta}}{\partial x}\Big)+\frac{\partial^2 v_{\theta}}{\partial y^2}\right)$$ again with homogeneous Neumann boundary conditions. Since $\alpha_{\theta}$ is complex valued, the operator $A_{\theta}$ is no longer selfadjoint. However, we use the same definition as above for $\sigma_{{\mathrm{ess}}}(A_{\theta})$, which is licit for this operator. We recall below the main spectral properties of $A_{\theta}$ [@Simo78]:
\[thmUsualPMLs\] i) There holds $$\label{essClassicalPMLs}
\sigma_{{\mathrm{ess}}}(A_{\theta})=\bigcup_{n\in{\mathbb{N}},\,t\ge0}\{n^2\pi^2+te^{-2i\theta}\}.$$ ii) The spectrum of $A_{\theta}$ satisfies $\sigma(A_{\theta})\subset\mathscr{R}_{{\mathrm{\theta}}}^-$ with $$\mathscr{R}_{{\mathrm{\theta}}}^-:=\{z\in{\mathbb{C}}\,|\,-2\theta\le{\mathrm{arg}}(z)\le 0\}.$$ iii) $\sigma (A_{\theta})\setminus \sigma_{{\mathrm{ess}}}(A_{\theta})$ is discrete and contains only eigenvalues of finite multiplicity.\
iv) Assume that $k^2\in\sigma(A_{\theta})\setminus\sigma_{{\mathrm{ess}}}(A_{\theta})$. Then $k^2$ is real if and only if $k^2\in\mathscr{K}_{{\mathrm{t}}}$. Moreover if $v_{\theta} $ is an eigenfunction associated to $k^2$ such that $\Im m\,k^2<0$, then $v_{\theta}\circ\mathcal{I}_{-\theta}$ is a solution of the original problem (\[PbInitial\]) whose amplitude is exponentially growing at $+\infty$ or at $-\infty$.
The interesting point is that now TMs correspond to isolated eigenvalues of $A_{\theta}$, and as such, they can be computed numerically as illustrated below. Note that the elements $k^2$ of $\sigma(A_{\theta})\setminus\sigma_{{\mathrm{ess}}}(A_{\theta})$ such that $\Im m\,k^2<0$, if they exist, correspond to complex resonances (quasi normal modes). Let us point out that the complex scaling is just a technique to reveal them. Indeed complex resonances are intrinsic objects defined as the poles of the meromorphic extension from $\{z\in{\mathbb{C}}\,|\,\Im m\,z>0\}$ to $\{z\in{\mathbb{C}}\,|\,\Im m\,z\le0\}$ of the operator valued map $z\mapsto (\Delta +z\gamma)^{-1}$. For more details, we refer the reader to [@AsPV00].
Conjugated complex scaling {#SectionUnusual}
==========================
Now, we show that replacing the classical complex scaling by an unusual *conjugated* complex scaling, and proceeding as in the previous section, we can define a new complex spectrum which contains the reflectionless values $k^2\in \mathscr{K}_{{\mathrm{r}}}$ we are interested in. We define the map $\mathcal{J}_{\theta}:{\Omega}\to{\mathbb{C}}\times(0;1)$ using the following complex change of variables $$\label{defDilatationConjugated}
\mathcal{J}_{\theta}(x)=\begin{array}{|ll}
-L+(x+L)\,\overline{\eta}&\mbox{ for }x\le- L\\
x&\mbox{ for }|x|< L\\
+ L+(x-L)\,\eta&\mbox{ for } x\ge L,
\end{array}$$ with again $\eta=e^{i\theta}$ ($0<\theta<\pi/2$). Note the important difference in the definitions of $\mathcal{J}_{\theta}$ and $\mathcal{I}_{\theta}$ for $x\le-L$: $\eta$ has been replaced by the conjugated parameter $\overline{\eta}$ to select the ingoing modes instead of the outgoing ones in accordance with (\[DefReflectionlessMode\]). Now, if $u$ is a RM associated to $k^2\in \mathscr{K}_{{\mathrm{r}}}$, setting $w_{\theta}=u\circ \mathcal{J}_{\theta}$, one has $w_{\theta}=u$ for $|x|< L$ and $w_{\theta}\in {{\mathrm{L}}}^2(\Omega)$ (which is not the case for $u$). The function $w_{\theta}$ satisfies the following equation in ${\Omega}$: $$\label{DefPbFortCPMLs}
\beta_{\theta}\frac{\partial}{\partial x}\Big(\beta_{\theta}\frac{\partial w_{\theta}}{\partial x}\Big)+\frac{\partial^2 w_{\theta}}{\partial y^2}+k^2\gamma w_{\theta}=0$$ with $\beta_{\theta}(x)=1$ for $|x|< L$, $\beta_{\theta}(x)=\eta$ for $x\le-L$ and $\beta_{\theta}(x)=\overline{\eta}$ for $x\ge L$. This leads us to define the unbounded operator $B_{\theta}$ of ${{\mathrm{L}}}^2({\Omega})$ such that $$\label{defOpConjugatedPMLs}
B_{\theta}w_{\theta}=-\cfrac{1}{\gamma}\left(\beta_{\theta}\frac{\partial}{\partial x}\Big(\beta_{\theta}\frac{\partial w_{\theta}}{\partial x}\Big)+\frac{\partial^2 w_{\theta}}{\partial y^2}\right)$$ with homogeneous Neumann boundary conditions. As $A_{\theta}$, the operator $B_{\theta}$ is not selfadjoint. Its spectral properties are summarized in the following theorem which is proved in the last section of this article.
\[thmConjugatedPMLs\] i) There holds $$\label{essConjPMLs}
\sigma_{{\mathrm{ess}}}(B_{\theta})=\bigcup_{n\in{\mathbb{N}},\,t\ge0}\{n^2\pi^2+te^{-2i\theta},\,n^2\pi^2+te^{+2i\theta}\}.$$ ii) The spectrum of $B_{\theta}$ satisfies $\sigma(B_{\theta})\subset\mathscr{R}_{{\mathrm{\theta}}}$ with $$\label{DefCPML}
\mathscr{R}_{{\mathrm{\theta}}}:=\{z\in{\mathbb{C}}\,|\,-2\theta\le{\mathrm{arg}}(z)\le 2\theta\}.$$ iii) Assume that $k^2\in\sigma(B_{\theta})\setminus\sigma_{{\mathrm{ess}}}(B_{\theta})$. Then $k^2$ is real if and only if $k^2\in\mathscr{K}_{{\mathrm{t}}}\cup\mathscr{K}_{{\mathrm{r}}}$. Moreover if $w_{\theta}$ is an eigenfunction associated to $k^2$ such that $\pm\Im m\,k^2<0$, then $w_{\theta}\circ\mathcal{J}_{-\theta}$ is a solution of (\[PbInitial\]) whose amplitude is exponentially growing at $\pm\infty$ and exponentially decaying at $\mp\infty$.
The important result is that isolated real eigenvalues of $B_{\theta}$ correspond precisely to TMs and RMs. The following proposition provides a criterion to determine whether an eigenfunction associated to a real eigenvalue of $B_{\theta}$ is a TM or a RM.
\[PropositionCarac\] Assume that $(k^2,w_{\theta})\in{\mathbb{R}}\times {{\mathrm{L}}}^2({\Omega})$ is an eigenpair of $B_{\theta}$ such that $k\in(N\pi;(N+1)\pi)$, $N\in{\mathbb{N}}$. Set $$\label{DefAlephFunction}
{\rho}(w_{\theta})=\sum_{n=0}^N\Big|\int_{0}^1w_{\theta}(-L,y)\varphi_n(y)\,dy\Big|^2$$ where $\varphi_n$ is defined in (\[defModes\]). If ${\rho}(w_{\theta})=0$ then $w_{\theta}\circ\mathcal{J}_{-\theta}$ is a TM ($k^2\in\mathscr{K}_{{\mathrm{t}}}$). If ${\rho}(w_{\theta})>0$ then $w_{\theta}\circ\mathcal{J}_{-\theta}$ is a RM ($k^2\in\mathscr{K}_{{\mathrm{r}}}$). In this case, the incident field $u_i$ defined in (\[DefIncidentField\]) with $$a_n=\int_{0}^1w_{\theta}(-L,y)\varphi_n(y)\,dy,\quad n=0,\dots,N,$$ yields a scattered field which decays exponentially for $x\le -L$.
The next proposition tells that $B_{\theta}$ satisfies the celebrated $\mathcal{PT}$ symmetry property when the obstacle is symmetric with respect to the $(Oy)$ axis. This ensures in particular the stability of simple real eigenvalues, with respect to perturbations of the obstacle satisfying the same symmetry constraint.
\[PropositionPTsym\] Assume that $\gamma$ satisfies $\gamma(x,y)=\gamma(-x,y)$ for all $(x,y)\in{\Omega}$. Then the operator $B_{\theta}$ is $\mathcal{PT}$-symmetric ($\mathcal{PT}B_{\theta}\mathcal{PT}=B_{\theta}$) with $\mathcal{P}\varphi(x,y)=\varphi(-x,y)$, $\mathcal{T}\varphi(x,y)=\overline{\varphi(x,y)}$ for $\varphi\in{{\mathrm{L}}}^2({\Omega})$. Therefore, we have $\sigma(B_{\theta})=\overline{\sigma(B_{\theta})}$.
The proof is straightforward observing that the $\beta_{\theta}$ defined after (\[DefPbFortCPMLs\]) satisfies $\beta_{\theta}(-x,y)=\overline{\beta_{\theta}(x,y)}$.\
Finally let us mention a specific difficulty which appears in the spectral analysis of $B_{\theta}$. While Theorem \[thmUsualPMLs\] guarantees that $\sigma(A_{\theta})\setminus\sigma_{{\mathrm{ess}}}(A_{\theta})$ is discrete, we do not write such a statement for the operator $B_{\theta}$ in Theorem \[thmConjugatedPMLs\]. A major difference between both operators is that ${\mathbb{C}}\setminus \sigma_{{\mathrm{ess}}}(A_{\theta})$ is connected whereas ${\mathbb{C}}\setminus \sigma_{{\mathrm{ess}}}(B_{\theta})$ has a countably infinite number of connected components. As a consequence, to prove that $\sigma (B_{\theta})\setminus \sigma_{{\mathrm{ess}}}(B_{\theta})$ is discrete using the Fredholm analytic theorem, it is necessary to find one $\lambda$ such that $B_{\theta}-\lambda$ is invertible in each of the components of ${\mathbb{C}}\setminus \sigma_{{\mathrm{ess}}}(B_{\theta})$. In general, in presence of an obstacle, such a $\lambda$ probably exists (proofs for certain classes of $\gamma$ can be obtained working as in [@BoCN15]). But for this problem, we can have surprising perturbation results. Thus, if there is no obstacle ($\gamma\equiv1$ in ${\Omega}$), then there holds $\sigma(B_{\theta})=\mathscr{R}_{{\mathrm{\theta}}}$ (see (\[DefCPML\])): all connected components of ${\mathbb{C}}\setminus \sigma_{{\mathrm{ess}}}(B_{\theta})$, except the one containing the complex half-plane $\Re e\,\lambda<0$, are filled with eigenvalues. To show this result, observe that for $k^2\in\sigma(B_{\theta})\setminus\sigma_{{\mathrm{ess}}}(B_{\theta})$, the function $u\circ \mathcal{J}_{\theta}$, with $u(x,y)=e^{ik x}$, is a non-zero element of ${{\mathrm{ker}}\,}B_{\theta}$. Notice that this pathological property is also true when ${\Omega}$ contains a family of sound hard cracks (homogeneous Neumann boundary condition) parallel to the $(Ox)$ axis.
Numerical experiments {#SectionNumerics}
=====================
Classical complex scaling: classical complex resonance modes
-------------------------------------------------------------
We first compute the spectrum of the operator $A_{\theta}$ defined in (\[defOpClassicalPMLs\]) with a classical complex scaling (complex resonance spectrum). For the numerical experiments, we truncate the computational domain at some distance of the obstacle and use finite elements. This corresponds to the so-called *Perfectly Matched Layers* (PMLs) method. We refer the reader to [@Kalv13] for the numerical analysis of the error due to truncation of the waveguide and discretization. The setting is as follows. We take $\gamma$ such that $\gamma=5$ in $\mathscr{O}=(-1;1)\times(0.25;0.75)$ and $\gamma=1$ in ${\Omega}\setminus\overline{\mathscr{O}}$ (see Figure \[figsetting\] (a)). In the definition of the maps $\mathcal{I}_{\theta}$, $\alpha_{\theta}$ (see (\[defDilatationx\]), (\[defEquationDilatation\])), we take $\theta=\pi/4$ (so that $\eta=e^{i\pi/4}$) and $L=1$. In practice, we use a ${\mathrm{P2}}$ finite element method in the bounded domain ${\Omega}_{12}=\{(x,y)\in{\Omega}\,|\,-12<x<12\}$ with Dirichlet boundary condition at $x=\pm12$. Finite element matrices are constructed with *FreeFem++*[^1].\
In Figure \[figClassicalPMLs\] and in the rest of the paper, we display the square root of the spectrum ($k$ instead of $k^2$). The vertical marks on the real axis correspond to the thresholds ($0$, $\pi$, $2\pi$, ...). In accordance with Theorem \[thmUsualPMLs\], we observe that $\sqrt{\sigma(A_{\theta})}$ is located in the region $\sqrt{\mathscr{R}_{\theta}^-}=\{z\in{\mathbb{C}}\,|\,-\theta\le{\mathrm{arg}}(z)\le 0\}$. Moreover, the discretisation of the essential spectrum $\sigma_{{\mathrm{ess}}}(A_{\theta})$ defined in (\[essClassicalPMLs\]) and forming branches starting at the threshold points appears clearly. Note that a simple calculation shows that $\sqrt{\{n^2\pi^2+te^{-2i\theta},\,t\ge0\}}$ is a half-line for $n=0$ and a piece of hyperbola for $n\ge1$. This is precisely what we get. Eigenvalues located on the real axis correspond to trapped modes ($k^2\in\mathscr{K}_{{\mathrm{t}}}$). In the chosen setting, which is symmetric with respect to the axis ${\mathbb{R}}\times\{0.5\}$, one can prove that trapped modes exist [@EvLV94]. On the other hand, the eigenvalues in the complex plane which are not the discretisation of the essential spectrum correspond to complex resonances.
![Classical complex resonances in the complex $k$ plane corresponding to the spectrum of $A_{\theta}$ for a symmetric obstacle (Figure \[figsetting\] (a)). The trapped modes are in red, the dashed lines represent the essential spectrum of $A_{\theta}$ (see (\[essClassicalPMLs\])). The picture on the right is a zoom-in of that on the left.[]{data-label="figClassicalPMLs"}](images/UsualPMLs "fig:"){width=".49\linewidth"} ![Classical complex resonances in the complex $k$ plane corresponding to the spectrum of $A_{\theta}$ for a symmetric obstacle (Figure \[figsetting\] (a)). The trapped modes are in red, the dashed lines represent the essential spectrum of $A_{\theta}$ (see (\[essClassicalPMLs\])). The picture on the right is a zoom-in of that on the left.[]{data-label="figClassicalPMLs"}](images/UsualPMLsZoom "fig:"){width=".49\linewidth"}
Conjugated complex scaling: reflectionless modes
------------------------------------------------
Now we compute the spectrum of the operator $B_{\theta}$ defined in (\[defOpConjugatedPMLs\]) with a conjugated complex scaling. First, we use exactly the same symmetric setting (see Figure \[figsetting\] (a)) as in the previous paragraph. In Figure \[figConjugatedPMLs\], we display the square root of the spectrum $\sqrt{\sigma(B_{\theta})}$. Since $\gamma$ satisfies $\gamma(x,y)=\gamma(-x,y)$, according to Proposition \[PropositionPTsym\] we know that $B_{\theta}$ is $\mathcal{PT}$-symmetric and that therefore its spectrum is stable by conjugation ($\sigma(B_{\theta})=\overline{\sigma(B_{\theta})}$). This is indeed what we obtain. Note that the mesh has been constructed so that $\mathcal{PT}$-symmetry is preserved at the discrete level. $\mathcal{PT}$-symmetry is an interesting property in our case because it guarantees that eigenvalues located close to the real axis which are isolated (no other eigenvalue in a vicinity) are real. Therefore, according to Theorem \[thmConjugatedPMLs\], they correspond to trapped modes or to reflectionless modes. Remark that, for the same geometry, the spectrum of $B_{\theta}$ (Figure \[figConjugatedPMLs\]) contains more elements on the real axis than the spectrum of $A_{\theta}$ (Figure \[figClassicalPMLs\]): the additional elements (green points in Figure \[figConjugatedPMLs\]) correspond to reflectionless modes.
![Reflectionless eigenvalues in the complex $k$ plane corresponding to the spectrum of $B_{\theta}$ for a symmetric obstacle (Figure \[figsetting\] (a)). The trapped modes in red are the same as in Figure \[figClassicalPMLs\]. The reflectionless modes are in green and the dashed lines represent the essential spectrum of $B_{\theta}$ (see (\[essConjPMLs\])). The picture on the right is a zoom-in of that on the left. []{data-label="figConjugatedPMLs"}](images/ConjugatedPMLs "fig:"){width=".49\linewidth"} ![Reflectionless eigenvalues in the complex $k$ plane corresponding to the spectrum of $B_{\theta}$ for a symmetric obstacle (Figure \[figsetting\] (a)). The trapped modes in red are the same as in Figure \[figClassicalPMLs\]. The reflectionless modes are in green and the dashed lines represent the essential spectrum of $B_{\theta}$ (see (\[essConjPMLs\])). The picture on the right is a zoom-in of that on the left. []{data-label="figConjugatedPMLs"}](images/ConjugatedPMLsZoom "fig:"){width=".4805\linewidth"}
![Top: real part of eigenmodes associated with real eigenvalues of $B_{\theta}$ from Figure \[figConjugatedPMLs\]. Bottom: value of $k$ and of the indicator function ${\rho}$ for each of these $7$ eigenmodes. The 3rd and the 5th eigenmodes are trapped modes, the five others are reflectionless modes.[]{data-label="figEigenfunctions"}](images/fonction1 "fig:"){width=".65\linewidth"}\
![Top: real part of eigenmodes associated with real eigenvalues of $B_{\theta}$ from Figure \[figConjugatedPMLs\]. Bottom: value of $k$ and of the indicator function ${\rho}$ for each of these $7$ eigenmodes. The 3rd and the 5th eigenmodes are trapped modes, the five others are reflectionless modes.[]{data-label="figEigenfunctions"}](images/fonction2 "fig:"){width=".65\linewidth"}\
![Top: real part of eigenmodes associated with real eigenvalues of $B_{\theta}$ from Figure \[figConjugatedPMLs\]. Bottom: value of $k$ and of the indicator function ${\rho}$ for each of these $7$ eigenmodes. The 3rd and the 5th eigenmodes are trapped modes, the five others are reflectionless modes.[]{data-label="figEigenfunctions"}](images/fonction3 "fig:"){width=".65\linewidth"}\
![Top: real part of eigenmodes associated with real eigenvalues of $B_{\theta}$ from Figure \[figConjugatedPMLs\]. Bottom: value of $k$ and of the indicator function ${\rho}$ for each of these $7$ eigenmodes. The 3rd and the 5th eigenmodes are trapped modes, the five others are reflectionless modes.[]{data-label="figEigenfunctions"}](images/fonction4 "fig:"){width=".65\linewidth"}\
![Top: real part of eigenmodes associated with real eigenvalues of $B_{\theta}$ from Figure \[figConjugatedPMLs\]. Bottom: value of $k$ and of the indicator function ${\rho}$ for each of these $7$ eigenmodes. The 3rd and the 5th eigenmodes are trapped modes, the five others are reflectionless modes.[]{data-label="figEigenfunctions"}](images/fonction5 "fig:"){width=".65\linewidth"}\
![Top: real part of eigenmodes associated with real eigenvalues of $B_{\theta}$ from Figure \[figConjugatedPMLs\]. Bottom: value of $k$ and of the indicator function ${\rho}$ for each of these $7$ eigenmodes. The 3rd and the 5th eigenmodes are trapped modes, the five others are reflectionless modes.[]{data-label="figEigenfunctions"}](images/fonction6 "fig:"){width=".65\linewidth"}\
![Top: real part of eigenmodes associated with real eigenvalues of $B_{\theta}$ from Figure \[figConjugatedPMLs\]. Bottom: value of $k$ and of the indicator function ${\rho}$ for each of these $7$ eigenmodes. The 3rd and the 5th eigenmodes are trapped modes, the five others are reflectionless modes.[]{data-label="figEigenfunctions"}](images/fonction7 "fig:"){width=".65\linewidth"}\
[|>m[1.2cm]{}|>m[0.7cm]{}|>m[0.7cm]{}|>m[1.5cm]{}|>m[0.7cm]{}|>m[1.5cm]{}|>m[0.7cm]{}|>m[0.7cm]{}|]{} $k$ & 0.9 & 1.8 & 2.4 & 2.6 & 2.8 & 3.3 & 3.9\
${\rho}(w_{\theta})$ & 0.14 & 0.14 & 8.0$10^{-10}$ & 0.14 & 4.3$10^{-9}$ & 0.14& 0.14\
In Figure \[figEigenfunctions\] top, we represent the real part of eigenfunctions associated with seven real eigenvalues of $B_{\theta}$. To obtain these pictures, we take $L=4$ in the definition of $\mathcal{J}_{\theta}$ in (\[defDilatationConjugated\]) and we display only the restrictions of the eigenfunctions to ${\Omega}_{L}=\{(x,y)\in{\Omega}\,|\,-L<x<L\}$. We recognize two trapped modes (images 3 and 5). The other modes are reflectionless modes. In Figure \[figEigenfunctions\] bottom, we provide the value of the indicator function ${\rho}$ defined in (\[DefAlephFunction\]) for the seven eigenmodes. We have to mention that eigenmodes are normalized so that their ${{\mathrm{L}}}^2$ norm is equal to one. The indicator function ${\rho}$ offers a clear criterion to distinguish between trapped modes and reflectionless modes. Moreover, in order to inspect the scattering coefficient, we remark that for reflectionless modes associated with wavenumbers $k$ smaller than $\pi$, the incident field $u_i$ in (\[DefIncidentField\]) decomposes only on the piston mode $w^{+}_0(x,y)=e^{ikx}/\sqrt{2k}$ (monomode regime). In this case the reflection matrix $R(k)$ in (\[reflection matrix\]) is nothing but the usual reflection coefficient. In Figure \[figBalayage\], we thus display the modulus of this coefficient $R_{00}(k)$ with respect to $k\in(0.1;3.1)$. As expected, we observe that $R_{00}$ vanishes for the values of $k$ obtained in Figure \[figEigenfunctions\] solving the spectral problem for $B_{\theta}$. Of course obtaining the curve $k\mapsto |R_{00}(k)|$ is relatively costly and it is precisely what we want to avoid by computing the reflectionless $k$ as eigenvalues. Here it is simply a way to check our results.
![Curve $k\mapsto |R_{00}(k)|$ (modulus of the reflection coefficient) for $k\in(0.1;3.1)$. The green and red dots represent respectively the reflectionless modes and the trapped modes computed in Figure \[figConjugatedPMLs\]. We indeed observe that $R_{00}(k)$ is null for reflectionless $k$.[]{data-label="figBalayage"}](images/CoeffRfreq){width=".49\linewidth"}
In Figure \[figBrokenSym\], we represent the modulus of reflectionless mode eigenfunctions of $B_{\theta}$ associated with one real eigenvalue and two complex conjugated eigenvalues. We observe, and this is true in general, a symmetry with respect to the $(Oy)$ axis for modes corresponding to real eigenvalues which disappears for complex ones. This is the so-called broken symmetry phenomenon which is well-known for $\mathcal{PT}$-symmetric operators (see e.g. the review [@Bend07]).
![Modulus of eigenfunctions of $B_{\theta}$ associated to the eigenvalues $k\approx 5.31$ (top), $k\approx 5.29 - 0.13i$ (middle) and $k\approx 5.29+0.13i$ (bottom) obtained in Figure \[figConjugatedPMLs\]. The symmetry $x\to-x$ for real $k$ (due to $\mathcal{PT}$-symmetry) disappears for complex $k$. []{data-label="figBrokenSym"}](images/PTsym "fig:"){width=".65\linewidth"}\
![Modulus of eigenfunctions of $B_{\theta}$ associated to the eigenvalues $k\approx 5.31$ (top), $k\approx 5.29 - 0.13i$ (middle) and $k\approx 5.29+0.13i$ (bottom) obtained in Figure \[figConjugatedPMLs\]. The symmetry $x\to-x$ for real $k$ (due to $\mathcal{PT}$-symmetry) disappears for complex $k$. []{data-label="figBrokenSym"}](images/PTsymBroken1 "fig:"){width=".65\linewidth"}\
![Modulus of eigenfunctions of $B_{\theta}$ associated to the eigenvalues $k\approx 5.31$ (top), $k\approx 5.29 - 0.13i$ (middle) and $k\approx 5.29+0.13i$ (bottom) obtained in Figure \[figConjugatedPMLs\]. The symmetry $x\to-x$ for real $k$ (due to $\mathcal{PT}$-symmetry) disappears for complex $k$. []{data-label="figBrokenSym"}](images/PTsymBroken2 "fig:"){width=".65\linewidth"}
Now, we use the non-symmetric setting (see Figure \[figsetting\] (b)), and we display the square root of the spectrum of $B_{\theta}$ (in Figure \[figSpectrumNonSym\]) for a coefficient $\gamma$ which is not symmetric in $x$ nor in $y$. More precisely, we take $\gamma$ such that $\gamma=5$ in $\mathscr{O}=(-1;0]\times(0.25;0.5)\cup[0;1)\times(0.25;0.75)$ and $\gamma=1$ in ${\Omega}\setminus\overline{\mathscr{O}}$. We observe that the spectrum is no longer stable by conjugation ($\sigma (B_{\theta})\ne\overline{\sigma (B_{\theta})}$) since the operator $B_{\theta}$ is not $\mathcal{PT}$-symmetric, and there is no “help” for the eigenvalues to be real. However, a closer look shows the presence of eigenvalues close to the real axis, in particular for $k\approx1.0 + 0.13i$, $k\approx1.9 + 0.005i$, $k\approx2.5 + 0.02i$, $k\approx2.8 + 0.08i$ and $k\approx3.0 - 0.008i$. In Figure \[figNonSymCurve\], we represent $k\mapsto|R_{00}(k)|$ for $k\in(0.1;3.1)$ where there is only one propagating mode in the leads. It is interesting to note that the above computed complex reflectionless modes (located close to the real axis) have an influence on this curve. More precisely, $k\mapsto|R_{00}(k)|$ attains minima for $k\in(0.1;3.1)$ close to the real part of these complex reflectionless modes. Therefore complex reflectionless modes also have significance for scattering at real frequncies.
![Spectrum of $B_{\theta}$ in the complex $k$ plane for a non symmetric obstacle (Figure \[figsetting\] (b)). The dashed lines represent the essential spectrum of $B_{\theta}$ (see (\[essConjPMLs\])). The spectrum is not stable by conjugation. The picture on the right is a zoom-in of that on the left.[]{data-label="figSpectrumNonSym"}](images/SpectreSansSym "fig:"){width=".49\linewidth"} ![Spectrum of $B_{\theta}$ in the complex $k$ plane for a non symmetric obstacle (Figure \[figsetting\] (b)). The dashed lines represent the essential spectrum of $B_{\theta}$ (see (\[essConjPMLs\])). The spectrum is not stable by conjugation. The picture on the right is a zoom-in of that on the left.[]{data-label="figSpectrumNonSym"}](images/SpectreSansSymZoom "fig:"){width=".4805\linewidth"}
![Curve $k\mapsto |R_{00}(k)|$ (modulus of the reflection coefficient) for $k\in(0.1;3.1)$ and a non symmetric obstacle. The blue dots and the vertical dashed lines correspond to the real parts of the eigenvalues of $B_{\theta}$ located close to the real axis computed in Figure \[figSpectrumNonSym\]. We observe that $|R_{00}(k)|$ is minimal for these particular $k$.[]{data-label="figNonSymCurve"}](images/CoeffRfreqSansSym){width=".49\linewidth"}
Proofs {#SectionProofs}
======
Finally we give the proofs of Theorem \[thmConjugatedPMLs\] and Proposition \[PropositionCarac\].\
**Proof of Theorem \[thmConjugatedPMLs\].** $i)$ First, let us explain how to show that for $t\ge0$ and $n\in{\mathbb{N}}$, $n^2\pi^2+te^{-2i\theta}$ belongs to $\sigma_{{\mathrm{ess}}}(B_{\theta})$. Consider a smooth cut-off function $\chi$ defined on ${\mathbb{R}}$ such that $\chi(x)=0$ for $|x|>L$ and $\|\chi\|_{{{\mathrm{L}}}^2({\mathbb{R}})}=1$. For $m\ge1$, set $v_+^{(m)}(x,y)=m^{-1/2}\chi\left((x-m^2L)/m\right)e^{i\sqrt{t}x}\varphi_n(y)$ where $\varphi_n$ is defined in (\[defModes\]). One can check that $(v_+^{(m)})$ is a singular sequence for $B_{\theta}$ at $n^2\pi^2+te^{-2i\theta}$ (note that the supports of $v_+^{(m)}$ and $v_+^{(m')}$ for $m\neq m'$ do not overlap). On the other hand, we prove that $(v_-^{(m)})$, with $v_-^{(m)}(x,y)=v_+^{(m)}(-x,y)$, is a singular sequence for $B_{\theta}$ at $n^2\pi^2+te^{+2i\theta}$. Summing up, we obtain $\cup_{n\in{\mathbb{N}},\,t\ge0}\{n^2\pi^2+te^{-2i\theta},\,n^2\pi^2+te^{+2i\theta}\}\subset \sigma_{{\mathrm{ess}}}(B_{\theta})$.\
The converse inclusion requires a bit more work. Observe that if $\lambda\in\sigma_{{\mathrm{ess}}}(B_{\theta})$, then one can prove by localization the existence of a corresponding singular sequence $(v^{(m)})$ supported either in $x>L$ or in $x<-L$. In the first case, it means that $(v^{(m)})$ is also a singular sequence for $A_{\theta}$, so that $\lambda\in\sigma_{{\mathrm{ess}}}(A_{\theta})$. In the second case, $(v^{(m)})$ is a singular sequence for the complex conjugate of $A_{\theta}$, which implies that $\overline{\lambda}\in\sigma_{{\mathrm{ess}}}(A_{\theta})$. Finally, the result follows from item $i)$ of Theorem \[thmUsualPMLs\].\
$ii)$ Define the form $b_{\theta}(\cdot,\cdot)$ such that $$b_{\theta}(u,v)=\int_{{\Omega}}\beta_{\theta}\partial_xu\,\partial_x\overline{v}+\beta^{-1}_{\theta}\partial_yu\,\partial_y\overline{v}-k^2\,\beta^{-1}_{\theta}u\,\overline{v}\,dxdy.$$ One can check that for $k^2\in{\mathbb{C}}\setminus\mathscr{R}_{\theta}$, the numbers $1$, $e^{\pm i\theta}$, $-k^2$, $-k^2\,e^{\pm i\theta}$ are all located in some region $\{z\in{\mathbb{C}}\,|\,\Re e\,(e^{i\kappa}z)\ge \mu\}$ for some $\kappa\in [0;2\pi)$ and $\mu>0$. We deduce that $$\Re e\,(e^{i\kappa}b_{\theta}(u,u))\ge \mu \int_{{\Omega}}|u|^2+|\partial_xu|^2+|\partial_yu|^2\,dxdy$$ which shows that $b_{\theta}(\cdot,\cdot)$ is coercive on the Sobolev space ${{\mathrm{H}}}^1(\Omega)$. As a consequence of the Lax-Milgram theorem, $B_{\theta}-k^2$ is invertible for $k^2\in{\mathbb{C}}\setminus\mathscr{R}_{\theta}$, which guarantees that $\sigma(B_{\theta})\subset\mathscr{R}_{\theta}$.\
$iii)$ If $k^2\in\mathscr{K}_{{\mathrm{t}}}\cup\mathscr{K}_{{\mathrm{r}}}$, then $k^2$ is real and by construction $k^2\in\sigma(B_{\theta})\setminus\sigma_{{\mathrm{ess}}}(B_{\theta})$. Conversely, assume that $k^2$ is real and that $k^2\in\sigma(B_{\theta})\setminus\sigma_{{\mathrm{ess}}}(B_{\theta})$. There is one $N\in{\mathbb{N}}$ such that $k\in(N\pi;(N+1)\pi)$. Consider a non zero $w_{\theta}\in{{\mathrm{ker}}\,}(B_{\theta}-k^2)$. Setting $u=w_{\theta}\circ\mathcal{J}_{-\theta}$, we find that $u$ satisfies $\Delta u+k^2\gamma u=0$ in ${\Omega}$ and expands as $$\label{DecompoEigenBtheta}
u=\begin{array}{|l}
{\displaystyle}\sum_{n=0}^{N} a^{-}_n\,w_n^{+}+\sum_{n=N+1}^{+\infty} a^{-}_n\,w_n^{-}\quad\mbox{ for } x\le -L\\[12pt]
{\displaystyle}\sum_{n=0}^{+\infty} a^{+}_n\,w_n^{+}\quad\mbox{ for } x\ge L,
\end{array}$$ with $(a^{\pm}_n)\in{\mathbb{C}}^{{\mathbb{N}}}$. If one of the $a^{-}_n$, for $n=0,\dots,N$, is non zero, it means that $u$ is a RM. If on the contrary $a^{-}_n=0$ for all $n=0,\dots,N$, then one can prove that $a^{+}_n=0$ for all $n=0,\dots,N$, so that $u$ is a TM. Indeed, integrating by parts, we show that the quantity $$\mathscr{F}=\int_{\Sigma_{-L}\cup\Sigma_L} (\partial_x u \overline{u}-u \partial_x \overline{u}) \,dy,$$ with $\Sigma_{\pm L}=\{\pm L\}\times(0;1)$, satisfies $\mathscr{F}=0$. On the other hand, a direct calculation using expansion (\[DecompoEigenBtheta\]) and the orthonormality of the $\varphi_n$ yields $
\mathscr{F}=\sum_{n=0}^N i(|a_n^+|^2+|a_n^-|^2)
$ and the result follows.\
Finally, consider some $k^2\in\sigma(B_{\theta})\setminus\sigma_{{\mathrm{ess}}}(B_{\theta})$ such that $\Im m\,k^2>0$ (the case $\Im m\,k^2<0$ is similar). There is a unique $N\in{\mathbb{N}}$ such that $2\pi-2\theta<{\mathrm{arg}}(k^2-(N\pi)^2)<2\pi$ and $\pi<{\mathrm{arg}}(k^2-((N+1)\pi)^2)<2\pi-2\theta$. Then if $w_{\theta}\in{{\mathrm{ker}}\,}(B_{\theta}-k^2)\setminus\{0\} $, expansion (\[DecompoEigenBtheta\]) for $u=w_{\theta}\circ\mathcal{J}_{-\theta}$ holds and one of the $a^{-}_n$, $n=0,\dots,N$, has to be non zero. Indeed, otherwise $u$ would be exponentially decaying for $\pm x\ge L$ and $k^2$ would be in $\sigma(A)$, which is impossible because $\sigma(A)=[0;+\infty)$. Therefore the amplitude of $u$ is exponentially decaying at $+\infty$ and exponentially growing at $-\infty$.$\square$\
**Proof of Proposition \[PropositionCarac\].** If $(k^2,w_{\theta})\in{\mathbb{R}}\times {{\mathrm{L}}}^2(\Omega)$ is an eigenpair of $B_{\theta}$, then $u=w_{\theta}\circ\mathcal{J}_{-\theta}$ expands as in (\[DecompoEigenBtheta\]). Moreover, we deduce from the orthogonality of the $\varphi_n$ that $$a_n^-=\int_{0}^1w_{\theta}(-L,y)\varphi_n(y)\,dy,\quad n=0,\dots,N,$$ which gives the result, using the same arguments as in the proof of Theorem \[thmConjugatedPMLs\], item $iii)$. $\square$
Concluding remarks {#SectionConclusion}
==================
It is often desirable to determine frequencies for which a wave can be completely transmitted through a structure, a task usually leading to the tedious work of evaluating the scattering coefficients for each frequency. Here, we have shown that reflectionless frequencies can be directly computed as the eigenvalues of a non-selfadjoint operator $B_{\theta}$ (see (\[defOpConjugatedPMLs\])) with conjugated complex scalings enforcing ingoing behaviour in the incident lead and outgoing behaviour in the other lead. The reflectionless spectrum of this operator $B_{\theta}$ provides a complementary information to the one contained in the classical complex resonance spectrum associated with leaky modes which decompose only on outgoing waves (see the operator $A_\theta$ in (\[defOpClassicalPMLs\])). Note that eigenvalues corresponding to trapped modes belong to both the reflectionless spectrum and to the classical complex resonance spectrum because trapped modes do not excite propagating waves.\
Moreover, since ingoing and outgoing complex scalings can be associated, respectively, with gain and loss, one observes that the non-selfadjoint operator $B_{\theta}$ leads to consider a natural $\mathcal{P}\mathcal{T}$ symmetric problem when the structure has mirror symmetry. Interestingly, a direct calculus shows that in the very simple case of a ${\mathrm{1D}}$ transmission problem through a slab of constant index, reflectionless frequencies are all real. This gives an example of a non-selfadjoint $\mathcal{P}\mathcal{T}$ symmetric operator with only real eigenvalues.\
In this work, we investigated scattering problems in waveguides with $N=2$ leads for which two reflectionless spectra exist: one associated with incident waves propagating from the left and another corresponding to incident waves propagating from the right. The more general case with $N$ ($N \ge 2$) leads can be considered as well. Among the total of $2^N$ different spectra with an ingoing or an outgoing complex scaling in each lead, two spectra correspond to eigenmodes which decompose on waves which are all outgoing or all ingoing. As a consequence, there are $2^N-2$ reflectionless spectra.
[^1]: *FreeFem++*, <http://www.freefem.org/ff++/>.
|
---
author:
- 'Jean-Pierre Aubin'
title: |
The Method of Characteristics Revisited\
A Viability Approach\
\
A Mini-Course[^1]
---
\[section\] \[Theorem\][Definition]{} \[Theorem\][Proposition]{} \[Theorem\][Lemma]{} \[Theorem\][Corollary]{} 0.0cm 0.0cm 0.0cm 20.5cm 15.0cm 1.5cm
[Jean-Pierre Aubin]{}
[Université de Paris-Dauphine]{}
[Centre de Recherche Viabilit‚, Jeux, Contr“le]{}
F-75775 Paris cx(16), France
[**Acknowledgments**]{}\
The author wishes to thank Roger Wets for giving him the opportunity to present theses lectures at University of California at Davis.
He is grateful to H‚lŠne Frankowska for her contributions about Hamilton-Jacobi equations and systems of first-order partial differential equations on which the main results presented here are based and daily discussions on these topics.
Typesetting: LaTeX
>From the same author\
- [Approximation of Elliptic Boundary-Value Problems]{} (1972) Wiley
- [Applied Abstract Analysis]{} (1977) Wiley-Interscience
- [Applied Functional Analysis]{} (1979) Wiley-Interscience, Second Edition, 1999 (Version Française: [Analyse Fonctionnelle Appliquée, Tomes 1 & 2.]{} (1987) Presses Universitaires de France)
- [Mathematical Methods of Game and Economic Theory]{} (1979) North-Holland Second Edition, 1982
- [Méthodes Explicites de l’Optimisation]{} (1982) Dunod (English translation: [Explicit Methods of Optimization]{}, Dunod, 1985)
- [L’Analyse non linéaire et ses motivations économiques]{} (1983) Masson (English translation: [Optima and Equilibria]{}, Springer Verlag, 1993, Second Edition, 1998)
- [Differential Inclusions]{} \[in collaboration with A. CELLINA\], (1984) Springer-Verlag
- [Applied Nonlinear Analysis]{} \[in collaboration with I. EKELAND\], (1984) Wiley-Interscience
- [Exercices d’analyse non linéaire]{} (1987) Masson
- [Set-Valued Analysis]{} \[with H. FRANKOWSKA\], (1990) Birkhäuser
- [Viability Theory]{}, (1991) Birkhäuser
- [Initiation à l’Analyse Appliquée]{} (1994) Masson (English Version to appear)
- [Neural Networks and Qualitative Physics: A Viability Approach]{} (1996) Cambridge University Press
- [Dynamic Economic Theory: A Viability Approach]{} (1997) Springer-Verlag (Studies in Economic Theory). Second Edition to appear
- [Mutational and Morphological Analysis: Tools for Shape Regulation and Morphogenesis]{} (1999), Birkhäuser
- [La mort du devin, l’émergence du démiurge. Essai sur la contingence et la viabilité des systèmes]{} (in preparation)
This mini-course provides a presentation of the method of characteristics to initial/boundary-value problems for systems of first-order partial differential equations and to Hamilton-Jacobi variational inequalities.
These results are indeed useful for the treatment of hybrid systems of control theory.
We use the tools forged by set-valued analysis and viability theory, which happen to be both efficient and versatile to cover many problems. They find here an unexpected relevance.
Indeed, since solutions to first-order systems $$\forall \; j=1, \ldots ,p, \; \; \frac{ \partial }{ \partial t}u (t,x) +
\sum_{i=1}^{n}\frac{ \partial }{ \partial x_{i}}u_{i}
(t,x) f_{i}(t,x,u (t,x)) - g_{j}(t,x,u (t,x)) \; = \; 0$$ may have “shocks”, i.e., may be set-valued maps (or multi-valued maps), it seems to us natural to use the concept of “graphical derivative” of a set-valued map from set-valued analysis instead of “distributional derivative” from the theory of distributions to give a meaning to a concept of solution to such systems of partial differential equations.
The basic concept useful in our framework is the concept of [*capture basin*]{} of a subset $C$ under a differential equation, which is the set of points from which a solution to the differential equation reaches $C$ in finite time.
Then we shall prove that [*the graph of the solution $ (t,x)
\leadsto U (t,x)$ to the above boundary value problem is the capture basin of the graph of the initial/boundary data under the characteristic system of differential equations $$\left\{ \begin{array}{ll}
i) & \tau ' (t) \; = \; -1 \\
ii) & x' (t ) \; = \; -f ( \tau (t),x (t),y (t))) \\
iii) & y' (t) \; = \; - g ( \tau (t),x (t),y (t))
\end{array} \right.$$ and that, under adequate assumptions, this solution is [**unique**]{} among the solutions with closed graph to this boundary value problem*]{}.
Such a solution is thus taken in a generalized — or weak — sense (Frankowska solutions), since such maps, even when they are single-valued, are not differentiable in the usual sense. But when they are, they naturally coincide with the above concept of solution thanks to the uniqueness property.
Existence and uniqueness of the solution is obtained
1. from a first characterization of the capture basin of $C$ as the union of reachable sets $ \vartheta _{-f} (t,C) := \{\vartheta _{-f}
(t,c)\}_{c \in C}$, where $\vartheta _{-f} (t,c)$ denotes the value at time $t$ of the solution to the differential equation $x'=-f (x)$ starting at $c$, allowing explicit computations of the capture basins in specific instances,
2. from a second characterization of the capture basin of $C$ of a closed backward invariant subset $M $ which is a repeller[^2] under a differential equation $x'=f (x)$. It states that the capture basin is the [**unique**]{} closed subset $K$ such that
1. $C\; \subset \; K \; \subset \; M$,
2. $K$ is [*backward invariant*]{} in the sense that any backward solution starting from $K$ is viable in $K$,
3. $K \backslash C$ is [*locally viable*]{} in the sense that for any $x \in K \backslash C$, there exist $T>0$ and a solution to the differential equation starting at $x$ viable in $K \backslash C$ on the interval $[0,T]$,
3. from the 1942 Nagumo Theorem stating that
1. the subset $K$ is backward invariant under a differential inclusion $x' =f (x)$ if and only if, for every $x \in K$, $f (x) \in -
T_{K} (x)$, or equivalently, if and only if, for every $x \in K $, $p \in
N_{K} (x)$, $ \langle p, f (x) \rangle \geq 0$,
2. the subset $K \backslash C$ is locally viable if and only if, for every $x \in K \backslash C$, $f (x)\in T_{K} (x)$, or equivalently, if and only if, for every $x \in K \backslash C$, $p \in N_{K} (x)$, $
\langle p, f (x) \rangle \leq 0$,
where the “contingent cone” $T_{K} (x)$ to a subset $K$ at a point $x \in K$, introduced in the early thirties independently by Bouligand and Severi, adapts to any subset the concept of tangent space to manifolds, and where the “normal cone” $N_{K} (x):= T_{K} (x)^{-}$, defined as the polar cone to the contingent cone, adapts to any subset the concept of normal space to manifolds,
4. from the definition of the contingent derivative $DU (t,x,y)$ of the set-valued map $U: (t,x) \leadsto U(t,x)$ at a point $ (t,x,y)$ of the graph of $U$ as the set-valued map from $ {\bf R} \times X$ to $Y$ the graph of which is the contingent cone to the graph of $U$ at the point $
(t,x,y)$.
The above results — which are interesting by themselves for other mathematical models of evolutionary economics, population dynamics, epidemiology — can be applied to many other problems. Dealing with subsets, they can be applied to graphs of single-valued maps as well as set-valued maps, to epigraphs and hypographs of (extended) real-valued functions, to graph of “impulse” maps (which take empty values except in a discrete sets, useful in the study of hybrid systems or inventory management), etc. Since these results are also valid for underlying differential inclusions, we are able to treat control problems for such boundary-value problems for systems of first-order partial differential equations.
We also illustrate the strategy of using the properties of the viability kernel — the subset of elements from which starts at least a solution to the differential equation viable in $K$ — and the capture basin of a subset advocated by H‚lŠne Frankowska[^3] for characterizing the value functions of some variational problems or stopping time problems as “contingent solutions” and/or “viscosity solutions to Hamilton-Jacobi “differential variational inequalities” $$\left\{ \begin{array}{ll}
i) & {\bf u} (x) \; \leq \; {\bf u} ^{\top } (x)\\
ii) & \displaystyle{\left\langle \frac{ \partial }{ \partial
x}{\bf u}
^{\top } (x),f (x)\right\rangle +l (x,f (x)) +a{\bf u} ^{\top } (x)\; \leq
\;
0}\\
iii) & \displaystyle{({\bf u} (x)-{\bf u} ^{\top } (x)) \left(
\left\langle
\frac{ \partial }{ \partial x}{\bf u} ^{\top } (x),f (x)\right\rangle +l
(x,f
(x)) +a{\bf u} ^{\top } (x) \right) \; = \; 0}
\end{array} \right.$$ and $$\left\{ \begin{array}{ll}
i) & 0 \; \leq \; {\bf u} ^{\bot }(x) \; \leq \; {\bf u} (x)\\
ii) & \displaystyle{\left\langle \frac{ \partial }{ \partial
x}{\bf u}
^{\bot } (x),f (x)\right\rangle +l (x,f (x)) +a{\bf u} ^{\bot } (x)\; \geq
\;
0}\\
iii) & \displaystyle{({\bf u} (x)-{\bf u} ^{\bot } (x)) \left(
\left\langle
\frac{ \partial }{ \partial x}{\bf u} ^{\bot } (x),f (x)\right\rangle +l
(x,f
(x)) +a{\bf u} ^{\bot } (x) \right) \; = \; 0}
\end{array} \right.$$ where
1. $f:{\bf R}^{n} \mapsto {\bf R}^{n}$ defines the dynamics of the differential equation $x'= f(x)$,
2. $l : (x,p) \in {\bf R}^{n} \times {\bf R}^{n} \mapsto l (x,p) \in
{\bf R}_{+}$ is a nonnegative “Lagrangian”,
3. ${\bf u}:{\bf R}^{n} \mapsto {\bf R}_{+} \cup \{+\infty \}$ is an extended nonnegative function (regarded as an obstacle, as in unilateral mechanics).
We shall observe that the epigraphs of the solutions are respectively the viability kernel and the capture basin of the epigraph of $ {\bf u}$ under the map $g: {\bf R}^{n}
\times {\bf R} \mapsto {\bf R}^{n} \times {\bf R}$ defined by $$g (x,y) \; := \; ( f (x), -ay-l (x,f (x)))$$
This allows us to compute the “solutions” to these Hamilton-Jacobi equations and to check that they are respectively defined by $${\bf u}^{\top} (x) \; := \; \sup_{t \geq 0} \left( e^{a t}{\bf u} (x
(t)) + \int_{0}^{t}e^{a \tau }l ( x( \tau ),x' ( \tau )) d \tau \right)$$ and $${\bf u}^{\bot} (x) \; := \; \inf_{t \geq 0} \left( e^{a t}{\bf u}
(x (t)) + \int_{0}^{t}e^{a \tau }l ( x( \tau ),x' ( \tau )) d \tau
\right)$$ where $x ( \cdot )$ is the solution to the differential equation $x'=f (x)$ starting at $x$.
Using the fact that the contingent cone to the epigraph of an extended lower semicontinuous function is the epigraph of its “contingent epiderivative”, the characterizations of viability kernel and capture basins in terms of tangential conditions allows to interpret these “solutions” as “contingent solutions” to the above Hamilton-Jacobi variational inequalities. Using the characterization in terms of normal cones, we obtain the equivalent interpretation of these solutions as viscosity solutions whenever the solution is continuous instead of being merely lower semicontinuous.
This this is the reason why this mini course begins with the Nagumo Theorem on closed subsets viable and/or invariant under a differential equation. Next, the notions of the “capture basin” of a closed subset, as well as its “viability kernel” are characterized and linked with the Nagumo Theorem. After recalling the concepts of contingent epiderivative and subdifferential of an extended function, these theorems are used for solving some Hamilton-Jacobi variational inequalities. They are also used for solving initial/boundary value problems for systems of first-order partial differential equations.
For the sake of simplicity, we restrict ourselves in this introductory mini-course to the simple case when the characteristic system is made of differential equations. However, the very same methods can be adapted to the case when the characteristic system is made of differential inclusions. Actually, the methods and the view points developed in these notes are more important than the examples presented here, since they can be easily generalized and efficiently used for solving many more difficult problems.
The Nagumo Theorem
==================
This chapter is a presentation of the basic Nagumo Theorem and its corollaries and consequence in the simple framework of ordinary differential equations $x' = f(x)$. These results — which are interesting by themselves as mathematical metaphors of evolutionary economics, population dynamics, epidemiology, biological evolution when they are extended to differential inclusions — can be applied to many other problems, such as control problems and, as we shall illustrate here, can be used as tools for solving other mathematical problems. Dealing with subsets, they can be applied to graphs of single-valued maps as well as set-valued maps, to epigraphs and hypographs of (extended) real-valued functions, and to be used as versatile and efficient tools for solving systems of first-order partial differential equations.
A function $[0,T] \ni t \rightarrow x(t)$ is said to be (locally) [*viable*]{} in a given subset $K$ on $[0,T]$ if, for any $t \in [0,T]$, the state $x(t)$ remains in $K$.
Therefore, if a continuous map $f: {\bf R}^{n} \mapsto {\bf R}^{n} $ describes the dynamics of the system $$\forall \; t \geq 0, \; \; x' (t) \; = \; f (x (t))$$ we shall say that $K$ is [*viable under*]{} $f$ if starting from any initial point of $K$, [*at least one solution*]{} to the differential equation is viable in $K$.
The Nagumo Theorem characterizes such a viability property for any locally compact subset $K$ in terms of contingent and normal cones.
Hence, we begin by recalling the concept of contingent cones and normal cones to a arbitrary subset of a finite dimensional vector space. For that purpose, we start with the notion of upper convergence of sets introduced by Painlevé: The upper limit of a sequence of subsets $K_{n} \subset X$ is the set of cluster points of sequences $x_{n} \in K_{n}$. With that concept, we can define [**the contingent cone**]{} $T_{K} (x)$ to a subset $K$ at $x$ is the upper limit of the “difference quotients” $
\frac{K-x}{h}$. The normal cone $N_{K} (x)$ is next defined as the polar cone $T_{K} (x)^{-}$ to the contingent cone.
The Nagumo Theorem characterizes such a viability property for any locally compact subset $K$ by stating that $K$ is viable under $f$ if and only if $$\forall \; x \in K, \; \; f (x) \; \in \; T_{K} (x)$$ or, equivalently, if and only if $$\forall \; x \in K, \; \forall \; p \in N_{K} (x)\; \; \langle p,f (x)
\rangle \; \leq \; 0$$
Since open subsets, closed subsets, intersection of open and closed subsets of $ {\bf R}^{n} $ are locally compact, we shall be able to specify this theorem in each of theses cases.
We then state and prove these viability theorems. Many proofs of viability theorems are now available: We chose the most elementary one (which is not the shortest) because it is the prototype of the extensions of the viability theorems. It is just a modification of the Euler method of approximating a solution by piecewise linear functions (polygonal lines) in order to force the solution to remain viable in $K$.
Viability & Invariance Properties
---------------------------------
Let $K $ be a subset of a finite dimensional vector-space ${\bf R}^{n}
$. We shall say that a function $x(\cdot ) $ from $[0,T] $ to ${\bf R}^{n} $ is [*viable in $K$ on $[0,T]$*]{} if for all $t \in [0,T],\;\; x(t) \in K $.
Let us describe the dynamics of the system by a map $f$ from ${\bf
R}^{n}$ to ${\bf R}^{n} $. We consider the initial value problem (or Cauchy problem) associated with the differential equation
$$\label{01A11}
\forall \; t \in [0,T],\; \;x'(t)\;=\;f(x(t ) )$$
satisfying the initial condition $x(0) = x_{0}$.
\[Viability & Invariance Properties\] \[02A11\] Let $K \subset$ ${\bf R}^{n} $. We shall say that $K$ [*is locally viable under $f$*]{} if for any initial state $x_{0 }$ of $K$, [*there exist* ]{} $T >0$ and a [*viable* ]{}solution on $[0,T]$ to differential equation (\[01A11\]) starting at $x_{0 }$. It is said to be (globally) viable under $f$ if we can always take $T = \infty$.
The subset $K $ is said to be locally [*invariant under $f$*]{} if for any initial state $ x_{0 }$ of $K $ and for [*all* ]{} solutions $x (
\cdot )$ to differential equation (\[01A11\]) (a priori defined on ${\bf
R}^{n} $) starting from $x_{0}$, there exists $T>0$ such that $x ( \cdot )$ is [*viable in $K$*]{} on $[0,T]$. It is said to be (globally) invariant under $f$ if we can always take $T = \infty$ for all solutions.
A subset $K$ is a [repeller]{} if from any initial element $x_{0} \in K$, all solutions to the differential equation (\[01A11\]) starting at $x_{0} \in K$ leave $K$ in finite time.
[**Remark**]{} — We should emphasize that the concept of invariance [*depends upon the behavior of $f $ on the domain ${\bf R}^{n} $ outside $K$*]{}. But we observe that viability property depends only on the behavior of $f $ on $K $. $\; \; \Box$
So, the viability property requires only the existence of at least one viable solution whereas the invariance property demands that all solutions, if any, are viable.
Observe also that whenever there exists a unique solution to differential equation $x'=f (x)$ starting from any initial state $x_{0}$, then viability and invariance properties of a closed subset $K$ are naturally equivalent.
We shall begin by characterizing the subsets $K $ which are viable under $f$. The idea is simple, intuitive and makes good sense: [*A subset $K $ is viable under $f$ if at each state $x $ of $K $, the velocity $f(x ) $ is “tangent” to $K$ at $x$, so to speak, for bringing back a solution to the differential equation inside $K $*]{}.
Contingent and Normal Cones
---------------------------
#### Limits of Sets
Limits of sets have been introduced by Paul Painlevé in 1902 without the concept of topology. They have been popularized by Kuratowski in his famous book [Topologie]{} and thus, often called [*Kuratowski lower and upper limits*]{} of sequences of sets. They are defined without the concept of a topology on the power space.
\[04A391\] Let $(K_{n})_{n \in {\bf N}}$ be a sequence of subsets of a finite dimensional vector space $X$. We say that the subset $$\mbox{\rm Limsup}_{n \rightarrow \infty}K_{n} \; := \; \left\{ x
\in X \;\; | \;\; \liminf_{n \rightarrow \infty}d(x,K_{n}) = 0 \right\}$$ is the [*upper limit*]{} or [*outer limit*]{}[^4] of the sequence $K_{n}$ and that the subset $$\mbox{\rm Liminf}_{n \rightarrow \infty}K_{n} \;
:= \; \left\{ x \in X \;\; | \;\; \mbox{\rm lim}_{n \rightarrow
\infty}d(x,K_{n}) = 0 \right\}$$ is its [*lower limit*]{} or [*inner limit*]{}. A subset $K$ is said to be the [*limit*]{} or the [*set limit*]{} of the sequence $K_{n}$ if $$K \; = \; \mbox{\rm Liminf}_{n \rightarrow \infty }K_{n} \;
= \; \mbox{\rm Limsup}_{n \rightarrow \infty } K_{n} \; =: \;
\mbox{\rm Lim}_{n \rightarrow \infty }K_{n}$$
Lower and upper limits are obviously [*closed*]{}. We also see at once that $$\mbox{\rm Liminf}_{n \rightarrow \infty }K_{n} \; \subset \;
\mbox{\rm Limsup}_{n \rightarrow \infty }K_{n}$$ and that the upper limits and lower limits of the subsets $K_{n}$ and of their closures $\overline{K}_{n}$ do coincide, since $d(x,K_{n}) =
d(x,\overline{K}_{n})$.
Any decreasing sequence of subsets $K_{n}$ has a limit: $$\mbox{\rm if} \; \; K_{n} \subset K_{m} \; \; \mbox{when} \; \;
n \geq m, \; \; \mbox{then} \; \; \mbox{\rm Lim}_{n \rightarrow \infty
}K_{n} \; = \; \bigcap_{n \geq 0}\overline{K_{n}}$$ An upper limit may be empty (no subsequence of elements $x_{n} \in
K_{n}$ has a cluster point.)
Concerning sequences of singletons $\{x_{n}\}$, the set limit, when it exists, is either empty (the sequence of elements $x_{n}$ is not converging), or is a singleton made of the limit of the sequence.
It is easy to check that:
\[02A391\] If $(K_{n})_{n \in {\bf N}}$ is a sequence of subsets of a finite dimensional vector space, then $\mbox{\rm Liminf}_{n \rightarrow \infty}K_{n}$ is the [*set of limits of sequences*]{} $x_{n} \in K_{n}$ and $\mbox{\rm Limsup}_{n
\rightarrow \infty}K_{n}$ is the set [*of cluster points of sequences*]{} $x_{n} \in K_{n}$, i.e., of limits of subsequences $x_{n'} \in K_{n'}$.
#### Contingent Cone
We reformulate the definition of contingent direction to a subset of a finite dimensional vector space introduced independently by Georges Bouligand and Francesco Severi [^5] in the 30’s:
When $K \subset X$ is a subset of a normed vector space $X$ and when $x \in K$, the set $T_{K} (x)$ $$T_{K}(x) \; := \; \mbox{\rm Limsup}_{h \rightarrow 0+}
\frac{K-x}{h}$$ of [*contingent directions*]{} to $K$ at $x$ is a closed cone, called the [*contingent cone*]{} or simply, the [*tangent cone*]{}, to $K$ at $x$.
Therefore, a direction $v \in X$ is [*contingent*]{} to $K$ at $x$ if and only if $$\liminf_{h \rightarrow 0+} \frac{d (x+hv,K)}{h} \; = \; 0$$ or, equivalently, if and only if there exists a sequence of elements $h_{n}
>0$ converging to $0$ and a sequence of $v_{n} \in X$ converging to $v$ such that $$\forall \; n \geq 0, \; \; x + h_{n}v_{n} \; \in \; K$$
The lemma below shows right away why these cones will play a crucial role: they appear naturally whenever we wish to differentiate viable functions.
\[derviablem\] Let $x(\cdot)$ be a differentiable [*viable*]{} function from $[0,T]$ to $K$. Then $$\forall \; t \in [0,T[,\; \; \; x'(t) \; \in \;
T_{K}(x(t))$$
[**Proof**]{} — Let us consider a function $x(\cdot) $ viable in $K$. It is easy to check that $x'(0)$ belongs to the contingent cone $T_{K}(x_{0})$ because $x(h)$ belongs to $K$ and consequently, $$\frac{ d_{K}(x_{0}+ hx'(0))}{h} \; \leq \; \frac{\|x(0)+h x'(0)
- x(h)\|}{h} \;\mbox{\rm converges to}\; 0$$ Hence $x' (0)$ belongs to the contingent cone to $K$ at $x_{0}$. $\; \; \Box$
For convex subsets $K$, the contingent cone coincides with the closed cone spanned by $K-x$:
\[01A955\] Let us assume that $K$ is convex. Then the contingent cone $T_{K}(x)$ to $K$ at $x$ is convex and $$T_{K}(x) \; = \; \overline{\bigcup_{h>0}^{} \frac{K-x}{h}}$$ We shall say in this case that it is the [*tangent cone*]{} to the convex subset $K$ at $x$.
[**Proof**]{} — We begin by stating the following consequence of convexity: If $0 <
h_{1} \leq h_{2}$, then $$\frac{K-x}{h_{2}} \; \subset \; \frac{K-x}{h_{1}}$$ because $x + h_{1}v = \frac{h_{1}}{h_{2}} (x+h_{2}v)+ \left(
1-\frac{h_{1}}{h_{2}} \right)x$ belongs to $K$ whenever $x+h_{2}v$ belongs to $K$. The sequence of the subsets $\frac{K-x}{h}$ being increasing, our proposition ensues. $\; \; \Box$
\[Viability Domain\] Let $K $ be a subset of ${\bf R}^{n} $. We shall say that $K $ is a [ *viability domain*]{} of the map $f :{\bf R}^{n} \mapsto {\bf R}^{n} $ if $$\forall \; x \in K , \; \; f(x) \; \in \; T_{K}(x)$$
We recall that a subset $K \subset {\bf R}^{n}$ is [*locally compact*]{} if there exists $r>0$ such that the ball $B_{K}(x_{0},r) := K \cap (x_{0}+rB)$ is [*compact*]{}. Closed subsets, open subsets and intersections of closed and open subsets of a finite dimensional vector space are locally compact.
#### Normals
\[normalcondef\] The polar cone $N_{K} (x) :=\left( T_{K} (x) \right)^{-}$ is called the [*normal cone*]{} to $K$ at $x$.
It is also called the [*Bouligand normal cone*]{}, or the [*contingent normal cone*]{}, or also, the [*sub-normal cone*]{} and more recently, the [*regular normal cone*]{} by R.T. Rockafellar and R. Wets. In this book, only (regular) normals are used, so that we shall drop the adjective “regular”. $\; \; \Box$
\[localsep\] Let $K \subset X$ be a closed subset. Let $y \notin K$ and $x \in
\Pi _{K} (y)$ a best approximation of $y$ by elements of $K$: $ \|y-x\| =d
(y,K)$. Then $$\forall \; y \notin K, \; \forall \; x \in \Pi_{K} (y), \; \;\; y-x
\; \in \; N_{K} (x)$$
[**Proof**]{} — Since $x \in \Pi _{K} (y)$ minimizes the distance $ z \mapsto
\frac{1}{2}\|y-z\|^{2}$ on $K$, we deduce that $$\forall \; v \in T_{K} (x), \; \; 0 \; \leq \; \left\langle
x-y , v \right\rangle$$ so that $y-x$ belongs to $T_{K} (x)^{-}=:N_{K} (x)$. $\; \; \Box$
When $K$ is convex, we deduce that $N_{K} (x)$ is the polar cone to $ K-x$ because the tangent cone is spanned by $K-x$:
\[03A955\] Let $K $ be a closed convex subset of a finite dimensional vector space. Then $$p \in N_{K} (x) \;\; \mbox{if and only if} \;\; \forall \; y \in K, \;
\; \langle p, y \rangle \; \leq \; \langle p, x \rangle$$ and the graph of the set-valued map $x \leadsto N_{K} (x)$ is closed in $X \times X^{\star}$.
[**Proof**]{} — Let us consider sequences of elements $x_{n} \in K$ converging to $x$ and $p_{n} \in N_{K}(x_{n})$ converging to $p$. Then inequalities $$\forall \; y \in K, \;\; \langle p_{n},y\rangle \; \leq
\; \langle p_{n},x_{n} \rangle$$ imply by passing to the limit inequalities $$\forall \; y \in K, \; \; \langle p,y \rangle \; \leq
\; \langle p,x \rangle$$ which state that $p$ belongs to $N_{K}(x)$. Hence the graph is closed. $\; \; \Box$
In the general case, we provide now the following characterization of the normal cone:
\[mormconechar\] Let $K$ be a subset of a finite dimensional vector-space $X$. Then $p \in N_{K} (x) $ if and only $$\left\{ \begin{array}{l} \label{boh}
\forall \; \varepsilon >0, \; \exists \; \eta >0 \;\; \mbox{\rm
such that} \;\;\forall \; y \in K\cap B(x,\eta ),\\
\\
\langle p,y-x \rangle \; \leq \; \varepsilon \|y- x\|
\end{array} \right.$$
[**Proof**]{} — Let $p$ satisfy above property (\[boh\]) and $v \in T_{K}
(x)$. Then there exist $h_{n}$ converging to $0$ and $v_{n}$ converging to $v$ such that $$y \; := \; x + h_{n} v_{n} \; \in \; K \cap B(x,\eta )$$ for $n$ large enough. Consequently, inequalities $\langle p,v_{n} \rangle
\leq \varepsilon $ imply by taking the limit that $\langle p,v \rangle
\leq \varepsilon $ for all $\varepsilon >0$. Hence $\langle p,v \rangle
\leq 0$, so that any element $p$ satisfying the above property belongs to the polar cone of $T_{K} (x)$.
Conversely, assume that $p$ violates property (\[boh\]): There exist $\varepsilon >0$ and a sequence of elements $x_{n} \in K$ converging to $x$ such that $$\left\langle p,x_{n}-x \right\rangle \; > \;
\varepsilon \|x_{n} - x\|$$ We set $h_{n} := \|x_{n}- x\|$, which converges to $0$, and $v_{n} := (x_{n}-x)/h_{n}$. These elements belonging to the unit sphere, a subsequence (again denoted) $v_{n}$ converges to some $v$. By definition, this limit belongs to $T_{K} (x)$, so that $\langle p,v
\rangle \leq 0$. But our choice implies that $\langle p,v_{n} \rangle
>\varepsilon $, so that $\langle p,v \rangle \geq \varepsilon $, a contradiction. $\; \; \Box$
We provide a useful characterization by duality of viability domains in terms of normal cones:
\[3russianfrank\] Let $K$ be a locally compact subset of a finite dimensional vector space ${\bf R}^{n}$ and $f:K \mapsto {\bf R}^{n}$ be a continuous single-valued map. Then $$\forall \; x \in K , \; \; f(x) \; \in \; T_{K}(x)$$ if and only if $$\forall \; x \in K , \; \; f(x) \; \in \; \overline{ \mbox{\rm
co}}(T_{K}(x))$$ or, equivalently, in terms of normal cone, if and only if $$\forall \; x \in K, \; \forall \; p \in N_{K} (x), \; \; \langle p,f
(x) \rangle \; \leq \; 0$$
[**Proof**]{} — Since the normal cone $N_{K} (x)$ is the polar cone to $T_{K} (x)$, and thus, to $\overline{ \mbox{\rm co}}(T_{K}(x))$, then $\overline{ \mbox{\rm
co}}(T_{K}(x))$ is the polar cone to $N_{K} (x)$, so that the two last statements are equivalent by polarity. Since the first statement implies the second one, it remains to prove that if for any $x \in K$, $f (x)$ belongs to $\overline{ \mbox{\rm co}}(T_{K}(x))$, then $f (x)$ is actually contingent to $K$ at $x$ for any $x \in K$. This follows from the following
\[01A142\] Let $K \subset {\bf R}^{n}$ be a locally compact subset of a finite dimensional vector space ${\bf R}^{n}$ and $f:K
\mapsto {\bf R}^{n}$ be a continuous single-valued map. Assume that there exists $ \alpha >0$ such that $$\forall \; x \in K \cap B (x_{0}, \alpha ), \; \; f (x)\; \in \;
\overline{
\mbox{\rm co}}(T_{K}(x))$$ Then, $f (x)$ is contingent to $K$ at elements $x$ of a neighborhood of $x_{0}$. Actually, for any $ \varepsilon >0$, there exists $ \eta (x_{0}, \varepsilon )\in ]0, \alpha ]$ such that $$\label{01A1421}
\forall \; x \in B (x_{0}, \eta (x_{0}, \varepsilon )), \; \forall \; h
\leq \eta (x_{0}, \varepsilon ), \; \; d \left( f (x), \frac{ \Pi _{K}
(x+hf (x))-x}{h} \right) \; \leq \; \varepsilon$$ where $ \Pi_{K} (y)$ denotes the set of best approximations $ \bar{x} \in
K$ of $y$, i.e., the solutions to $ \| \bar{x}-y\|=d (y,K)$.
[**Proof**]{} — It is sufficient to check the Lemma when $f (x) \ne 0$. Let us set $$g (t) \; := \; \frac{1}{2}d (x+tf (x),K)^{2} \; = \; \|x+tf
(x)-x_{t}\|^{2}$$ where $x_{t} \in \Pi_{K} (x+tf (x))$ is a best approximation of $x+tf (x)$ by elements of $K$. We take $ \alpha $ small enough for $K \cap B (x_{0}, \alpha )$ to be compact. We observe that there exists $ \beta \in ]0, \alpha ]$ such that for all $x
\in B (x_{0}, \beta )$, $ \|f (x)\| \leq 2 \|f (x_{0})\|$ because $f$ is continuous at $x_{0}$. Furthermore, $$\|x+tf (x)-x_{t}\| \; = \; d (x+tf (x),K) \; \leq \; t \|f (x)\| \; \leq
\; 2 t\|f (x_{0})\|$$ because $x$ belongs to $K \cap B (x_{0}, \beta )$, so that $ \|x-x_{t}\|
\leq
2t \|f (x)\|\leq 4t \|f (x_{0})\|$ converges to $0$ with $t$.
On the other hand, for every $v_{t} \in T_{K} (x_{t})$, there exists a sequence of $h_{n}>0$ converging to $0$ and $v_{t}^{n}$ converging to $v_{t}$ such that $x_{t}+h_{n}v_{t}^{n}$ belongs to $K$. Therefore, $$g (t+h_{n})-g (t) \; \leq \; \frac{1}{2} \left( \|x+tf (x)-x_{t}+ h_{n}(f
(x)-v_{t}^{n})\|^{2}- \|x+tf (x)-x_{t}\|^{2} \right)$$ and thus, dividing by $h_{n}>0$ and letting $h_{n}$ converge to $0$, that $$\forall \; v_{t} \in T_{K} (x_{t}), \; \; g' (t) \; \leq \;
\left\langle x+tf (x)-x_{t}, f (x)-v _{t} \right\rangle$$ Since it is true for any $v_{t} \in T_{K} (x_{t})$ and since the right-hand side is affine with respect to $v_{t}$, we deduce that this inequality remains true for any $v_{t} \in \overline{ \mbox{\rm co}}(T_{K} (x_{t}))$, and thus, by assumption, for $f (x_{t})\in\overline{ \mbox{\rm co}}(T_{K}
(x_{t}))$: $$g' (t) \, \leq \, \left\langle x+tf (x)-x_{t}, f (x)-f
(x_{t})\right\rangle \leq 2t \|f (x_{0})\| (\|f (x)-f (x_{0})\|
+\|f (x_{0})-f (x_{t})\|)$$ For any $ \varepsilon $, let $ \gamma >0$ such that $\|f (y)-f (x_{0})\|
\leq \frac{\varepsilon^{2}}{8 \|f (x_{0})\|}$ whenever $y \in B (x_{0}, 2
\gamma ) $. Since $ \|x_{0}-x_{t}\| \leq \|x- x_{0}\|+4t \|f (x_{0})\|
\leq 2\gamma $ whenever $ \|x-x_{0}\| \leq \gamma $ and $t \leq \frac{
\gamma }{4 \|f (x_{0})\|}$, then, setting $ \eta (x_{0}, \varepsilon ):=
\min \left( \gamma , \frac{ \gamma }{4 \|f (x_{0})\|} \right)$, we obtain $$\forall \; x \in K \cap B (x_{0}, \eta (x_{0}, \varepsilon )), \; \forall
\; t \in ]0, \eta (x_{0}, \varepsilon )], \; \; g' (t) \; \leq \;
\frac{t}{2} \varepsilon^{2}$$
Therefore, after integration from $0$ to $h$, we obtain $$\forall \; h \in [0, \eta (x_{0}, \varepsilon )], \; \; g (h) -g (0) \;
\leq \; h^{2} \varepsilon^{2}$$ Observing that $g (0)=0$, we derive the conclusion of the Lemma: $\forall
\; x_{h} \in \Pi _{K} (x +hf (x))$, $$d\left( f (x), \frac{ \Pi _{K} (x+hf (x))-x}{h} \right) \; \leq \; \frac{
\|x+h f (x)-x_{h}\|}{h} \; = \; \frac{d (x+hf (x),K)}{h} \; \leq \;
\varepsilon \; \; \Box$$
Statement of the Viability Theorems
-----------------------------------
Nagumo was the first one to prove the viability theorem for ordinary differential equations in 1942. This theorem was apparently forgotten, for it was rediscovered many times during the next twenty years.
We shall prove it when the subset $K \subset {\bf R}^{n}$ is [*locally compact*]{}:
\[Nagumo\] \[01A43\] Let us assume that $$\left\{ \begin{array}{ll}
i) & K \; \mbox{is locally compact} \\
ii) & f \;\mbox{is continuous from $K $ to ${\bf R}^{n} $ }
\end{array} \right.$$ Then $K $ is locally viable under $f$ if and only if $K $ is a viability domain of $f$ in the sense that $$\forall \; x \in K, \; \; f (x) \; \in \; T_{K} (x)$$ or, equivalently, in terms of normal cones, $$\forall \; x \in K, \; \forall \; p \in N_{K} (x), \; \; \langle p,f
(x) \rangle \; \leq \; 0$$
Since the contingent cone to an open subset is equal to the whole space, an open subset is a viability domain of any map. So, it is viable under any continuous map because any open subset of a finite dimensional vector space is locally compact. The Peano existence theorem is then a consequence of Theorem \[01A43\].
\[03A13\] Let $\Omega $ be an open subset of a finite dimensional vector space ${\bf R}^{n}$ and $f :\Omega \mapsto {\bf R}^{n} $ be a continuous map.
Then, for every $x_{0} \in \Omega$, there exists $T > 0$ such that differential equation (\[01A11\]) has a solution on the interval $[0,T] $ starting at $x_{0}$.
If $C \subset K \subset {\bf R}^{n}$ is a closed subset of a closed subset $K$ of a finite dimensional vector space, then $K \backslash C$ is locally compact, because for any $x \in K \backslash C$, there exists $r>0$ such that $K
\cap B (x,r) \subset {\bf R}^{n} \backslash C$. On the other hand, $$\forall \; x \in K \backslash C, \; \; T_{K \backslash C} (x) \; = \;
T_{K} (x)$$
Therefore, Theorem \[01A43\] implies
\[03A131\] Let $C \subset K \subset {\bf R}^{n}$ is a closed subset of a closed subset $K$ of a finite dimensional vector space ${\bf R}^{n}$ and $f : K \backslash C \mapsto {\bf R}^{n} $ be a continuous map. Then $K \backslash C$ is locally viable under $f$ if and only if $$\forall \; x \in K \backslash C, \; \; f (x) \; \in \; T_{K} (x)$$ or, equivalently, in terms of normal cones, $$\forall \; x \in K \backslash C, \; \forall \; p \in N_{K} (x), \; \;
\langle p,f (x) \rangle \; \leq \; 0$$
The interesting case from the viability point of view is the one when the viability subset is [*closed*]{}. This is possible because any closed subset of a finite dimensional vector space is locally compact. However, in this case, we derive from Theorem \[01A43\] a more precise statement.
\[03A43\] Let us consider a [*closed*]{} subset $K $ of a finite dimensional vector space $ {\bf R}^{n}$ and a [*continuous* ]{} map $f $ from $K $ to ${\bf R}^{n} $. Then $K$ is locally viable under $f$ if and only if $$\forall \; x \in K , \; \; f (x) \; \in \; T_{K} (x)$$ If such is the case, then for every initial state $x_{0} \in K$, there exist a positive $T $ and a [*viable*]{} solution on $[0,T[ $ to differential equation (\[01A11\]) starting at $x_{0}$ such that $$\left\{ \begin{array}{cll}
\mbox{either} & T = \infty & \\
\mbox{or} & T < \infty & \mbox{and} \;\; \limsup_{t\rightarrow T-}\|
x(t ) \| = \infty
\end{array} \right.$$
Further adequate information — a priori estimates on the growth of $f $ — allows us to exclude the case when .
This is the case for instance when $f$ is bounded on $K $, and, in particular, when $K $ is bounded.
More generally, we can take $T = \infty $ when $f $ enjoys linear growth:
\[04A43\] Let us consider a subset $K $ of a finite dimensional vector space $ {\bf
R}^{n}$ and a map $f $ from $K $ to ${\bf R}^{n} $. We assume that the map $f $ is [*continuous*]{} from $K$ to ${\bf R}^{n} $, that it has [*linear growth*]{} in the sense that $$\exists \; c > 0 \;\; \mbox{ such that} \;\;\forall \; x \in K,\;\;
\|f(x ) \| \; \leq \; c(\| x\|+1 )$$ If $$\forall \; x \in K , \; \; f (x) \; \in \; T_{K} (x)$$ then $K$ is globally viable under $f$: For every initial state $x_{0} \in
K $, there exists a [*viable solution on*]{} $[0,\infty] $ to differential equation (\[01A11\]) starting at $x_{0}$ and satisfying $$\forall \; t \geq 0, \; \; \|x (t)\| \; \leq \; \|x_{0}\| e^{ ct}
+e^{ct}-1$$
Proofs of the Viability Theorems
--------------------------------
We shall begin by proving Theorem \[01A43\]. The necessary condition follows from Lemma \[derviablem\].
For proving the sufficient condition, we begin by constructing approximate solutions by modifying the classical the Euler method to take into account the viability constraints, we then deduce from available estimates that a subsequence of these solutions converges uniformly to a limit, and finally check that this limit is a viable solution to differential equation (\[01A11\].)
1\. —
Since $K$ is locally compact, there exists $r>0$ such that the ball $B_{K}(x_{0},r) := K \cap (x_{0}+rB)$ is [*compact*]{}. When $C$ is a subset, we set $$\|C\| \; := \; \sup_{v \in C}\|v\|$$ and $$K_{0} \; := \;K \cap B(x_{0},r), \;C \;:= \; B(f(K_{0}),1), \;
T \; := \; \frac{r}{\|C\|}$$ We observe that [*$C$ is bounded*]{} since [*$K_{0}$ is compact*]{}.
Let us consider the balls $B (x, \theta (x, \varepsilon) )$ defined in Lemma \[01A142\] with $ \varepsilon := \frac{1}{m}$. The compact subset $K_{0}$ can be covered by $q$ balls $B (x_{i},
\eta (x_{i},\frac{1}{m}))$. Taking $ \theta := \min_{i=1, \ldots ,q}\eta
(x_{i},\frac{1}{m})
>0$, $J$ the smallest integer larger than or equal to $ \frac{T}{ \theta }$ and setting $h:= \frac{T}{J} \leq \theta $, we infer that $$\forall \; x \in K_{0}, \; \; d \left( f (x), \frac{ \Pi _{K}
(x+hf (x))-x}{h} \right) \; \leq \; \frac{1}{m}$$ Starting from $x_{0}$, instead of defining recursively the sequence of elements $y_{j+1}:=y_{j}+hf (y_{j})$ as in the classical Euler method, we define recursively a sequence of elements $$x_{j+1} \; := \; x_{j}+hu_{j} \; \in \;\Pi _{K} (x_{j}+hf (x_{j}))$$ where $f (x_{j})$ is replaced by $u_{j}$: $$u_{j} \; \in \; \frac{\Pi _{K} (x_{j}+hf
(x_{j}))-x_{j}}{h} \; \subset \; C \; \mbox{\rm satisfies} \; \|f
(x_{j})-u_{j}\| \; \leq \; \frac{1}{m}$$ for keeping the elements $x_{j} \in K$.
The elements $x_{j}$ belong to $K_{0}$, since they belong to $K$ and $$\|x_{j}-x_{0}\| \;\leq \;
\sum_{i=0}^{i=j-1}\|x_{i+1}-x_{i}\| \; \leq \; jh\|C\| \; \leq
\;T\|C\| = r$$ whenever $j \leq J$. We interpolate the sequence of elements $x_{j}$ at the nodes $jh$ by the piecewise linear functions $x_{m}(t)$ defined on each interval $[jh, (j+1)h[$ by $$\forall \; t \in [jh, (j+1)h[, \;\; x_{m}(t) \;:= \; x_{j} +
(t - jh)u_{j}$$ We observe that this sequence satisfies the following estimates $$\left\{ \begin{array}{ll} \label{02A142}
i) & \forall \; t \in [0,T], \;\; x_{m}(t) \in \; B (K_{0},
\varepsilon _{m}) \\
& \\
ii) & \forall \; t \in [0,T], \;\; \|x'_{m}(t)\| \leq \|C\|
\end{array} \right.$$ Let us fix $t \in [\tau_{m}^{j}, \tau_{m}^{j+1}[$. Since $\|x_{m}(t) - x_{m}(\tau_{m}^{j})\|$ = $h_{j}\|u_{j}\| \leq \|C\|/m$, and since $(x_{j},u_{j})$ belongs to $B(\mbox{\rm Graph}(f),\frac{1}{m})$ by Lemma \[01A142\], we deduce that these functions are approximate solutions in the sense that $$\left\{ \begin{array}{ll} \label{03A142}
i) & \forall \; t \in [0,T], \;\; x_{m}(t) \in
B(K_{0},\varepsilon_{m})\\
& \\
ii) & \forall \; t \in [0,T], \; (x_{m}(t),x'_{m}(t)) \in B(
\mbox{\rm Graph}(f), \varepsilon_{m})
\end{array} \right.$$ where $\varepsilon_{m} := \frac{\|C\|+1}{m}$ converges to $0$.
2\. —
Estimates (\[02A142\]) imply that for all $t \in [0,T]$, the sequence $x_{m}(t)$ remains in the compact subset $B(K_{0},1)$ and that the sequence $x_{m}(\cdot)$ is [*equicontinuous*]{}, because the derivatives $x'_{m}(\cdot)$ are bounded. We then deduce from Ascoli’s Theorem[^6] that it remains in a compact subset of the Banach space ${\cal C}(0,T;{\bf R}^{n})$, and thus, that a subsequence (again denoted) $x_{m}(\cdot)$ converges uniformly to some function $x(\cdot)$.
3\. —
Condition (\[03A142\])i) implies that $$\forall \; t \in [0,T], \;\; x(t) \in K_{0}$$ i.e., that $x(\cdot)$ is viable.
Property (\[03A142\])ii) implies that for almost every $t \in
[0,T]$, there exist $u_{m}$ and $v_{m}$ converging to $0$ such that $$x_{m}' (t) \; = \; f (x_{m} (t)-u_{m}) +v_{m}$$ We thus deduce that for almost $t \geq 0$, $x_{m}' (t)$ converges to $f(x (t))$. On the other hand, $$x_{m} (t) -x_{m} (s) \; = \int_{s}^{t}x'_{m} ( \tau )d \tau$$ implies that $x'_{m} (t)$ converges almost every where to $x' (t)$. We thus infer that $x ( \cdot )$ is a solution to the differential equation. $ \; \; \Box $
[**Proof of Theorem \[03A43\]**]{} — First, $K$ is locally compact since it is closed and the dimension of ${\bf R}^{n}$ is finite.
Second, we claim that starting from any $x_{0}$, there exists a maximal solution. Indeed, denote by ${\cal S}_{[0,T[}( x_{0})$ the set of solutions to the differential equation defined on $[0,T[$.
We introduce the set of pairs $ \{(T,x(\cdot ))\}_{T>0, \; x(\cdot ) \in {\cal
S}_{[0,T[}(x_{0})}$ on which we consider the order relation $ \prec $ defined by $$(T,x(\cdot )) \prec (S, y(\cdot )) \; \mbox{if and only if} \;
T \; \leq \; S \;\; \& \;\;\forall \; t \in [0,T[, \; \; x(t) = y(t)$$
Since every totally ordered subset has obviously a majorant, Zorn’s Lemma implies that any solution $y(\cdot ) \in {\cal S}_{[0,S[}(x_{0})$ defined on some interval $[0,S[$ can be extended to a solution $x(\cdot )
\in {\cal S}_{[0,T[}(x_{0})$ defined on a maximal interval $[0,T[$.
Third, we have to prove that if $T$ is finite, we cannot have $$c \; := \; \limsup_{t \rightarrow T-}\|x(t)\| \; < \; +\infty$$ Indeed, if $c <+\infty$, there would exist a constant $\eta \in
]0,T[$ such that $$\forall \; t \in [T-\eta,T[, \;\;\; \|x(t)\| \;\leq \; c+1$$ Since $f$ is continuous images on the compact subset $K \cap
(c+1)B$, we infer that there exists a constant $ \rho $ such that for all $ s \in [T-\eta,T[$, $ \|f(x (s))\| \leq \rho $.
Therefore, for all $\tau, \sigma \in [T-\eta,T[$, we obtain: $$\|x(\tau)-x(\sigma)\| \; \leq \; \int_{\sigma}^{\tau}\|f(x(s))\|
ds
\; \leq \; \rho |\tau - \sigma|$$ Hence the Cauchy criterion implies that $x(t)$ has a limit when $t
\rightarrow T-$. We denote by $x(T)$ this limit, which belongs to $K$ because it is closed. Equalities $$x(T_{k}) \; = \; x_{0} + \int_{0}^{T_{k}} f(x(\tau))d\tau$$ imply that by letting $k \rightarrow
\infty$, we obtain: $$x(T) \; = \; x_{0} + \int_{0}^{T}f(x(\tau))d\tau$$ This means that we can extend the solution up to $T$ and even beyond, since Theorem \[01A43\] allows us to find a viable solution starting at $x(T)$ on some interval $[T,S]$ where $S>T$. Hence $c$ cannot be finite. $ \; \; \Box$
[**Proof of Theorem \[04A43\]**]{} — Since the growth of $f$ is linear, $$\exists \; c \geq 0, \;\; \mbox{ such that} \;\; \forall \; x \in
{\bf R}^{n}, \;\; \|f(x)\| \leq c(\|x\|+1)$$ Therefore, any solution to differential equation (\[01A11\]) satisfies the estimate: $$\|x'(t)\| \; \leq \; c(\|x(t)\|+1)$$ The function $t \rightarrow \|x(t)\|$ being locally Lipschitz, it is almost everywhere differentiable. Therefore, for any $t$ where $x(t)$ is different from $0$ and differentiable, we have $$\frac{d}{dt}\|x(t)\| \; = \; \left\langle
\frac{x(t)}{\|x(t)\|},x'(t) \right\rangle \; \leq \; \|x'(t)\|$$ These two inequalities imply the estimates: $$\label{xA445}
\|x(t)\| \; \leq \; ( \|x_{0}\|+1)e^{ct} \; \; \& \; \;
\|x'(t)\| \; \leq \; c( \|x_{0}\|+1)e^{ct}$$ Hence, for any $T > 0$, we infer that $$\limsup_{t \rightarrow
T-}\|x(t)\| \; < \; + \infty$$ Theorem \[03A43\] implies that we can extend the solution on the interval $[0,\infty[$. $ \; \; \Box$
The Solution Map
----------------
\[02A44\] We denote by ${\cal S}_{f}(x_{0})$ \[01A44\] the set of solutions to differential equation (\[01A11\]) and call the set-valued map ${\cal S}_{f}: x \leadsto {\cal S}_{f} (x)$ the [*solution map*]{} of $f$ (or of differential inclusion (\[01A11\]).)
\[03A44\] Let us consider a finite dimensional vector space $ {\bf R}^{n}$ and a [*continuous map $f :{\bf R}^{n} \mapsto {\bf R}^{n}$*]{} with linear growth. Then the graph of the restriction of ${\cal S}_{f}|_{L}$ to any compact subset $L$ is compact in ${\bf R}^{n} \times {\cal C}(0, \infty ;{\bf
R}^{n})$ where the space ${\cal C}(0,\infty ;{\bf R}^{n})$ is supplied with the compact convergence topology.
[**Proof**]{} — We shall show that the graph of the restriction ${\cal S}_{f}|_{L}$ of the solution map ${\cal S}_{f}$ to a compact subset $ L $ is compact. Let us choose a sequence of elements $(x_{0_{n}},x_{n}(\cdot))$ of the graph of the solution map ${\cal S}_{f}$. They satisfy: $$x'_{n}(t) =f (x_{n}(t)) \; \; \& \; \; x_{n}(0) =
x_{0_{n}} \in L$$ A subsequence (again denoted) $x_{0_{n}}$ converges to some $x_{0}
\in L$ because $L$ is compact. By Theorem \[04A43\], $$\forall \; n \geq 0, \; \; \|x_{n}(t)\| \; \leq \; (
\|x_{0_{n}}\|+1) e^{ct} \; \; \& \; \; \|x'_{n}(t)\| \; \leq \; c(
\|x_{0_{n}}\|+1) e^{ct}$$ Thus, by Ascoli’s Theorem, the sequence $x_{n}(\cdot)$ is relatively compact in ${\cal C}(0,\infty;{\bf R}^{n})$. We deduce from this, that a subsequence (again denoted) $x_{n} ( \cdot)$ converges to a continuous function $x ( \cdot )$ uniformly on compact intervals. Therefore, passing to the limit in equalities $$x_{m} (t) \; = \; x_{0_{n}} + \int_{0}^{t}f (x_{m} ( \tau ))d \tau$$ we deduce that $x ( \cdot )$ is a solution to the differential equation starting at $x_{0}$. $ \; \; \Box$
Uniqueness Criteria
-------------------
Whenever there exists a unique solution to differential equation $x'=f (x)$ starting from any initial state $x_{0}$, then viability and invariance properties of a closed subset $K$ are naturally equivalent. This is one of the motivations for providing uniqueness criteria.
We shall say that a map $f:{\bf R}^{n} \mapsto {\bf R}^{n}$ is [*monotone*]{} if there exists there exists $ \mu \in {\bf R}$ such that $$\label{monassg}
\langle f(x_{1}) - f (x_{2}) ,x_{1}-x_{2} \rangle \; \leq \; -\mu
\|x_{1}-x_{2}\|^{2}$$
The interesting case is obtained when $ \mu >0$. When $f$ is Lipschitz with constant $ \lambda $, then it is monotone with $ \mu =-
\lambda $.
\[05A13m\] Let us consider a subset $K $ of a finite dimensional vector space $ {\bf
R}^{n}$ and a continuous and monotone map $f $ from ${\bf R}^{n}$ to ${\bf R}^{n} $. The solution to differential equation $x'=f (x)$ starting from $x_{0}$ is unique. If $x_{i}
( \cdot )$ are two solutions to the differential equation $x'=f (x)$,then $$\|x_{1} (t)-x_{2} (t)\| \; \leq \; e^{- \mu t} \|x_{1} (0)-x_{2} (0)\|$$
[**Proof**]{} — Indeed, integrating the two sides of inequality $$\frac{d}{dt} \|x_{1} (t)-x_{2} (t)\|^{2} \; = \; 2 \langle f(x _{1}
(t))-f (x_{2} (t)),x_{1} (t)-x_{2} (t) \rangle \; \leq \; - 2 \mu \|x_{1}
(t)-x_{2} (t)\|^{2}$$ yields $$\|x_{1} (t)-x_{2} (t)\|^{2} \; \leq \; e^{-2 \mu t} \|x_{1} (0)-x_{2}
(0)\|^{2}$$
Backward Viability
------------------
The subset $K$ is [locally backward viable]{} under $ f$ if for any $x\in K $, for any $t>0$, there exist $s \in [0,t[$ and a solution $x (
\cdot )$ to differential equation (\[01A11\]) such that $$\forall \; \tau \in [s,t], \; x( \tau ) \in K \; \& \; x (t)=x$$ It is (globally) backward viable if we can take $s=0$ in the above statement, and locally (resp. globally) backward invariant if for any $x\in K $, for any $t>0$, for all solutions $x ( \cdot )$ to differential equation (\[01A11\]), there exist $s \in [0,t[$ such that $$\forall \; \tau \in [s,t], \; x( \tau ) \in K \; \& \; x (t)=x$$
We now compare the invariance of a subset and the backward invariance of its complement:
\[bacwinvcomp\] A subset $K$ is invariant under a map $ f$ if and only if its complement $K^{c}:= {\bf R}^{n} \backslash K$ is backward invariant under $ f $.
[**Proof**]{} — To say that $K$ is not invariant under $f$ amounts to saying that there exist a solution $x ( \cdot )$ to differential equation (\[01A11\]) and $T>0$ such that $$x (0) \; \in \; K \; \& \; x (T) \; \in \; {\bf R}^{n}
\backslash K$$ and to say that ${\bf R}^{n} \backslash K$ is not backward invariant amounts to saying that there exist a solution $y( \cdot )$ to differential equation (\[01A11\]), $T >0$ and $S \in [0,T[$ such that $$y (S) \; \in \; K \; \& \; y (T) \; \in \; {\bf R}^{n}
\backslash K$$ It is obvious that the first statement implies the second one by taking $y ( \cdot )=x ( \cdot )$ and $S=0$. Conversely, the second statement implies the first one by taking $x (t):= y (t+S)$ and replacing $T$ by $T-S >0$ since $x (0)=y (S)$ belongs to $K$ and $x (T-S)=y (T)$ belongs to ${\bf R}^{n} \backslash K$. $\; \; \Box$
It is also useful to relate backward viability and invariance under $ f$ to viability and invariance under $ - f$:
\[backinvab\] Let us assume that $f$ is continuous with linear growth. Then $K$ is locally backward viable (resp. invariant) under $ f$ if and only if $f$ is locally viable (resp. invariant) under $ - f $.
[**Proof**]{} — Let us check this statement for local viability. Assume that $K$ is locally backward viable and infer that $K$ is locally invariant under $ - f$. Indeed, let $x \in K$. Then, for any $T >0$, there exists $S \in [0,T[$ and a solution $ x( \cdot ) $ to differential equation (\[01A11\]) viable in $K$ on the interval $[S,T]$ and satisfying $x (T)=x$. Let $y (
\cdot ) $ be a solution to the differential equation $y'=-f (y)$ starting at $y (0)=x (S)$. Then the function $ z ( \cdot )$ defined by $$z (t) \; = \left\{ \begin{array}{lll}
x (T-t) & \mbox{\rm if} & t \in [0,T-S] \\
y (t+T-S) & \mbox{\rm if} & t \geq T-S
\end{array} \right.$$ is a solution to the differential equation $z'=-f (z)$ starting at $z (0)=x (T)=x$ and viable in $K$ on the interval $[0,T-S]$.
Conversely, assume that $K$ is locally viable under $ -f
$ and check that $K$ is locally backward invariant. Let $x \in
K$, $T >0$ and one solution $x ( \cdot )$ to differential equation $x'=-f (x)$ viable in $K$ on $[0,R]$ where $R>0$. Let be any solution $y ( \cdot )$ to $y' (t)=f (y (t))$ starting at $x
$ and set $$z (t) \; = \left\{ \begin{array}{lll}
x (T-t) & \mbox{\rm if} & t \in [0,T] \\
y (t-T) & \mbox{\rm if} & t \geq T
\end{array} \right.$$ Hence $ z ( \cdot )$ to differential equation (\[01A11\]) satisfying $x (T) =x \in K$ and viable in $K$ on the interval $[S,T]$ where $S := \max (T-R,0)$. $\; \; \Box$
Time-Dependent Differential Equations
-------------------------------------
\[04A43t\] Let us consider a subset $K $ of a finite dimensional vector space $ {\bf
R}^{n}$ and a map $f $ from $ {\bf R}_{+} \times K $ to ${\bf R}^{n} $. We assume that the map $f $ is [*continuous*]{} from $ {\bf R}_{+} \times K$ to ${\bf R}^{n} $, that it has [*uniform linear growth*]{} in the sense that $$\exists \; c > 0 \;\; \mbox{ such that} \;\;\forall \; t \geq 0, \; x
\in K,\;\; \|f(t,x ) \| \; \leq \; c(\| x\|+1 )$$ If $$\forall \; t \geq 0, \forall \; x \in K , \; \; f (t,x) \; \in \; T_{K}
(x)$$ then $K$ is globally viable under $f$: for every initial state $x_{0} \in
K $, there exists a [*viable solution on*]{} $[0,\infty] $ to differential equation $$x' (t) \; := \; f (t,x (t))$$ starting at $x_{0}$ and satisfying $$\forall \; t \geq 0, \; \; \|x (t)\| \; \leq \; \|x_{0}\| e^{ ct}
+e^{ct}-1$$ Assume moreover that $f$ is [*uniformly monotone*]{} in the sense that[^7] there exists $ \mu
\in {\bf R}$ such that $$\label{monassgt}
\langle f(t,x_{1}) - f (t,x_{2}) ,x_{1}-x_{2} \rangle \; \leq \; -\mu
\|x_{1}-x_{2}\|^{2}$$ If $x_{1}$ and $x_{2}$ are two initial states, then the solutions $x_{i} (
\cdot )$ starting from $x_{i}$, $ (i=1,2)$, satisfy $$\|x_{1} (t)-x_{2} (t)\| \; \leq \; e^{- \mu t} \|x_{1} (0)-x_{2} (0)\|$$
[**Proof**]{} — We deduce the first statement from the standard trick which amounts to observing that a solution $x ( \cdot )$ to the time-dependent differential equation $x'=f (t,x)$ starting at time $0$ from the initial state $x_{0}$ if and only if $ ( \tau ( \cdot ),x ( \cdot ))$ is a solution to the system of differential equations $$\left\{ \begin{array}{ll}
i) & \tau ' (t) \; = \; 1 \\
ii) & x' (t) \; = \; f ( \tau (t),x (t))
\end{array} \right.$$ starting at time $0$ from $ (0,x_{0})$. The solution $x ( \cdot )$ is viable in $K$ under $f$ if and only if $ ( \tau ( \cdot ),x ( \cdot ))$ is viable in $ {\bf R}_{+} \times K$. By the Nagumo Theorem, this is equivalent to require that $ (1,f (t,x)) \in T_{ {\bf R}_{+} \times K}
(t,x)$, i.e., that $f (t,x)$ belongs to $T_{K} (x)$.
The proof of the second statement is same than the one of Theorem \[05A13m\]. $\; \; \Box$
Viability Kernels and Capture Basins
====================================
When a closed subset $K$ is not viable under a dynamical economy, then two questions arise naturally:
1. find solutions starting from $K$ which [*remain viable in $K$ as long as possible, hopefully, forever*]{},
2. and starting outside of $K$, find solutions which [*return to $K$ as soon as possible, hopefully, in finite time*]{}
Studying these questions leads to the concepts of
1. [*viability kernel*]{} of a subset $K$ under a dynamical system, as the set of elements of $K$ from which starts a solution viable in $K$,
2. [*capture basin*]{} of $C$, which is the set of points of $K$ from which a solution reaches $C$ in finite time,
3. when $C \subset K$, [*viable-capture basin*]{}, which is the subset of points of $K$ from which starts a solution reaching $C$ before leaving $K$.
We shall provide characterizations of these concepts and derive their properties we shall use later for solving some Hamilton-Jacobi equations and boundary-value problems for systems of first-order partial differential equations. For instance, if $K$ is backward invariant and a repeller, the capture basin of $C$ is the unique closed subset $D$ satisfying $$\left\{ \begin{array}{ll}
i) & C \; \subset \; D \; \subset \; K\\
ii) & \forall \; x \in D \backslash C, \; \;f(x) \; \in \;T_{D} (x)
\\
iii) & \forall \; x \in D, \; \; -f(x) \; \in \; T_{D} (x)\\
\end{array} \right.$$ or, equivalently, by duality, the “normal conditions” $$\left\{ \begin{array}{ll}
i) & C \; \subset \; D \; \subset \; K\\
ii) & \forall \; x \in D \backslash C, \; \forall \; p \in N_{D} (x),
\; \; \langle p,f (x) \rangle \; = \; 0 \\
iii) & \forall \; x \in D, \; \forall \; p \in N_{D} (x), \; \;
\langle p,f (x) \rangle \; \geq \; 0\\
\end{array} \right.$$
Reachable, Viability and Capture Tubes
--------------------------------------
Let $f:{\bf R}^{n} \mapsto {\bf R}^{n}$ be a map and $C \subset {\bf
R}^{n}$ be a subset. The [reachable map]{} $\vartheta_{f} ( \cdot ,x)$ is defined by $$\forall \; x \in {\bf R}^{n}, \; \; \vartheta_{f} (t,x) \; := \; \left\{
x(t)
\right\}_{ x(\cdot ) \in{\cal
S}_{f}(x)}$$ We associate with it the [reachable tube]{} $ t \leadsto
\vartheta_{f} ( t,C)$ defined by $$\vartheta_{f} (t,C) \; := \; \left\{ x(t) \right\}_{ x(\cdot ) \in{\cal
S}_{f}(C)}$$
We derive the following properties:
The reachable map $t\leadsto \vartheta_{f}(t,x)$ enjoys the semi-group property: $\forall t,s\geq 0, \; \;\vartheta_{f}(t+s,x)=\vartheta_{f}
(t,\vartheta_{f} (s,x))$.
Furthermore, $$( \vartheta _{f } (t, \cdot ))^{-1} \; := \; \vartheta _{-f } (t, \cdot )$$ Therefore, the subset $ \vartheta _{-f} (t,C)$ is the subset of elements $x
\in {\bf R}^{n}$ which reach the subset at the prescribed time $t$.
If $f$ is continuous with linear growth and $K \subset {\bf R}^{n}$ is closed, the graph of the reachable map $ t \leadsto \vartheta_{f} ( t,K)$ is closed.
[**Proof**]{} — The semi-group property is obvious. Let us prove the second one: If $ y \in \vartheta_{f } (t,x)$, there exists a solution $x ( \cdot )$ to the differential equation $x'= f(x)$ starting at $x$ such that $y = x (t)$. We set $y (s) := x (t-s)$ if $ s \in
[0,t]$ and we choose any solution $y ( \cdot )$ to the differential equation $y' \in - f(y)$ starting at $x$ at time $t$ for $s \geq t$. Then such a function $y ( \cdot )$ is a solution to the differential equation $y' \in - f(y)$ starting at $y$ and satisfying $y (t)=x$. This shows that $x \in \vartheta _{-f } (t,y)$.
The last statement is a consequence of Theorem \[04A43\]. $\; \; \Box$
Let $f:{\bf R}^{n} \mapsto {\bf R}^{n}$ be a map and $C \subset
{\bf R}^{n}$ be any subset.
1. The subset $\mbox{\rm Viab}_{f}(C,T)$ of initial states $x_{0} \in C$ such that one solution $x(\cdot )$ to differential equation $x'= f(x)$ starting at $x_{0}$ is viable in $C$ for all $t \in [0,T]$ is called the [*$T$-viability kernel*]{} and the subset $\mbox{\rm Viab}_{f}(C) := \mbox{\rm Viab}_{f}(C, \infty )$ is called the [viability kernel]{} of $C$ under $f$. A subset $C$ is a repeller if its viability kernel is empty.
2. The subset $ \mbox{\rm Capt}_{f}(C,T)$ of initial states $x_{0} \in {\bf R}^{n}$ such that $C$ is reached before $T$ by one solution $x(\cdot
)$ to differential equation $x'= f(x)$ starting at $x_{0}$ is called the [*$T$-capture basin*]{} and $$\mbox{\rm Capt}_{f}(C) \; := \; \bigcup_{T > 0}^{} \mbox{\rm
Capt}_{f}(C,T)$$ is said to be the [capture basin]{} of $C$.
3. When $C \subset K$, the [viable-capture basin]{} $ \mbox{\rm Capt}_{f}^{K}(C)$ of $C$ in $K$ by $f$ is the set of initial states $x_{0} \in
{\bf R}^{n}$ from which starts at least one solution to the $x'= f (x)$ viable in $K$ until it reaches $C$ in finite time.
[**Remarks**]{} — We observe that if $T_{1} \leq T_{2}$,
$$\left\{ \begin{array}{ccccc}
\mbox{\rm Viab}_{f}(C) & \subset & \mbox{\rm Viab}_{f}(C,T_{2}) &
\subset & \mbox{\rm Viab}_{f}(C,T_{1}) \\
\cap & & \cap & & \cap \\
\mbox{\rm Capt}_{f}(C,0) & = & C & = & \mbox{\rm Viab}_{f}(C,0) \\
\cap & & \cap & & \cap \\
\mbox{\rm Capt}_{f}(C) & \supset & \mbox{\rm Capt}_{f}(C,T_{2}) &
\supset & \mbox{\rm Capt}_{f}(C,T_{1}) \\
\end{array} \right.$$
One can write $$\mbox{\rm Capt}_{f}(C,T) \; = \; \bigcup_{t \in [0,T]}^{} \vartheta_{-f}
(t,C)$$
We point out the following obvious properties:
Let $C \subset K$ be closed subsets. The capture basin $\mbox{\rm
Capt}_{f}(C)$ is smallest backward invariant containing $C$ and $\mbox{\rm Capt}_{f}(C) \backslash C$ is locally viable. The capture basin of any union of subsets $C_{i} \; (i \in I)$ is the union of the capture basins of the $C_{i}$. When $C \subset K$ where $K$ is assumed to be backward invariant, then the viable-capture basin satisfies $$\mbox{\rm Capt}_{f}^{K}(C) \; = \; \mbox{\rm Capt}_{f}(C) \; \subset \; K$$
The complement $\mbox{\rm Capt}_{f}^{K}(C) \backslash C$ of $C$ in the viable-capture basin $\mbox{\rm Capt}_{f}^{K}(C)$ is locally viable.
[**Proof**]{} — Indeed, whenever $K$ is backward invariant, each backward reachable set $\vartheta_{-f} (t,C) $ is contained in $K$, so that $$\mbox{\rm Capt}_{f}(C) \; = \; \bigcup_{t \geq 0}^{} \vartheta_{-f} (t,C)
\; \subset \; K$$ Since the intersection of backward invariant subsets is backward invariant, the capture basin is contained in the smallest backward invariant subset containing $C$. The semi group property implies that the capture basin, which is the union of the backward reachable subsets, is backward invariant.
When $C \subset K$ where $K$ is assumed to be backward invariant, then the capture basin $$\bigcup_{t \geq 0}^{} \vartheta_{-f} (t,C)$$ is contained in $K$.
If $x$ belongs to $\mbox{\rm Capt}_{f}^{K}(C) \backslash C$, then there exists a solution to the differential equation $x'=f (x)$ starting from $x$ which reaches $C$ before leaving $K$, and thus, which is viable in $\mbox{\rm Capt}_{f}^{K}(C) \backslash C$ on some nonempty interval. $\; \;
\Box$
\[largestvaibinv\] The viability kernel $\mbox{\rm Viab}_{f}(C)$ of $C$ under $f$ is the largest subset of $C$ viable under $f$.
Furthermore, $C \backslash \mbox{\rm Viab}_{f}(C)$ is a repeller and $\mbox{\rm Viab}_{f}(C) \backslash \partial C$ is locally backward invariant.
[**Proof**]{} — Every subset $L \subset C$ viable under $f$ is obviously contained in the viability kernel $\mbox{\rm Viab}_{f}(C)$ of $C$ under $f$.
On the other hand, if $x(\cdot)$ is a solution to the differential equation $x'= f(x)$ viable in $C$, then for all $t >0$, the function $y(\cdot)$ defined by $y(\tau) := x(t+ \tau)$ is also a solution to the differential equation, starting at $x(t)$, viable in $C$.
Therefore, for any element $x_{0} \in \mbox{\rm Viab}_{f}(C)$, there exists a viable solution $x(\cdot)$ to the differential equation starting from $x_{0}$, and thus, for all $t \geq 0$, $ x(t) \in \mbox{\rm
Viab}_{f}(C)$, so that it is viable under $f$.
Let us assume that $\mbox{\rm Viab}_{f}(C) \backslash \partial C$ is not locally backward invariant: There would exist $x \in \mbox{\rm
Viab}_{f}(C) \backslash \partial C$, $T >0$ and a solution $x ( \cdot )$ to the differential equation $x'=f (x)$ satisfying $x (T)=x$ such that for all $S <T$, there exist $S ' \in [S,T]$ such that $x (S')$ belong to the union of $ \partial C$ and of the complement of the viability kernel $\mbox{\rm Viab}_{f}(C)$. Since $x (T)=x$ does not belong to the boundary $
\partial C$ of $C$, we know that for $S$ close enough to $T$, $x ([S,T])
\cap \partial C = \emptyset $. Hence $x (S')$ does not belong to the boundary of $C$, so that it belongs to complement of the viability kernel $\mbox{\rm Viab}_{f}(C)$, and thus, the solution $x ( \cdot )$ starting from $x (S') \in \mbox{\rm Viab}_{f}(C)$ at time $S' $ should leave $C$ in finite time, a contradiction. $\; \; \Box$
Hitting and Exit Times
----------------------
We say that the [hitting functional]{} (or [minimal time functional]{}) associating with $x (\cdot) $ its [hitting time]{} $\omega_{C}(x(\cdot ))$ is defined by $$\omega_{C}(x(\cdot )) \; := \; \inf \left\{ t \in [0,+\infty [ \; | \; x(t)
\in C \right\}$$ and the function $\omega_{C}^{f ^{\flat }}: C \mapsto {\bf R}_{+} \cup
\{+\infty \}$ defined by $$\omega_{C}^{f ^{\flat }}(x) \; := \; \inf_{ x(\cdot ) \in {\cal
S}_{f}(x)}\omega_{C}(x(\cdot ))$$ is called the (lower) [*hitting function*]{} or [*minimal time function*]{}. In the same way, the [exit functional]{} is defined by $$\tau _{C} (x ( \cdot )) \; := \; \omega_{{\bf R}^{n} \backslash C}(x(\cdot
)) \; :=
\; \inf \left\{ t \in [0,+\infty [ \; | \; x(t) \notin C \right\}$$ and the function $ \tau _{C}^{f ^{\sharp }}:C \mapsto {\bf R}_{+} \cup
\{+\infty \}$ defined by $$\tau _{C}^{f ^{\sharp }}(x) \; := \; \sup _{ x(\cdot ) \in {\cal
S}_{f}(x)}\tau _{C}(x(\cdot ))$$ is called the (upper) [*exit function*]{}.
Let $C \subset K \subset {\bf R}^{n}$ be two closed subsets. We also introduce the function $$\gamma _{ (K,C)}^{f ^{\flat }} (x) \; := \; \inf_{ x ( \cdot ) \in
{\cal S}_{f} (x)} ( \omega _{C} (x ( \cdot ))- \tau _{K} (x ( \cdot
)))$$
We observe that if $C=K$, $\omega _{C} (x ( \cdot )) =0$ for all solutions $x ( \cdot ) \in {\cal S}_{f} (x)$ starting from $K$ and thus, that $$\gamma _{ (K,K)}^{f ^{\flat }} (x) \; = \; - \tau _{K}^{f ^{\sharp
}} (x)$$
To say that $K \backslash C$ is a repeller amounts to saying that for every solution $ x ( \cdot ) \in {\cal S}_{f} (x)$ starting from $x
\in K \backslash C$, $\min ( \omega _{C} (x ( \cdot ), \tau _{K} (x (
\cdot )))) < + \infty $ and to say that $K$ is a repeller under $f$ amounts to saying that the exit function $ \tau _{K}^{f ^{\sharp }}$ is finite on $K$.
\[Picard02\] When $K$ is closed, the exit functional $ \tau _{K}$ is upper semicontinuous and the hitting functional $\omega_{K}$ is lower semicontinuous when ${\cal C}(0,\infty ;{\bf R}^{n})$ is supplied with the compact convergence topology.
[**Proof**]{} — For proving that the exit functional $ \tau_{K}$ is upper semicontinuous, we shall check that the subsets $ \{x ( \cdot ) \; | \; \tau _{K} ( x(\cdot
)) < T \}$ are open for the pointwise convergence, and thus, the compact convergence. Let $x_{0} ( \cdot )$ belong to such a set when it is not empty. Since $x_{0}(T)$ does not belong $K$ which is closed, there exist $
\alpha >0$ such that $B (x_{0}(T), \alpha ) \cap K = \emptyset $. Then the set of continuous functions $x ( \cdot )$ such that $x (T) \in
\stackrel{\circ}{B} (x_{0}(T), \alpha )$ is open and satisfy $\tau _{K} (
x(\cdot )) < T$.
For proving that the hitting functional is lower semicontinuous, we shall check that the subsets $ \{x ( \cdot ) \; | \; \omega _{K} ( x(\cdot )) \leq T
\}$ are closed. Let $x_{n} ( \cdot )$ satisfying $\omega _{K} ( x_{n}(\cdot
))
\leq T $ converge to $x ( \cdot )$ uniformly on compact intervals. For any $ \varepsilon >0$, one can find $t_{n} \leq T+ \varepsilon $ such that $x_{n} (t_{n})$ belongs to $K$. A subsequence (again denoted by) $t_{n}$ converges to some $t$. Since $x_{n} ( \cdot )$ converges uniformly to $x (
\cdot )$ on $[t - \varepsilon ,T + \varepsilon ]$, we deduce that $x (t)$ is the limit of $x_{n} (t_{n}) \in K$ and thus, that $x (t)$ belongs to the closed subset $K$ and thus, that $ \omega _{K} (x ( \cdot )) \leq T +
\varepsilon $. Letting $ \varepsilon $ converge to $0$, we infer that $
\omega _{K} (x ( \cdot )) \leq T$. $\; \; \Box$
We deduce the following properties of these hitting and exit functions:
\[Picard03se\] Let $f:{\bf R}^{n} \mapsto {\bf R}^{n}$ be a continuous map with linear growth and $C \subset
K \subset {\bf R}^{n}$ be two closed subsets. Assume that $K$ is a repeller.
The function $\gamma _{ (K,C)}^{f ^{\flat }}$ is lower semicontinuous and for any $x \in K$, there exists a solution $x_{ (K,C)} ( \cdot )
\in {\cal S}_{f} (x)$ satisfying $$\gamma _{ (K,C)}^{f ^{\flat }} (x) \; := \; \omega _{C} (x_{
(K,C)}( \cdot ))- \tau _{K} (x _{ (K,C)}( \cdot )))$$
In the same way, the hitting function $\omega_{K}^{f ^{\flat }}$ is lower semicontinuous and the exit function $\tau _{K}^{f ^{\sharp }}$ is upper semicontinuous. Furthermore, for any $x \in \mbox{\rm Dom}(\omega_{K}^{f
^{\flat }})$, there exists one solution $x ^{\flat } ( \cdot ) \in {\cal
S}_{f} (x)$ which hits $K$ as soon as possible $$\omega_{K}^{f ^{\flat }} (x) \; = \; \omega_{K} (x ^{\flat } ( \cdot ))$$ and for any $x \in \mbox{\rm Dom}(\tau _{K}^{f ^{\sharp }})$, there exists one solution $x ^{\sharp} ( \cdot ) \in {\cal S}_{f} (x)$ which remains viable in $K$ as long as possible: $$\tau _{K}^{f ^{\sharp }} (x) \; = \; \tau _{K} (x ^{\sharp} ( \cdot ))$$
[**Proof**]{} — Since the function $ x ( \cdot ) \mapsto \omega _{C} (x ( \cdot ))-
\tau _{K} (x ( \cdot ))$ is lower semicontinuous on $ {\cal C} (0,
\infty ,{\bf R}^{n})$ supplied with the compact convergence by Theorem \[Picard02\], we deduce first from Theorem \[03A44\] that the infimum is reached by a solution $x_{ (K,C)} ( \cdot ) \in
{\cal S}_{f} (x)$ because the set $ {\cal S}_{f} (x)$ is compact and second, that this function $\gamma _{ (K,C)}^{f ^{\flat }}$ is lower semicontinuous, by checking that the subsets $ \{x \in K \; | \;
\gamma _{ (K,C)}^{f ^{\flat }} (x) \leq T\}$ are closed. Indeed, let us consider a sequence of elements $x_{n}$ of such a subset converging to $x$. There exist solutions $x_{n} ( \cdot ) \in {\cal S}_{f} (x)$ such that $$\gamma _{ (K,C)} (x_{n} ( \cdot )) \; \leq \; \gamma _{ (K,C)}^{f
^{\flat }} (x_{n}) + \frac{1}{n} \; \leq \; T+ \frac{1}{n}$$ On the other hand, since $x_{n}$ belongs to the compact ball $B (x,
1)$, Theorem \[03A44\] implies that a subsequence (again denoted by) $x_{n} ( \cdot )$ converges to some solution $x ( \cdot ) \in {\cal S}_{f}
(x)$ uniformly on compact intervals. Since the functional $\gamma _{
(K,C)}$ is lower semicontinuous, we infer that $$\gamma _{ (K,C)} ^{f ^{\flat }} (x) \; \leq \; \gamma _{ (K,C)}(x (
\cdot )) \; \leq \; \liminf_{n \rightarrow + \infty } \gamma _{
(K,C)} (x_{n} ( \cdot )) \; \leq \; T$$
In particular, taking $C:=K$, we observe that $ \gamma _{ (K,K)}^{f
^{\flat }} =- \tau _{K}^{f ^{\sharp }}$, and thus, we deduce the upper semicontinuity of the exit function. The same proof shows that the hitting function $ \omega _{C}^{f
^{\flat }}$ is lower semicontinuous. $\; \; \Box$
Viability kernels and capture basins can be characterized in terms of exit and hitting functionals:
\[characviabkern\] If $f:{\bf R}^{n} \mapsto {\bf R}^{n}$ is continuous with linear growth and $C \subset K
\subset {\bf R}^{n}$ are closed subsets, then $$\left\{ \begin{array}{ll}
\mbox{\rm Capt}_{f}(K,T) & = \; \left\{ x \in {\bf R}^{n} \; | \; \omega
_{K}^{f
^{\flat }}(x) \; \leq \; T\right\} \; \; \& \; \;\mbox{\rm Capt}_{f}(K,T)
\; = \; \mbox{\rm Dom}(\omega _{K}^{f ^{\flat }})\\
& \\
\mbox{\rm Capt}_{f}^{K}(C,T) & = \; \left\{ x \in {\bf R}^{n} \; | \;
\gamma _{
(K,C)}^{f ^{\flat }}(x) \; \leq \; 0\right\} \\
& \\
\mbox{\rm Viab}_{f}(K,T) & := \; \left\{ x \in {\bf R}^{n}\; | \;\tau
_{K} ^{\sharp
}(x) \; \geq \; T\right\} \\
\end{array} \right.$$ In particular, the $T$-viability kernels $ \mbox{\rm Viab}_{f}(K,T)$ of a closed subset $K \subset {\bf R}^{n}$, the $T$-capture basins of $K$ under $f$ and the viable-capture basin $\mbox{\rm Capt}_{f}^{K}(C,T) $ are closed.
[**Proof**]{} — The subset of initial states $x \in {\bf R}^{n} $ such that $K$ is reached before $T$ by a solution $x(\cdot )$ to the differential equation $x'=f (x)$ starting at $x$ is obviously contained in the subset $\left\{ x \in {\bf
R}^{n} \; |
\; \omega_{K} ^{f ^{\flat }}(x) \; \leq \; T\right\}$.
Conversely, consider an element $x $ satisfying $ \omega_{K} ^{f ^{\flat
}}(x) \; \leq \; T$. Hence the solution $x ^{\flat}( \cdot ) \in {\cal
S}_{f} (x)$ such that $ \omega _{K} (x ^{\flat } ) = \omega _{K}^{f ^{\flat
}} (x) \leq T$ belongs to the $T$-capture basin.
Now, to say that a solution $x ( \cdot ) \in {\cal S}_{f} (x)$ is viable in $K$ until it reaches the target $C$ means that $ \omega
_{C} (x ( \cdot )) \leq \tau _{K} (x ( \cdot ))$. Therefore, $x$ belongs to the viable-capture basin $ \mbox{\rm Capt}_{f}^{K} (C)$ if and only if $\gamma _{ (K,C)}^{f ^{\flat }} (x) \leq 0$.
The proof of the characterization of the $T$-viability kernel $\mbox{\rm Viab}_{f}(K,T)$ as upper sections of the exit time function $\tau _{K} ^{\sharp }$ is analogous.
The topological properties then follow from the semicontinuity properties of the above functions stated in Proposition \[Picard03se\]. $\; \; \Box$
Characterization of the Viability Kernel
----------------------------------------
We deduce at once the following consequence:
\[viablbascharthm\] Let $f:{\bf R}^{n} \mapsto {\bf R}^{n}$ be a continuous map with linear growth and $K \subset
{\bf R}^{n}$ be a closed subset. Then the viability kernel is the largest closed subset $D \subset K$ viable under $f$, or, equivalently, the largest closed subset of $K$ satisfying $$\left\{ \begin{array}{ll}
i) & D \; \subset \; K\\
ii) & \forall \; x \in D, \; \; f (x) \; \in \; T_{D} (x)
\end{array} \right.$$ or, equivalently, in terms of normal cones, the largest closed subset of $K$ satisfying $$\left\{ \begin{array}{ll}
i) & D \; \subset \; K\\
ii) & \forall \; x \in D, \; \forall \; p \in N_{D} (x)\; \langle p,f
(x) \rangle \; \leq \; 0
\end{array} \right.$$ Furthermore, the viability kernel satisfies the following properties $$\forall \; x \in \mbox{\rm Viab}_{f} (K) \backslash \partial K, \; \;
f (x) \; \in \; T_{\mbox{\rm Viab}_{f} (K)} (x) \cap - T_{\mbox{\rm
Viab}_{f} (K)} (x)$$ or, equivalently, in terms of normal cones, $$\forall \; x \in \mbox{\rm Viab}_{f} (K) \backslash \partial K, \;
\forall \; p \in N_{\mbox{\rm Viab}_{f} (K)} (x), \; \; \langle p,f (x)
\rangle \; = \; 0$$
[**Proof**]{} — The first property follows from the Nagumo Theorem characterizing viable subsets in terms of tangential and/or normal conditions. The second property translates the fact that $\mbox{\rm Viab}_{f} (K) \backslash
\partial K$ is locally backward invariant, and thus, locally backward viable. $\; \; \Box$
\[absfinitime\] Let $f:{\bf R}^{n} \mapsto {\bf R}^{n}$ be a continuous map with linear growth and $K \subset
{\bf R}^{n}$ be a closed subset. If $M\subset {\bf R}^{n} \backslash
\mbox{\rm Viab}_{f}(K)$ is compact, then, for every $x \in M$ and every solution $x(\cdot ) \in
{\cal S}_{f}(x)$, there exists $t \in [0, \sup_{x \in M} \tau _{K}^{f
^{\sharp }} (x)]$ such that $x(t) \notin K$.
[**Proof**]{} — Indeed, $M$ being compact and the exit function being upper semicontinuous, then $\sup_{x \in M} \tau _{K}^{f ^{\sharp }} (x)$ is finite because, for each $x \in M$, $ \tau _{K}^{f ^{\sharp }} (x)$ is finite. $\; \; \Box$
In particular,
Let us assume that $K$ is a compact and that $f:{\bf R}^{n}
\mapsto {\bf R}^{n}$ is continuous with linear growth. Then either the viability kernel of $K$ is not empty or $K$ is a repeller, and in this case, $\overline{T} := \sup_{x \in K} \tau _{K}^{f ^{\sharp }} (x)$ is finite and satisfies $$\mbox{\rm Viab}_{f}(K,\overline{T}) \; \ne \; \emptyset \; \; \&
\; \; \forall \; T>\overline{T}, \; \; \mbox{\rm Viab}_{f}(K,T) \; = \;
\emptyset$$
[**Proof**]{} — When $K$ is a repeller, the exit function is finite. Being compact, $\overline{T}:=\sup_{x \in K} \tau _{K}^{f ^{\sharp }} (x)$ is thus finite and achieves its maximum at some $ \bar{x}$. By Theorem \[Picard03se\], there exists a solution $
\bar{x} ( \cdot ) \in {\cal S}_{f} ( \bar{x})$ such that $ \tau _{K} (
\bar{x} ( \cdot )) = \tau _{K}^{f ^{\sharp }} ( \bar{x}) =\overline{T}$. $\; \; \Box$
In other words, when $K$ is a compact repeller, there exists a smallest nonempty $T$-viability kernel of $K$, the “viability core”, so to speak, because it is the subset of initial states from which one solution which enjoys the longest “life expectation” $\overline{T} $ in $K$. The viability kernel, when it is nonempty, is the viability core with infinite life expectation.
Characterization of Viable-Capture Basins
-----------------------------------------
Let $C \subset K$ be a closed subset of a closed subset $K \subset
{\bf R}^{n}$.
\[viablcaptbascharthm\] Let us assume that $f$ is continuous with linear growth and that $K$ is a closed repeller under $f$. Then the viable-capture basin $
\mbox{\rm Capt}_{f}^{K} (C)$ is the largest closed subset $D$ satisfying $$\left\{ \begin{array}{ll}
i) & C \; \subset \; D \; \subset \; K\\
ii) & D \backslash C \; \mbox{\rm is locally viable under $f$}
\end{array} \right.$$ or, equivalently, is the largest closed subset $D$ satisfying $$\left\{ \begin{array}{ll} \label{wonderfulthmeq27}
i) & C \; \subset \; D \; \subset \; K\\
ii) & \forall \; x\in D \backslash C, \; \; f(x) \in T_{D} (x)
\end{array} \right.$$ or, equivalently, in terms of normal cones, is the largest closed subset $D$ satisfying $$\left\{ \begin{array}{ll} \label{wonderfulthmeq28}
i) & C \; \subset \; D \; \subset \; K\\
ii) & \forall \; x \in D \backslash C, \; \forall \; p \in N_{D}
(x), \; \; \langle p,f (x) \rangle \; \leq \; 0
\end{array} \right.$$
[**Proof of Theorem \[viablcaptbascharthm\]** ]{} — Assume that a closed subset $D$ such that $C \subset D \subset K$ is a repeller under $f$ such that $D \backslash C$ is locally viable under $f$ and let us check that it is contained in $ \mbox{\rm Capt}_{f}^{K} (C)$. Since $ C \subset \mbox{\rm Capt}_{f}^{K} (C)$, let $x$ belong to $D
\backslash C$ and show that it belongs to $ \mbox{\rm Capt}_{f}^{K} (C)$. Since $K$ is a repeller, all solutions starting from $x$ leave $D
\backslash C$ in finite time. At least one of them, the solution $x ^{\sharp } ( \cdot )\in
{\cal S}_{f} (x)$ which maximizes $ \tau _{D} (x ( \cdot ))$: $$\tau _{D}^{f ^{\sharp }}(x) \; := \; \sup_{ x(\cdot ) \in {\cal
S}_{f}(x)}\tau _{D}^{f ^{\sharp }}(x(\cdot )) \; = \; \tau _{D}^{f ^{\sharp
}}(x ^{\sharp } (x))$$ leaves $D \backslash C$ through $C$. This solution exists by Proposition \[Picard03se\] since $D$ is closed and $f$ is continuous with linear growth. Then we claim that $ x ^{\sharp
}:=x ^{\sharp } (\tau _{D}^{f ^{\sharp }}(x))$ belongs to $C$. If not, $D
\backslash C$ being locally viable, one could associate with $ x ^{\sharp
}\in D \backslash C$ a solution $y ( \cdot ) \in {\cal S}_{f} (x ^{\sharp
} )$ and $T>0$ such that $y ( \tau ) \in D \backslash C$ for all $ \tau \in
[0,T]$. Concatenating this solution to $x ^{\sharp } ( \cdot )$, we obtain a solution viable in $D$ on an interval $[0, \tau _{D}^{f ^{\sharp
}}(x)+T]$, which contradicts the definition of $x ^{\sharp } ( \cdot )$. Furthermore, $x ^{\sharp } ( \cdot )$ is viable in $K$ since $D \subset K$. This implies that $D \subset \mbox{\rm Capt}_{f}^{K} (C)$.
The viable-capture basin being a closed subset such that $\mbox{\rm
Capt}_{f}^{K} (C) \backslash C$ is locally viable, we conclude that it is the largest closed subset $D$ of $K$ containing $C$ such that $D \backslash
C$ is locally viable under $f$.
Since $f$ is continuous and since $D \backslash C$ is locally compact, the Viability Theorem \[03A131\] states that $D \backslash C$ is locally viable if and only if (\[wonderfulthmeq27\])ii) or (\[wonderfulthmeq28\])ii) holds true. $\; \; \Box$
Characterization of Capture Basins
----------------------------------
\[wonderfulthmthmbis\] Let us assume that the closed subset $K \subset {\bf R}^{n}$ is a repeller under $f$ and backward invariant, that $C \subset K$ is closed and that $f$ is continuous with linear growth.
Then the capture basin $ \mbox{\rm Capt}_{f} (C)$ is the [*unique*]{} closed subset $D $ which satisfies $$\left\{ \begin{array}{ll}
i) & C \; \subset \; D \; \subset \; K\\
ii) & D\backslash C \;\; \mbox{\rm is locally viable under $f$}\\
iii) & D \;\; \mbox{\rm is backward invariant under $f$}\\
\end{array} \right.$$ If we assume furthermore that $f$ is Lipschitz, it is the unique closed subset satisfying the “tangential conditions” $$\left\{ \begin{array}{ll}
i) & C \; \subset \; D \; \subset \; K\\
ii) & \forall \; x \in D \backslash C, \; \;f(x) \; \in \;T_{D} (x)
\\
iii) & \forall \; x \in D, \; \; -f(x) \; \in \; T_{D} (x)\\
\end{array} \right.$$ or, equivalently, by duality, the “normal conditions” $$\left\{ \begin{array}{ll}
i) & C \; \subset \; D \; \subset \; K\\
ii) & \forall \; x \in D \backslash C, \; \forall \; p \in N_{D} (x),
\; \; \langle p,f (x) \rangle \; = \; 0 \\
iii) & \forall \; x \in D, \; \forall \; p \in N_{D} (x), \; \;
\langle p,f (x) \rangle \; \geq \; 0\\
\end{array} \right.$$
[**Proof**]{} — Since $C \subset K$ and $K$ is backward invariant, we already observed that $$\mbox{\rm Capt}_{f}^{K}(C) \; = \; \mbox{\rm Capt}_{f}(C) \; \subset \; K$$
Since $K$ is a repeller, the capture basin $\mbox{\rm
Capt}_{f}(C)$ is closed: For that purpose, let $x_{n} \in \mbox{\rm
Capt} _{f}(C)$ converge to $x$ and infer that $x$ belongs to the capture basin of $C$. There exist $t_{n}$ and $c_{n}:= \vartheta _{f}
(t_{n},x_{n})$ which belongs to $C$. Since $K$ is a repeller, the exit function $ \tau _{K}^{f ^{\sharp }}$ is finite, and, being upper semicontinuous thanks to Lemma \[Picard02\], $T:= \sup_{x_{n}\in B
(x,1)}\tau _{K}^{f ^{\sharp }} (x_{n}) <+ \infty $. Since $C \subset K$, we infer that $t_{n} \leq T <+ \infty $. Hence we can extract a subsequence (again denoted by) $t_{n}$ converging to some $t \in [0, T]$, so that, the reachable map being continuous, $c_{n}$ converges to $c= \vartheta _{f}
(t,x)$ which belongs to $C$. Hence $x $ belongs to $ \mbox{\rm Capt}
_{f}(C)$, which is then closed.
Therefore, the capture basin $\mbox{\rm Capt}_{f}(C)$ is a closed subset, backward invariant and locally viable.
If $D$ is closed and backward invariant, we infer that $\mbox{\rm
Capt}_{f}(C) \subset D$. By Theorem \[viablcaptbascharthm\], if $D
\backslash C$ is locally viable, we know that $$\mbox{\rm Capt}_{f}(C) \; = \; \mbox{\rm Capt}_{f}^{K}(C) \; \subset \; D$$ Hence, the capture basin $\mbox{\rm Capt}_{f}(C)$ is the unique closed subset, backward invariant and locally viable between $C$ and $K$. $\; \; \Box$
Stability Properties
--------------------
Let us consider now a sequence of closed subsets $K_{n}$ viable under a map $f$. [*Is the upper limit of these closed subsets still viable under $f$?*]{} The answer is positive.
\[01A452\] Let us assume that $f:{\bf R}^{n} \mapsto {\bf R}^{n}$ is continuous with linear growth. Then the upper limit of a sequence of subsets viable under $f$ is still viable under $f$ and the upper limit of the viability kernels of $K_{n}$ is contained in the viability kernel of the upper limit: $$\label{liminfviabsup}
\mbox{\rm Limsup}_{n \rightarrow + \infty } \mbox{\rm Viab}_{f}(C_{n})
\; \subset \;
\mbox{\rm Viab} _{f}(\mbox{\rm Limsup}_{n \rightarrow + \infty }C_{n})$$
In particular, the intersection of a decreasing family of closed viability domains is a closed viability domain.
[**Proof**]{} — We shall prove that the upper limit $K^{\sharp}$ of a sequence of subsets $K_{n}$ viable under $f$ is still viable under $f$.
Let $x$ belong to $K^{\sharp}$. It is the limit of a subsequence $x_{n'} \in K_{n'}$. Since the subsets $K_{n}$ are viable under $f$, there exist solutions $y_{n'}(\cdot)$ to differential equation $x' =f(x)$ starting at $x_{n'}$ and viable in $K_{n'}$. Theorem \[03A44\] implies that a subsequence (again denoted) $y_{n'}(\cdot)$ converges uniformly on compact intervals to a solution $y(\cdot)$ to differential equation $x'
=f(x)$ starting at $x$. Since $y_{n'}(t)$ belongs to $K_{n'}$ for all $n'$, we deduce that $y(t)$ does belong to $K^{\sharp}$ for all $t > 0$. $\; \; \Box$
\[convgraivenvthm\] Let us consider a sequence of closed subsets $C_{n}$.
1. If the map $f$ is continuous with linear growth and if the subsets $C_{n}$ are contained in a closed repeller $K$, then $$\label{liminfenvsup}
\mbox{\rm Limsup}_{n \rightarrow + \infty } \mbox{\rm Capt}_{f}^{K}(C_{n})
\; \subset \;
\mbox{\rm Capt} _{f}^{K}(\mbox{\rm Limsup}_{n \rightarrow + \infty
}C_{n})$$
2. If the map $f$ is furthermore monotone and $K$ is backward invariant, then $$\label{liminfenvinv}
\mbox{\rm Capt}_{f}(\mbox{\rm Liminf}_{n \rightarrow + \infty }C_{n}) \;
\subset \; \mbox{\rm Liminf}_{n \rightarrow + \infty } \mbox{\rm
Capt}_{f}(C_{n})$$
[**Proof**]{} — For proving inclusion (\[liminfenvsup\]), we consider the limit $x:=\lim_{n \rightarrow + \infty }x_{n}$ of elements $x_{n} $ of $\mbox{\rm
Capt}_{f}^{K}(C_{n})$. Let us consider solutions $x_{n} ( \cdot )$ satisfying $$t_{n} \; := \; \omega _{C_{n}} (x_{n} ( \cdot )) \; \leq \; \tau _{K}
(x_{n} ( \cdot )) \; \leq \; T \; := \; \sup_{y \in B (x,1) \cap K} \tau
_{K}^{f ^{\sharp }} (y)$$ where $T$ is finite because $K$ is a repeller.
Therefore, a subsequence (again denoted by) $t_{n}$ converges to some $t \leq T$ and another subsequence (again denoted by) $x_{n} ( \cdot )$ converges uniformly on compact intervals to some solution $x ( \cdot )$ starting from $x$. Since $x_{n} (t_{n})$ belongs to $C_{n}$ and converges to $x (t)$, we infer that $x (t)$ belongs to the upper limit $C ^{\sharp }$ of the $C_{n}$. Hence $$\omega_{C ^{\sharp }} (x) \; \leq \; \lim_{n \rightarrow + \infty }t_{n}
\; \leq \; \limsup_{n \rightarrow + \infty } \tau _{K}
(x_{n} ( \cdot )) \; \leq \; \tau _{K}
(x ( \cdot ))$$ and thus, $x$ belongs to the viable-capture basin of $C ^{\sharp }$.
We now prove (\[liminfenvinv\]). Let $C ^{\flat }$ denote the lower limit of the subsets $C_{n}$ and let us consider $x $ an element of $\mbox{\rm Capt}_{f}(C ^{\flat })$, a solution $x ( \cdot )$ starting from $x$ and reaching $C ^{\flat }$ at time $T$ at $c:=x (T)$. Hence $y (t):=x (T-t)$ is a solution to the backward differential equation $y'=-f (y)$ starting at $c$ and satisfying $y (T)=x$. Since $c= \lim_{n \rightarrow + \infty }c_{n}$ where $c_{n} \in C_{n}$, Theorem \[05A13m\] states that the solutions $y_{n} ( \cdot )$ to the differential equation $y'=-f (y)$ starting from $c_{n}$ satisfy $$\|y (t)-y_{n} (t)\| \; \leq \; e^{ - \mu t} \|c-c_{n}\|$$ Then $x_{n} := y_{n} (T) \in \mbox{\rm Capt}_{f}(C_{n})$ converges to $x=y
(T) \in \mbox{\rm Capt}_{f}(C ^{\flat })$. $\; \; \Box$
Epiderivatives and Subdifferentials
===================================
For reasons motivated by optimization theory, Lyapunov stability, control theory, Hamilton-Jacobi equations and mathematical morphology, the order relation on $ {\bf R}$ is involved. This leads us to associate with an extended functions $ {\bf u}: X \mapsto {\bf R} \cup \{+ \infty \}$ its epigraph instead of its graph. It actually happens that the properties of the extended functions $ {\bf u}: X \mapsto {\bf R} \cup \{+
\infty \}$ are actually properties of their [*epigraphs*]{}. This “epigraphical point of view” is the key to “variational analysis” and to our treatment of Hamilton-Jacobi inequalities.
In particular, one can define the following concepts:
1. The epigraph of the [*lower epilimit*]{} of a sequence of extended functions $ {\bf u}_{n}:X \mapsto {\bf R} \cup \{+\infty \}$ is the upper limit of the epigraphs of the $f_{n}$,
2. [*The contingent epiderivative*]{} $D _{\uparrow } {\bf u} (x)$ at $x$ is the lower epigraphical limit of the difference quotients $ \nabla
_{h} {\bf u}(x)$, so that the epigraph of the contingent epiderivative is the contingent cone to the epigraph of $ {\bf u}$.
By duality, the normal cones to the epigraph of an extended function yields the concept of [**subdifferential**]{}, which is in particular used in the concepts of viscosity solutions to Hamilton-Jacobi equations.
Extended Functions and their Epigraphs
--------------------------------------
A function ${\bf v}: X \mapsto {\bf R} \cup \{+ \infty \}$ is called an [*extended (real-valued) function*]{}. Its [*domain*]{} is the set of points at which ${\bf v}$ is finite: $$\mbox{\rm Dom}({\bf v}) \; := \; \{x \in X \;\; | \;\; {\bf
v}(x) < +\infty \}$$ A function is said to be if its domain is not empty. Any function ${\bf v}$ defined on a subset $K \subset
X$ can be regarded as the extended function ${\bf v}_{K}$ equal to ${\bf
v}$ on $K$ and to $+ \infty $ outside of $K$, whose domain is $K$.
Since the order relation on the real numbers is involved in the definition of the Lyapunov property (as well as in minimization problems and other dynamical inequalities), we no longer characterize a real-valued function by its graph, but rather by its [*epigraph*]{} index[epigraph]{} $${\cal E}p({\bf v}) \; := \; \{(x,\lambda ) \in X \times {\bf R} \;
| \; {\bf v}(x) \leq \lambda \}$$ The [*hypograph*]{} of a function ${\bf v}: X \mapsto {\bf R} \cup \{-\infty \}$ is defined in a symmetric way by $${\cal H}yp({\bf v}) \; := \; \{(x,\lambda ) \in X \times {\bf R} \;
| \; {\bf v}(x) \geq \lambda \} \; = \; - {\cal E}p(-{\bf v})$$
[*The graph of a real-valued (finite) function is then the intersection of its epigraph and its hypograph*]{}.
We also remark that some properties of a function are actually properties of their epigraphs. For instance, [*an extended function ${\bf v}$ is convex (resp. positively homogeneous) if and only if its epigraph is convex (resp. a cone).*]{} The epigraph of ${\bf v}$ is closed if and only if ${\bf v}$ is lower semicontinuous: $$\forall \; x \in X, \; \; {\bf v}(x) \; = \; \liminf_{y
\rightarrow x}{\bf v}(y)$$
We recall the convention $\inf (\emptyset) := +\infty $.
Consider a function ${\bf v} :X \mapsto {\bf R} \cup \{\pm \infty
\} $. Its epigraph is closed if and only if $$\forall \; x \; \in \; X, \; \; {\bf v} (x) \; = \;
\liminf_{x' \rightarrow x} {\bf v} (x')$$ Assume that the epigraph of ${\bf v} $ is a closed cone. Then the following conditions are equivalent: $$\left\{ \begin{array}{ll}
i) & \forall \; x \; \in \; X, \; \; {\bf v} (x) \; > \;
-\infty \\
ii) & {\bf v} (0) \; = \; 0 \\
iii) & (0,-1) \; \notin \; {\cal E}p({\bf v} )
\end{array} \right.$$
[**Proof**]{} — Assume that the epigraph of ${\bf v} $ is closed and pick $x \in
X$. There exists a sequence of elements $x_{n}$ converging to $x$ such that $$\lim_{n \rightarrow \infty } {\bf v} (x_{n}) \; = \; \liminf_{x'
\rightarrow x} {\bf v} (x')$$ Hence, for any $\lambda > \liminf_{x' \rightarrow x} {\bf v} (x')$, there exist $N$ such that, for all $n \geq N$, ${\bf v} (x_{n}) \leq
\lambda $, i.e., such that $(x_{n},\lambda ) \in {\cal E}p({\bf v} )$. By taking the limit, we infer that ${\bf v} (x) \leq \lambda $, and thus, that ${\bf v} (x) \leq \liminf_{x' \rightarrow x} {\bf v} (x')$. The converse statement is obvious.
Suppose next that the epigraph of ${\bf v} $ is a cone. Then it contains $(0,0)$ and ${\bf v}(0) \leq 0$. The statements $ii)$ and $iii)$ are clearly equivalent.
If $i)$ holds true and ${\bf v}(0)<0$, then $$(0,-1) \; = \; \frac{1}{-{\bf v}(0)}(0,{\bf v}(0))$$ belongs to the epigraph of ${\bf v}$, as well as all $(0,-\lambda
)$, and (by letting $\lambda \rightarrow +\infty $) we deduce that ${\bf
v}(0)=-\infty $, so that $i)$ implies $ii)$.
To end the proof, assume that ${\bf v} (0) =0$ and that for some $x$, ${\bf v} (x) =-\infty $. Then, for any $\varepsilon >0$, the pair $(x,
-1/\varepsilon )$ belongs to the epigraph of ${\bf v} $, as well as the pairs $(\varepsilon x,-1)$. By letting $\varepsilon $ converge to $0$, we infer that $(0,-1)$ belongs also to the epigraph, since it is closed. Hence ${\bf v}(0)<0$, a contradiction. $\; \; \Box$
[*Indicators $\psi _{K}$ of subsets $K$*]{} are cost functions defined by $$\psi _{K}(x) := 0 \;\;\mbox{\rm if}\;\; x \in K \;\;\mbox{\rm
and}\;\; + \infty \;\;\mbox{\rm if not}$$ which characterize subsets (as [*characteristic functions*]{} do for other purposes) provide important examples of extended functions. It can be regarded as a [*membership cost*]{}[^8] to $K$: it costs nothing to belong to $K$, and $+ \infty $ to step outside of $K$.
Since $${\cal E}p ( \psi K) \; = \; K \times {\bf R}_{+}$$ we deduce that the indicator $\psi _{K}$ is lower semicontinuous if and only if $K$ is closed and that $\psi _{K}$ is convex if and only if $K$ is convex. One can regard the sum ${\bf v} + \psi _{K}$ as the restriction of ${\bf v}$ to $K$.
We recall the convention $\inf (\emptyset) := +\infty $.
Epilimits
---------
The epigraph of the [*lower epilimit*]{} $\mbox{\rm
lim}_{\uparrow}^{\sharp}\mbox{}_{n \rightarrow \infty }{\bf u}_{n}$ of a sequence of extended functions $ {\bf u}_{n} :X \mapsto {\bf
R} \cup \{+\infty \}$ is the upper limit of the epigraphs: $${\cal E}p(\mbox{\rm lim}_{\uparrow}^{\sharp}\mbox{}_{n \rightarrow
\infty }{\bf u}_{n}) \; := \; \mbox{\rm Limsup}_{n \rightarrow \infty
} {\cal E}p({\bf u}_{n})$$ The function $\mbox{\rm lim}_{\uparrow}^{\flat}\mbox{}_{n
\rightarrow \infty }{\bf u}_{n}$ whose epigraph is the lower limit of the epigraphs of the functions $ {\bf u}_{n}$ $${\cal E}p( \mbox{\rm lim}_{\uparrow}^{\flat}\mbox{}_{n \rightarrow
\infty }{\bf u}_{n}) \; := \; \mbox{\rm Liminf}_{n \rightarrow \infty
} {\cal E}p({\bf u}_{n})$$ is the [*upper epilimit*]{} of the functions $ {\bf u}_{n}$
One can check that $$\mbox{\rm lim}_{\uparrow}^{\sharp}\mbox{}_{n \rightarrow \infty
}{\bf u}_{n}(x_{0})
\; = \; \liminf_{n \rightarrow \infty, x \rightarrow x_{0}}{\bf
u}_{n}(x)$$
Contingent Epiderivatives
-------------------------
When $ {\bf u}$ is an extended function, we associate with it its epigraph and the contingent cones to this epigraph. This leads to the concept of epiderivatives of extended functions.
Let $ {\bf u}: X \mapsto {\bf R} \cup \{\pm \infty \}$ be a nontrivial extended function and $x$ belong to its domain.
We associate with it the [*differential quotients*]{} $$u \; \leadsto \; \nabla _{h}{\bf u}(x)(u) \; := \; \frac{{\bf
u}(x+hu)-{\bf u}(x)}{h}$$
The contingent epiderivative $D _{\uparrow }{\bf u}(x)$ of $ {\bf
u}$ at $ x \in \mbox{\rm Dom}( {\bf u})$ is the lower epilimit of its differential quotients: $$D _{\uparrow }{\bf u}(x) \; = \; \mbox{\rm
lim}_{\uparrow}^{\sharp}\mbox{}_{h \rightarrow 0+}\nabla _{h}{\bf u}(x)$$ We shall say that the function $ {\bf u}$ is [*contingently epidifferentiable*]{} at $x$ if for any $u \in X$, $D_{\uparrow }{\bf u}(x)(u) >-\infty $ (or, equivalently, if $D_{\uparrow }{\bf u}(x)(0)=0$).
\[cgepider\] Let $ {\bf u}: X \mapsto {\bf R} \cup \{\pm \infty \}$ be a nontrivial extended function and $x$ belong to its domain. Then the contingent epiderivative $D _{\uparrow }{\bf u}(x)$ satisfies $$\forall \; u \in X, \; \; D_{\uparrow }{\bf u}(x)(u) \; = \;
\liminf_{h \rightarrow 0+, u'\rightarrow u}\frac{{\bf u}(x+hu')-{\bf
u}(x)}{h}$$ and the epigraph of the contingent epiderivative $D _{\uparrow
}{\bf u}( \cdot )$ is equal to the contingent cone to the epigraph of $
{\bf u}$ at $(x,{\bf u}(x))$ is $${\cal E}p (D_{\uparrow}{\bf u}(x)) \; = \; T_{{\cal E}p ( {\bf
u})}(x,{\bf u}(x))$$
[**Proof**]{} — The first statement is obvious. For proving the second one, we recall that the contingent cone $$T_{{\cal E}p ( {\bf u})}(x,{\bf u}(x)) \;
= \; \mbox{\rm Limsup}_{h \rightarrow 0+} \frac{{\cal E}p ( {\bf u})
-(x,{\bf u}(x))}{h}$$ is the upper limit of the differential quotients $
\displaystyle{ \frac{{\cal E}p ( {\bf u})-(x,{\bf u}(x))}{h}}$ when $h
\rightarrow 0+$. It is enough to observe that $${\cal E}p(D _{\uparrow }{\bf u}(x)) := T_{{\cal E}p ( {\bf
u})}(x,y)
\; \; \& \; \; {\cal E}p(\nabla _{h}{\bf u}(x)) = \frac{{\cal E}p
( {\bf u}) -(x,{\bf u}(x))}{h}$$ to conclude. $\; \; \Box$
Consequently, [*the epigraph of the contingent epiderivative at $x$ is a closed cone. It is then lower semicontinuous and positively homogeneous whenever $ {\bf u}$ is contingently epidifferentiable at $x$.*]{}
We observe that the contingent epiderivative of the indicator function $ \psi _{K}$ at $x \in K$ is the indicator of the contingent cone to $K$ at $x$: $$D _{\uparrow } \psi _{K} (x) \; = \; \psi _{T_{K} (x)}$$ making precise the intuition stating that the contingent cone $T_{K} (x)$ plays the role of a “derivative of a set”, as the limit of differential quotients $ \displaystyle{ \frac{K-x}{h}}$ of sets.
The hypoderivatives of an extended function are defined in a analogous way: The contingent hypoderivative $D _{\downarrow }{\bf u}(x)$ of $ {\bf u}$ at $ x \in \mbox{\rm Dom}( {\bf u})$ is the upper hypolimit of its differential quotients: $$D _{ _{\downarrow } }{\bf u}(x) \; = \; \mbox{\rm lim}_{
_{\downarrow } }^{\sharp}\mbox{}_{h \rightarrow 0+}\nabla _{h}{\bf u}(x)$$ We observe that it is equal to $$\forall \; u \in X, \; \; D_{\downarrow }{\bf u}(x)(u) \; = \;
\limsup_{h \rightarrow 0+, u'\rightarrow u}\frac{{\bf u}(x+hu')-{\bf
u}(x)}{h}$$ and that [*the hypograph of the [*contingent hypoderivative*]{} $D_{\downarrow }{\bf
u}(x) $ of $ {\bf u}$ at $x$ is the contingent cone to the hypograph of $
{\bf u}$ at $(x,{\bf u}(x))$*]{}: $${\cal E}p (D_{\downarrow}{\bf u}(x)) = T_{{\cal H}yp( {\bf
u})}(x,{\bf u}(x))$$
We shall say that $ {\bf u}: X \mapsto W$ is [*differentiable from the right*]{} at $x$ if the contingent epiderivative and hypoderivative coincide: $$\forall \; v \in X, \; \; D _{\uparrow }{\bf u}(x) (v) \; = \; D
_{\downarrow } {\bf u}(x) (v)$$
Let $K \subset X$ be a closed subset and $ {\bf u}:X \mapsto {\bf
R} \cup \{+\infty \}$ be an extended function. We denote by $ {\bf
u}|_{K}:=f + \psi _{K}$ the restriction to $ {\bf u}$ at $K$. Inequality $$D _{\uparrow }{\bf u}(x) |_{T_{K}} (x) \; \leq \; D _{\uparrow
}{\bf u}|_{K} (x)$$ always holds true. It is an equality when $ {\bf u}$ is differentiable from the right: [*the contingent derivative of the restriction of $ {\bf u}$ to $K$ is the restriction of the derivative to the contingent cone*]{}.
[**Proof**]{} — Indeed, let $x \in K \cap \mbox{\rm Dom}( {\bf u})$. If $ u $ belongs to $T_{K} (x)$, there exist $h_{n} \rightarrow 0+$, $ \varepsilon
_{n} \rightarrow 0+$ and $x_{n}:=x+h_{n}u_{n} \in K$ such that $$D _{\uparrow }{\bf u}(x) ( u ) \; \leq \; \liminf_{n \rightarrow
+ \infty } \frac{{\bf u}(x_{n})-{\bf u}(x)}{h_{n}} \; = \;\liminf_{n
\rightarrow + \infty } \frac{{\bf u}|_{K}(x_{n})-{\bf u}|_{K}(x)}{h_{n}}$$ which implies the inequality. If $ {\bf u}$ is differentiable from the right, the differential quotient converges to the common value $ D
_{\uparrow }{\bf u}(x) = D _{\uparrow }{\bf u}|_{K} (x)= D _{\downarrow }
{\bf u}|_{K} (x)$. $\; \; \Box$
For locally Lipschitz functions, the contingent epiderivatives are finite:
\[04A9627\] If $ {\bf u} :X \mapsto {\bf R} \cup \{+\infty \}$ is Lipschitz around $x \in \mbox{\rm Int}( \mbox{\rm Dom}( {\bf u}))$, then the contingent epiderivative $D _{\uparrow }{\bf u}(x)$ is Lipschitz: there exists $ \lambda >0$ such that $$\forall \; u \in X, \; \; D _{\uparrow }{\bf u}(x) (u) \; = \;
\liminf_{h \rightarrow 0+} \frac{{\bf u}(x+hu)-{\bf u}(x)}{h} \; \leq \;
\lambda \|u\|$$
[**Proof**]{} — Since $ {\bf u}$ is Lipschitz on some ball $B (x, \eta )$, the above inequality follows immediately from $$\forall \; u \in \eta B, \; \; \frac{{\bf u}(x+hu)-{\bf u}(x)}{h}
\; \leq \; \frac{{\bf u}(x+hu')-{\bf u}(x)}{h} + \lambda (\|u\|+
\|u'-u\|)$$ by taking the liminf when $h \rightarrow 0+$ and $u' \rightarrow
u$. $\; \; \Box$
For convex functions, we obtain:
\[04A962\] When the function $ {\bf u}:X\mapsto {\bf R}\cup \{+\infty \}$ is convex, the contingent epiderivative is equal to $$D_{\uparrow }{\bf u}(x)(u) \; = \; \liminf_{u' \rightarrow
u}\left( \inf_{h>0}\frac{{\bf u}(x+hu')-{\bf u}(x)}{h} \right)$$
[**Proof**]{} — Indeed, Proposition \[01A955\] implies that if $0< h_{1} \leq
h_{2}$, $${\cal E}p(\nabla _{h_{2}}{\bf u}(x)) \; = \; \frac{{\cal E}p (
{\bf u}) -(x,{\bf u}(x))}{h_{2}} \; \subset \; {\cal E}p(\nabla
_{h_{1}}{\bf u}(x))$$ i.e., $$\forall \; u \in X, \; \;\nabla _{h_{1}}{\bf u}(x)(u) \; \leq \;
\nabla _{h_{2}}{\bf u}(x)(u)$$ Therefore, $$\forall \; u\in X, \; \; D{\bf u}(x) (u) \; := \; \lim_{h
\rightarrow 0+} \frac{{\bf u}(x+hu)-{\bf u}(x)}{h} \; = \;
\inf_{h>0}\frac{{\bf u}(x+hu)-{\bf u}(x)}{h}$$ and this function $D{\bf u}(x)$ is convex with respect to $u$. Since the epigraph of $D{\bf u}(x)$ is the increasing union of the epigraphs of the differential quotients $\nabla _{h}{\bf u}(x)$, we infer that $$D _{\uparrow }{\bf u}(x) (u) \; := \; \liminf_{u' \rightarrow
u}D{\bf u}(x) (u')$$
We recall the following important property of convex functions defined on finite dimensional vector spaces:
An extended convex function $ {\bf u}$ defined on a finite dimensional vector-space is locally Lipschitz and subdifferentiable on the interior of its domain. Therefore, when $x$ belongs to the interior of the domain of $ {\bf u}$, there exists a constant $ \lambda _{x}$ such that $$\forall \; u \; \in \; X, \; \; D_{\uparrow }{\bf u}(x)(u) \;
= \; \inf_{h>0}\frac{{\bf u}(x+hu)-{\bf u}(x)}{h} \; \leq \; \lambda_{x}
\|u\|$$
The second statement follows from Proposition \[04A9627\]. $\; \; \Box$
Generalized Gradients
---------------------
Let $ {\bf u}:X \mapsto {\bf R} \cup \{+\infty \}$ be a nontrivial extended function. The continuous linear functionals $p \in X^{\star}$ satisfying $$\forall \; v \in X, \; \; \langle p,v \rangle \; \leq \; D_{\uparrow
}{\bf u}(x)(v)$$ are called the [*(regular) subgradients*]{} of $ {\bf
u}$ at $x$, which constitute the (possibly empty) closed convex subset $$\partial _{-}{\bf u}(x) \; := \; \{p \in X^{\star}\;|\;\forall \; v
\in X, \;\; \langle p,v \rangle \; \;
\leq \; D_{\uparrow }{\bf u}(x)(v)\}$$ called the [*(regular) subdifferential*]{} of $ {\bf u}$ at $x_{0}$.
In a symmetric way, the [*superdifferential*]{} $\partial
_{+}{\bf u}(x)$ of $ {\bf u}$ at $x$ is defined by $$\partial _{+}{\bf u}(x) \; := \; - \partial
_{-}(- {\bf u})(x)$$
Naturally, when $ {\bf u}$ is Fréchet differentiable at $x$, then $$D_{\uparrow }{\bf u}(x)(v) \; = \; \langle f'(x),v \rangle$$ so that [*the subdifferential $\partial _{-}{\bf u}(x)$ is reduced to the gradient $ {\bf u}'(x)$.*]{}
We observe that $$\partial _{-} {\bf u}(x) + N_{K} (x) \; \subset \; \partial ({\bf
u}|_{K}) (x)$$
If $ {\bf u}$ is differentiable at a point $x \in K$, then the [*subdifferential of the restriction is the sum of the gradient and the normal cone*]{}: $$\partial_{-}({\bf u}|_{K})(x) \; = \; {\bf u}'(x) + N_{K}(x)$$
We also note that the subdifferential of the indicator of a subset is the normal cone: $$\partial_{-}\psi _{K}(x) \; = \; N_{K}(x)$$ that $$\left\{ \begin{array}{ll}
i) & (p,-1) \in N_{ {\cal E}p ( {\bf u})} (x,{\bf u}(x)) \;\mbox{\rm
if and only if}\;p \in \partial_{-}{\bf u}(x)\\
ii) & (p,0) \in N_{ {\cal E}p ( {\bf u})} (x,{\bf u}(x)) \;\mbox{\rm
if and only if}\; p \in \mbox{\rm Dom}(D _{\uparrow }{\bf u}(x))^{-}
\end{array} \right.$$ so that We also deduce that $$N_{ {\cal E}p ( {\bf u})} (x,{\bf u}(x)) \; = \;
\{ \lambda (q,-1)\}_{q \in \partial_{-}{\bf u}(x), \; \lambda >0}
\bigcup_{}^{} \{ (q,0)\}_{q \in \mbox{\rm Dom}(D _{\uparrow }{\bf
u}(x))^{-}}$$ The subset $ \mbox{\rm Dom}(D _{\uparrow }{\bf u}(x))^{-} = \{0\}$ whenever the domain of the contingent epiderivative $D _{\uparrow }{\bf
u}(x)$ is dense in $X$. This happens when $ {\bf u}$ is locally Lipschitz and when the dimension of $X$ is finite:
Let $X$ be a finite dimensional vector space, $ {\bf u}:X \mapsto
{\bf R} \cup \{\pm \infty \}$ be a nontrivial extended function and $x_{0} \in \mbox{\rm Dom}( {\bf u})$. Then the subdifferential $\partial _{-}{\bf u}(x)$ is the set of elements $p \in
X^{\star}$ satisfying $$\label{af-064-27}
\liminf_{x \rightarrow
x_{0}}\frac{{\bf u}(x)-{\bf u}(x_{0})- \langle p,x-x_{0} \rangle
}{\|x-x_{0}\|} \; \geq \; 0$$ is the .
In a symmetric way, the [*superdifferential*]{} $\partial
_{+}{\bf u}(x_{0})$ of $ {\bf u}$ at $x_{0}$ is the subset of elements $p
\in X^{\star}$ satisfying $$\limsup_{x \rightarrow
x_{0}}\frac{{\bf u}(x)-{\bf u}(x_{0})- \langle p,x-x_{0} \rangle
}{\|x-x_{0}\|} \; \leq \; 0$$
[**Proof**]{} — This is an easy consequence of proposition \[mormconechar\]. $\; \; \Box$
The equivalent formulation (\[af-064-27\]) of the concept of subdifferential has been introduced by Crandall & P.-L. Lions for defining [*viscosity solutions*]{} to Hamilton-Jacobi equations.
Moreau-Rockafellar Subdifferentials
-----------------------------------
When $ {\bf u}$ is convex, the generalized gradient coincides with the subdifferential introduced by Moreau and Rockafellar for convex functions in the early 60’s:
Consider a nontrivial function $ {\bf u}:X\mapsto {\bf R}\cup
\{+\infty \}$ and $x \in \mbox{\rm Dom}( {\bf u})$. The closed convex subset $ \partial {\bf u}(x)$ defined by $$\partial {\bf u}(x) \; = \; \{p \; \in \; X^{\star}\;| \;
\;\forall \; y \; \in \; X, \; \langle p,y-x \rangle \; \leq \; {\bf
u}(y)-{\bf u}(x)\}$$ (which may be empty) is called the [*Moreau-Rockafellar subdifferential*]{} of $ {\bf u}$ at $x$. We say that $ {\bf u}$ is [*subdifferentiable at $x$*]{} if $
\partial {\bf u}(x) \ne \emptyset $.
Proposition \[03A955\] implies that in the convex case
Let $ {\bf u}:X \mapsto {\bf R}_{+} $ be a nontrivial extended convex function. Then the subdifferential $ \partial_{-}{\bf u}(x)$ coincides with Moreau-Rockafellar subdifferential $ \partial {\bf u}(x)$.
Furthermore, the graph of the subdifferential map $x \leadsto
\partial {\bf u}(x)$ is closed.
Let us mention the following simple — but useful — remark:
\[lopsidfrematruleprp\] Assume that $ {\bf u}:={\bf v}+{\bf w}$ is the sum of a differentiable function ${\bf v}$ and a convex function $ {\bf w}$. If $
\bar{x}$ minimizes $ {\bf u}$, then $$-{\bf v}' ( \bar{x}) \; \in \; \partial {\bf w} ( \bar{x})$$
[**Proof**]{} — Indeed, for $h>0$ small enough, $\bar{x} +h (y-\bar{x}) =
(1-h)\bar{x} +hy$ so that $$0 \; \leq \; \frac{{\bf u}(\bar{x} +h (y-\bar{x}))- {\bf u}
(\bar{x})}{h} \; \leq \; \frac{{\bf u}(\bar{x} +h (y-\bar{x}))- {\bf u}
(\bar{x})}{h} + {\bf w} (y)-{\bf w} ( \bar{x})$$ thanks to the convexity of ${\bf w}$. Letting $h$ converge to $0$ yields $$0 \; \leq \; \langle {\bf v}' (\bar{x}), y-\bar{x} \rangle +
{\bf w} (y) - {\bf w} (\bar{x})$$ so that $-{\bf v}' ( \bar{x}) $ belongs to $ \partial {\bf w} (
\bar{x})$. $\; \; \Box$
Some Hamilton-Jacobi Equations
==============================
Let us introduce
1. a differential equation $x'= f(x)$, where $f:{\bf R}^{n} \mapsto
{\bf R}^{n}$ is continuous and has linear growth,
2. a nonnegative continuous “Lagrangian” $$l : (x,p) \in {\bf R}^{n} \times {\bf R}^{n} \mapsto l (x,p) \in {\bf
R}_{+}$$
3. an extended nonnegative lower semicontinuous function ${\bf u}:{\bf
R}^{n} \mapsto {\bf R}_{+} \cup
\{+\infty \}$
We consider the problem $${\bf u}^{\top} (x) \; := \; \alpha _{ (f,l)} ^{\top} ({\bf u}) (x) \; :=
\; \inf_{x (
\cdot ) \in {\cal S}_{f} (x)} \left( \sup_{t \geq 0} \left( e^{a t}{\bf
u} (x
(t)) + \int_{0}^{t}e^{a \tau }l ( x( \tau ),x' ( \tau )) d \tau \right)
\right)$$ and the “stopping time” problem $${\bf u}^{\bot} (x) \; := \; \alpha _{ (f,l)} ^{\bot } ({\bf u}) (x) \; :=
\; \inf_{x
( \cdot ) \in {\cal S}_{f} (x)} \left( \inf_{t \geq 0} \left( e^{a
t}{\bf u}
(x (t)) + \int_{0}^{t}e^{a \tau }l ( x( \tau ),x' ( \tau )) d \tau
\right) \right)$$ whenever the graph of the function ${\bf u}$ is regarded as an obstacle, as in unilateral mechanics. Taking $l \equiv 0$, we obtain the $a$-Lyapunov function $$\alpha _{ (f,0)} ^{\top} ({\bf u}) (x) \; := \; \inf_{x ( \cdot ) \in
{\cal
S}_{f} (x)} \left( \sup_{t \geq 0} e^{a t}{\bf u} (x (t)) \right)$$ as an example of the first problem and taking ${\bf u} \equiv 0$, we obtain the variational problem $$\alpha _{ (f,l)} ^{\top} (0) (x) \; := \; \inf_{x ( \cdot ) \in {\cal
S}_{f} (x)} \int_{0}^{ \infty }e^{a \tau }l ( x( \tau ),x' ( \tau )) d
\tau$$
We shall prove that these functions are “generalized” solutions to Hamilton-Jacobi “differential variational inequalities” $$\left\{ \begin{array}{ll}
i) & {\bf u} (x) \; \leq \; {\bf u} ^{\top } (x)\\
ii) & \displaystyle{\left\langle \frac{ \partial }{ \partial
x}{\bf u}
^{\top } (x),f (x)\right\rangle +l (x,f (x)) +a{\bf u} ^{\top } (x)\; \leq
\;
0}\\
iii) & \displaystyle{({\bf u} (x)-{\bf u} ^{\top } (x)) \left(
\left\langle
\frac{ \partial }{ \partial x}{\bf u} ^{\top } (x),f (x)\right\rangle +l
(x,f
(x)) +a{\bf u} ^{\top } (x) \right) \; = \; 0}
\end{array} \right.$$ and $$\left\{ \begin{array}{ll}
i) & 0 \; \leq \; {\bf u} ^{\bot }(x) \; \leq \; {\bf u} (x)\\
ii) & \displaystyle{\left\langle \frac{ \partial }{ \partial
x}{\bf u}
^{\bot } (x),f (x)\right\rangle +l (x,f (x)) +a{\bf u} ^{\bot } (x)\; \geq
\;
0}\\
iii) & \displaystyle{({\bf u} (x)-{\bf u} ^{\bot } (x)) \left(
\left\langle
\frac{ \partial }{ \partial x}{\bf u} ^{\bot } (x),f (x)\right\rangle +l
(x,f
(x)) +a{\bf u} ^{\bot } (x) \right) \; = \; 0}
\end{array} \right.$$ respectively.
For this type of partial differential equations, the concept of distributional derivatives happens to be much less adequate than other concepts of “contingent epiderivatives”, as it was shown by Hélène Frankowska, or subdifferentials, as they appear in the concept of “viscosity solutions” introduced by Michael Crandall and Pierre-Louis Lions for a general class of nonlinear Hamilton-Jacobi equations.
Indeed, for those Hamilton-Jacobi type equations derived from the calculus of variations, control theory or differential games, one can derive from the properties of the viability kernels and capture basins of auxiliary systems that the various value functions involved in these problems are solutions to such Hamilton-Jacobi equations or differential variational inequalities in an adequate sense.
We illustrate here this general approach for the two preceding problems only. Namely, we shall prove that the epigraphs of the two “value functions” $
{\bf u}^{\top}$ and $ {\bf u}^{\bot}$ are respectively the viability kernel and the capture basin of the epigraph of the function ${\bf u}$ under the map $g:
{\bf R}^{n}
\times {\bf R} \mapsto {\bf R}^{n} \times {\bf R}$ defined by $$g (x,y) \; := \; ( f (x), -ay-l (x,f (x)))$$ which is a continuous map with linear growth whenever $f$ and the Lagrangian $l$ are continuous with linear growth. The basic observation allowing this transfer of properties is summarized in the statement of the following:
\[tranprplemm\] If a closed subset $ {\cal V} \subset {\bf R}^{n} \times {\bf R}_{+} $ is locally viable under $g$, so is the epigraph of the associated extended function ${\bf v}
:{\bf R}^{n} \mapsto {\bf R}_{+} \cup \{+\infty \}$ defined by $${\bf v} (x) \; := \; \inf_{ (x,y) \in {\cal V}}y$$ where we set as usual ${\bf v} (x) :=+ \infty $ whenever the subset $ \{y
\; | \;
(x,y) \in {\cal V}\}$ is empty. As a consequence, for any $x \in \mbox{\rm
Dom}({\bf v})$, there exist a solution to the differential equation $x'=f
(x)$ and $T>0$ satisfying $$\label{dynineeq}
\forall \; t \in [0,T], \; \; e^{a t} {\bf v} (x (t)) + \int_{0}^{t}e^{a
\tau
}l ( x( \tau ),x' ( \tau )) d \tau \; \leq \; {\bf v} (x)$$ (where $T=+ \infty $ whenever $ {\cal V} $ is (globally) viable under $g$).
[**Proof**]{} — If $x$ belongs to the domain of ${\bf v}$, then $ (x,{\bf v} (x))$ belongs to $ {\cal V}$ so that there exist a solution $ x ( \cdot ) \in {\cal
S}_{ f} (x)$ and $T>0$ such that $$\forall \; t \in [0,T], \; \;\left( x (t) , y (t):=e^{-at}{\bf v} (x)-
\int_{0}^{t}e^{-a(t - \tau }l ( x( \tau ),x' ( \tau )) d \tau \right) \;
\in \; {\cal V}$$ i.e., if and only if $y (t) \geq {\bf v} (x (t))$, which can be written in the form (\[dynineeq\]). If $y_{0} >{\bf v} (x)$, then we observe that for all $t
\geq 0$, $$y_{0} (t) \; := \; e^{-at}y_{0} -\int_{0}^{t}e^{-a(t - \tau }l ( x( \tau
),x' ( \tau )) d \tau \; \geq \; y (t)\; \geq \; {\bf v} (x (t))$$ and thus, that $ (x (t), y_{0} (t))$ is a solution to the differential equation $ (x' (t),y' (t))=g (x (t),y (t))$ starting at $ (x,y_{0})$ and viable in the epigraph of ${\bf v}$. $\; \; \Box$
Value Function
--------------
We associate with ${\bf u}:{\bf R}^{n} \mapsto {\bf R}_{+} \cup
\{+\infty \}$ the problem $${\bf u}^{\top} (x) \; := \; \inf_{x ( \cdot ) \in {\cal S}_{f} (x)} \left(
\sup_{t \geq 0} \left( e^{a t}{\bf u} (x (t)) + \int_{0}^{t}e^{a \tau
}l (
x( \tau ),x' ( \tau )) d \tau \right) \right)$$ The function $ {\bf u}^{\top}:=\alpha _{ (f,l)} ^{\top }({\bf u})$ is called the [value function associated with ${\bf u}$]{}.
If $l=0$, the above problem can be written $$\alpha _{ (f,0)} ^{\top} ({\bf u}) (x) \; := \; \inf_{x ( \cdot ) \in
{\cal
S}_{f} (x)} \left( \sup_{t \geq 0} e^{a t}{\bf u} (x (t)) \right)$$ and if ${\bf u} \equiv 0$, the above problem boils down to $$\alpha _{ (f,l)} ^{\top} (0) (x) \; := \; \inf_{x ( \cdot ) \in {\cal
S}_{f} (x)} \int_{0}^{ \infty }e^{a \tau }l ( x( \tau ),x' ( \tau )) d
\tau$$
Before investigating further these examples, we begin by characterizing the epigraph of ${\bf u}^{\top}$:
\[optvalfctstoptimepbvkthm\] Let us assume that $f$ and $l$ are continuous with linear growth and that ${\bf u} : {\bf R}^{n} \mapsto {\bf R}_{+} \cup \{+\infty \}$ is nontrivial, non negative and lower semicontinuous.
Then the epigraph of $ {\bf u}^{\top}:=\alpha _{ (f,l)} ^{\top }({\bf u})$ is the viability kernel $ \mbox{\rm Viab}_{g} ( {\cal E}p ({\bf u})) $ of the epigraph of ${\bf u}$ under $g$.
Consequently, the function $ {\bf u}^{\top}$ is characterized as the smallest of the lower semicontinuous functions ${\bf v}:{\bf R}^{n} \mapsto {\bf R}
\cup \{+\infty \}$ larger than or equal to ${\bf u}$ such that for any $x \in \mbox{\rm
Dom}({\bf v})$, there exists a solution $x ( \cdot )$ to the differential equation $x'=f
(x)$ satisfying property (\[dynineeq\]): $$\forall \; t \geq 0, \; \; e^{a t}{\bf v} (x (t))+ \int_{0}^{t}e^{a
\tau }l
(x ( \tau ),x' ( \tau ))d \tau \; \leq \; {\bf v} (x)$$
[**Proof**]{} — Indeed, to say that a pair $ (x,y)$ belongs to the viability kernel $
\mbox{\rm Viab}_{g} ( {\cal E}p ({\bf u})) $ means that there exists a solution $ x ( \cdot ) \in {\cal S}_{ f} (x)$ such that $$\forall \; t \geq 0, \; \;\left( x (t) , e^{-at}y-\int_{0}^{t}e^{-a(t -
\tau }l ( x( \tau ),x' ( \tau )) d \tau \right) \; \in \; {\cal E}p ({\bf
u})$$ i.e., if and only if $$\forall \; t \geq 0, \; \; e^{a t} {\bf u} (x (t)) + \int_{0}^{t}e^{a
\tau }l
( x( \tau ),x' ( \tau )) d \tau \; \leq \; y$$ This implies that $${\bf u}^{\top} (x) \; := \; \inf_{x ( \cdot ) \in {\cal S}_{f} (x)}
\left( \sup_{t \geq 0} \left( {\bf u} (x (t)) + \int_{0}^{t}l ( x( \tau
),x' (
\tau )) d \tau \right) \right) \; \leq \; y$$ and thus, that $ \mbox{\rm Viab}_{g} ( {\cal E}p ({\bf u}))$ is contained in $
{\cal E}p ( {\bf u}^{\top } )$.
Since the set $ {\cal S}_{f} (x)$ of solutions is compact in the space $
{\cal C} (0, \infty ;{\bf R}^{n})$ thanks to Theorem \[03A44\] and since the function $x ( \cdot ) \mapsto \int_{0}^{t} l (x ( \tau ),f (x ( \tau )))$ is continuous on $ {\cal C} (0, \infty ;{\bf R}^{n})$, the infimum $${\bf u}^{\top }(x) \; := \; \sup_{t \geq 0} \left( e^{a t}{\bf u} (
\bar{x} ( t))
+ \int_{0}^{ t} e^{a \tau }l ( \bar{x}( \tau ), \bar{x}' ( \tau )) d
\tau \right)$$ is reached by a solution $ \bar{x} ( \cdot ) \in {\cal S}_{f} (x)$. Consequently, the function $$\left( \bar{x} (t), e^{-at}{\bf u}^{\top } (x)-\int_{0}^{t}e^{-a(t - \tau
}l (
\bar{x}( \tau ), \bar{x}' ( \tau )) d \tau \right) \; \in \; {\cal E}p
({\bf u}^{\top })$$ is viable in the epigraph of ${\bf u} ^{\top }$. Therefore, $ (x, {\bf u} ^{\top } (x))$ belongs to the viability kernel of the epigraph of ${\bf u}$.
By Lemma \[tranprplemm\], the epigraph of ${\bf u} ^{\top }$ being the viability kernel, contains the epigraph of any lower semicontinuous functions ${\bf v}:{\bf R}^{n} \mapsto {\bf R} \cup \{+\infty \}$ larger than or equal to ${\bf u}$ viable under $g$, i.e., satisfying property (\[dynineeq\]). Therefore, the function $ {\bf u}^{\top }$ is the smallest of the lower semicontinuous functions ${\bf v}:{\bf R}^{n} \mapsto {\bf R} \cup
\{+\infty \}$ larger than or equal to ${\bf u}$ satisfying property (\[dynineeq\]). $\; \; \Box$
\[optvalfctstoptimepbvkthm02\] We posit the assumptions of Theorem \[optvalfctstoptimepbvkthm\]. Then the value function $ {\bf u}^{\top }$ is characterized as the smallest of the nonnegative lower semicontinuous functions ${\bf v}:{\bf R}^{n}
\mapsto {\bf R}
\cup \{+\infty \}$ satisfying for every $x$ $$\left\{ \begin{array}{ll}
i) & {\bf u} (x) \; \leq \; {\bf v}(x)\\
ii) & D _{\uparrow }{\bf v} (x) (f (x))+l (x,f (x)) +a{\bf v} (x)\; \leq
\; 0
\end{array} \right.$$ Furthermore, it satisfies the property $$\left\{ \begin{array}{l}
\forall \; x \; \mbox{ such that} \; {\bf u} (x) \; < \; {\bf
u}^{\top } (x),\\
D _{\uparrow }{\bf u}^{\top }({\bf u}) (x) (-f (x))-l (x,f (x)) - a {\bf
u}^{\top } (x)\;
\leq \; 0
\end{array} \right.$$
[**Remark**]{} — If the function ${\bf u} ^{\top }$ is differentiable, then the contingent epiderivative coincides with the usual derivatives, so that ${\bf u} ^{\top
}$ is a solution to the linear Hamilton-Jacobi “differential variational inequalities” $$\left\{ \begin{array}{ll}
i) & {\bf u} (x) \; \leq \; {\bf u} ^{\top } (x)\\
ii) & \displaystyle{\left\langle \frac{ \partial }{ \partial
x}{\bf u}
^{\top } (x),f (x)\right\rangle +l (x,f (x)) +a{\bf u} ^{\top } (x)\; \leq
\;
0}\\
iii) & \displaystyle{({\bf u} (x)-{\bf u} ^{\top } (x)) \left(
\left\langle
\frac{ \partial }{ \partial x}{\bf u} ^{\top } (x),f (x)\right\rangle +l
(x,f
(x)) +a{\bf u} ^{\top } (x) \right) \; = \; 0}
\end{array} \right.$$
[**Proof**]{} — By the Nagumo Theorem, the epigraph of ${\bf v}$ is viable under $g$ if and only if $$\forall \; (x,y) \in {\cal E}p ({\bf v}), \; \; (f (x), -ay-l (x,f
(x)))
\; \in \; T_{ {\cal E}p ({\bf v})} (x,y)$$ When $y={\bf v} (x)$, we deduce from the fact that the contingent cone to the epigraph of ${\bf v}$ at $ (x,{\bf v} (x))$ $$T_{ {\cal E}p ({\bf v})} (x,{\bf v} (x)) \; := \; {\cal E}p (D
_{\uparrow }{\bf v}
(x))$$ that if the epigraph of ${\bf v}$ is viable under $g$, then $$\forall
\; x
\in \mbox{\rm Dom} ({\bf v}), \; \;D _{\uparrow }{\bf v} (x) (f (x))+l
(x,f (x)) +a{\bf v}
(x)\; \leq \; 0$$ Conversely, this inequality implies that $ (f (x), -a{\bf v} (x)-l
(x,f
(x))) \in T_{ {\cal E}p ({\bf v})} (x,{\bf v} (x))$. It also implies that if $y >{\bf v} (x)$, $ (f (x), -ay-l (x,f (x)))$ belongs to $T_{ {\cal E}p
({\bf v})}
(x,y)$. Indeed, we know that there exist sequences $h_{n}>0$ converging to $0$, $v_{n}$ converging to $f (x)$ and $ \varepsilon _{n}$ converging to $0$ such that $$(x+h_{n}v_{n}, {\bf v} (x) - h_{n}(a{\bf v} (x)+l (x,f (x)) + h_{n}
\varepsilon_{n}) \; \in \; {\cal E}p ({\bf v})$$ We thus deduce that $$\left\{ \begin{array}{l}
(x+h_{n}v_{n}, y - h_{n}(ay+l (x,f (x)) + h_{n} \varepsilon_{n}) \\ =
\; (x+h_{n}v_{n}, {\bf v} (x) - h_{n}(a{\bf v} (x)+l (x,f (x)) + h_{n}
\varepsilon_{n})
+ (0, (1-h_{n})(y-{\bf v} (x)) )\\ \in \; {\cal E}p ({\bf v})+ \{0\}
\times {\bf
R}_{+} \; = \; {\cal E}p ({\bf v}) \\
\end{array} \right.$$ and thus, that $ (f (x), -ay-l (x,f (x)))$ belongs to $T_{ {\cal E}p ({\bf
v})}
(x,y)$.
Finally, Theorem \[viablbascharthm\] states that ${\bf u} ^{\top }$ satisfies $$\label{locbacinv}
\forall \, (x,{\bf u} ^{\top }(x)) \in {\cal E}p ({\bf u} ^{\top })
\backslash
\partial \left( {\cal E}p ({\bf u}) \right), \; (-f (x), a{\bf u}
^{\top } (x) +l
(x,f (x))) \, \in \, T_{ {\cal E}p ({\bf u} ^{\top })} (x,{\bf u} ^{\top
} (x))$$ which, joined to the other properties, can be translated as the “Frankowska” solution to the variational inequalities. $\; \; \Box$
Lyapunov Functions
------------------
Consider a differential equation $x'= f(x)$ and a nontrivial nonnegative lower semicontinuous extended function ${\bf u}:{\bf R}^{n} \mapsto {\bf
R}_{+} \cup
\{+\infty \}$.
The function $\alpha _{ (f,0)} ^{\top} ({\bf u}):{\bf R}^{n} \mapsto {\bf
R}_{+} \cup
\{+\infty \}$ defined by $$\alpha _{ (f,0)} ^{\top} ({\bf u}) (x) \; := \; \inf_{x ( \cdot ) \in
{\cal
S}_{f} (x)} \left( \sup_{t \geq 0} e^{a t}{\bf u} (x (t)) \right)$$ is said [*to enjoy the $a$-Lyapunov property*]{} because for any initial state $x_{0}$, there exists a solution to the differential equation $x'=f (x)$ satisfying $$\label{lyap03A82}
\forall \; t \geq 0, \;\; {\bf u} (x(t)) \; \leq \; e^{-a
t}\alpha
_{ (f,0)} ^{\top} ({\bf u}) (x_{0})$$ Such inequalities allow us to deduce many properties on the asymptotic behavior of ${\bf v}$ along the solutions to the differential equation. This may be quite useful when ${\bf u}$ is the distance function $d_{M}(\cdot)$ to a subset. The domain of this Lyapunov function $\alpha _{
(f,0)} ^{\top} (d_{M})
$ provides the $a$-basin of attraction of $M$, which is the set of states from which a solution $x ( \cdot )$ to the differential equation converges exponentially to $M$: $$\forall \; x_{0} \in \mbox{\rm Dom}(\alpha _{ (f,0)} ^{\top}
(d_{M})), \; \; d_{M}(x(t)) \; \leq \; e^{-a t} \alpha _{ (f,0)} ^{\top}
(d_{M}) (x_{0})$$
The main question we face is [*to characterize this Lyapunov function*]{}. Ever since Lyapunov proposed in 1892 his second method for studying the behavior of a solution around an equilibrium, finding Lyapunov functions for such and such differential equation has been a source of numerous problems requiring most often many clever tricks.
We deduce from Theorems \[optvalfctstoptimepbvkthm\] and \[optvalfctstoptimepbvkthm02\] with $l=0$ the following characterization of Lyapunov functions:
\[optvalfctstoptimepbvkthm023\] Let us assume that $f$ is continuous with linear growth and that ${\bf u} :
{\bf R}^{n}
\mapsto {\bf R}_{+} \cup \{+\infty \}$ is nontrivial, non negative and lower semicontinuous. Then the epigraph of the Lyapunov function $ \alpha _{ (f,0)} ^{\top
}({\bf u})$ is the viability kernel $ \mbox{\rm Viab}_{g} ( {\cal E}p ({\bf u})) $ of the epigraph of ${\bf u}$ under $g$. Therefore, the Lyapunov function $ \alpha _{ (f,0)} ^{\top }({\bf
u})$ is the smallest of the nonnegative lower semicontinuous functions ${\bf
v}:{\bf R}^{n}
\mapsto {\bf R} \cup \{+\infty \}$ enjoying the $a$-Lyapunov property, i.e., such that from any $x_{0}\in \mbox{\rm Dom}({\bf v})$ starts at least one solution to the differential equation $x'=f (x)$ satisfying $$\forall \; t \geq 0, \; \; {\bf v} (x (t)) \; \leq \; {\bf v}
(x_{0})e^{-at}$$ or, equivalently, $$D _{\uparrow }{\bf v} (x) (f (x)) +a{\bf v} (x)\; \leq \; 0$$ Furthermore, if ${\bf u} (x) < \alpha _{ (f,0)} ^{\top }({\bf u}) (x)$, it satisfies $$D _{\uparrow }\alpha _{ (f,0)} ^{\top }({\bf u}) (x) (-f (x)) - a \alpha
_{
(f,0)} ^{\top }({\bf u}) (x)\; \leq \; 0$$
Finite Length Solutions
-----------------------
We define now $l (x,p):= \|p\|$, so that $l (x,f (x))= \|f (x)\|$, and take $a:=0$ and ${\bf u} (x):=0$. Then $$\alpha _{ (f, \|f\|)} ^{\top }(0) (x_{0}) \; = \; \inf_{x ( \cdot ) \in
{\cal S}_{f} (x)} \int_{0}^{+ \infty } \|x' ( \tau )\| d \tau$$ is the minimal length of the trajectories of the solutions $x ( \cdot )$ to the differential equation $x'=f (x)$ starting from $x_{0}$.
Its epigraph is the viability kernel of ${\bf R}^{n} \times {\bf R}_{+} $ under the system of differential equations $ (x',y') = (f (x), - \|f (x)\|)$. The minimal length is the smallest of the nonnegative lower semicontinuous functions ${\bf v}:{\bf R}^{n} \mapsto {\bf R} \cup \{+\infty \}$ satisfying for every $x$ $$D _{\uparrow }{\bf v} (x) (f (x)) + \|f (x)\| \; \leq \; 0$$ and satisfies whenever the length $\alpha _{ f, \|f\|} ^{\top }(0) (x)>0$ is strictly positive $$D _{\uparrow }\alpha _{ f, \|f\|} ^{\top }(0) (x) (-f (x)) - \|f (x)\|
\leq \; 0$$
Stopping Time Problem
---------------------
We still consider the ${\bf u}:{\bf R}^{n} \mapsto {\bf R}_{+} \cup
\{+\infty \}$, regarded as an “obstacle” in problems of unilateral mechanics. We associate with it the stopping time problem $${\bf u}^{\bot} (x) \; := \; \alpha _{ (f,l)} ^{\bot } ({\bf u}) (x) \; :=
\; \inf_{x
( \cdot ) \in {\cal S}_{f} (x)} \left( \inf_{t \geq 0} \left( e^{a
t}{\bf u}
(x (t)) + \int_{0}^{t}e^{a \tau }l ( x( \tau ),x' ( \tau )) d \tau
\right) \right)$$
We begin by characterizing its epigraph:
\[optvalfctstoptimepbthm\] Let us assume that $f$ and $l$ are continuous with linear growth and that ${\bf u} : {\bf R}^{n} \mapsto {\bf R}_{+} \cup \{+\infty \}$ is nontrivial, non negative and lower semicontinuous.
Then the epigraph of $ {\bf u}^{\bot}:=\alpha _{ (f,l)} ^{\bot }({\bf u})$ is the capture basin $ \mbox{\rm Capt}_{g} ( {\cal E}p ({\bf u})) $ of the epigraph of ${\bf u}$ under $g$.
[**Proof**]{} — To say that a pair $ (x,y)$ belongs to the capture basin $ \mbox{\rm
Capt}_{g} ( {\cal E}p ({\bf u})) $ means that there exist a solution $ ( x
(
\cdot )) \in {\cal S}_{ (f)} (x)$ and $t \geq 0$ such that $$\left( x (t) , e^{-at}y-\int_{0}^{t}e^{-a(t - \tau }l ( x( \tau ),x' ( \tau
)) d \tau \right) \; \in \; {\cal E}p ({\bf u})$$ i.e., if and only if $$e^{a t} {\bf u} (x (t)) + \int_{0}^{t}e^{a \tau }l ( x( \tau ),x' ( \tau
)) d
\tau \; \leq \; y$$ This implies that $$\left\{ \begin{array}{l}
{\bf u}^{\bot}(x) \\ \displaystyle{:= \; \inf_{x ( \cdot ) \in {\cal
S}_{f} (x)}
\left( \inf_{t \geq 0} \left( e^{a t} {\bf u} (x (t)) + \int_{0}^{t} e^{a
\tau }l ( x( \tau ),x' ( \tau )) d \tau \right) \right) \; \leq \; y}
\\
\end{array} \right.$$ and thus, that $ \mbox{\rm Capt}_{g} ( {\cal E}p ({\bf u}))$ is contained in $
{\cal E}p ( {\bf u}^{\bot } )$.
Since the infimum $${\bf u}^{\bot} (x) \; := \; e^{a \bar{t}}{\bf u} ( \bar{x} (
\bar{t})) +
\int_{0}^{ \bar{t}}e^{a \tau }l ( \bar{x}( \tau ), \bar{x}' ( \tau )) d
\tau$$ is reached by a solution $ \bar{x} ( \cdot ) \in {\cal S}_{f} (x)$ at a time $ \bar{t}$, this states that $ (x, {\bf u}^{\bot}(x))$ belongs to the capture basin of the epigraph of ${\bf u}$. $\; \; \Box$
In order to apply the properties of capture basins, we need to check that $X \times {\bf R}_{+} $ is a repeller under the auxiliary system $ g $.
\[awlem01\] Let us assume that there exist real constants $ \gamma _{-}$ and $ \delta ^{-}$ such that $$\left\{ \begin{array}{ll} \label{aweq02}
i) & \inf_{x \in X} \frac{ \langle x,f (x) \rangle }{ \|x\|} \;
\geq \; \gamma_{-} ( \|x\|+1)\\
ii) & \inf_{x \in X} l (x,f (x))\; \geq \; \delta _{-} (
\|x\|+1)
\end{array} \right.$$ and if $ a+ \gamma _{-} >0$, then $X \times {\bf R}_{+} $ is a repeller.
[**Proof**]{} — Whenever $l$ is nonnegative, the backward solutions $ ( x ( \cdot
),y ( \cdot ))$ starting from $X \times {\bf R}_{+} $ are viable in $X \times {\bf R}_{+}$ because $y' (t)= -ay (t)+l (x
(t)) \geq 0$.
Let $ (x ( \cdot ),y ( \cdot ))$ be the solution to the differential equation $ (x',y') = g (x,y)$ starting from $
(x_{0},y_{0})$.
Therefore $$\frac{d}{dt} \|x (t)\| \; = \; \left\langle x' (t), \frac{x
(t)}{ \|x (t)\|}\right\rangle \; = \; \left\langle f(x (t))),
\frac{x (t)}{ \|x (t)\|}\right\rangle \; \geq \; \gamma _{-} (
\|x (t)\|+1)$$ so that $$\forall \; t \geq 0, \; \; \|x (t) \| \; \geq \; e^{ \gamma
_{-}t} ( \|x_{0}\|+1) -1$$ Furthermore, since $$l (x ( \tau ),x' ( \tau )) \; \geq \; \delta_{-} ( \|x ( \tau
)\|+1) \; \geq \; \delta_{-} ( \|x_{0}\|+1)e^{ \gamma _{-}t}$$ and since $$e^{at} y (t) \; = \; y_{0} - \int_{0}^{t}e^{a \tau } l (x ( \tau
))d \tau$$ we infer that $$e^{at} y (t) \; \leq \; y_{0} - \delta_{-} (
\|x_{0}\|+1)\int_{0}^{t}e^{ \gamma_{-}+ a)\tau } \; = \; y_{0} -
\frac{\delta_{-} ( \|x_{0}\|+1)}{ \gamma_{-}+a} \left(e^{
(\gamma_{-}+a)t}-1 \right)$$ Consequently, if $ \gamma {-}+ a>0$ $$e^{at}y (t) \; \leq \; y_{0}+\frac{\delta_{-} ( \|x_{0}\|+1)}{
\gamma_{-}+a} - \frac{\delta_{-} ( \|x_{0}\|+1)}{ \gamma_{-}+a}
e^{ (\gamma _{-}-a)t}$$ so that $y (t)$ becomes negative in finite time. $\; \; \Box$
\[optvalfctstoptimepbthm28\] We posit the assumptions of Proposition \[optvalfctstoptimepbthm\] and we assume that $$\forall \; x \in K, \; \;\forall \; x ( \cdot ) \in {\cal S}_{f} (x),
\; \; \int_{0}^{+ \infty }e^{a \tau }l (x ( \tau ),x' ( \tau ))d \tau
\; = \; + \infty$$
Then ${\bf u}^{\bot }$ is characterized as the [**unique**]{} nonnegative lower semicontinuous functions ${\bf v}:{\bf R}^{n} \mapsto {\bf R} \cup
\{+\infty \}$ such that from any $x $ satisfying ${\bf v} (x)<{\bf u} (x)$ starts a solution $x
( \cdot ) \in {\cal S}_{f} (x)$ satisfying, for some time $T>0$ $$\forall \; t \in [0,T], \; \; e^{a t} {\bf v} (x (t)) +
\int_{0}^{t}e^{a \tau }l ( x( \tau ),x' ( \tau )) d \tau \; \leq
\; {\bf v} (x)$$ and that, for any $T>0$ and any $x_{T} \in \mbox{\rm Dom}({\bf u})$, all solutions $x ( \cdot )$ to the differential equation $x'=f (x)$ arriving at $x_{T}$ at time $T$ satisfy $$\forall \; t \in [0,T], \; \; e^{a t} {\bf v} (x (t)) +
\int_{0}^{t}e^{a \tau }l ( x( \tau ),x' ( \tau )) d \tau \; \leq
\; {\bf v} (x ( 0))$$ The function ${\bf u} ^{\bot }$ is also the smallest of the lower semicontinuous functions ${\bf v}$ satisfying $$\left\{ \begin{array}{ll}
i) & 0 \; \leq \; {\bf v} (x) \; \leq \; {\bf u} (x)\\
ii) & \mbox{\rm if} \; 0 \leq {\bf v} (x) < {\bf u} (x), \; \; D
_{\uparrow }{\bf v} (x) (f
(x))+l (x,f (x)) +a{\bf v} (x)\; \leq \; 0
\end{array} \right.$$ If we assume furthermore that $f$ and $l$ are Lipschitz, then the function ${\bf u} ^{\bot }$ is the [**unique**]{} solution ${\bf v} \geq 0$ to the system of “differential inequalities”: for every $x \in \mbox{\rm Dom}({\bf v})$, $$\left\{ \begin{array}{ll}
i) & 0 \; \leq \;{\bf v} (x) \; \leq \; {\bf u} (x)\\
ii) & \mbox{\rm if} \; 0 \leq {\bf v} (x) < {\bf u} (x) , \; \;
D _{\uparrow }{\bf v} (x) (f (x))+l (x) + a{\bf v} (x) \; \leq \; 0\\
iii) & \mbox{\rm if} \; 0 \leq {\bf v} (x) \leq {\bf u} (x) , \; \; D
_{\uparrow }{\bf v}
(x) (-f (x) -l (x),f (x)) - a {\bf v}(x)\; \leq \;0\\
\end{array} \right.$$
Knowing the function ${\bf u} ^{\bot }$, the stopping time is the first time $
\bar{t}\geq 0$ when ${\bf v} (x ( \bar{t}))= {\bf u} ^{\bot }( x (
\bar{t}))$.
[**Remark**]{} — If the function ${\bf u} ^{\bot }:= \alpha _{ (f,l)} ^{\bot }({\bf u})$ is differentiable, then the contingent epiderivative coincides with the usual derivatives, so that ${\bf u} ^{\bot }$ is a solution to the linear Hamilton-Jacobi “differential variational inequalities” $$\left\{ \begin{array}{ll}
i) & 0 \; \leq \; {\bf u} ^{\bot }(x) \; \leq \; {\bf u} (x)\\
ii) & \displaystyle{\left\langle \frac{ \partial }{ \partial
x}{\bf u}
^{\bot } (x),f (x)\right\rangle +l (x,f (x)) +a{\bf u} ^{\bot } (x)\; \geq
\;
0}\\
iii) & \displaystyle{({\bf u} (x)-{\bf u} ^{\bot } (x)) \left(
\left\langle
\frac{ \partial }{ \partial x}{\bf u} ^{\bot } (x),f (x)\right\rangle +l
(x,f
(x)) +a{\bf u} ^{\bot } (x) \right) \; = \; 0}
\end{array} \right.$$
[**Proof**]{} — Since the Lagrangian is nonnegative, the closed subset ${\bf R}^{n}
\times {\bf
R}_{+} $ is backward invariant under $g$. It is a repeller under $g$ whenever $$\forall \; x \in {\bf R}^{n}, \; \forall \; x ( \cdot ) \in {\cal S}_{f}
(x), \; \;
\int_{0}^{+ \infty }e^{a \tau }l (x ( \tau ),x' ( \tau ))d \tau \; = \;
+ \infty$$ By Theorem \[wonderfulthmthmbis\], $ {\cal E}p ({\bf u} ^{\bot }):=
\mbox{\rm Capt}_{g} ( {\cal E}p ({\bf u}))$ is the unique closed subset $
{\cal
{\bf v}}$, and in particular, the unique epigraph ${\cal E}p ({\bf v})$ of lower semicontinuous function ${\bf v}$, which satisfies $$\left\{ \begin{array}{ll}
i) & {\cal E}p ({\bf u}) \; \subset \; {\cal E}p ({\bf v})\; \subset \;
{\bf R}^{n} \times
{\bf R}_{+} \\
ii) & {\cal E}p ({\bf v}) \backslash {\cal E}p ({\bf u}) \;\; \mbox{\rm is
locally viable under $g$}\\
iii) & {\cal E}p ({\bf v}) \;\; \mbox{\rm is backward invariant under
$g$}\\
\end{array} \right.$$ The first condition means that for any $x \in {\bf R}^{n}$, $0 \leq {\bf
v} (x) \leq {\bf u}
(x)$ and we observe that $(x,y) \in {\cal E}p ({\bf v} ) \backslash {\cal
E}p ({\bf u})$ if and only if $y \in [{\bf v} (x),{\bf u} (x)[$. Hence, the first statement of the theorem ensues.
By Theorem \[viablcaptbascharthm\], $ {\cal E}p ({\bf u} ^{\bot }):=
\mbox{\rm Capt}_{g} ( {\cal E}p ({\bf u}))$ is also the smallest of the nonnegative lower semicontinuous ${\bf v}$ satisfying $$\left\{ \begin{array}{ll}
i) & {\cal E}p ({\bf u}) \; \subset \; {\cal E}p ({\bf v}) \; \subset \;
{\bf R}^{n} \times
{\bf R}_{+} \\
ii) & \forall \; (x,y) \in {\cal E}p ({\bf v} ) \backslash {\cal E}p
({\bf u}), \;
\; (f (x),-l (x,f (x))-ay) \; \in \; T_{ {\cal E}p ({\bf v})} (x,y)
\end{array} \right.$$
When $y={\bf v} (x)$, the second condition can be written $$\mbox{\rm if} \; {\bf v} (x) <{\bf u} (x), \; \; D _{\uparrow }{\bf
v} (x)+a{\bf v} (x)+ l
(x,f (x)) \; \leq \; 0$$ Conversely, this condition implies that for any $y \in ]{\bf v}
(x),{\bf u}
(x)[$, $(f (x),-l (x,f (x))-ay) $ also belongs to $ T_{ {\cal E}p ({\bf
v})}
(x,y)$ as in the proof of Theorem \[optvalfctstoptimepbvkthm02\].
Since $ {\cal E}p ({\bf u} ^{\bot }) = \mbox{\rm Capt}_{g} ( {\cal E}p
({\bf u}))$ is the unique closed subset satisfying the above properties and being backward invariant under $g$, this implies that $$\mbox{\rm if} \; {\bf v} (x) \leq {\bf u} (x) , \; \; D _{\uparrow
}{\bf v} (x) (-f
(x) -l (x),f (x)) - a {\bf v}(x)\; \leq \;0$$ The converse is true when $f$ and $l$ are Lipschitz, or whenever the solution to the system $ (x',y')=g (x,y)$ is unique. In this case, the function ${\bf u} ^{\bot }$ is the unique solution satisfying the two properties. $\; \; \Box$
Minimal Time and Minimal Length Solutions
-----------------------------------------
Let us consider a closed subset $K \subset {\bf R}^{n}$ and $ \psi_{K}$ its indicator and take $a=0$.
We observe that the hitting time (or minimal time) function $ \omega_{K}^{f
^{\flat }}$ is equal to $$\omega_{K}^{f ^{\flat }} \; = \; \alpha_{f,1}^{\bot} ( \psi_{K})$$
In the same way, we introduce the [*minimal length functional*]{} associating with $x ( \cdot )$ $$\lambda_{K}(x ( \cdot )) \; := \; \inf_{ \{t \; | \;x (t) \in
K\}}\int_{0}^{t} \|x' (s)\| ds$$ the minimal length of the curve $s \mapsto x (s)$ from $0$ to $t$ such that $x (t) \in K$. We next define the “minimal length” function $ \lambda_{K}^{f ^{\flat }}$ by $$\lambda_{K}^{f ^{\flat }} (x) \; := \; \inf_{x ( \cdot ) \in {\cal S}_{f}
(x)}\lambda_{K}(x ( \cdot ))$$
We note that $$\lambda _{K}^{f ^{\flat }} \; = \; \alpha_{f, \|f\|}^{\bot} ( \psi_{K})$$
So, these two functions enjoy the properties proved above. For instance:
1. The minimal time function $ \omega_{K}^{f ^{\flat }}$ is the smallest nonnegative lower semicontinuous ${\bf v}$ function vanishing on $K$ such that, $$\forall \; x \notin K, \;\; D _{\uparrow } {\bf v} (x) (f (x)) +1 \; \leq
\;0$$ If $f$ is assumed furthermore to be Lipschitz, it is the [**unique**]{} nonnegative lower semicontinuous solution vanishing on $K$ satisfying the above inequalities and $$\forall \; x \in {\bf R}^{n}, \; \; D _{\uparrow } {\bf v} (x) (-f (x))
- 1 \; \leq \;0$$
2. Assume that $$\forall \; x \in {\bf R}^{n}, \; \; \inf_{x ( \cdot ) \in {\cal S}_{f}
(x)} \int_{0}^{+ \infty } \|x' (s)\|ds \; = \; + \infty$$ Then the minimal length function $ \lambda_{K}^{f ^{\flat }}$ is the smallest nonnegative lower semicontinuous ${\bf v}$ function vanishing on $K$ such that, $$\forall \; x \notin K, \;\; D _{\uparrow } {\bf v} (x) (f (x)) + \|f
(x)\| \; \leq \;0$$ If $f$ is assumed furthermore to be Lipschitz, it is the [**unique**]{} nonnegative lower semicontinuous solution vanishing on $K$ satisfying the above inequalities and $$\forall \; x \in {\bf R}^{n}, \; \; D _{\uparrow } {\bf v} (x) (-f (x))
- \|f (x)\| \; \leq \;0$$
Since the minimal time and minimal length functions coincide with the indicator of $ \psi_{K}$ on $K$, the above conditions imply that $K$ is backward invariant under $f$ whenever $f$ is Lipschitz.
Viscosity Type Solutions
------------------------
We now use the characterizations in terms of normal cones for deriving the formulations in terms of subgradients instead of contingent epiderivatives.
\[optvalfctstoptimepbvkthm02gg\] We posit the assumptions of Theorem \[optvalfctstoptimepbvkthm\]. Then the value function $ {\bf u}^{\top }$ is the solution to $$\left\{ \begin{array}{ll}
i) & {\bf u} (x) \; \leq \; {\bf u} ^{\top } (x)\\
ii) & \displaystyle{ \forall \; p \in \partial _{-}{\bf u}^{\top}
(x),
\; \; \left\langle p,f (x)\right\rangle +l (x,f (x)) +a{\bf u} ^{\top }
(x)\;
\leq \; 0}\\ & \mbox{\rm and} \\
& \displaystyle{\forall \; p \in \mbox{\rm Dom}(D _{\uparrow
}{\bf u}^{\top} (x))^{-}, \; \; \langle p,f (x) \rangle \; \leq 0 \;}\\
iii) & \displaystyle{\forall \; p \in \partial _{-}{\bf u}^{\top}
(x),
\; \;({\bf u} (x)-{\bf u} ^{\top } (x)) \left( \left\langle p,f
(x)\right\rangle +l
(x,f (x)) +a{\bf u} ^{\top } (x) \right) \; = \; 0}\\
& \mbox{\rm and} \\
& \displaystyle{\forall \; p \in \mbox{\rm Dom}(D _{\uparrow
}{\bf u}^{\top} (x))^{-}, \; \; ({\bf u} (x)-{\bf u} ^{\top } (x))\langle
p,f (x) \rangle
\; = 0 \;}\\
\end{array} \right.$$
Such a solution, recently discovered independently by Frankowska and Barron & Jensen, are sometime called “bilateral solutions” to Hamilton-Jacobi equation $$\left\langle \frac{ \partial }{ \partial x}{\bf u} (x),f (x)\right\rangle
+l
(x,f (x)) +a{\bf u} (x) \; = \; 0$$ The method we present here is due to Frankowska.
[**Proof**]{} — By Theorem \[optvalfctstoptimepbvkthm02\], we know that whenever ${\bf u}
(x) \leq {\bf u}^{\top} (x)$, $$(f (x),-a{\bf u}^{\top} (x)-l (x,f (x))) \; \in \; T_{ {\cal E}p
({\bf u}^{\top})} (x,{\bf u}^{\top} (x))$$ and that whenever ${\bf u} (x) < {\bf u}^{\top} (x)$, $$(f (x),-a{\bf u}^{\top} (x)-l (x,f (x))) \; \in \; T_{ {\cal E}p
({\bf u}^{\top})} (x,{\bf u}^{\top} (x)) \cap - T_{ {\cal E}p ({\bf
u}^{\top})} (x,{\bf u}^{\top}
(x))$$ Theorem \[3russianfrank\] implies that these conditions are equivalent to $$\forall \; (p, \lambda ) \in N_{ {\cal E}p ({\bf u}^{\top})} (x,{\bf
u}^{\top}
(x)), \; \; \langle p,f (x) \rangle - \lambda (a{\bf u}^{\top} (x)+l (x,f
(x)))
\; \leq \; 0$$ whenever ${\bf u} (x) \leq {\bf u}^{\top} (x)$, and to $$\forall \; (p, \lambda ) \in N_{ {\cal E}p ({\bf u}^{\top})} (x,{\bf
u}^{\top}
(x)), \; \; \langle p,f (x) \rangle - \lambda (a{\bf u}^{\top} (x)+l (x,f
(x)))
\; = \; 0$$ whenever ${\bf u} (x) < {\bf u}^{\top} (x)$.
It remains now to recall that $$\left\{ \begin{array}{ll}
i) & (p,-1) \in N_{ {\cal E}p ({\bf u}^{\top})} (x,{\bf u}^{\top}
(x))
\;\mbox{\rm if and only if}\;p \in \partial_{-}{\bf u}^{\top} (x)\\
ii) & (p,0) \in N_{ {\cal E}p ({\bf u}^{\top})} (x,{\bf u}^{\top}
(x))
\;\mbox{\rm if and only if}\; p \in \mbox{\rm Dom}(D _{\uparrow }{\bf
u}^{\top}
(x))^{-}
\end{array} \right.$$ Taking $ \lambda =-1$, we obtain $$\forall \; p \in \partial _{-}{\bf u}^{\top} (x), \; \;\langle p,f
(x)
\rangle - \lambda (a{\bf u}^{\top} (x)+l (x,f (x))) \; \leq \; 0$$ and whenever ${\bf u} (x) < {\bf u}^{\top} (x)$, $$\forall \; p \in \partial _{-}{\bf u}^{\top} (x), \; \; \langle p,f
(x)
\rangle - \lambda (a{\bf u}^{\top} (x)+l (x,f (x))) \; = \; 0$$ Taking $ \lambda =0$ yields that for all $p \in \mbox{\rm Dom}(D
_{\uparrow }{\bf u}^{\top} (x))^{-}$, $ \langle p,f (x) \rangle \leq 0$ if ${\bf u}
(x) \leq {\bf u}^{\top} (x)$ and $ \langle p,f (x) \rangle =0$ if ${\bf
u} (x) <
{\bf u}^{\top} (x)$. This means that $f (x)$ belongs to the closure of $\mbox{\rm Dom}(D _{\uparrow }{\bf u}^{\top} (x))$ in the general case and that $x$ belongs to the vector space spanned by $\mbox{\rm Dom}(D _{\uparrow
}{\bf u}^{\top} (x))$ when ${\bf u} (x) < {\bf u}^{\top} (x)$. $\; \; \Box$
We obtain an analogous statement for the function ${\bf u}^{\bot}$:
\[optvalfctstoptimepbthm28gg\] We posit the assumptions of Theorem \[optvalfctstoptimepbthm28\]. Then the value function $ {\bf u}^{\bot }$ is the solution to $$\left\{ \begin{array}{ll}
i) & 0 \; \leq \; {\bf u}^{\bot}(x) \; \leq \; {\bf u} (x)\\
ii) & \displaystyle{ \forall \; p \in \partial _{-}{\bf u}^{\bot}
(x),
\; \; \left\langle p,f (x)\right\rangle +l (x,f (x)) +a{\bf u} ^{\bot }
(x)\;
\geq \; 0}\\ & \mbox{\rm and} \\
& \displaystyle{\forall \; p \in \mbox{\rm Dom}(D{\bf u}
_{\downarrow }
^{\top} (x))^{-}, \; \; \langle p,f (x) \rangle \; \geq 0 \;}\\
iii) & \displaystyle{\forall \; p \in \partial _{-}{\bf u}^{\bot}
(x),
\; \;({\bf u} (x)-{\bf u} ^{\bot } (x)) \left( \left\langle p,f
(x)\right\rangle +l
(x,f (x)) +a{\bf u} ^{\bot } (x) \right) \; = \; 0}\\
& \mbox{\rm and} \\
& \displaystyle{\forall \; p \in \mbox{\rm Dom}(D _{\downarrow }
{\bf u}^{\top} (x))^{-}, \; \; ({\bf u} (x)-{\bf u} ^{\top } (x))\langle
p,f (x) \rangle \;
= 0 \;}\\
\end{array} \right.$$
[**Proof**]{} — By Theorem \[optvalfctstoptimepbthm28\], we know that whenever ${\bf u}
(x) \geq {\bf u}^{\bot} (x)$, $$(-f (x),a{\bf u}^{\bot} (x) + l (x,f (x))) \; \in \; T_{ {\cal E}p
({\bf u}^{\bot})} (x,{\bf u}^{\bot} (x))$$ and that whenever ${\bf u} (x) > {\bf u}^{\bot} (x)$, $$(f (x),-a{\bf u}^{\bot} (x)-l (x,f (x))) \; \in \; T_{ {\cal E}p
({\bf u}^{\bot})} (x,{\bf u}^{\bot} (x)) \cap - T_{ {\cal E}p ({\bf
u}^{\bot})} (x,{\bf u}^{\bot}
(x))$$ Theorem \[3russianfrank\] implies that these conditions are equivalent to whenever ${\bf u} (x) \geq {\bf u}^{\bot} (x)$, $$\forall \; (p, \lambda ) \in N_{ {\cal E}p ({\bf u}^{\bot})} (x,{\bf
u}^{\bot}
(x)), \; \; \langle p,f (x) \rangle - \lambda (a{\bf u}^{\bot} (x)+l (x,f
(x)))
\; \geq \; 0$$ and that whenever ${\bf u} (x) > {\bf u}^{\bot} (x)$, $$\forall \; (p, \lambda ) \in N_{ {\cal E}p ({\bf u}^{\bot})} (x,{\bf
u}^{\bot}
(x)), \; \; \langle p,f (x) \rangle - \lambda (a{\bf u}^{\bot} (x)+l (x,f
(x)))
\; = \; 0$$ It remains now to translate these statements in terms of subgradients. $\; \; \Box$
[**Remark: Viscosity Solutions**]{} — When we know [*a priori*]{} that the solution ${\bf u}^{\bot}$ is continuous, we can prove that it is also a “viscosity solution” the the Hamilton-Jacobi variational inequalities:
\[optvalfctstoptimepbthm28visc\] We posit the assumptions of Theorem \[optvalfctstoptimepbthm28\], and we assume that $f$ and $l$ are Lipschitz and the function ${\bf u}^{\bot}$ is continuous. Then the value function $ {\bf u}^{\bot }$ is the solution to $$\left\{ \begin{array}{ll}
i) & 0 \; \leq \; {\bf u}^{\bot}(x) \; \leq \; {\bf u} (x)\\
ii) & \displaystyle{ \forall \; p \in \partial _{+}{\bf u}^{\bot}
(x),
\; \; \left\langle -p,f (x)\right\rangle +l (x,f (x)) +a{\bf u} ^{\bot
}
(x)\; \geq \; 0}\\
\\ & \mbox{\rm and} \\
& \displaystyle{\forall \; p \in \mbox{\rm Dom}(D _{\downarrow }
{\bf u}^{\bot})^{-}, \; \; \langle p,f (x) \rangle \; \leq 0 \;}\\
iii) & \displaystyle{\forall \; p \in \partial _{-}{\bf u}^{\bot}
(x),
\; \; \left\langle p,f (x)\right\rangle +l (x,f (x)) +a{\bf u} ^{\bot }
(x)
\; \leq \; 0}\\& \mbox{\rm and} \\
& \displaystyle{\forall \; p \in \mbox{\rm Dom}(D _{\uparrow }
{\bf u}^{\bot} (x))^{-}, \; \; \langle p,f (x) \rangle \; \leq 0 \;}\\
\end{array} \right.$$
A solution to such a system of inequalities is called a ‘viscosity solution’ to Hamilton-Jacobi equation $$\left\langle \frac{ \partial }{ \partial x}{\bf u} (x),f (x)\right\rangle
+l
(x,f (x)) +a{\bf u} ^{\bot } (x) \; = \; 0$$ by Michael Crandall and Pierre-Louis Lions.
[**Proof**]{} — We know that epigraph of ${\bf u}^{\bot}$ is backward invariant under $g$. By Lemmas \[bacwinvcomp\] and \[backinvab\], this implies that its complement is forward invariant under $g$, and, since $f$ and $l$ are Lipschitz, that its closure is also forward invariant. Since we assumed that ${\bf u}^{\bot}$ is continuous, the closure of the complement of the epigraph of ${\bf u}^{\bot}$ is the hypograph of ${\bf u}^{\bot}$. Therefore, $$\forall \; (x,{\bf u} ^{\bot }(x)) \in \mbox{\rm Graph}({\bf u} ^{\bot })
, \; \; (f
(x), -a{\bf u} ^{\bot } (x) -l (x,f (x))) \; \in \; T_{ {\cal H}yp ({\bf
u} ^{\bot
})} (x,{\bf u} ^{\bot } (x))$$ and thus, $$\forall \; (p, \lambda ) \in N_{ {\cal H}yp ({\bf u}^{\bot})} (x,{\bf
u}^{\bot}
(x)), \; \; \langle p,f (x) \rangle - \lambda (a{\bf u}^{\bot} (x)+l (x,f
(x)))
\; \leq \; 0$$ It remains now to recall that $$\left\{ \begin{array}{ll}
i) & (p,1) \in N_{ {\cal H}yp ({\bf u}^{\bot})} (x,{\bf u}^{\bot}
(x))
\;\mbox{\rm if and only if}\; p \in \partial_{+}{\bf u}^{\bot} (x)\\
ii) & (p,0) \in N_{ {\cal H}yp ({\bf u}^{\bot})} (x,{\bf u}^{\bot}
(x))
\;\mbox{\rm if and only if}\; \forall \; p \in \mbox{\rm Dom}(D
_{\downarrow }{\bf u}^{\bot} (x))^{-}
\end{array} \right.$$ for deducing that the above condition is equivalent to $$\forall \; p \in \partial_{+}{\bf u}^{\bot} (x), \; \; \langle
-p,f (x)
\rangle +a{\bf u}^{\bot} (x)+ l (x,f (x)) \; \geq 0 \;$$ and that $f (x)$ belongs to the closure of the domain of $D
_{\downarrow } {\bf u}^{\bot} (x)$ for achieving the proof. $\; \; \Box$
Systems of First-Order Partial Differential Equations
=====================================================
We study here Dirichlet boundary value problems for systems of first-order partial differential equations of the form $$\forall \; j=1, \ldots ,p, \; \; \frac{ \partial }{ \partial t}u (t,x) +
\sum_{i=1}^{n}\frac{ \partial }{ \partial x_{i}}u_{i}
(t,x) f_{i}(t,x,u (t,x)) - g_{j}(t,x,u (t,x)) \; = \; 0$$ on $ {\bf R}_{+} \times \Omega $, where $ \Omega \subset {\bf R}^{n} $ is an open subset, $ \Gamma $ its boundary and $K:=\overline{ \Omega } $ its closure.
It is known that the solution $U$ to the above system can be set-valued, describing “shocks”. This is considered as a pathology whenever the solution is regarded as a map from the input space $ {\bf R}_{+} \times K$ to the output space ${\bf R}^{p}$, but is quite natural when the solution $U$ is considered as a graph, i.e., a subset of ${\bf R}_{+} \times K \times {\bf R}^{p}$ and when the tools of set-valued analysis are used. Since we are looking for set-valued map solutions, we begin by introducing
1. the graph of the [*upper graphical limit*]{} of a sequence of maps $U_{n}:X\leadsto Y$ (single-valued or set-valued) is the upper limit of the graphs of $U_{n}$,
2. [*the contingent derivative*]{} $DU (x,y)$ at point $ (x,y)$ of the graph of $U$ is the upper graphical limit of the difference quotients $
\nabla _{h} U (x,y)$, so that the graph of the contingent derivative is the contingent cone to the graph of $U$: $$\mbox{\rm Graph}(DU (x,y)) \; = \; T_{ \mbox{\rm Graph}(U) (x,y)}$$
Introducing
1. an [*initial data*]{} $u_{0}: K \mapsto {\bf R}^{p}$,
2. a [*boundary data*]{} $v_{ \Gamma}: {\bf R}_{+} \times \partial
K \mapsto {\bf R}^{p}$.
we shall prove the [*existence and the uniqueness*]{} of a set-valued solution $:{\bf
R}_{+} \times K \leadsto {\bf R}^{p} $ to the system of first-order partial differential equations satisfying the initial/boundary-value conditions
$$\left\{ \begin{array}{ll} \label{initboucd001}
i) & \forall \; x \in K, \; \; u_{0} (x) \; \in \; U(0,x)\\
ii) & \forall \; t>0, \; \forall \; x \in \Gamma, \; \; v_{ \Gamma}
(t,x) \; \in \; U (t,x)
\end{array} \right.$$
Again, the strategy is the same than the one we followed for Hamilton-Jacobi variational inequalities. It is enough to revive the method of characteristics by observing that the graph of the solution is the capture basin of the graph of the initial/boundary-value data under an auxiliary system (the “characteristic system”), use the characterizations derived from the Nagumo Theorem and the fact that the contingent cone to the graph is the graph of the contingent derivative of a set-valued map.
However, the solution becomes single-valued when the maps $f_{i}$ depend only on the variables $x$. In this case, we even obtain explicit formulas.
Contingent Derivatives of Set-Valued Maps
-----------------------------------------
### Set-Valued Maps
Let $X $ and $Y $ be two spaces. A set-valued map $F $ from $X $ to $Y $ is characterized by its $Graph(F)
$, the subset of the product space $X \times Y $ defined by $$\mbox{\rm Graph}(F) \; := \; \{ (x,y) \in X \times Y \; |\;\;
y \in F(x) \}$$
We shall say that $F(x) $ is the or the of $F $ at $x $.
A set-valued map is said to be if its graph is not empty, i.e., if there exists at least an element $x \in X$ such that $F(x)$ is not empty.
We say that $F$ is if all images $F(x)$ are not empty. The of F is the subset of elements $x
\in X$ such that $F(x)$ is not empty: $$\mbox{\rm Dom}(F) \; := \; \{x \in X \;\; |\;\;F(x) \ne
\emptyset \}$$ The of $F$ is the union of the images (or values) $F(x)$, when $x$ ranges over $X$: $$\mbox{ \rm Im}(F) \; := \; \bigcup_{x \in X}F(x)$$ The $F^{-1 }$ of $ F$ is the set-valued map from $Y $ to $X $ defined by $$x \in F^{-1}(y) \; \Longleftrightarrow \; y \in F(x) \;
\Longleftrightarrow \; (x,y) \in \mbox{\rm Graph}(F)$$
We shall emphasize the characterization of a set-valued map (as well as a single-valued map) by its graph. This point of view has been coined the [*graphical approach*]{} by R.T. Rockafellar.
The domain of $F$ is thus the image of $F^{-1 }$ and coincides with the projection of the graph onto the space $X $ and, in a symmetric way, the image of $F $ is equal to the domain of $F^{-1 } $ and to the projection of the graph of $F $ onto the space $Y $.
Sequences of subsets can be regarded as set-valued maps defined on the set ${\bf N}$ of integers.
### Graphical Convergence of Maps
Since the graphical approach consists in regarding closed set-valued maps as graphs, i.e., as closed subsets of the product space, ranging over the space $ {\cal F} (X \times Y)$, one can supply this space with upper and lower limits, providing the concept of upper and lower graphical convergence:
\[01A392\] Let us consider metric spaces $X, \; Y$ and a sequence of set-valued maps $F_{n} : X \leadsto Y$. The set-valued maps $\mbox{\rm
Lim}^{\sharp}\mbox{}_{n \rightarrow \infty }F_{n}$ and $\mbox{\rm
Lim}^{\flat}\mbox{}_{n \rightarrow \infty }F_{n}$ from $X$ to $Y$ defined by $$\left\{ \begin{array}{llll}
i) & {\rm Graph}( \mbox{\rm Lim}^{\sharp}\mbox{}_{n \rightarrow
\infty }F_{n}) & := & \mbox{\rm Limsup}_{n \rightarrow \infty}{\rm
Graph}(F_{n}) \\
& & & \\
ii) & {\rm Graph}(\mbox{\rm Lim}^{\flat}\mbox{}_{n \rightarrow
\infty }F_{n}) & := & \mbox{\rm Liminf}_{n \rightarrow \infty}{\rm
Graph}(F_{n})
\end{array} \right.$$ are called the [*(graphical) upper and lower limits*]{} of the set-valued maps $F_{n}$ respectively. \[02A392\]
Even for single-valued maps, this is a weaker convergence than the pointwise convergence:
1. If $f_{n}:X \mapsto Y$ converges pointwise to $f$, then, for every $x \in X$, $f (x) \in f ^{\sharp } (x)$. If the sequence is equicontinuous, then $f ^{\sharp } (x) = \{f (x)\}$.
2. Let $ \Omega \subset {\bf R}^{n} $ be an open subset. If a sequence $f_{n} \in L^{p} ( \Omega) $ converges to $f$ in $L^{p} ( \Omega
)$, then $$\mbox{\rm for almost all} \; x \in \Omega, \; \; f (x) \in f
^{\sharp } (x)$$
### Contingent Derivatives
Let $F:X \leadsto Y$ be a set-valued map. We introduce the [*differential quotients*]{} $$u \; \leadsto \; \nabla _{h}F(x,y)(u) \; := \; \frac{F(x+hu)-y}{h}$$ of a set-valued map $F:X \leadsto Y$ at $ (x,y) \in \mbox{\rm
Graph}(F)$.
The contingent derivative $DF(x,y)$ of $F$ at $ (x,y) \in
\mbox{\rm Graph}(F)$ is the graphical upper limit of differential quotients: $$DF (x,y) \; := \; \mbox{\rm Lim}^{\sharp}\mbox{}_{h \rightarrow
0+}\nabla _{h}F(x,y)$$
In other words, $v $ belongs to $ DF(x,y)(u)$ if and only if there exist sequences $h_n\rightarrow 0^+$, $u_n\rightarrow u$ and $v_n\rightarrow v$ such that $\forall n\geq 0, ~~y+h_nv_n\in F(x+h_nu_n)$.
In particular, if $f:X\mapsto Y$ is a single valued function, we set $Df(x)=Df(x,f(x))$.
We deduce the fundamental formula on the graph of the contingent derivative:
The graph of the contingent derivative of a set-valued map is the contingent cone to its graph: for all $(x,y) \in \mbox{\rm Graph}(F)$, $$\mbox{\rm Graph}(DF (x,y)) = T_{ \mbox{\rm Graph}(F)} (x,y)$$
[**Proof**]{} — Indeed, we know that the contingent cone $$T_{ \mbox{\rm
Graph}(F)}(x,y) \; = \; \mbox{\rm Limsup}_{h \rightarrow 0+} \frac{
\mbox{\rm Graph}(F) -(x,y)}{h}$$ is the upper limit of the differential quotients $ \frac{\mbox{\rm Graph}(F) -(x,y)}{h}$ when $h \rightarrow 0+$. It is enough to observe that $$\mbox{\rm Graph}(\nabla _{h}F(x,y)) = \frac{\mbox{\rm Graph}(F)
-(x,y)}{h}$$ and to take the upper limit to conclude. $\; \; \Box$
We can easily compute the derivative of the inverse of a set-valued map $F$ (or even of a noninjective single-valued map): [*The contingent derivative of the inverse of a set-valued map $F$ is the inverse of the contingent derivative*]{}: $$D(F^{-1}) (y,x) \;\; = \;\; DF(x,y)^{-1}$$
If $K$ is a subset of $X$ and $f$ is a single-valued map which is Fréchet differentiable around a point $x \in K$, then [*the contingent derivative of the restriction of $f$ to $K$ is the restriction of the derivative to the contingent cone*]{}: $$D(f|_{K})(x) = D(f|_{K})(x,f(x)) = f'(x)|_{T_{K}(x)}$$
Frankowska Solutions to First-Order Partial Differential Equations
------------------------------------------------------------------
We consider two finite dimensional vector spaces ${\bf R}^{n}$ and ${\bf
R}^{p}$, an open subset $ \Omega \subset {\bf R}^{n} $, its closure $K
:=\overline{ \Omega } $ closed, its boundary $ \Gamma := \partial \Omega
= \Gamma$, two time-dependent maps $f: {\bf R}_{+} \times K
\times {\bf R}^{p} \mapsto {\bf R}^{n}$ and $g: {\bf R}_{+} \times K
\times {\bf R}^{p} \mapsto {\bf R}^{p}$.
We shall study the system of first-order partial differential equations $$\label{frankpde01}
\frac{ \partial }{ \partial t}u (t,x) + \frac{ \partial }{ \partial x}u
(t,x) f (t,x,u (t,x)) - g (t,x,u (t,x)) \; = \; 0$$ on $ {\bf R}_{+} \times K$.
It is known that the solution $U$ to the above system can be set-valued, describing “shocks”. This is considered as a pathology whenever the solution is regarded as a map from the input space $ {\bf R}_{+} \times K$ to the output space ${\bf R}^{p}$, but is quite natural when the solution $U$ is considered as a graph, i.e., a subset of ${\bf R}_{+} \times K \times {\bf R}^{p}$ and when the tools of set-valued analysis are used.
Introducing
1. an [initial data]{} $u_{0}: K \mapsto {\bf R}^{p}$,
2. a [boundary data]{} $v_{ \Gamma}: {\bf R}_{+} \times \Gamma
\mapsto {\bf R}^{p}$.
we shall prove the existence and the uniqueness of a solution $:{\bf
R}_{+} \times K \leadsto $ to the system of first-order partial differential equations (\[frankpde01\]) satisfying the initial/boundary-value conditions
$$\left\{ \begin{array}{ll} \label{initboucd003}
i) & \forall \; x \in K, \; \; u_{0} (x) \; \in \; U(0,x)\\
ii) & \forall \; t>0, \; \forall \; x \in \Gamma, \; \; v_{ \Gamma}
(t,x) \; \in \; U (t,x)
\end{array} \right.$$
Actually, we associate with the initial data $u_{0}: K \mapsto {\bf
R}^{p}$ and the boundary data $v_{ \Gamma}: {\bf R}_{+} \times \partial
K \mapsto {\bf R}^{p}$ the “extended” boundary data $ \Psi (u_{0},v_{
\Gamma}): {\bf R}_{+} \times K \leadsto {\bf R}^{p}$ defined by $$\Psi (u_{0}, v_{ \Gamma})(s, x ) :=\left\{ \begin{array}{cllll}
u_{0}( x) & \mbox{\rm if} & s=0 & \& & x\in K \\
v_{ \Gamma} (s , x ) & \mbox{\rm if} & s \geq 0 & \& & x \in \partial
K\\
\emptyset & \mbox{\rm if} & s>0 & \& & x \in \mbox{\rm Int}(K) \\
\end{array} \right.$$ which is a set-valued map since it takes (empty) set values, the domain of which is $\mbox{\rm Dom}( \Psi (u_{0}, v_{ \Gamma})):= \partial (
{\bf R}_{+} \times K) = (\{0\} \times K) \cup ({\bf R}_{+} \times
\Gamma)$.
The set-valued map $ \Psi $ encapsulates or replaces initial/boundary-value data. Hence initial and boundary conditions (\[initboucd003\]) can be written in the form $$\forall \; (t,x) \; \in \; {\bf R}_{+} \times K, \; \; \Psi (u_{0},
v_{ \Gamma})(t, x ) \; \subset \; U (t,x)$$
By the way, we can study as well the case when $ \Psi : {\bf R}_{+}
\times {\bf R}^{n} \leadsto {\bf R}^{p}$ is any set-valued map, which allows to study other problems than initial/boundary-value problems associated with the system of first-order partial differential equations (\[frankpde01\]).
So, in the general case, we introduce two set-valued maps $ \Psi :{\bf
R}_{+} \times K \leadsto {\bf R}^{p}$ and $ \Phi : {\bf R}_{+} \times K
\leadsto {\bf R}^{p}$ satisfying $$\forall \; (t,x) \; \in \; {\bf R}_{+} \times K, \; \; \Psi (t,x) \;
\subset \; \Phi (t,x)$$
We shall prove the existence and the uniqueness of a solution $:{\bf
R}_{+} \times K \leadsto $ to the system of first-order partial differential equations (\[frankpde01\]) satisfying the conditions $$\label{frankbdvc01}
\forall \; (t,x) \in {\bf R}_{+} \times K, \; \; \Psi (t,x) \; \subset
\; U (t,x) \; \subset \; \Phi (t,x)$$
The set-valued map $ \Phi $ describes “viability constraints” on the solution the solution $U$ to the above system. The particular case without constraints is naturally obtained when $ \Phi (t,x):={\bf R}^{p}$.
[**Example: Impulse Boundary Value Problems**]{} This is the case when we provide boundary condition $v_{
\Gamma}^{i}$ only at impulse times $t_{i}$ of an increasing sequence of impulse times $t_{0}=0 < t_{1} \cdots < t_{n} < \cdots$.
We associate with them the map $ \Psi : {\bf R}_{+} \times K \leadsto {\bf R}^{p}$ only derived from the initial data $u_{0}: K \mapsto {\bf R}^{p}$ and the boundary data $v_{
\partial
K}^{i}$ by $$\Psi (u_{0}, \{v_{ \Gamma}\}^{i})(s, x ) :=\left\{
\begin{array}{cllll}
u_{0}( x) & \mbox{\rm if} & s=0 & \& & x\in K \\
v_{ \Gamma}^{i}( x) & \mbox{\rm if} & s=t_{i}, \; i >0, & \& & x\in
\Gamma \\
\emptyset & \mbox{\rm if} & s \in ]t_{i},t_{i+1}[, i > 0, & \& & x \in K
\\
\end{array} \right.$$ defined on $\mbox{\rm Dom}( \Psi (u_{0}, \{v_{ \partial
K}\}^{i})):= (\{0\} \times K) \cup \bigcup_{i}^{} ( \{t_{i}\} \times
\Gamma)$. $\; \; \Box$
We denote by $ h: {\bf R}_{+} \times K \times {\bf R}^{p} \mapsto {\bf
R}^{n} \mapsto {\bf
R}_{+} \times K \times {\bf R}^{p} \mapsto {\bf R}^{n} $ the map defined by $$h ( \tau ,x,y) \; := \; (1,f ( \tau ,x,y),g ( \tau ,x,y))$$ and the associated system $ ( \tau ',x',y')=h ( \tau ,x,y)$ of differential equations $$\left\{ \begin{array}{ll} \label{syschardeqeq}
i) & \tau ' (t) \; = \; 1 \\
ii) & x' (t) \; = \; f ( \tau (t),x (t),y (t))
\\
iii) & y' (t) \; = \; g ( \tau (t),x (t),y (t))
\end{array} \right.$$ often called the associated “characteristic system”.
Given two time-dependent maps $f: {\bf R}_{+} \times K \times {\bf R}^{p}
\mapsto {\bf R}^{n}$ and $g: {\bf R}_{+} \times K \times {\bf R}^{p} \mapsto {\bf R}^{p}$ and two set-valued maps $ \Psi :{\bf R}_{+} \times K \leadsto {\bf R}^{p}$ and $ \Phi :{\bf R}_{+} \times K \leadsto {\bf R}^{p}$ satisfying $$\forall \; (t,x) \; \in \; {\bf R}_{+} \times K, \; \; \Psi (t,x) \;
\subset \; \Phi (t,x)$$ we shall denote by $U := {\cal A}_{ (f,g)}^{ \Phi } ( \Psi ): {\bf R}_{+}
\times K \leadsto {\bf R}^{p}$ the set-valued map defined by $$\label{defsolmapueq01}
\mbox{\rm Graph}(U) \; := \; \mbox{\rm Capt}_{-h}^{ \mbox{\rm Graph}( \Phi
)}( \mbox{\rm Graph}( \Psi ))$$ the graph of which is the viable-capture basin under $-h$ of the graph of $
\Psi $ in the graph of $ \Phi $.
Even when $ \Psi $ is single-valued on its domain, this map $U$ can take several values, defined as “shocks” in the language of physicists.
Indeed, even though the trajectories of the solutions $ (x (t),y
(t))$ to the [system of differential equation]{} $$\left\{ \begin{array}{ll}
i) & x' (t) \; = \; f (t,x (t),y (t)) \\
ii) & y' (t) \; = \; g(t,x (t),y (t)) \\
\end{array} \right.$$ never intersect in ${\bf R}^{n} \times {\bf R}^{p}$, their [projections]{} $x (t)$ onto ${\bf R}^{n}$ — privileged in his role of input space — may do so. In other words, if $x_{1} \ne x_{2}$, the solutions $(x_{i} (t),y_{i} (t))$ starting from the initial conditions $ (x_{i}, u_{0}(x_{i}))$ ($i=1,2$) never intersect, but one cannot exclude the case when for some $t$, we may have $x_{1} (t)=x_{2} (t)$. At this time, the solution $U$ takes (at least) the values $y_{1} (t) \ne y_{2} (t)$ associated with the common input $x :=
x_{1} (t)=x_{2} (t)$.
However, $U$ is single-valued whenever $f (t,x,y) \equiv f (t,x)$ is independent of $y$ and the above system has a unique solution for any initial condition.
Theorems \[viablcaptbascharthm\] and \[wonderfulthmthmbis\] can be translated in terms of invariant manifold:
We shall say that a set-valued map $ V: {\bf R}_{+}
\times K \leadsto {\bf R}^{p}$ defines an [invariant manifold]{} under the pair $ (f,g)$ if for every $t_{0} \geq 0$, $ x_{0} \in K $ and $y_{0} \in V (t_{0},x_{0})$, every solution $ (x ( \cdot ),y ( \cdot ))$ to the system of differential equations $$\left\{ \begin{array}{ll}
i) & x' (t) \; = \; f (t,x (t),y (t)) \\
ii) & y' (t) \; = \; g (t,x (t),y (t))
\end{array} \right.$$ starting at $ (x_{0},y_{0})$ at time $t_{0}$ satisfies $$\forall \; t \geq t_{0},\; \; y (t) \; \in \; V (t, x (t))$$
We observe that $V$ defines an invariant manifold under $ (f,g)$ if and only if the graph of $ V $ is invariant under $h$.
\[wontrackbistd\] Let us assume that the maps $f$ and $g$ are continuous with linear growth and that the graphs of the set-valued maps $ \Psi \subset \Phi $ are closed. The associated set-valued map $U := {\cal A}_{ (f,g)}^{ \Phi } (
\Psi ): {\bf R}_{+} \times K \leadsto {\bf R}^{p}$ defined by (\[defsolmapueq01\]) is the [**largest**]{} closed set-valued map enjoying the following properties:
1. $ \forall \; t \geq 0, \; \forall \; x \in K, \; \; \Psi (t,x) \;
\subset \;U(t,x) \; \subset \; \Phi (t,x)$
2. for every $t,x$ and $y \in U (t,x) \backslash \Psi (t,x)$, there exist $s \in [0,t[$, $x_{s} \in K$ and $y_{s} \in U (s,x_{s})$ such that the solution to the above system of differential equations starting at $
(x_{s},u(s,x (s)))$ at time $s$ satisfies $$\forall \; \tau \in [s,t],\; \; y ( \tau ) \; \in \; U ( \tau , x ( \tau
)), \; \; x (t) \; = \; x \; \& \; y (t)=y$$
If we assume furthermore that the set-valued map $ \Phi $ defines an invariant manifold under $ (f,g)$, then $U$ is the [**unique**]{} set-valued map satisfying the above conditions, which also defines an invariant manifold under $ (f,g)$.
[**Proof**]{} — By assumption, $ \mbox{\rm Graph}( \Psi )$ is contained in $ \mbox{\rm
Graph}( \Phi )$. The graphs of the maps $ \Phi $ and $ \Psi $ are closed subsets and repellers since they are contained in $ {\bf R}_{+} \times K
\times {\bf R}^{p}$, which is obviously a repeller under the map $-h$.
Therefore, by Theorem \[viablcaptbascharthm\], $ \mbox{\rm
Graph}(U):= \mbox{\rm Capt}_{-h}^{ \mbox{\rm Graph}( \Phi )} ( \Psi )$ is the largest closed subset $D:= \mbox{\rm Graph}(U)$ satisfying $$\left\{ \begin{array}{ll}
i) & \mbox{\rm Graph}( \Psi ) \; \subset \; \mbox{\rm Graph}(U)\; \subset
\; \mbox{\rm Graph}( \Phi )\\
ii) & \mbox{\rm Graph}(U) \backslash \mbox{\rm Graph}( \Psi ) \;\;\mbox{\rm
is locally viable under } \; -h \\
\end{array} \right.$$ which are translated into the first properties of the above theorem. If we assume furthermore that $ \Phi $ defines an invariant manifold under $ (f,g)$, Theorem \[wonderfulthmthmbis\] implies that $ \mbox{\rm Graph}(U) =
\mbox{\rm Capt}_{-h} ( \Psi )$ is the unique closed subset satisfying the above properties and being backward invariant under $-h$, i.e., invariant under $h$. $\; \; \Box$
Theorem \[viablcaptbascharthm\] and \[wonderfulthmthmbis\] also allow us to derive the existence and the uniqueness of a set-valued map solution to the system of first-order partial differential equations (\[frankpde01\]) $$\frac{ \partial }{ \partial t}u (t,x) + \frac{ \partial }{ \partial x}u
(t,x) f (t,x,u (t,x)) - g (t,x,u (t,x)) \; = \; 0$$ satisfying the conditions (\[frankbdvc01\]) $$\forall \; (t,x) \in {\bf R}_{+} \times K, \; \; \Psi (t,x) \; \subset
\; U (t,x) \; \subset \; \Phi (t,x)$$
We say that a set-valued map $U: {\bf R}_{+} \times K \mapsto {\bf
R}^{p}$ is a Frankowska solution to the problem ((\[frankpde01\]),(\[frankbdvc01\])) if the graph of $U$ is closed and if $$\left\{ \begin{array}{ll}
i)& \forall \; (t,x) \in {\bf R}_{+} \times K, \; \forall \; y \in U (t,x)
\backslash \Psi (t,x), \; \; \\
& 0 \; \in \; DU (t,x,y) ( -1, -f (t,x,y)) +g (t,x,y) \\
& \mbox{\rm and} \\
ii) & \forall \; (t,x) \in {\bf R}_{+} \times K, \; \forall \; y \in U
(t,x), \;
0 \; \in \; DU (t,x,y) ( 1, f (t,x,y)) -g (t,x,y)\\ \end{array} \right.$$
Naturally, if $ \Psi := \psi $ and $U:=u$ are single-valued on their domains, then a Frankowska solution can be written $$\left\{ \begin{array}{l}
\forall \, (t,x) \in \mbox{\rm Dom}(u) \backslash \mbox{\rm Dom}( \psi ),
\; 0 \in Du (t,x) ( -1, -f (t,x,u (t,x))) +g (t,x,u (t,x))
\\\mbox{\rm and} \\
\forall \, t \geq 0, \, \forall \, x \in K,
0 \, \in \, Du (t,x) ( 1,f (t,x,u (t,x)) -g (t,x,u (t,x))\\
\end{array} \right.$$
If $Du (t,x) (-1,- \xi )=-Du (t,x) (1, \xi )$, these two equations boil down to only one of them on $\mbox{\rm Dom}(u) \backslash \mbox{\rm Dom}(
\psi )$. If $u$ is differentiable in the usual sense, it satisfies the above first order partial differential equation in the usual sense outside the domain of $ \psi $.
\[frankwssolstatstrpbbistd\] Let us assume that the maps $f$ and $g$ are continuous with linear growth and that the graphs of the set-valued maps $ \Psi \subset \Phi $ are closed.
Then the set-valued map $U$ defined by (\[defsolmapueq01\]) is the [**largest**]{} set-valued map with closed graph satisfying the condition $$\forall \; (t,x) \in {\bf R}_{+} \times K, \; \; \Psi (t,x) \; \subset
\; U (t,x) \; \subset \; \Phi (t,x)$$ and solution to $$\forall \, (t,x) \in {\bf R}_{+} \times K, \; \forall \, y \in U (t,x)
\backslash \Psi (t,x), \; 0 \, \in \, DU (t,x,y) ( -1, -f (t,x,y))
+g (t,x,y)$$ If we assume furthermore that $f$ and $g$ are uniformly Lipschitz with respect to $x$ and $y$ and that $ \Phi $ defines an invariant manifold under $ (f,g)$, then $U$ is the [**unique**]{} Frankowska solution to the problem ((\[frankpde01\]),(\[frankbdvc01\])).
[**Proof**]{} — By Theorem \[viablcaptbascharthm\], $ \mbox{\rm Graph}(U):=
\mbox{\rm Capt}_{-h}^{ \mbox{\rm Graph}( \Phi )} ( \mbox{\rm Graph}( \Psi
))$ is the largest closed subset $D:= \mbox{\rm Graph}(U)$ satisfying $$\left\{ \begin{array}{ll}
i) & \mbox{\rm Graph}( \Psi ) \; \subset \; \mbox{\rm Graph}(U)\; \subset
\; \mbox{\rm Graph}( \Phi )\\
ii) & \forall \; (t,x,y) \in \mbox{\rm Graph}(U) \backslash \mbox{\rm
Graph}( \Psi ),\\
& -(1,f (t,x,t), g(t,x,t)) \; \in \; T_{ \mbox{\rm Graph}(U)} (t,x,y) \; =
\; \mbox{\rm Graph}(DU) (t,x,y)
\end{array} \right.$$ which can be translated $$\forall \; (t,x), \; \; \forall \; y \in U (t,x) \backslash \Psi
(t,x), \; \; -g (t,x,y) \; \in \; DU (t,x,y) (-1,-f (t,x,y))$$ If we assume furthermore that $ \Phi $ defines an invariant manifold under $ (f,g)$, Theorem \[wonderfulthmthmbis\] implies that $ \mbox{\rm Graph}(U) =
\mbox{\rm Capt}_{-h} ( \mbox{\rm Graph}( \Psi ))$ is the unique closed subset satisfying the above properties and being backward invariant under $-h$, i.e., invariant under $h$. This can be translated by stating that $$\left\{ \begin{array}{l}
\forall \; (t,x,y) \in \mbox{\rm Graph}(U), \\
(1,f (t,x,t), g(t,x,t)) \; \in \; T_{ \mbox{\rm Graph}(U)} (t,x,y) \; = \;
\mbox{\rm Graph}(DU) (t,x,y)\\
\end{array} \right.$$ i.e., $$\forall \; (t,x) \in {\bf R}_{+} \times K, \; \forall \; y \in U
(t,x), \; \;
g (t,x,y)\; \in \; DU (t,x,y) (1,f (t,x,y))$$
Single-Valued Frankowska Solutions
----------------------------------
We already mentioned that even when $ \Psi $ is single-valued on its domain, the solution $U$ can take several values, defined as “shocks” in the language of physicists.
However, single-valuedness is naturally preserved whenever $$h ( \tau ,x,y) \; := \; ( 1,\varphi (x), g( \tau ,x,y))$$ when the second component of the map $h$ does not depend upon the second variable $y$ and the differential equation $ ( \tau ',x',y')=h ( \tau ,x,y)
$ has a unique solution for any initial condition.
Therefore, we proceed with the specific case when $f (t,x,y) \equiv \varphi
(x)$ depends only upon the variable $x$.
When $ (t,x) \in {\bf R}_{+} \times K$ is chosen, we introduce the function $x ( \cdot ) := \vartheta_{ \varphi }( \cdot -t,x)$ the solution to the differential equation $x' = \varphi (x)$ starting at time $0$ at $
\vartheta_{ \varphi } (-t,x)$, or arriving at $x$ at time $t$. We associate with it the map $g_{ (t,x)} : {\bf R}_{+} \times {\bf R}^{p} \mapsto {\bf
R}^{p}$ defined by $$\forall \; \tau \geq 0, \; y \in {\bf R}^{p}, \; \; g_{ (t,x)} ( \tau
,y) \; := \;
g ( \tau ,\vartheta _{ \varphi } ( \tau -t,x),y)$$
We denote by $ \vartheta _{g_{ (t,x)} } (t,s,y (s))$ the value at $t$ of the solution to the differential equation $$y' ( \tau ) \; = \; g_{ (t,x)} ( \tau ,y ( \tau )) \; := \; g ( \tau ,
\vartheta_{ \varphi }( \tau -t,x), y ( \tau ))$$ starting at $y (s)$ associated with the evolution $x ( \tau ):= \vartheta
_{ \varphi } ( \tau -t,x )$ starting at $x (s)=\vartheta _{ \varphi } (
s-t,x )$ at initial time $s$.
We associate with the backward exit function the map $ \Theta_{K}^{ -
\varphi }$ defined by
$$\forall \; x \in K, \; \;
\Theta_{K}^{- \varphi } (x) \; := \; \vartheta _{- \varphi } ( \tau _{K}^{-
\varphi } (x),x)$$
and we say that $ \Theta_{K}^{- \varphi }$ is the “exitor” (for exit projector) of $K$. It maps $K$ to its boundary $ \Gamma$ and satisfies $ \Theta_{K}^{- \varphi } (x)=x$ for every $x \in \Theta
_{K}^{- \varphi } (K)$.
It will be very convenient to extend the function $ \tau_{K}^{- \varphi }$ defined on $K$ to the function (again denoted by) $\tau_{K}^{- \varphi }$ defined on $ {\bf R}_{+} \times K$ by $ \tau _{K}^{- \varphi }(t,x) :=
\min (t, \tau _{K}^{- \varphi }(x))$: $$\tau_{K}^{- \varphi } (t,x) \; := \left\{ \begin{array}{cll}
t & \mbox{\rm if} & t \in [0, \tau _{K}^{- \varphi } (x) ] \\
\tau _{K}^{- \varphi } (x) & \mbox{\rm if} & t \in ] \tau _{K}^{- \varphi
} (x), \infty [
\end{array} \right.$$ so that we can also extend the exitor map by setting $$\Theta_{K}^{ -\varphi } (t,x) \; := \; \vartheta _{ -\varphi } (
\tau_{K}^{- \varphi } (t,x),x) \; \in \; K$$ because we observe that $\Theta_{K}^{ -\varphi }(t,x) $ is equal to
$$\left\{ \begin{array}{cll}
\vartheta _{ - \varphi } (t,x) & \mbox{\rm if} & t \in [0, \tau _{K}^{-
\varphi } (x) ] \\
\Theta_{K}^{ - \varphi } (x) & \mbox{\rm if} & t \in ] \tau _{K}^{-
\varphi } (x), \infty [
\end{array} \right.$$
\[propgrencpropbistd\] We posit assumption $$\label{assumKonceandforall}
K \;\;\mbox{\rm is closed and (forward) invariant under} \; \; \varphi$$ and $$\left\{ \begin{array}{ll} \label{assumKonceandforall27}
i) & \varphi \;\;\mbox{\rm is Lipschitz and that $g$ is continuous} \\
ii) & \mbox{\rm $ ( \tau ',x',y')=h ( \tau ,x,y) $ has a unique solution
for any initial condition}\\
\end{array} \right.$$ Let us introduce
1. an [initial data]{} $u_{0}: K \mapsto {\bf R}^{p}$,
2. a [boundary data]{} $v_{ \Gamma}: {\bf R}_{+} \times \partial
K \mapsto {\bf R}^{p}$
The solution $ u := {\bf {\cal A}}_{ ( \varphi ,g)} ( \Psi (u_{0},v_{
\Gamma}))$ is the single-valued map with closed graph defined by $$\label{magmath077}
u (t,x) \; = \; \vartheta_{ g_{ (t,x)} }( t, t- \tau _{K}^{ -\varphi }
(t,x), (\Psi (u_{0}, v_{ \Gamma}) ( t- \tau _{K}^{ -\varphi } (t,x),
\Theta_{K}^{- \varphi } (t,x)))$$ or, more explicitly, by $$\left\{ \begin{array}{cll}
\vartheta_{ g_{ (t,x)} }( t,0, u_{0}( \vartheta _{- \varphi } (t,x))) &
\mbox{\rm if} & t \in [0, \tau _{K}^{- \varphi } (x) ] \\ & & \\
\vartheta_{ g_{ (t,x)} }( t, t- \tau _{K}^{ -\varphi } (x), v_{ \partial
K} (t- \tau _{K}^{ -\varphi } (x), \Theta_{K}^{ -\varphi } (x))) &
\mbox{\rm if} & t \in ] \tau _{K}^{- \varphi } (x), \infty [
\end{array} \right.$$
Furthermore, if we assume the following viability assumptions on $\Phi $ $$\left\{ \begin{array}{ll} \label{vainonautconstrpr}
i) & \forall \; x \in K, \; \forall \; t \geq 0, \; \; \forall \; y \in
\Phi (t,x), \; \; g (t,x,y) \; \in \; D\Phi (t,x,y) (1, \varphi (x))\\
ii) & \forall \; \xi \in \Gamma, \; \forall \; t>0, \; \; v_{
\Gamma} (t, \xi ) \; \in \; \Phi ( t,\xi )\\
iii)& \forall \; x \in K, \; \; u_{0} (x) \; \in \; \Phi (0,x)\\
\end{array} \right.$$ then $$\forall \; (t,x) \in {\bf R}_{+} \times K, \; \; u (t,x) \; \in \;
\Phi (t,x)$$
[**Remark:**]{} — We stress the fact that the solution $u (t,x)$ [depends only]{}
1. upon the initial condition $u_{0} (x)$ when $t \leq \tau _{K}^{-
\varphi } (x)$,
2. upon the boundary condition $v_{ \Gamma}$ when $t > \tau _{K}^{-
\varphi } (x)$.
The second property proves a general principle concerning demographic evolution stating the state of the system [eventually forgets its initial condition $u_{0} ( \cdot )$]{}.
[**Proof**]{} — We first take $ \Phi (t,x) \equiv {\bf R}^{p}$, which is invariant by assumption (\[assumKonceandforall\]). Then the graph of $u$ is defined by (\[defsolmapueq01\]) is equal to $$\mbox{\rm Graph}(U) \; := \; \mbox{\rm Capt}_{-h}( \mbox{\rm Graph}(
\Psi(u_{0},v_{ \Gamma}) ))$$ An element $ (t,x,y)$ of the graph of $U$ is the value at some $ h \geq 0$ of the solution $ ( \tau ( \cdot ),x ( \cdot ),y ( \cdot ))$ to the system of differential equations $$\left\{ \begin{array}{ll}
i) & \tau ' (t) \; = \; 1 \\
ii) & x' (t) \; = \; \varphi (x (t)) \\
iii) & y' (t) \; = \; g ( \tau (t),x (t),y (t))
\end{array} \right.$$ starting from $ (s,c, \Psi (u_{0},v_{ \Gamma})(s,c)))$, assumed to be unique.
This implies that $t= s + h$ and $x = \vartheta _{ \varphi } ( t,c)$. If $s =0$, then $t=h$, $x= \vartheta_{ \varphi } (t,c)$ and $$y \; = \; \vartheta_{ g_{ (t,x)} }( t,0, u (0, \vartheta _{- \varphi }
(t,x)))$$ If $s>0$, then $c \in \Gamma$, so that $h= \tau _{K}^{- \varphi }
(x)$, $s=t-\tau _{K}^{- \varphi } (x)$ and $c= \Theta_{K}^{- \varphi } (x)$ and $$y \; = \; \vartheta_{ g_{ (t,x)} }( t, t- \tau _{K}^{ -\varphi } (x), v_{
\Gamma} (t- \tau _{K}^{ -\varphi } (x), \Theta_{K}^{ -\varphi } (x)))$$ Therefore, $y=:u (t,x)$ is uniquely determined by $t$ and $x$ so that $U
=:u$ is single-valued.
Assumptions (\[vainonautconstrpr\]) imply that the graph of the map $
(t,x) \leadsto \Phi (t,x)$ is invariant under $ ( 1, \varphi ,g)$ and that $ \mbox{\rm Graph} (\Psi _{u_{0},v_{ \Gamma}}) $ is contained in the graph of $\Phi $. Hence the graph of $u$ is contained in the graph of $\Phi $. $\; \; \Box$
[**Example**]{} Let us consider a $x$-dependent matrix $A (x) \in {\cal L} ({\bf
R}^{p},{\bf R}^{p})$. We associate the differential equation $$y' ( \tau ) \; = \; -A (x ( \tau )) y ( \tau )$$ Then $u(t,x)$ is equal to
$$\left\{ \begin{array}{cll}
e^{- \int_{0}^{t} A ( \vartheta _{ \varphi } ( \tau -t,x ) d \tau }u_{0}(
\vartheta _{- \varphi } (t,x))
& \mbox{\rm if} & t \in [0, \tau _{K}^{- \varphi } (x)] \\
e^{- \int_{t - \tau _{K}^{- \varphi } (x)}^{t} A ( \vartheta _{ \varphi }
( \tau - t,x ) d \tau } v_{ \Gamma} (t- \tau _{K}^{ -\varphi } (x),
\Theta_{K}^{- \varphi } (x)) & \mbox{\rm if} & t \in ] \tau _{K}^{-
\varphi } (x), \infty [
\end{array} \right.$$
[**Example**]{} Let us consider the case when $ K \; := \; K_{1} \times \cdots \times
K_{n}$ here the $K_{i} \subset {\bf R}^{n}_{i}$ are close subsets and set ${\bf R}^{n} :=
\prod_{j=1}^{n}{\bf R}^{n}_{j}$.
\[maurinlemma001\] Assume that $K := \prod_{j=1}^{n}K_{j}$ is the product of $n$ closed subsets $K_{j} \subset {\bf R}^{n}_{j}$. We posit assumptions (\[assumKonceandforall\]) on $K$. Denoting by $$\tau _{K_{j}}^{- \varphi } (x) \; := \; \tau _{K_{j}} ( ( \vartheta_{ -
\varphi } ( \cdot ,x))_{j})$$ the [partial backward exit time functions]{}, the backward exit time function can then be written $$\tau _{K}^{- \varphi } (x) \; := \; \min_{j=1, \ldots ,n} \tau _{K_{j}}^{-
\varphi } (x)$$
[**Proof**]{} — We observe that $$\prod_{j=1}^{n} {\bf R}^{n}_{j} \backslash \left( \prod_{j=1}^{n}
K_{j}\right) \; =
\; \bigcup_{j=1}^{n} \left( \prod_{i=1}^{j-1}{\bf R}^{n}_{i} \times ({\bf
R}^{n}_{j} \backslash
K_{j}) \times \prod_{l=j+1}^{n}{\bf R}^{n}_{l} \right)$$ and thus, that for any function $t \mapsto x (t)= (x_{1} (t), \ldots ,x_{n}
(t))$, $$\tau_{K} (x ( \cdot )) \; := \; \inf_{x (t) \in {\bf R}^{n} \backslash
K}t \; = \;
\min_{j=1, \ldots ,n} \left( \inf_{x_{j}(t) \in {\bf R}^{n}_{j} \backslash
K_{j}}t
\right) \; = \; \min_{j=1, \ldots ,n} \tau _{K_{j}} (x_{j} ( \cdot ))$$ since the infimum on a finite union of subsets is the minimum of the infima on each subsets. $\; \; \Box$
In this case, $$\partial \left( \prod_{j=1}^{n} K_{j}\right) \; = \; \bigcup_{j=1}^{n}
\left( \prod_{i=1}^{j-1}K_{i} \times \Gamma_{j} \times
\prod_{l=j+1}^{n}K_{l} \right)$$ so that the boundary data defined on $ \Gamma$ are defined by $n$ maps $$v_{ \Gamma}^{j} : \prod_{i=1}^{j-1}K_{i} \times \Gamma_{j} \times
\prod_{l=j+1}^{n}K_{l} \mapsto {\bf R}^{p}$$
[**Example**]{} For instance, let us consider the case when the four-dimensional causal variable $x:= (x_{1},x_{2},x_{3},x_{4})$ ranges over the product $
K:= \prod_{i=1}^{4} K_{i} $ with $ K_{1}:={\bf R}_{+} $, $
K_{2}:=[0,r_{2}] $, $K_{3}:={\bf R}_{+} $ and $ K_{4}:=[0,b] $.
We are looking for solutions to the system of first-order partial differential equations $$\left\{ \begin{array}{l}
\displaystyle{ \frac{ \partial }{ \partial t}u (t,x) + \frac{
\partial }{ \partial x_{1}}u(t,x) - \rho \frac{ \partial }{ \partial
x_{2}}u (t,x)+ \sigma \frac{ \partial }{ \partial x_{3}}u (t,x)+ \beta
(b-x_{4})x_{4} \frac{ \partial }{ \partial x_{4}}u(t,x)
}\\ = \; g (t,x,u(t,x))
\end{array} \right.$$ (where the scalars functions $ \rho , \; \sigma $ and $ \beta $ are positive) satisfying the initial and boundary conditions $$\left\{ \begin{array}{ll}
i) & \forall \; x_{1}\geq 0, \; \forall \; x_{2}\in [0,r_{2}], \;
x_{3} \geq 0, \; x_{4} \in [0,b],\\ & u (0,x_{1},x_{2},x_{3},x_{4}) \; =
\; u_{0}(x_{1},x_{2},x_{3},x_{4}) \\
ii) & \forall \; t \geq 0, \; \forall \; x_{2}\in [0,r_{2}], \;
x_{3} \geq 0, \; x_{4} \in [0,b], \\ & u (t,0,x_{2},x_{3},x_{4}) \; = \;
v_{1}(t,x_{2},x_{3},x_{4}) \\
i) & \forall \; t \geq 0, \; \forall \; x_{1}\geq 0, \; x_{3} \geq
0, \; x_{4} \in [0,b], \\ & u (t,x_{1},r_{2},x_{3},x_{4}) \; = \;
v_{r_{2}}(t,x_{1},x_{3},x_{4}) \\
\end{array} \right.$$
Hence, we derive the existence and the uniqueness of the Frankowska solution to the above system of partial differential equations satisfying an initial condition, a boundary condition for $x_{1}=0$ (births) and a boundary condition for $x_{2}=r_{2}$.
For computing it, we need to know the backward exit time and the exitor for the associated characteristic system given by $$\left\{ \begin{array}{ll}
i) & x'_{1} (t) \; = \; 1 \\
ii) & x'_{2} (t) \; = \; - \rho x_{2} (t)\\
iii)& x'_{3} (t) \; = \; \sigma x_{3} (t)\\
iv)& x'_{4} (t) \; = \; \beta (b -x_{4} (t))x_{4} (t)
\end{array} \right.$$ where $ \varphi (x) := ( \varphi _{i} (x))_{i=1, \ldots ,4}$ with $
\varphi_{1} (x_{1}):=1$, $ \varphi _{2} (x_{2}):=- \rho x_{2}$, $\varphi_{3} (x_{3}):= \sigma x_{3}$ and $ \varphi _{4} (x_{4}):= \beta
(b-x_{4})x_{4}$.
We recall that the solution to the purely logistic equation $y' (t)= \beta
(t) (b-y (t))y (t)$ starting at $y_{s}$ at time $s$ is given by $$y (t) \; := \; \frac{b}{1 + \left( \frac{b}{y_{s}}-1 \right)e^{-b
\int_{s}^{t} \beta ( \tau ) d \tau } }$$
The closed subset $K$ is obviously (forward) invariant under $ \varphi $ defined by $ \varphi $ and one can observe easily that $$\tau _{K_{3}}^{- \varphi } (x) \; = \; + \infty \; \& \; \tau _{K_{4}}^{-
\varphi } (x) \; = \; + \infty$$ so that $$\tau _{K}^{- \varphi } (t,x_{1},x_{2},x_{3},x_{4}) \; = \; \min \left( t,
x_{1}, \frac{1}{ \rho }\log \left( \frac{r_{2}}{x_{2}} \right) \right)$$ Therefore,
1. if $t \leq \min \left( x_{1}, \frac{1}{ \rho }\log \left(
\frac{r_{2}}{x_{2}} \right) \right)$, then $$\Theta_{K}^{- \varphi } (t,x) \; = \; \left( 0,x_{1}-t,e^{ \rho
t}x_{2},e^{- \sigma t}x_{3}, \frac{b}{1+ \left( \frac{b}{x_{4}}-1
\right)e^{ \beta bt}} \right)$$
2. if $x_{1} \leq \min \left( t, \frac{1}{ \rho }\log \left(
\frac{r_{2}}{x_{2}} \right) \right)$, then $$\Theta_{K}^{- \varphi } (t,x) \; = \; \left( t-x_{1},0,e^{ \rho
x_{1}}x_{2},e^{- \sigma x_{1}}x_{3}, \frac{b}{1+ \left( \frac{b}{x_{4}}-1
\right)e^{ \beta bx_{1}}} \right)$$
3. if $ \frac{1}{ \rho }\log \left( \frac{r_{2}}{x_{2}} \right)\leq
\min \left( t, x_{1} \right)$, then $$\left\{ \begin{array}{l}
\Theta_{K}^{- \varphi } (t,x) \; = \\ \displaystyle{ \left( t-\frac{1}{
\rho }\log \left( \frac{r_{2}}{x_{2}} \right),x_{1}-\frac{1}{ \rho }\log
\left( \frac{r_{2}}{x_{2}} \right),r_{2}, e^{ \left( \frac{x_{2}}{r_{2}}
\right)^{ \frac{ \sigma }{ \rho }}}x_{3}, \frac{b}{1+ \left(
\frac{b}{x_{4}}-1 \right) \left( \frac{r_{2}}{x_{2}} \right)^{ \frac{ \beta
b}{ \rho }}} \right)} \\
\end{array} \right.$$
If the right-hand side $g (t,x,y) :=-A(t,x)y$ where $A (t,x) \in {\cal L}
({\bf R}^{p},{\bf R}^{p})$ is linear, then we set $$A (t,x; \tau ) \; := \; A \left( \tau , x_{1}-t+ \tau , e^{- \rho (t- \tau
)}x_{2}, e^{ \sigma (t- \tau )}x_{3}, \frac{b}{1+ \left( \frac{b}{x_{4}}-1
\right) e^{ \beta b (t- \tau )}}
\right)$$ Then the solution is given by
1. if $t \leq \min \left( x_{1}, \frac{1}{ \rho }\log \left(
\frac{r_{2}}{x_{2}} \right) \right)$, then $$u (t,x) \; = \; e^{- \int_{0}^{t}A (t,x; \tau )d \tau } u_{0}
\left( x_{1}-t,e^{ \rho t}x_{2},e^{- \sigma t}x_{3}, \frac{b}{1+ \left(
\frac{b}{x_{4}}-1 \right)e^{ \beta bt}} \right)$$
2. if $x_{1} \leq \min \left( t, \frac{1}{ \rho }\log \left(
\frac{r_{2}}{x_{2}} \right) \right)$, then $$u (t,x) \; = \; e^{- \int_{t-x_{1}}^{t}A (t,x; \tau )d \tau } v_{1} \left(
t-x_{1},e^{ \rho x_{1}}x_{2},e^{- \sigma x_{1}}x_{3}, \frac{b}{1+ \left(
\frac{b}{x_{4}}-1 \right)e^{ \beta bx_{1}}} \right)$$
3. if $ \frac{1}{ \rho }\log \left( \frac{r_{2}}{x_{2}} \right)\leq
\min \left( t, x_{1} \right)$, then $$\left\{ \begin{array}{l}
\displaystyle{u (t,x) \; = \; e^{- \int_{t- \frac{1}{ \rho }\log \left(
\frac{r_{2}}{x_{2}} \right)}^{t}A (t,x; \tau )d \tau } }\\
\displaystyle{v_{r_{2}}
\left( t-\frac{1}{ \rho }\log \left( \frac{r_{2}}{x_{2}}
\right),x_{1}-\frac{1}{ \rho }\log \left( \frac{r_{2}}{x_{2}} \right), e^{
\left( \frac{x_{2}}{r_{2}} \right)^{ \frac{ \sigma }{ \rho }}}x_{3},
\frac{b}{1+ \left( \frac{b}{x_{4}}-1 \right) \left( \frac{r_{2}}{x_{2}}
\right)^{ \frac{ \beta b}{ \rho }}} \right)} \\
\end{array} \right.$$
Regularity Properties
---------------------
We begin by proving that the operator $ {\bf {\cal A}}_{ ( \varphi ,g)}$ preserves continuity and boundedness:
\[propgrencpropbis2td\] We posit assumptions (\[assumKonceandforall\]) and (\[assumKonceandforall27\]) and the regularity property $$\label{assumKonceandforallbis}
\tau _{K}^{- \varphi }: K \mapsto \Gamma \;\;\mbox{\rm is
continuous}\;\;$$ Assume also that $g$ enjoys uniform linear growth with respect to $y$ in the sense that there exists a positive constant $c$ such that $$\label{lingrassg}
\forall \; x \in K, \; \forall \; y \in {\bf R}^{p}, \; \; \|g (
t,x,y)\| \; \leq
\;c (1+ \|y\|)$$ Then the ${\bf {\cal A}}_{ (\varphi ,g)} ( \Psi (u_{0},v_{ \Gamma}))$ is continuous and bounded whenever $u_{0}$ and $v_{ \Gamma}$ are continuous and bounded.
[**Proof**]{} — By assumption (\[assumKonceandforallbis\]), $ \tau _{K} (t,x):= \min (t,
\tau _{K}^{- \varphi } (x))$ is also continuous, so that $ \Theta_{K}^{-
\varphi }(t,x) := \vartheta_{- \varphi } ( \tau _{K}^{- \varphi }
(t,x),x)$ is also continuous, and thus $$u (t,x) \; = \; \vartheta _{g_{ (t,x)}} (t,t- \tau _{K} (t,x), \Psi
(u_{0},v_{ \Gamma}) (t- \tau _{K} (t,x),\Theta_{K}^{- \varphi }(t,x)
))$$ is also continuous. When $ \|g (t, x,y)\| \; \leq \;c (1+ \|y\|)$, we also infer that $$\|u (t,x)\| \; \leq \; e^{c \tau _{K}^{- \varphi } (t,x)} \max ( \|u_{0}
(x)\|, \|v_{ \Gamma} ( \Theta_{K}^{- \varphi} (x))\|)$$ and thus, that $$\|u (t, \cdot )\|_{ \infty } \; \leq \; e^{ct} \max ( \|u_{0}\|_{ \infty
}, \|v_{ \Gamma}\|_{ \infty })$$
Next, we prove that under monotonicity assumptions, the operator $ {\bf {\cal A}}_{ ( \varphi ,g)}$ is Lipschitz:
We say that $g$ is [uniformly monotone]{} with respect to $t$ and $y$ if there exists $ \mu \in {\bf R}$ such that $$\label{monassg37}
\langle g (t,x, y_{1}) - g (t,x,y_{2}) ,y_{1}-y_{2} \rangle \; \leq \;
-\mu \|y_{1}-y_{2}\|^{2}$$
The interesting case is obtained when $ \mu >0$. When $g$ is uniformly Lipschitz with respect to $t$ and $y$ with constant $ \lambda $, then it is uniformly monotone with $ \mu =- \lambda $.
\[propgrencpropasybistdmon\] We posit assumption (\[assumKonceandforall\]) and assume that $ \varphi
$ is Lipschitz and that $g$ is continuous and that $g$ is [uniformly monotone]{} with respect to $y$.
Then, for each $t>0$, $$\left\{ \begin{array}{l}
\sup_{ x \in K}\| ({\bf {\cal A}}_{ ( \varphi ,g)}\Psi (u^{1}_{0}, v^{1}_{
\Gamma})) ( (t,x)- ({\bf {\cal A}}_{ ( \varphi ,g)} \Psi (u^{2}_{0},
v^{2}_{ \Gamma}) (t,x)\| \\ \\
\leq \; e^{- \mu t} \max ( \|u^{1}_{0}-u^{2}_{2}\|_{ \infty } , \|v^{1}_{
\Gamma} (t, \cdot )-v^{2}_{ \Gamma}) (t, \cdot )\|_{ \infty } )\\
\end{array} \right.$$
Consequently, if we posit assumptions (\[assumKonceandforallbis\]) and (\[lingrassg\]), we deduce that for any $T>0$ $${\bf {\cal A}}_{ ( \varphi ,g)} :{\cal C}_{ \infty }(K ,{\bf R}^{p}
)\times {\cal
C}_{ \infty }([0,T] \times \Gamma,{\bf R}^{p}) \mapsto {\cal C}_{ \infty
}([0,T]
\times K,{\bf R}^{p})$$ is a Lipschitz operator from the space $ {\cal C}_{ \infty }(K ,{\bf
R}^{p} )\times
{\cal C}_{ \infty }([0,T] \times \Gamma,{\bf R}^{p})$ of pairs of continuous and bounded initial and boundary data to the space $ {\cal C}_{ \infty }([0,T]
\times K,{\bf R}^{p})$ of continuous and bounded maps from $[0,T] \times
K$ to ${\bf R}^{p}$.
[**Proof**]{} — Indeed, setting $u^{i} (t,x):= ({\bf {\cal A}}_{ (\varphi ,g)} ( \Psi
(u_{0}^{i}, v_{ \Gamma}^{i})))(x) $, $i=1,2$, we know that $$\forall \; x \in K , \; \;u^{i} (t,x)\; := \; \vartheta _{g_{t,x}} (t,t-
\tau _{K}^{- \varphi } (t,x), \Psi ( u_{0}^{i}, v_{ \Gamma}^{i}) (
\Theta_{K}^{- \varphi } (t,x)) )$$ is the value $ y^{i}(t):=u^{i} (t,x)$ at time $ t$ of the solution $ y (
\cdot )$ to the differential equation $y' ( \tau )=g ( \tau ,x ( \tau ), y
( \tau ))$ starting from $ \Psi (u_{0}^{i},v_{ \Gamma}^{i}) ( \Theta
_{K}^{ - \varphi } (t,x))$ at time $t-\tau _{K}^{- \varphi } (t,x)$, where $x ( \cdot )$ is the solution to the differential equation $x' = \varphi
(x)$ starting from $ \Theta_{K}^{- \varphi } (t,x)$ at time $t-\tau
_{K}^{- \varphi } (t,x)$.
Recalling that $ \tau _{K}^{- \varphi } (t,x)= \min (t, \tau _{K}^{-
\varphi } (x)) \leq t$, we infer from assumption (\[monassg37\]) that[^9] $$\left\{ \begin{array}{l}
\|y^{1}(t)- y^{2}(t)\| \\ \leq \; e^{ - \mu \tau _{K}^{- \varphi }
(t,x)} \| \Psi (u_{0}^{1},v_{ \Gamma}^{1})( \Theta_{K}^{ - \varphi }
(t,x))- \Psi (u_{0}^{2},v_{ \Gamma}^{2})( \Theta_{K}^{ - \varphi }
(t,x))\|\\
\leq \; e^{ - \mu t} \max (\sup_{x \in K} \|u_{0}^{1} (x)-u_{0}^{2}
(x)\|,
\sup_{ \xi \in \Gamma}\|v_{ \Gamma}^{1} ( \xi )-v_{ \partial
K}^{2} ( \xi )\| ) \\ = \; e^{ - \mu t } \max ( \|u_{0}^{1}-u_{0}^{2}\|_{
\infty }+\|v^{1}_{ \Gamma}-v^{2}_{ \Gamma}\|_{ \infty })
\end{array} \right.$$
[abc99xys]{}
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[^1]: This is the first preliminary draft of lecture notes of a mini-course at University of California at Davis in 1999. Any critical comment and correction will be gratefully acknowledged.
[^2]: This means that all solutions to the differential equation $x'=f (x)$ starting from $M$ leave $M$ in finite time.
[^3]: Hélène Frankowska proved that the epigraph of the value function of an optimal control problem — assumed to be only lower semicontinuous — is invariant and backward viable under a (natural) auxiliary system. Furthermore, when it is continuous, she proved that its epigraph is viable and its hypograph invariant ([@f89hjc; @f89hje; @HJB92 Frankowska]). By duality, she proved that the latter property is equivalent to the fact that the value function is a viscosity solution of the associated Hamilton-Jacobi equation in the sense of M. Crandall and P.-L. Lions. See also [@bj90hj Barron & Jensen], [@ba93hj Barles] and [@bcd98vis Bardi & Capuzzo-Dolcetta] for more details. Such concepts have been extended to solutions of systems of first-order partial differential equation without boundary conditions by Hélène Frankowska and the author (see [@af90cm1; @af90hyp; @af91hyp; @af90cm; @af94cpdi Aubin & Frankowska] and chapter 8 of [@avt Aubin]). See also [@adp90hyp; @adp92hyp Aubin & Da Prato]. This point of view is used here in the case of boundary value problems.
[^4]: The terms outer and inner limits of sets have been proposed by R.T Rockafellar and R. Wets.
[^5]: The concepts of [*semitangenti*]{} and of [*corde improprie*]{} to a set at a point of its closure had been introduced by the Italian geometer Francesco Severi (1879-1961) and are equivalent to the concepts of [*contingentes*]{} and [*paratingentes*]{} introduced independently by the French mathematician Georges Bouligand, slightly later. Severi explains for the second time that he had discovered these concepts developed by Bouligand in [*“ suo interessante libro recente”*]{} and comments: [*“All’egregio geometra è evidentemente sfuggito che le sue ricerche in proposito sono state iniziate un po’ più tardi delle mie ... Ma non gli muovo rimprovero per questo, perché neppur io riesco a seguire con cura minuziosa la bibliografica e leggo più volontieri una memoria o un libro dopo aver pensato per conto mio all’argomento.”*]{} I am grateful to M. Bardi for this information about Severi.
[^6]: Let us recall that a subset ${\cal H}$ of continuous functions of ${\cal C}(0,T;{\bf R}^{n})$ is [*equicontinuous*]{} if and only if $$\forall \; t \in [0,T], \; \forall \; \varepsilon > 0, \;
\exists \; \eta \; := \; \eta({\cal H},t, \varepsilon ) \; | \;
\forall \; s \in [t-\eta,t+\eta], \; \sup_{x(\cdot) \in {\cal
H}}\|x(t)-x(s)\| \leq \varepsilon$$ Locally Lipschitz functions with the same Lipschitz constant form an equicontinuous set of functions. In particular, a subset of differentiable functions satisfying $$\sup_{t \in [0,T]} \|x'(t)\| \leq c < + \infty$$ is equicontinuous.
[*Ascoli’s Theorem*]{} states that a subset ${\cal H}$ of functions is [*relatively compact*]{} in ${\cal
C}(0,T;{\bf R}^{n})$ if and only if it is equicontinuous and satisfies $$\forall \; t \in [0,T], \; {\cal H}(t) \; := \;
\{x(t)\}_{x(\cdot) \in {\cal H}} \; \mbox{\rm is compact.}$$
[^7]: The interesting case is obtained when $ \mu >0$. When $f$ is uniformly Lipschitz with constant $ \lambda $, then it is uniformly monotone with $ \mu =- \lambda $.
[^8]: Functions ${\bf v}:X \mapsto [0, + \infty ]$ can be regarded as some kind of [*fuzzy sets*]{}, called [*toll sets*]{}.
[^9]: Indeed, integrating the two sides of inequality $$\frac{d}{dt} \|y_{1} (t)-y_{2} (t)\|^{2} \; = \; 2 \langle g (t,x
(t),y_{1} (t))-g (t,x (t), y_{2} (t)),y_{1} (t)-y_{2} (t) \rangle \; \leq
\; - 2 \mu \|y_{1} (t)-y_{2} (t)\|^{2}$$ yields $$\|y_{1} (t)-y_{2} (t)\|^{2} \; \leq \; e^{-2 \mu t} \|y_{1} (0)-y_{2}
(0)\|^{2}$$
|
---
abstract: 'Motivated by biological questions, we study configurations of equal-sized disks in the Euclidean plane that neither pack nor cover. Measuring the quality by the probability that a random point lies in exactly one disk, we show that the regular hexagonal grid gives the maximum among lattice configurations.'
author:
- 'Herbert Edelsbrunner[^1]'
- 'Mabel Iglesias-Ham[^2]'
- 'Vitaliy Kurlin[^3]'
title: 'Relaxed Disk Packing[^4]'
---
Introduction {#sec1}
============
High-resolution microscopic observations of the DNA organization inside the nucleus of a human cell support the *Spherical Mega-base-pairs Chromatin Domain model* [@Crem00; @Kreth01]. It proposes that inside the chromosome territories in eukaryotic cells, DNA is compartmentalized in sequences of highly interacting segments of about the same length [@Dixon12]. Each segment consists of roughly a million base pairs and resembles a round ball. The balls are tightly arranged within a restricted space, tighter than a packing since they are not rigid, and less tight than a covering to allow for external access to the DNA needed for gene expression.
Motivated by these biological findings, [@Ham14] considered configurations in which the overlap between the balls is limited and the quality is measured by the density, which we define as the expected number of balls containing a random point. We introduce a new measure that favors configurations between packing and covering without explicit constraints on the allowed overlap. Specifically, we measure a configuration by *the probability a random point is contained in exactly one ball*. Since empty space and overlap between disks are both discouraged, the optimum lies necessarily between packing and covering. The interested reader can find references for traditional packing and covering in two and higher dimensions in [@Conway99; @FejesToth53]. In this paper, we restrict attention to equal-sized disks in the plane whose centers form a lattice, leaving three and higher dimensions as well as non-lattice configurations as open problems. Our main result is the following non-surprising fact.
Among all lattice configurations in ${{\ifmmode{{\mathbb R}}\else{\mbox{\({\mathbb R}\)}}\fi}}^2$, the regular hexagonal grid in which each disk overlaps the six neighboring circles in $30^\circ$ arcs maximizes the probability that a random point lies exactly in one disk.
For obvious reasons, we call this the *$12$-hour clock configuration*. We prove its optimality in four sections: preparing the background in Section \[sec2\], proving an equilibrium condition in Section \[sec3\], developing the main argument in Section \[sec4\], and giving the technical details in Appendix \[appA\].
Background {#sec2}
==========
In this section, we introduce notation for lattices, Voronoi domains, and Delaunay triangulations.
#### Lattices.
Depending on the context, we interpret an element of ${{\ifmmode{{\mathbb R}}\else{\mbox{\({\mathbb R}\)}}\fi}}^2$ as a point or a vector in the plane. Vectors $a, b \in {{\ifmmode{{\mathbb R}}\else{\mbox{\({\mathbb R}\)}}\fi}}^2$ are *linearly independent* if $\alpha a + \beta b = 0$ implies $\alpha = \beta = 0$. A *lattice* is defined by two linearly independent vectors, $a, b \in {{\ifmmode{{\mathbb R}}\else{\mbox{\({\mathbb R}\)}}\fi}}^2$, and consists of all integer combinations of these vectors: $$\begin{aligned}
L(a,b) &= \{ ia + jb \mid i, j \in {{\ifmmode{{\mathbb Z}}\else{\mbox{\({\mathbb Z}\)}}\fi}}\} .\end{aligned}$$ Its *fundamental domain* is the parallelogram of points $\alpha a + \beta b$ with real numbers $0 \leq \alpha, \beta \leq 1$. Writing ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}$ for the length of the vector and $\gamma$ for the angle between $a$ and $b$, the area of the fundamental domain is $\det L(a,b) = {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} \sin(\gamma)$. The same lattice is generated by different pairs of vectors, and we will see shortly that at least one of these pairs defines a non-obtuse triangle. We will be more specific about this condition shortly, as it is instrumental in our proof of the optimality of the regular hexagonal grid.
#### Voronoi domain.
Given a lattice $L$, the *Voronoi domain* of a point $p \in L$ is the set of points for which $p$ is the closest: $$\begin{aligned}
V(p) &= \{ x \in {{\ifmmode{{\mathbb R}}\else{\mbox{\({\mathbb R}\)}}\fi}}^2 \mid {{\ifmmode{\|{x}-{p}\|}\else{\mbox{\(\|{x}-{p}\|\)}}\fi}} \leq {{\ifmmode{\|{x}-{q}\|}\else{\mbox{\(\|{x}-{q}\|\)}}\fi}},
\forall q \in L \} .\end{aligned}$$ It is a convex polygon that contains $p$ in its interior. Any two Voronoi domains have disjoint interiors but may intersect in a shared edge or a shared vertex. The lattice looks the same from every one of its points, which implies that all Voronoi domains are translates of each other: $V(p) = p + V(0)$. Similarly, central reflection through the origin preserves the lattice, which implies that $V(0)$ is centrally symmetric.
The *Voronoi diagram* of $L$ is the collection of Voronoi domains of its points. It is *primitive* if the maximum number of Voronoi domains with non-empty common intersection is $3$. In this case, the Voronoi domain is a centrally symmetric hexagon; see Figure \[fig:hexsq\]. In the non-primitive case, there are generators that enclose a right angle, and the Voronoi domains are rectangles. To the first order of approximation, the area of any sufficiently simple and sufficiently large subset of ${{\ifmmode{{\mathbb R}}\else{\mbox{\({\mathbb R}\)}}\fi}}^2$ is the number of lattice points it contains times the area of $V(0)$. Similarly, it is the number of lattice points times the area of the fundamental domain. It follows that the area of $V(0)$ is equal to $\det L$.
#### Packing and covering.
For $\varrho > 0$, we write $B(p,\varrho)$ for the closed disk with center $p$ and radius $\varrho$. The *packing radius* is the largest radius, $r_L$, and the *covering radius* is the smallest radius, $R_L$, such that $B(0,r_L) \subseteq V(0) \subseteq B(0,R_L)$. The *density* of the configuration of disks with radius $\varrho$ centered at the points of $L$ is the area of a disk divided by the area of the Voronoi domain: $$\begin{aligned}
\delta_L (\varrho) &= \tfrac{\varrho^2 \pi}{\det L} .\end{aligned}$$ It is also the expected number of disks containing a random point in ${{\ifmmode{{\mathbb R}}\else{\mbox{\({\mathbb R}\)}}\fi}}^2$. The *packing density* is $\delta_L (r_L)$, which is necessarily smaller than $1$. It is maximized by the regular hexagonal grid, $H$, for which we have $\delta_H (r_H) = 0.906\ldots$. The *covering density* is $\delta_L (R_L)$, which is necessarily larger than $1$. It is minimized by the regular hexagonal grid for which we have $\delta_H (R_H) = 1.209\ldots$. More generally, it is known that $H$ maximizes the density among all configuration of congruent disks whose interiors are pairwise disjoint [@Thue10], and it minimizes the density among all configurations that cover the entire plane [@Kershner39]. Elegant proofs of both optimality results can be found in Fejes Tóth [@FejesToth53].
#### Delaunay triangulations.
Drawing a straight edge between points $p$ and $q$ in $L$ iff $V(p)$ and $V(q)$ intersect along a shared edge, we get the *Delaunay triangulation* of $L$. In the primitive case, the edges decompose the plane into triangles. Among the six triangles sharing $0$ as a vertex, three are translates of each other and, going around $0$, they alternate with their central reflections. It follows that all six triangles are congruent and, in particular, they have equally large circumcircles that all pass through $0$. Since their centers are vertices of the Voronoi domain of $0$, we have the following result.
The vertices of $V(0)$ all lie on the circle bounding $B(0,R_L)$.
The discussion above proves the Inscribed Voronoi Domain Lemma in the primitive case. It is also true in the simpler, non-primitive case in which $V(0)$ is a rectangle. Returning to the primitive case, we note that the two angles opposite to a shared edge in the Delaunay triangulation add up to less than $180^\circ$. In a lattice, these two angles are the same and therefore both acute. The two types of triangles in the Delaunay triangulation of a lattice can be joined across a shared edge in three different ways. We can therefore make the same argument three times and conclude that all angles are less than $90^\circ$. A slightly weaker bound holds in the non-primitive case.
Every lattice $L$ in ${{\ifmmode{{\mathbb R}}\else{\mbox{\({\mathbb R}\)}}\fi}}^2$ has vectors $a, b \in {{\ifmmode{{\mathbb R}}\else{\mbox{\({\mathbb R}\)}}\fi}}^2$ with $L = L(a,b)$ such that
1. in the primitive case $0, a, b$ are the vertices of an acute triangle,
2. in the non-primitive case $0, a, b$ are the vertices of a non-obtuse triangle with a right angle at $0$.
Assuming $a, b$ satisfy the Non-obtuse Generators Lemma, the triangle $0ab$ has edges of length ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}$, ${{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}$, and ${{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}} = {{\ifmmode{\|{a}-{b}\|}\else{\mbox{\(\|{a}-{b}\|\)}}\fi}}$. In the non-primitive case (ii), ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}$ and ${{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}$ are the lengths of the sides of the rectangle $V(0)$, and ${{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}$ is the length of a diagonal. We have ${{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}^2 = {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}^2 + {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}^2$, and therefore ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} \leq {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} \leq {{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}$, possibly after swapping $a$ and $b$. In the primitive case (i), we can choose $a$, $b$, and $c = a-b$ such that ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} \leq {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} \leq {{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}$. In this case, $V(0)$ is a centrally symmetric hexagon with distances ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}, {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}, {{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}$ between antipodal edge pairs.
Equilibrium Configurations {#sec3}
==========================
Given a lattice in ${{\ifmmode{{\mathbb R}}\else{\mbox{\({\mathbb R}\)}}\fi}}^2$, we are interested in the radius of the disks for which the probability that a random point lies inside exactly one disk is maximized. Further maximizing this probability over all lattices, we get the main result of this paper.
#### Partial disks.
Fix a lattice $L$ in ${{\ifmmode{{\mathbb R}}\else{\mbox{\({\mathbb R}\)}}\fi}}^2$. For a radius $\varrho > 0$, consider the set of points that belong to the disk centered at the origin but not to any other disk centered at a point of $L$: $$\begin{aligned}
D(0, \varrho) &= B(0, \varrho) \setminus
\bigcup_{0 \neq p \in L} B(p, \varrho) .\end{aligned}$$ As illustrated in Figure \[fig:partialdisk\], for radii strictly between the packing radius and the covering radius, this set is partially closed and partially open. We distinguish between the *convex boundary* that belongs to the circle bounding $B(0, \varrho)$, and the *concave boundary* that belongs to other circles: $$\begin{aligned}
\partial_x D(0, \varrho) &= \partial B(0, \varrho) \cap D(0, \varrho) , \\
\partial_v D(0, \varrho) &= \partial D(0, \varrho) \setminus
\partial_x D(0, \varrho) . \end{aligned}$$ We note that $\partial_x D(0, \varrho) = \partial B(0, \varrho) \cap V(0)$. By the Inscribed Voronoi Domain Lemma, the vertices of $V(0)$ are all at the same distance from $0$. This implies that for $r_L < \varrho < R_L$, the convex boundary consists of $2$, $4$, or $6$ circular arcs that alternate with the same number of circular arcs in the concave boundary.
#### Angles.
Recall that the convex boundary consists of at most three pairs of arcs, and let $\varphi_i (\varrho)$ be the angle of each of the two arcs in the $i$-th pair, for $i = 1,2,3$. The total angle of the convex boundary is $$\begin{aligned}
\Phi_L (\varrho) &= \sum_{i=1}^3 2 \varphi_i (\varrho),\end{aligned}$$ and the total angle of the concave boundary is $2 \pi - \Phi_L (\varrho)$. We have $\Phi_L (r_L) = 2 \pi$ and $\Phi_L (R_L) = 0$, and between these two limits, the function is continuous and monotonically decreasing.
Let $L$ be a lattice in ${{\ifmmode{{\mathbb R}}\else{\mbox{\({\mathbb R}\)}}\fi}}^2$. Then $\Phi_L \colon [r_L, R_L] \to [0, 2 \pi]$ is continuous, with $\Phi_L (\varrho_1) > \Phi_L (\varrho_2)$ whenever $\varrho_1 < \varrho_2$.
The continuity of the function follows from the fact that $\partial B(0, \varrho)$ intersects $\partial V(0)$ in at most a finite number of points.
To prove monotonicity, we recall that $\Phi_L (\varrho)$ is the total angle of $\partial_x D(0,\varrho) = \partial B(0, \varrho) \cap V(0)$. The Voronoi domain is a convex polygon with $0 \in V(0)$. Drawing circles with radii $\varrho_1 < \varrho_2$ centered at $0$, we let $0 \leq \theta < 2 \pi$ and write $p_1 (\theta)$ and $p_2 (\theta)$ for the points on the circles in direction $\theta$. Either both points belong to $V(0)$, both points do not belong to $V(0)$, or $p_1 (\theta) \in V(0)$ but $p_2 (\theta) \not\in V(0)$. The fourth combination is not possible, which implies $\Phi_L (\varrho_1) \geq \Phi_L (\varrho_2)$. To prove the strict inequality, we just need to observe that there is an arc of non-zero length in $\partial_x D(0, \varrho_1)$ such that the corresponding arc in $\partial B(0, \varrho_2)$ lies outside $V(0)$ and therefore does not belong to $\partial_x D(0, \varrho_2)$.
#### Area.
The probability that a random point belongs to exactly one disk is the area of $D(0, \varrho)$ over the area of $V(0)$. The latter is a constant independent of the radius. We will prove shortly that the former is a unimodal function in $\varrho$ with a single maximum at the radius $\varrho = \varrho_L$ that balances the lengths of the two kinds of boundaries; see Figure \[fig:equilibrium\]. We call $\varrho_L$ the *equilibrium radius* of $L$. Write $A_L \colon [r_L, R_L] \to {{\ifmmode{{\mathbb R}}\else{\mbox{\({\mathbb R}\)}}\fi}}$ for the function that maps $\varrho$ to the area of $D(0, \varrho)$.
Let $L$ be a lattice in ${{\ifmmode{{\mathbb R}}\else{\mbox{\({\mathbb R}\)}}\fi}}^2$. The function $A_L \colon [r_L, R_L] \to {{\ifmmode{{\mathbb R}}\else{\mbox{\({\mathbb R}\)}}\fi}}$ is strictly concave, with a unique maximum at the equilibrium radius $\varrho_L$ that satisfies $\Phi_L (\varrho_L) = \pi$.
Recall that $\Phi_L (\varrho)$ is the total angle of the convex boundary of $D(0, \varrho)$, and $2 \pi - \Phi_L (\varrho)$ is the total angle of the concave boundary. When we increase the radius, the partial disk grows along the convex boundary and shrinks along the concave boundary. Indeed, the derivative is the difference between the two lengths: $$\begin{aligned}
\tfrac{\partial A_L}{\partial \varrho} (\varrho)
&= \varrho [\Phi_L (\varrho) - (2 \pi - \Phi_L (\varrho)] \\
&= 2 \varrho [\Phi_L (\varrho) - \pi].
\end{aligned}$$ The derivative vanishes when $\Phi_L (\varrho) = \pi$, is positive when $\Phi_L (\varrho) > \pi$ and negative when $\Phi_L (\varrho) < \pi$, as claimed.
Optimality of the Regular Hexagonal Grid {#sec4}
========================================
In this section, we present the proof of our main result. After writing the probability that a random point lies in exactly one disk as a function of the radius, we distinguish between three cases, showing that the maximum is attained at the regular hexagonal grid.
#### Probability.
Given a lattice $L$ in ${{\ifmmode{{\mathbb R}}\else{\mbox{\({\mathbb R}\)}}\fi}}^2$, we write $r_L < \varrho_L < R_L$ for the packing, equilibrium, and covering radii. Recall that the probability in question is $$\begin{aligned}
P_L (\varrho_L) &= \tfrac{A_L (\varrho_L)}{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} \sin \gamma} ,\end{aligned}$$ in which $L = L(a,b)$ and $\gamma$ is the angle between $a$ and $b$. Recall furthermore that the convex boundary consists of at most six arcs, two each with angle $\varphi_1$, $\varphi_2$, $\varphi_3$, in which we set the angle to zero if the arc degenerates to a point or is empty.
Let $L = L(a,b)$ be a lattice in ${{\ifmmode{{\mathbb R}}\else{\mbox{\({\mathbb R}\)}}\fi}}^2$ with angle $\gamma$ between $a$ and $b$. Then $$\begin{aligned}
P_L (\varrho_L) &= \tfrac{2 \varrho_L^2}{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} \sin \gamma}
\cdot \sum_{i=1}^3 \sin \varphi_i .
\label{eqn:probability}
\end{aligned}$$
Recall that ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} \sin \gamma$ is the area of $V(0)$. Let $A_{\rm in}$ be the area of $B(0, \varrho_L) \cap V(0)$, let $A_{\rm out}$ be the area of $B(0, \varrho_L) \setminus V(0)$, and note that $A_{\rm in} - A_{\rm out}$ is the area of $D(0, \varrho_L)$. Since $A_{\rm in} + A_{\rm out} = \varrho_L^2 \pi$, we have $A_{\rm in} - A_{\rm out} = \varrho_L^2 \pi - 2 A_{\rm out}$. The portion of $B(0, \varrho_L)$ outside the Voronoi domain consists of up to three symmetric pairs of disk segments, with total area $$\begin{aligned}
A_{\rm out} &= 2 \sum_{i=1}^3 \tfrac{\varrho_L^2}{2}
(\varphi_i - \sin \varphi_i) \\
&= \varrho_L^2 \left( \tfrac{\pi}{2}
- \sum_{i=1}^3 \sin \varphi_i \right) ,
\label{eqn:areaout}
\end{aligned}$$ in which the second line is obtained using $\sum_{i=1}^3 \varphi_i = \tfrac{\pi}{2}$ from the Equilibrium Radius Lemma. The probability is $A_{\rm in} - A_{\rm out}$ divided by the area of the Voronoi domain: $$\begin{aligned}
P_L (\varrho_L) &= \tfrac{\varrho_L^2 \pi - 2 A_{\rm out}}
{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} \sin \gamma}.
\end{aligned}$$ Together with this implies the claimed relation.
#### Case analysis.
We focus on the primitive case in which the Voronoi domain is a hexagon, considering the non-primitive case a limit situation in which two of the edges shrink to zero length. Let $a, b \in {{\ifmmode{{\mathbb R}}\else{\mbox{\({\mathbb R}\)}}\fi}}^2$ be generators of the lattice satisfying the condition in the Non-obtuse Generators Lemma, set $c = a-b$, and assume ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} \leq {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} \leq {{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}$. Recall that these three lengths are the distances between parallel edges of the hexagon. Further notice that we have $r_L = \tfrac{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}}{2}$ for the packing radius and $R_L > \tfrac{{{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}}{2}$ for the covering radius. As before, we write $\varphi_i$ for the angles of the arcs of $\partial_x D(0, \varrho)$, and we index such that $\varphi_1 \geq \varphi_2 \geq \varphi_3$.
[Case 1:]{}
: $\tfrac{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}}{2} < \varrho_L \leq \tfrac{{{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}}{2}$. Then $\varphi_1 > 0$ and $\varphi_2 = \varphi_3 = 0$.
[Case 2:]{}
: $\tfrac{{{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}}{2} < \varrho_L \leq \tfrac{{{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}}{2}$. Then $\varphi_1 \geq \varphi_2 > 0$ and $\varphi_3 = 0$.
[Case 3:]{}
: $\tfrac{{{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}}{2} < \varrho_L < R_L$. Then $\varphi_1 \geq \varphi_2 \geq \varphi_3 > 0$.
For example the configuration depicted in Figure \[fig:partialdisk\] falls into Case 2. Using the expression for the probability in the Equilibrium Area Lemma, we determine the maximum for each of the three cases. Here we state the results, referring to Appendix \[appA\] for the proofs. By *the probability* we mean of course the probability that a random point belongs to exactly one disk.
In Case 1, the maximum probability is attained for ${{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} = \sqrt{2} {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}$, $\gamma = \arccos \tfrac{1}{2 \sqrt{2}}$, and $\varrho_L = {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}/\sqrt{2}$, which gives $P_L (\varrho_L) = 0.755\ldots$.
In Case 2, the maximum probability is attained for ${{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} = {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}$, $\gamma = \arccos (\sqrt{2} - 1)$, and $\varrho_L = {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}\sqrt{1 - 1/\sqrt{2}}$, which gives $P_L (\varrho_L) = 0.910\ldots$.
In Case 3, the maximum probability is attained for ${{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} = {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}$ = , $\gamma = \tfrac{\pi}{3}$ and $\varrho_L = {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}/(2\cos\tfrac{\pi}{12})$, which gives $P_L (\varrho_L) = 0.928\ldots$.
Note that the lattice in the Six Arcs Lemma is the regular hexagonal grid. Comparing the three maximum probabilities, we see that the regular hexagonal grid gives the global optimum; see Figures \[fig:prob\].
For this lattice, we get $\varrho_H$ such that each disk overlaps with six others and in each case covers $30^\circ$ of the bounding circle: the $12$-hour clock configuration in the plane. This implies the Main Theorem stated in Section \[sec1\]. We further illustrate the result by showing the graph of the function that maps ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} / {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}$ and the angle $\gamma$ to the probability at the equilibrium radius; see Figure \[fig:capregions\].
![The graph of the function that maps $0 \leq {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}/{{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} \leq 1$ and $\arccos \tfrac{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}}{2 {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}} \leq \gamma \leq 90^\circ$ to the probability that a random point lies in exactly one disk. The thus defined domain resembles a triangle and decomposes into three regions corresponding to Cases 1, 2, 3. The regular hexagonal grid is located at the lower left corner of the domain.[]{data-label="fig:capregions"}](Surf.png){width="12cm"}
Discussion {#sec5}
==========
The main result of this paper is a proof that the $12$-hour clock configuration of disks in the plane maximizes the probability that a random point lies in exactly one of the disks. Other criteria favoring configurations between packing and covering can be formulated, see [@Ham14], and it would be interesting to decide which one fits the biological data about DNA organization within the nucleus best. There are also concrete mathematical questions related to the work in this paper:
- Is the $12$-hour clock configuration optimal among all configurations of congruent disks in the plane?
- What is the optimal lattice configuration of balls in ${{\ifmmode{{\mathbb R}}\else{\mbox{\({\mathbb R}\)}}\fi}}^3$?
To appreciate the difficulty of the second question, we note that the FCC lattice gives the densest packing [@Gauss63], while the BCC lattice gives the sparsest covering [@Bambah54]. Does one of them also maximize the probability that a random point lies inside exactly one ball?
Acknowledgements
================
Vitaliy Kurlin is grateful to Herbert Edelsbrunner for hosting a short visit at IST Austria in March 2015, which was partially funded by the ACAT network. The authors are grateful for the collaboration of Michael Kerber on proof checking the maple file that supports the computations on this paper.
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*Sphere Packings, Lattices and Groups.* Third edition, Springer-Verlag, New York, New York, 1999.
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Topological domains in mammalian genomes identified by analysis of chromatin interactions. *Nature* [**485**]{} (2012) 376–80.
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Proofs {#appA}
======
In this appendix, we give detailed proofs of the three Arc Lemmas. As described in Section \[sec4\], the three lemmas add up to a proof of the Main Theorem stated in Section \[sec1\] of this paper. We begin with a few relations that will be useful in all three proofs. Given a triangle with edges of lengths ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}, {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}, {{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}$ and angle $\gamma$ opposite the edge $c$, the *law of cosines* implies $$\begin{aligned}
{{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}^2 &= {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}^2 + {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}^2 - 2 {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} \cos \gamma .
\label{eqn:lawofcosines}\end{aligned}$$ Assuming ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} \leq {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} \leq {{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}$, the angle $\gamma$ is at least as large as each of the other two angles. From together with ${{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} \leq {{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}$, we get $\cos \gamma \leq \tfrac{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}}{2 {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}}$. As justified by the Non-obtuse Generators Lemma, we may assume the triangle is non-obtuse, which implies $$\begin{aligned}
\arccos \tfrac{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}}{2 {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}} &\leq \gamma \leq \tfrac{\pi}{2} .\end{aligned}$$ The range of possible angles is largest for ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} = {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}$ where we get $60^\circ \leq \gamma \leq 90^\circ$. Furthermore, we write the angles $\varphi_i$ of the arcs in the convex boundary of the partial disk in terms of the edge lengths and the radius: $$\begin{aligned}
\cos \tfrac{\varphi_1}{2} &= \tfrac{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}}{2 \varrho} ,
\label{eqn:varphi1} \\
\cos \tfrac{\varphi_2}{2} &= \tfrac{{{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}}{2 \varrho} ,
\label{eqn:varphi2} \\
\cos \tfrac{\varphi_3}{2} &= \tfrac{{{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}}{2 \varrho} .
\label{eqn:varphi3}\end{aligned}$$ The first relation holds provided $\tfrac{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}}{2} \leq \varrho < R_L$, and similar for the second and third relations. Finally, we note that scaling does not affect the density of a configuration. We can therefore set ${{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} = 1$, which we will do to simplify computations.
#### Proof of the Two Arcs Lemma.
Case 1 is defined by $\tfrac{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}}{2} < \varrho_L \leq \tfrac{{{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}}{2}$, which implies $\varphi_1 > 0$ and $\varphi_2 = \varphi_3 = 0$. Since $\sin \varphi_2 = \sin \varphi_3 = 0$, the probability at the equilibrium radius simplifies to $$\begin{aligned}
P_L (\varrho_L) &= \tfrac{2 \varrho_L^2}{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} \sin \gamma}
\cdot \sin \varphi_1
\label{eqn:TwoProb1} \\
&= \tfrac{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}}{{{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} \sin \gamma} ,
\label{eqn:TwoProb2}\end{aligned}$$ where we get the second line by combining $\varphi_1 = \tfrac{\pi}{2}$ with to imply $\varrho_L = {{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}} / {\sqrt{2}}$. For the remainder of this proof, we normalize by setting ${{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} = 1$. To maximize the probability, we choose ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}$ as large as possible and $\gamma$ as small as possible. From ${{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}} / {\sqrt{2}} = \varrho_L \leq \tfrac{1}{2}$, we get ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} \leq 1 / \sqrt{2}$, and from ${{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} \leq {{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}$ we get $\gamma \geq \arccos \tfrac{1}{2 \sqrt{2}}$. The two parameters can be optimized simultaneously, which gives $$\begin{aligned}
P_L (\varrho_L) &= \tfrac{1}{\sqrt{2}
\sin \left( \arccos \tfrac{1}{2 \sqrt{2}} \right)}
= 0.755\ldots .\end{aligned}$$
#### Proof of the Four Arcs Lemma.
Case 2 is defined by $\tfrac{{{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}}{2} < \varrho_L \leq \tfrac{{{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}}{2}$, which implies $\varphi_1 \geq \varphi_2 > 0$ and $\varphi_3 = 0$. The probability at the equilibrium radius is therefore $$\begin{aligned}
P_L (\varrho_L) &= \tfrac{2 \varrho_L^2}{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} \sin \gamma}
\cdot ( \sin \varphi_1 + \sin \varphi_2 ) .
\label{eqn:FourProb}\end{aligned}$$ To get a handle on the maximum of this function, we first write the the sum of $\sin \varphi_1$ and $\sin \varphi_2$ and second the equilibrium radius in terms of other parameters. Using $\cos 2 \alpha = \cos^2 \alpha - \sin^2 \alpha$ and $\cos^2 \alpha + \sin^2 \alpha = 1$, we get $\cos \varphi_2 = 2 \cos^2 \tfrac{\varphi_2}{2} - 1$, and since $\varphi_1 + \varphi_2 = \frac{\pi}{2}$, we have $\sin \varphi_1 = \cos \varphi_2$. Recalling , we get $\sin \varphi_1 = {{{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}^2} / {(2 \varrho_L^2)} - 1$, and recalling , we get $\sin \varphi_2 = {{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}^2} / {(2 \varrho_L^2)} - 1$. Adding the two relations gives $$\begin{aligned}
\sin \varphi_1 + \sin \varphi_2
&= \tfrac{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}^2 + {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}^2}{2 \varrho_L^2} - 2 .
\label{eqn:sinplussin}\end{aligned}$$ To find a substitution for the equilibrium radius, we begin with , use $\tfrac{\varphi_2}{2} = \tfrac{\pi}{4} - \tfrac{\varphi_1}{2}$, and finally apply $\cos (\alpha+\beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$: $$\begin{aligned}
\tfrac{{{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}}{2 \varrho_L}
&= \cos \left( \tfrac{\pi}{4} - \tfrac{\varphi_1}{2} \right) \\
&= \tfrac{1}{\sqrt{2}} \left( \cos \tfrac{\varphi_1}{2}
+ \sin \tfrac{\varphi_1}{2} \right) .\end{aligned}$$ Next, we substitute the two trigonometric functions using and $\sin^2 \alpha = 1 - \cos^2 \alpha$. Simplifying the resulting relation and squaring it, we get $$\begin{aligned}
\varrho_L^2 &= \tfrac{1}{2}
\left( {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}^2 + {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}^2 - \sqrt{2} {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} \right) .
\label{eqn:fourarcsradius}\end{aligned}$$ Plugging and into the equation for the probability and normalizing by setting ${{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} = 1$, we get $$\begin{aligned}
P_L (\varrho_L) &= \tfrac{2 \sqrt{2} {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} - {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}^2 - 1}
{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} \sin \gamma} \\
&= \tfrac{2 \sqrt{2} {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} - {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}^2 - 1}
{{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}}\sqrt{1 -
\left( \sqrt{2}-\tfrac{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}^2+1}{2 {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}} \right)^2 }}
\label{eqn:fourarcsprob}\end{aligned}$$ where we maximize to get the second line by choosing $\gamma$ as small as possible. Specifically, $\gamma$ is implicitly restricted by $\tfrac{{{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}}{2} < \varrho_L \leq \tfrac{{{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}}{2}$, so we can use $4 \varrho_L^2 \leq {{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}^2$ together with and to get $$\begin{aligned}
\cos \gamma &\leq \sqrt{2} - \tfrac{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}^2 + {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}^2}
{2 {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}} .
\label{eqn:fourarcsgamma}\end{aligned}$$ Checking with the Maple software [@maple], we find that the right-hand-side of increases in $[0,1]$ attaining its maximum at ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} = 1$. We therefore get $\gamma = \arccos \left( \sqrt{2} - 1 \right)$ from , $\varrho_L^2 = 1 - 1/\sqrt{2}$ from , and $$\begin{aligned}
P_L (\varrho_L) &= \sqrt{ 2 \sqrt{2} - 2 }
= 0.910\ldots \end{aligned}$$ from .
#### Proof of the Six Arcs Lemma.
Case 3 is defined by $\tfrac{{{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}}{2} < \varrho_L < R_L$, which implies $\varphi_1 \geq \varphi_2 \geq \varphi_3 > 0$. Starting with the expression for the probability given in the Equilibrium Area Lemma, we first express the $\sin \varphi_i$ in terms of the other parameters: $$\begin{aligned}
\sin \varphi_1 &= \tfrac{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} \sqrt{4 \varrho_L^2 - {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}^2}}
{2 \varrho_L^2} ,
\label{eqn:sixarcsvarphi1} \\
\sin \varphi_2 &= \tfrac{{{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} \sqrt{4 \varrho_L^2 - {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}^2}}
{2 \varrho_L^2} ,
\label{eqn:sixarcsvarphi2} \\
\sin \varphi_3 &= 1 -
\tfrac{\left[ {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} \sqrt{4 \varrho_L^2 - {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}^2}
+ {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} \sqrt{4 \varrho_L^2 - {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}^2} \right]^2}
{8 \varrho_L^4} .
\label{eqn:sixarcsvarphi3}\end{aligned}$$ To get , we use $\sin 2 \alpha = 2 \sin \alpha \cos \alpha$ with $\alpha = \tfrac{\varphi_1}{2}$, together with . To get , we use the same trigonometric identity with $\alpha = \tfrac{\varphi_2}{2}$, together with . To get , we use the equilibrium condition together with $\sin \varphi_3 = \sin (\tfrac{\pi}{2} - \varphi_1 - \varphi_2) = \cos (\varphi_1 + \varphi_2)
= 1 - 2 \sin^2 \tfrac{\varphi_1 + \varphi_2}{2}$, and finally substitute $\sin (\alpha + \beta) = \sin \alpha \cos \beta + \sin \beta \cos \alpha$, with $\alpha = \tfrac{\varphi_1}{2}$ and $\beta = \tfrac{\varphi_2}{2}$.
To do the same for $\sin \gamma$, we take the cosine of both sides of the equilibrium condition, which is $\tfrac{\varphi_3}{2} = \tfrac{\pi}{4}
- \tfrac{\varphi_1}{2} - \tfrac{\varphi_2}{2}$. Writing $c_i = \cos \tfrac{\varphi_i}{2}$ and $s_i = \sin \tfrac{\varphi_i}{2}$, for $i = 1,2,3$, and applying standard trigonometric identities, we get $$\begin{aligned}
c_3 &= \tfrac{1}{\sqrt{2}} [ c_1 c_2 - s_1 s_2 + s_1 c_2 + c_1 s_2 ] . \\
c_3^2 &= \tfrac{1}{2} + (2 c_1 c_2^2 - c_1) \sqrt{1-c_1^2} \nonumber \\
&~~~~~~~~ + (2 c_1^2 c_2 - c_2) \sqrt{1-c_2^2} .\end{aligned}$$ Using , , and substituting ${{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}^2$ using , we get the following relation after a few rearrangements: $$\begin{aligned}
\cos \gamma
&= \tfrac{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}^2 + {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}^2 - 2}{2 {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}}
- \tfrac{{{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}^2-2\varrho_L^2}{4 \varrho_L^2 {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}} \sqrt{4\varrho_L^2-{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}^2}
\nonumber \\
& ~~~~~~~~~~~~~~~~~~~~~~~~~
- \tfrac{{{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}^2-2\varrho_L^2}{4 \varrho_L^2 {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}} \sqrt{4\varrho_L^2-{{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}^2} .
\label{eqn:squared}\end{aligned}$$ Using $\cos^2 \gamma = 1 - \sin^2 \gamma$, we can substitute $\sin \gamma$ in the formula for $P_L (\varrho_L)$. We thus arrived at a relation that gives the probability in terms of ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}$, ${{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}}$, and $\varrho_L$ only. While being lengthy, this relation is readily obtained by plugging , , , and into . We therefore take the liberty to omit the formula here and refer the interested reader to the website of the second author of this paper[^5].
It remains to determine the parameters that maximize the probability. To simplify this task, we normalize by setting ${{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} = 1$. The probability is thus a function of two variables, ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}$ and $\varrho_L$. Using the Maple software, we compute the two partial derivatives, $\partial P_L / \partial {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}}$ and $\partial P_L / \partial \varrho_L$. Setting both to zero, we get ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} = 1$ matched up with three radius values: $$\begin{aligned}
\varrho_L &= \tfrac{1}{2} \left( \sqrt{6}-\sqrt{2} \right) = 0.517\ldots,
\label{eqn:firstroot} \\
\varrho_L &= \tfrac{1}{2} \sqrt{C^\frac{1}{3}-1+C^{-\frac{1}{3}}} = 0.582\ldots,
\label{eqn:secondroot} \\
\varrho_L &= \tfrac{1}{2} \left( \sqrt{6}+\sqrt{2} \right) = 1.931\ldots,
\label{eqn:thirdroot}\end{aligned}$$ with $C = 3 + 2 \sqrt{2}$. Setting ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} = {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} = 1$, the covering radius depends only on the angle $\gamma$, which ranges from $\tfrac{\pi}{3}$ to $\tfrac{\pi}{2}$. It is largest for $\gamma = \tfrac{\pi}{2}$, where $R_L = \sqrt{2}/2 = 0.707\ldots$. The radius we get at the third root is larger than that and can therefore be excluded.
![The quadrangle in the plane defined by the length of $a$ and the radius is shaded. Along its boundary, we encounter three local minima (two corners and a point along the right edge) and three local maxima (a corner and a point each along the lower edge and the right edge).[]{data-label="fig:superset"}](superset.png){width="9cm"}
The maximum probability is attained at one of the two remaining roots or along the boundary of the domain region that corresponds to Case 3. As illustrated in Figure \[fig:superset\], we simplify the computation by taking a quadrangle that contains this region. The quadrangle is defined by $$\begin{aligned}
\tfrac{\sqrt{2}}{2} &\leq {{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} \leq 1 \\
\tfrac{1}{2} &\leq \varrho~~~~ \leq \tfrac{{{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}}}{2\sin \gamma_{max}} , \end{aligned}$$ in which the first interval follows from the bounds that define Case 3, and the second interval is obtained by limiting the radius by the covering radius of the configuration in which $a$ and $b$ enclose its maximum angle. The maximum angle for Case 3, $\gamma_{max}$, corresponds to the minimal angle for Case 2 derived from . The upper boundary on $\varrho$ is a convex curve and so we replace it with the straight line connecting its extremes.
We evaluate the probability at the four vertices and, using the Maple software, at the roots of the derivatives along the four edges. As shown in Figure \[fig:superset\], we find three local minima alternating with three local maxima along the boundary of the quadrangle. The maximum and minimum in the interior of the right edge coincide with the roots in and . Among the three local maxima, the probability is largest at , which is characterized by ${{\ifmmode{\|{a}\|}\else{\mbox{\(\|{a}\|\)}}\fi}} = {{\ifmmode{\|{b}\|}\else{\mbox{\(\|{b}\|\)}}\fi}} = 1$ and $\varrho_L = \tfrac{1}{2} \left( \sqrt{6} - \sqrt{2} \right) = 1 / (2 \cos \tfrac{\pi}{12})$. This gives $P_L (\varrho_L) = 0.928\ldots$, as claimed in the Six Arcs Lemma. Indeed, plugging the values into the equilibrium condition of Case 3, we get ${{\ifmmode{\|{c}\|}\else{\mbox{\(\|{c}\|\)}}\fi}} = 1$, which shows that the probability is maximized by the regular hexagonal grid.
[^1]: IST Austria (Institute of Science and Technology Austria), Klosterneuburg, Austria, `edels@ist.ac.at`.
[^2]: IST Austria (Institute of Science and Technology Austria), Klosterneuburg, Austria, `mabel.iglesias-ham@ist.ac.at`.
[^3]: Microsoft Research Cambridge and Mathematical Sciences, Durham University, United Kingdom, `vitaliy.kurlin@gmail.com`.
[^4]: This work is partially supported by the [Toposys]{} project FP7-ICT-318493-STREP, and by ESF under the ACAT Research Network Programme.
[^5]: A Maple file with the main steps in the formulas related with this paper is available at <http://mabelih9.wix.com/mabelhome#!publications/cee5>.
|
---
abstract: 'The spectrum and dominant strong decay properties of the doubly heavy baryons are revisited by using a chiral effective model with chiral partner structure. By regarding the doubly heavy baryons in the ground states and light angular momentum $j_l = 1/2$ sector of the first orbitally excited states as chiral partners to each other, we estimate the mass splitting of the chiral partners which arises from the spontaneous breaking of chiral symmetry to be about $430$ MeV for baryons including an unflavored light quark and about $350$ MeV for that including a strange quark. We point out that, similar to the heavy-light meson sector, the intermultiplet decay from a baryon with negative parity to its chiral partner and a pion is determined by the mass splitting throught the generalized Goldberger-Treiman relation. Furthermore, the isospin violating decay of $\Omega_{cc}$ baryon, $((1/2)^-, (3/2)^-)_s \to ((1/2)^+, (3/2)^+)_s + \pi^0$ through the $\eta$-$\pi^0$ mixing is the dominant decay channel of the doubly heavy baryons including a strange quark.'
author:
- 'Yong-Liang Ma'
- Masayasu Harada
title: Doubly heavy baryons with chiral partner structure
---
The heavy hadron spectroscopy has drown extensive attention since last decade because of the observation of the large amount of heavy hadrons in particle colliders. It is reasonable to expect that more and heavier resonances, such as the doubly heavy baryons (DHBs) concerned in this work, could be observed by the ongoing and future scientific facilities such as LHCb and Belle II.
The existence of DHBs is an immediate prediction of QCD. The theoretical discussion of these baryons has been done for a long time [@Moinester:1995fk; @Brodsky:2011zs; @Sun:2014aya]. Meanwhile, several experimental efforts have been made to detect such states and the positive results from SELEX shows that the mass of the doubly charmed baryons $\Xi_{cc}^+$ is about $3520$ MeV [@Agashe:2014kda]. In this paper, we investigate some properties of the DHBs based on the chiral partner structure. Since there is only one light quark in a DHB, unlike the light baryons, its chiral behaviour is quite simple.
The chiral partner structure of hadrons including heavy quark has been studied by several groups. In the heavy-light meson sector, the pioneer idea was proposed by Nowak [*et al.*]{} in Ref. [@Nowak:1992um] and then Bardeen and Hill [@Bardeen:1993ae]. In this picture, the heavy-light meson doublets in the heavy quark limit with quantum numbers ($1^-, 0^-$) and ($1^+, 0^+$) are regarded as chiral partners to each other and the mass splitting of them is induced by the dynamical breaking of the chiral symmetry so that the magnitude is about the constituent quark mass. This was confirmed by the spectrum of the relevant particles: $m_{D_0^{\ast}} - m_D \simeq m_{D_1}- m_{D^{\ast}} \simeq 450~$MeV is at the same order of $m_{D_{s0}}(2317) - m_{D_s} \simeq
m_{D_{s1}}(2460)- m_{D_{s}^{\ast}} \simeq 350~$MeV (see, e.g., Refs. [@Bardeen:2003kt; @Nowak:2003ra; @Nowak:2004jg]). In the sector of heavy baryons including a heavy quark, the chiral partner structure are mainly accessed based on the bound state approach (see, e.g., Ref. [@Harada:2012dm] and references therein). In this sector, there are some disagreements about the chiral partner structure. Normally, the chiral partner of the ground state baryon $\Lambda_c(2268)(J^P = \frac{1}{2}^+)$ is regarded as $\Lambda_c(2595)$ with $J^P = \frac{1}{2}^-$. However, a recent analysis with including the lowest lying vector mesons $\rho$ and $\omega$ meson effects shows that the chiral partner of $\Lambda_c(\frac{1}{2}^+,2286)$ is the $\Lambda_c(\frac{1}{2}^-,\frac{3}{2}^-)$ heavy quark doublet with a mass of about $3.1$GeV [@Harada:2012dm]. This disagreement might arise from the fact that, in contrast to the heavy-light meson sector and also the DHBs considered in this work, in the heavy baryons including one heavy quark, there are two light quarks so their chiral properties are not so simple.
Schematically, the quark contents of a DHB can be written as $QQq$ with $Q$ and $q$ being the heavy quark and light quark constituents, respectively. Since the DHB is a colorless object, the two heavy quarks in it form an anti-color triplet [@Bardeen:2003kt]. Because the heavy quarks in the DHB have a large mass, it takes a much larger energy to orbitally excite the heavy constituent quark than to orbitally excite the light one, it is reasonable to regard the two constituent quarks as an static compact object without orbital excitation and denote the constituent of the DHB as $\bar{\bm{\Phi}}q$ with $\bar{\Phi}$ being the heavy quark component which should be a bosonic quantity. With such an intuitive picture in mind, one can define the chiral partner structure similar to that in the heavy-light meson sector [@Nowak:1992um; @Bardeen:1993ae].
Since two heavy quarks in a DHB is antisymmetric in color space, they should have the total spin $J_Q = 1$ in $s$-wave and, therefore DHBs in the ground states can form a heavy quark doublet $D_\textbf{Q}^\mu$ whose components have quantum numbers $J^P = \frac{1}{2}^+,\frac{3}{2}^+$. For the first orbital excitation with relative angular momentum between the light quark and heavy quark source $l = 1$, the light angular momentum could be $j_l = \frac{1}{2}, \frac{3}{2}$. Combing the $j_l = \frac{1}{2}$ one can form another heavy quark doublet $N_\textbf{Q}^\mu$ with quantum numbers $J^P = \frac{1}{2}^-,\frac{3}{2}^-$. We regard the doublets $D_\textbf{Q}^\mu$ and $N_\textbf{Q}^\mu$ as chiral partners to each other and DHBs constructed from $j_l = \frac{3}{2}$ can be regarded as chiral partners of some states from $l =2$ baryons [@Bardeen:2003kt].
Similar to the heavy-light meson case, since there is only one light quark in a DHB, we can write the DHB doublets $D_\textbf{Q}^\mu$ and $N_\textbf{Q}^\mu$ in the chiral basis by introducing the fields $D_{\textbf{Q};L,R}^\mu$ which at the quark level are schematically written as $D_{\textbf{Q};L,R}^\mu \sim \bar{\bm{\Phi}}^\mu q_{L, R}$. Since the heavy quark component of the DHB is a boson, $D_{\textbf{Q};L,R}^\mu$ should be Lorentz spinors and, under chiral transformation, transform as $$\begin{aligned}
D_{\textbf{Q};L,R}^\mu & \to & g_{L,R} D_{\textbf{Q};L,R}^\mu ,
\label{eq:chiralDLR}\end{aligned}$$ where $g_{L,R} \in SU(3)_{L,R}$. In terms of the $D_{\textbf{Q}}$ and $N_{\textbf{Q}}$, we can write $$\begin{aligned}
D_{\textbf{Q};L}^\mu & = & \frac{1}{\sqrt{2}}\left(D_{\textbf{Q}}^\mu - i N_{\textbf{Q}}^\mu \right) , \nonumber\\
D_{\textbf{Q};R}^\mu & = & \frac{1}{\sqrt{2}}\left(D_{\textbf{Q}}^\mu + i N_{\textbf{Q}}^\mu \right),
\label{eq:ChiralPhys}\end{aligned}$$ which transform as $D_{\textbf{Q};L, R}^\mu \leftrightarrow \gamma_0 D_{\textbf{Q};\mu;R, L}$ under parity transformation and satisfy $v\hspace{-0.17cm}\slash D_{\textbf{Q};L,R}^\mu = D_{\textbf{Q};L,R}^\mu$ and $v_\mu D_{\textbf{Q};L,R}^\mu = 0$ for preserving the heavy quark symmetry and keeping the transversality. And, for later convenience, following the procedure given in Ref. [@Falk:1991nq], we write the DHB doublets $D_\textbf{Q}^\mu$ and $N_\textbf{Q}^\mu$ in terms of the physical states as $$\begin{aligned}
D_{\textbf{Q}}^\mu & = & \frac{1 + v \hspace{-0.17cm}\slash}{2}\Psi_{QQ}^\mu + \sqrt{\frac{1}{3}}\left(\gamma^\mu + v^\mu\right)\gamma^5\frac{1 + v \hspace{-0.17cm}\slash}{2}\Psi_{QQ} , \nonumber\\
N_{\textbf{Q}}^\mu & = & \frac{1 + v \hspace{-0.17cm}\slash}{2}\Psi_{QQ}^{\prime \mu} + \sqrt{\frac{1}{3}}\left(\gamma^\mu + v^\mu\right)\gamma^5\frac{1 + v \hspace{-0.17cm}\slash}{2}\Psi_{QQ}^{\ast} ,\end{aligned}$$ which is the same as that for the heavy baryons including one heavy quark [@Georgi:1990cx; @Cho:1992gg] and $\Psi_{QQ}^{(\prime)\mu}$ is the spin-$\frac{3}{2}$ Rarita-Schwinger field. One can easily check that these spinors satisfy $v\hspace{-0.17cm}\slash D_{\textbf{Q}}^\mu = D_{\textbf{Q}}^\mu$ and $v\hspace{-0.17cm}\slash N_{\textbf{Q}}^\mu = N_{\textbf{Q}}^\mu$. We have imposed the intrinsic parity behaviour $$\begin{aligned}
{\rm P} &:&~\Psi_{QQ}^\mu \to{}- \gamma_0 \Psi_{QQ,\mu},~~ \Psi_{QQ} \to \gamma_0 \Psi_{QQ} , \nonumber\\
& & ~ \Psi_{QQ}^{\prime \mu} \to \gamma_0\Psi_{QQ, \mu}^{\prime},~~ \Psi_{QQ}^{\ast} \to {} - \gamma_0\Psi_{QQ}^{\ast}.\end{aligned}$$ When the heavy quark in the DHB is $c$ quark and the light quark is either of $u, d$ and $s$ quarks, the DHB field, for example $\Psi_{QQ}$ stands for $\Xi_{cc}^{++}, \Xi_{cc}^+$ and $\Omega_{cc}^+$, respectively.
Now, we are in the position to construct the chiral effective theory of DHBs in the chiral basis. We note that the quark-diquark symmetry [@Savage:1990di] relates the doubly heavy baryons with the heavy mesons having the same Brown muck [@Hu:2005gf]. For relating the parameters based on the quark-diquark symmetry, we first write an effective Lagrangian for the heavy-light mesons with the chiral partner structure by introducing chiral fields $\mathcal{H}_{L,R}$ [@Nowak:1992um; @Bardeen:1993ae]. These chiral fields relate to the heavy-light meson doublets $H$ and $G$ with quantum numbers $(0^-,1^-)$ and $(0^+,1^+)$, respectively, through $$\begin{aligned}
\mathcal{H}_R = \frac{1}{\sqrt{2}}\left[ G - i H \gamma_5 \right] \ ,\ \ \mathcal{H}_L = \frac{1}{\sqrt{2}}\left[ G + i H \gamma_5 \right]\ ,
\label{eq:HLRGH}\end{aligned}$$ where $G$ and $H$ are heavy-light meson fields with the positive and negative parity, respectively. In terms of the physical states, they are expressed as $$\begin{aligned}
H & = & \frac{1 + v\hspace{-0.17cm}\slash }{2} \left[ D^{\ast\mu}\gamma_\mu + i D \gamma_5 \right] \ , \notag\\
G & = & \frac{1 + v\hspace{-0.17cm}\slash }{2} \left[{} - D_1^{\prime\mu} \gamma_\mu \gamma_5 + D_0^\ast \right] \ .
\label{eq:HGphys}\end{aligned}$$ It should be noticed that, since in the heavy-light meson fields, the heavy component is a heavy quark and the light component is a light antiquark, not the chiral fields $\mathcal{H}_{L,R}$ but their conjugates $\bar{\mathcal{H}}_{L,R} \equiv \gamma_0 \mathcal{H}_{L,R} \gamma_0$ transform as the chiral quark fields $q_{L,R}$ under chiral transformation, i.e., the same as Eq. . Here, we consider only the terms which survive in the heavy quark limit and including the terms up till one derivative. For the light mesons, we consider the chiral field $M$ which transforms as $M \to g_L M g_R^\dagger$ under chiral transformation. The effective Lagrangian is written as [@Harada:2012km; @Suenaga:2014sga] $$\begin{aligned}
{\mathcal L}_{\rm M} & = & {\rm tr}\left[\mathcal{H}_L(iv\cdot\partial)\bar{\mathcal{H}}_L] + {\rm tr}[\mathcal{H}_R(iv\cdot\partial)\bar{\mathcal{H}}_R\right] \nonumber\\
& &{} - \Delta \mbox{tr} \left[ \mathcal{H}_L \bar{\mathcal{H}}_L + \mathcal{H}_R \bar{\mathcal{H}}_R\right] \nonumber\\
& &{} - \frac{1}{2}\, g_\pi \mbox{tr}
\left[\mathcal{H}_L M \bar{\mathcal{H}}_R + \mathcal{H}_R M^{\dagger} \bar{\mathcal{H}}_L\right] \nonumber\\
& &{} + i \frac{g_{A}}{f_\pi}\mbox{tr}
\left[\mathcal{H}_L\gamma_5\gamma^{\mu}\partial_{\mu} M \bar{\mathcal{H}}_R
- \mathcal{H}_R\gamma_5\gamma^{\mu}\partial_{\mu}M^\dag \bar{\mathcal{H}}_L\right] \ , \label{pionlagrangian}\end{aligned}$$ where $\Delta$ provides the mass shift to both $G$ and $H$ in the same direction. After a suitable choice of the potential sector of the light meson Lagrangian which will not be specified here, one can realize the chiral symmetry in the Nambu-Goldstone phase. In such a case, after the spontaneous breaking of the chiral symmetry, the meson field $M$ can be replaced by $\bar{M} + \tilde{M}$ with $\bar{M} = {\rm diag} (v,v, v_3)$ being the vacuum expectation value of the chiral field in the isospin limit, which corresponds to the quark condensate, and $\tilde{M}$ being the fluctuation fields. Then, this $g_\pi$ term provides the mass difference between $G$ and $H$ as $$\Delta M_i = m_{G,i} - m_{H,i} = g_\pi \bar{M}_{ii} \ ,
\label{massdif:meson}$$ where the sub-indices $i$ stand for the light flavor with $i = 1,2$ and $3$ being $u, d$ and $s$ quark, respectively. Here we use $v = f_\pi =$92.4MeV, so that we obtain $g_\pi = 4.65$ from $\Delta M_{u,d} = 430\,$MeV. Note that the $g_\pi$ term also gives the interaction for the pionic transition between $G$ and $H$. The relation between these two quantities are known as generalized Goldberger-Treiman relation [@Nowak:1992um; @Bardeen:1993ae]. On the other hand, the $g_A$ term gives the interaction of the pionic transition within $G$ or $H$. The value of $g_A$ is determined from the experimental value of $D^\ast \to D + \pi$ decay as $g_A = 0.56$ (see e.g. [@Harada:2012km]).
Now, let us consider the effective Lagrangian for the doubly heavy baryons. As we stated above, the quark-diquark symemtry relates the Lagrangian to the above Lagrangian for the heavy mesons. The resultant effective Lagrangian is expressed as $$\begin{aligned}
{\cal L}_{\rm B} & = & \bar{D}_{\textbf{Q};L}^\mu i v\cdot \partial D_{\textbf{Q};\mu;L} + \bar{D}_{\textbf{Q};R}^\mu i v\cdot \partial D_{\textbf{Q};\mu;R} \nonumber\\
& &{} - \Delta\left( \bar{D}_{\textbf{Q};L}^\mu D_{\textbf{Q};\mu;L} + \bar{D}_{\textbf{Q};R}^\mu D_{\textbf{Q};\mu;R} \right) \nonumber\\
& &{} - \frac{1}{2}g_\pi\left( \bar{D}_{\textbf{Q};L}^\mu M D_{\textbf{Q};\mu; R} + \bar{D}_{\textbf{Q};R}^\mu M^\dagger D_{\textbf{Q};\mu;L} \right) \nonumber\\
& &{} + \frac{ig_A}{f_\pi}\left[ \bar{D}_{\textbf{Q};L}^\mu\gamma_5\gamma^\nu \partial_\nu M D_{\textbf{Q};\mu;R} \right. \nonumber\\
& & \left. \qquad\quad\;\; {} + \bar{D}_{\textbf{Q};R}^\mu\gamma_5\gamma^\nu \partial_\nu M^\dagger D_{\textbf{Q};\mu;L}\right] .
\label{eq:EffecL}\end{aligned}$$ By substituting into the Lagrangian and considering the spontaneous chiral symmetry breaking, one obtains the Lagrangian
$$\begin{aligned}
{\cal L}_{\rm B} & = & \bar{D}_{\textbf{Q}}^\mu i v\cdot \partial D_{\textbf{Q};\mu} + \bar{N}_{\textbf{Q}}^\mu i v\cdot \partial N_{\textbf{Q};\mu} - \Delta\left( \bar{D}_{\textbf{Q}}^\mu D_{\textbf{Q};\mu} + \bar{N}_{\textbf{Q}}^\mu N_{\textbf{Q};\mu} \right)
- \frac{1}{2} g_\pi\left( \bar{D}_{\textbf{Q}}^\mu \bar{M} D_{\textbf{Q};\mu} - \bar{N}_{\textbf{Q}}^\mu\bar{M} N_{\textbf{Q};\mu} \right) \nonumber\\
& &{} - \frac{1}{2} g_\pi\left( \bar{D}_{\textbf{Q}}^\mu S D_{\textbf{Q};\mu} - \bar{N}_{\textbf{Q}}^\mu S N_{\textbf{Q};\mu} \right)
+ \frac{1}{2} g_\pi\left( \bar{D}_{\textbf{Q}}^\mu \Phi N_{\textbf{Q};\mu} + \bar{N}_{\textbf{Q}}^\mu\Phi D_{\textbf{Q};\mu} \right) \nonumber\\
& &{} - \frac{g_A}{f_\pi}\left[ \bar{D}_{\textbf{Q}}^\mu\gamma_5\gamma_\nu \partial_\nu\Phi D_{\textbf{Q};\mu} - \bar{N}_{\textbf{Q}}^\mu\gamma_5\gamma_\nu\partial_\nu\Phi N_{\textbf{Q};\mu} + \bar{D}_{\textbf{Q}}^\mu\gamma_5\gamma_\nu \partial_\nu S N_{\textbf{Q};\mu} + \bar{N}_{\textbf{Q}}^\mu\gamma_5\gamma_\nu \partial_\nu S D_{\textbf{Q};\mu}\right] \ ,
\label{eq:LagDN}\end{aligned}$$
where $S$ and $\Phi$ are defined as $M = S + i \Phi = S + 2 i \left( \pi^a T^a \right)$ with $\pi^a$ being the pion fields and $\mbox{tr} \left( T_a T_b \right) = (1/2) \delta^{ab}$.
As for the case of heavy mesons, the $\Delta$ term shifts the masses of the DHBs to the same direction, and $g_\pi$ term provides the mass difference between the chiral partners as $$\begin{aligned}
\Delta M_{B;i} =
m_{D_{\textbf{Q},i}} - m_{N_{\textbf{Q},i}} = g_\pi\bar{M}_{ii},
\label{eq:MassDif}\end{aligned}$$ which is exactly same as that for the heavy-light mesons in Eq, (\[massdif:meson\]). Then, the mass difference for the non-strange doubly heavy baryon is determined as $$\begin{aligned}
m_{D_{\textbf{Q},q}} - m_{N_{\textbf{Q},q}} & = & 430~{\rm MeV}.\end{aligned}$$ When we identify the quantum numbers of the $\Xi_{cc}^+$ observed in Ref. [@Agashe:2014kda] as $\frac{1}{2}^+$ one can estimate the mass of the state $\Xi_{cc}^{\ast +}$ as $3950$ MeV.
We next consider the intermultiplet one-pion decays of the DHBs in the isospin symmetry limit. The relevant partial widths are expressed as $$\begin{aligned}
\Gamma \left( \Xi_{cc}^{\ast ++} \to \Xi_{cc}^{++} + \pi^0 \right) & = & \Gamma \left( \Xi_{cc}^{\prime ++ \mu } \to \Xi_{cc}^{++ \mu } + \pi^0 \right) \nonumber\\
& = & \frac{(\Delta M_{B;u,d})^2}{8\pi f_\pi^2} \,\vert p_\pi \vert \ .\end{aligned}$$ where $\vert p_\pi \vert $ is the three momentum of $\pi$ in the rest frame of the decaying DHB. Other partial widths of different possible charged states can be obtained by using the isospin relation. Our numerical results are given in Table. \[tab:sum\].
-----------------------------------------------------------------------------------------------------------------------------------------
Spectrum Prediction (MeV) Decay channel Partial width (MeV)
-------------------------------- ------------------ --------------------------------------------------------------- ---------------------
$m_{\Xi_{cc}}^{\ast}$ $ 3950 $ $\Xi_{cc}^{\ast ++} \to \Xi_{cc}^{++} + \pi^0$ $ %380!332
331$
$m_{\Xi_{cc}}^\mu$ $ 3625 $ $ \Xi_{cc}^{\ast ++} \to \Xi_{cc}^{+} + \pi^+$ $ %760 !666
662$
$m_{\Xi_{cc}}^{\prime \mu}$ $ 4055 $ $ \Xi_{cc}^{\prime ++} \to \Xi_{cc}^{++ \mu} + \pi^0$ $ %760
%662
332 $
$m_{\Omega_{cc}}^{\ast}$ $ 4028 $ $ \Xi_{cc}^{\prime ++} \to \Xi_{cc}^{+ \mu} + \pi^+$ $ %1520 !1332
%662
663 $
$m_{\Omega_{cc}}^{\mu}$ $ 3783 $ $ \Omega_{cc}^{\ast +} \to \Omega_{cc}^{+} + \pi^0 $ $ %17
20 \times 10^{-3} $
$m_{\Omega_{cc}}^{\prime \mu}$ $ 4133 $ $ \Omega_{cc}^{\prime + \mu} \to \Omega_{cc}^{+ \mu} + \pi^0$ $ %34
20 \times 10^{-3} $
-----------------------------------------------------------------------------------------------------------------------------------------
We next consider the DHBs including a strange quark. In such a case, by using the spectrum of the heavy-light meson including a strange quark, one predicts [@Bardeen:2003kt; @Nowak:2003ra; @Nowak:2004jg] $$\begin{aligned}
m_{D_{\textbf{Q},s}} - m_{N_{\textbf{Q},s}} & = & m_{G_{s}} - m_{H_{s}} = 350~{\rm MeV}.\end{aligned}$$ In this sector, due to the conservation of isospin, one might naively expect the dominant transition channel of $\Omega_{cc,s}^{\ast +}$ is $\Omega_{cc,s}^{\ast +} \to \Omega_{cc,s}^+ + \eta$. However, since the mass splitting $350~$MeV is smaller than the eta meson mass $m_\eta = 548~$MeV, this channel is forbidden due to the kinetic reason and dominant channel should be $\Omega_{cc,s}^{\ast +} \to \Omega_{cc,s}^+ + \pi^0$ arising from the $\eta$-$\pi^0$ mixing. The partial decay widths are expressed as $$\begin{aligned}
\Gamma \left( \Omega_{cc}^{\ast +} \to \Omega_{cc}^+ + \pi^0 \right) & = & \Gamma \left( \Omega_{cc}^{\prime\mu + } \to \Omega_{cc}^{\mu +} + \pi^0 \right) \nonumber\\
& = &
%\frac{(\Delta M_{B;s})^2}{8 \pi f_\pi^2} \theta_{\eta-\pi^0}^2\,\vert p_\pi \vert .
\frac{ (\Delta M_{B;s})^2}{2\pi f_\pi^2} \Delta_{\pi^0\eta}^2\,\vert p_\pi \vert .\end{aligned}$$ where $\Delta_{\pi^0\eta} = -5.32 \times 10^{-3}$ is the magnitude of the $\eta$-$\pi^0$ mixing estimated in Ref. [@Harada:2003kt] based on the two-mixing angle scheme (see, e.g. Ref. [@Harada:1995sj] and references therein). Since magnitude of the isospin breaking $\eta$-$\pi^0$ mixing is very small, the partial width of decay $\Omega_{cc,s}^{\ast +} \to \Omega_{cc,s}^+ + \pi^0$ is small. This situation is very similar to what happens in the heavy-light meson system in which the $D_{s0}(2317)$ is regarded as the chiral partner of $D_s$ and the dominant decay channel of the former is the isospin violating process $D_{s0}(2317) \to D_s + \pi^0$.
We further make a comment on the mass splitting of the baryons in a doublet which beyond the scope of the Lagrangian we constructed. Here we just quote the result obtained in Ref. [@Brambilla:2005yk], $$\begin{aligned}
m_{\Psi_{QQ}^{(\prime)\mu}} - m_{\Psi_{QQ}^{(\ast)}} = \frac{3}{4} \left( m_{D^{\ast}} - m_D \right) \simeq 105~{\rm MeV},
\label{eq:massdiffinter}\end{aligned}$$ which is smaller than the pion mass. So that, in contrast to the heavy-light meson sector, the one-pion intramultiplet decays are forbidden in the DHB sector due to the kinetic reason. Note that it is reasonable to expect $60\%$ correction from $\mathcal{O}(1/m_c)$ to result for doubly charmed baryons [@Hu:2005gf] so that the one-pion decay channel could open for the intermultiplet decay.
Since the mass splitting between the chiral partners are from the spontaneous breaking of chiral symmetry, a particularly relevant problem is what will happen for this splitting in QCD under extreme condition. From the lessons in the heavy-light meson sector [@Sasaki:2014asa; @Suenaga:2014sga], it is reasonable to expect that the magnitude of the mass spitting will be reduced in hot/dense matter. Such a scenario might be tested in a future scientific facility.
We finally want to stress that the present work mainly concerns the spectrum and dominant strong decay channels of the DHBs. Some other quantities such as the weak transitions of the DHBs through changing a heavy flavor are also interesting phenomenologically. These physics will be reported elsewhere.
In summary, we studied the spectrum and the dominant strong decay properties of the DHBs based on the chiral dynamics. We point out that the mass spitting between the lowest lying DHBs and their chiral partners is about $450$ MeV for the unflavored DHBs and $350$ MeV for the stranged DHBs. Moreover, we predicted that, due to the kinetic reason, the dominant decay channel of the parity odd strange DHB is isospin violating process therefore the partial width is small.
The work of Y.-L. M. was supported in part by National Science Foundation of China (NSFC) under Grant No. 11475071 and the Seeds Funding of Jilin University. M. H. was supported in part by the JSPS Grant-in-Aid for Scientific Research (S) No. 22224003 and (c) No. 24540266.
[99]{}
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|
---
abstract: 'The evidence are given that the radio source GB2 0909+353 (GB2 catalogue: Machalski 1978; ICRS 2000.0 coordinates: 09 12 51.7, +35 10 10) is likely one of the largest classical doubles known, though its optical identification is not certain. Our deep VLA observations at 5 GHz did not reveal a radio core brighter than 0.5 mJy/beam at this frequency. Thus a distance to the source is evaluated using photometric – redshift estimates of the faint galaxies in the optical field. The equipartition magnetic field and energy density in the source is calculated and compared with corresponding parameters of other ‘giant’ radio sources known, showing extremely low values of both physical parameters of the source investigated. On the other hand, the age of relativistic electrons, and the advance speed of the ‘hot spot’ in the source are typical for much smaller and brighter 3CR sources.'
---
[**GB2 0909+353: One of the Largest Double Radio Source**]{}
by
M. Jamrozy & J. Machalski
Astronomical Observatory, Jagellonian University, ul. Orla 171,\
30–244 Cracow, Poland\
e–mail: jamrozy@oa.uj.edu.pl machalsk@oa.uj.edu.pl
[**Key words:**]{} radio continuum: galaxies – galaxies: active – galaxies: individual: GB2 0909+353
Introduction
============
Recently, the largest and low-brightness classical radio sources are of a special interest for researching a number of astrophysical problems. Especially, their statistical studies are of a great importance for: (1) investigations of the time evolution of radio sources (cf. Kaiser et al. 1997; Chyży 1997; Blundell et al. 1999), (2) testing the orientation-dependent unification scheme (e.g. Barthel 1989; Urry & Padovani 1995), and (3) their usefulness to probe the low-density intergalactic and intercluster medium (cf. Strom & Willis 1980; Subrahmanyan & Saripalli 1993; Mack et al. 1998).
Active nucleus of the largest ‘giant’ radio sources is likely approaching an endstage of its activity, therefore their studies can provide important informations on the properties of old AGN. The identification of a number of quasars among these giant sources rises a problem for the unified scheme of AGN, where a quasar-appearance should characterize the AGN whose jets are oriented closer to the observer’s line of sight than those in radio galaxies. Thus, quasars should not have large projected linear sizes. Finally, the large angular size of giants allows detailed studies of their spectral evolution, rotation measure (RM), and the depolarization towards these sources. All the above can provide useful data on density of the medium at very large distances from the AGN.
In former decades, giant radio sources were often undetectable in radio surveys because they lay below the survey surface-brightness limit, even though they had total flux densities exceeding the survey flux limit. The new deep radio surveys WENSS (Rengelink et al. 1997), NVSS (Condon et al. 1998) offer a possibility to select unbiased samples of giant radio sources owing to their low surface-brightness limits. Using the NVSS survey, we have selected a sample of FRII-type (Fanaroff & Riley 1974) giant candidates, which further study is in progress. The source GB2 0909+353 is one of them.
In this paper we show that this source is one of the largest FRII-type sources with extremely low values of the equipartition magnetic field and energy density. The 1.4-GHz high- and low-resolution radio observations, available for this source, and its optical field are summarized in Sec. 2. Also our special, deep VLA observation to detect its radio core, is described there. The radio spectrum of the source is analysed in Sec. 3. In Sec. 4, a possible optical counterpart, the redshift estimate and projected size of the source are discussed. Finally, in Sec. 5, the equipartition magnetic field and energy density within the source are calculated, and compared with corresponding values found for other giants, as well as for much smaller 3CR sources.
The radio 1.4 GHz map and optical field
=======================================
The source has been mapped in three recent sky surveys: FIRST (VLA B–array 1.4 GHz; Becker, White & Helfand 1995), NVSS (VLA D–array 1.4 GHz; Condon et al. 1998), and WENSS (WSRT 325 MHz; Rengelink et al. 1997). The low-resolution VLA map at 1.4 GHz, reproduced from NVSS survey, is shown in Fig. 1 with ‘grey-scale’ optical image from the Digitized Sky Survey (hereafter DSS) overlaid. The VLA high-resolution FIRST map confirms the FRII-type (Fanaroff & Riley 1974) of the source, and gives 6.3 arc min angular separation of the brightest parts of its lobes. Unfortunately, no radio core brighter than about 1 mJy at 1.4 GHz was detected during the FIRST survey. Therefore, we made a follow-up, deep VLA observations to detect the core at a higher frequency. For this purpose, the sky field centred at J091252.0+351000 was mapped at 4885 MHz with the B-array. That observed frequency and the array configuration allow to map the brightness distribution within a radius of about 4 arc min. With the integration time of 35 min, the rms brightness fluctuations were about 0.025 mJy beam$^{-1}$.Unfortunately, no radio core brighter than 0.5 mJy at 5 GHz has been detected in vicinity of the target position. This makes the upper limit to the core–total flux ratio of about 0.023, which is exactly the median ratio determined at 8 GHz for radio galaxies larger than 1 Mpc by Ishwara-Chandra & Saikia (1999). Thus, our observations confirm very low brightness of the investigated radio source at high frequencies.
The radio spectrum
==================
Previously the source was detected during the low-frequency sky surveys at 151 and 408 MHz (6C2: Hales, Baldwin & Warner 1988, and 7C: Waldram et al., 1996; B2.3: Colla et al., 1973), respectively, and at 1400 MHz (GB2: Machalski 1978). It is present, as well, in the GB6 catalogue (Gregory et al., 1996), but its 4.85 GHz flux density may be underestimated there. Table 1 gives the flux densities available at the frequencies from 151 MHz to 5 GHz.
88888888 8888888899 11111111111 1010101010 1010101010 = Freq. Survey Total flux Soúth lobe Noŕth lobe Core\
(MHz) /Tel.\
\
151 6C2 800$\pm$ 80\
151 7C 820$\pm$ 60\
325 WENSS 512$\pm$ 40 351$\pm$ 42 161$\pm$ 49\
408 B2.3 511$\pm$ 72$^{a)}$\
1400 GB2 128$\pm$ 28\
1400 NVSS 162$\pm$ 5 104$\pm$ 5 58$\pm$ 4\
4850 GB6 22$\pm$ 4\
4885 VLA $<$0.5
Note: a) original B2.3 flux density (Colla et al., 1973) is multiplied by 1.065, i.e. adjusted to the common flux density scale of Baars et al. (1977).
The radio spectrum of the total source and its lobes is shown in Fig. 2. In order to calculate its total radio luminosity, the spectrum is expressed by a functional form. The best fit of the data in column 3 of Table 1 has been achieved with a parabola $S(x)$\[mJy\]$=ax^{2}+bx+c$, where $x=\log\nu$\[GHz\]; $a=-0.493\pm 0.085,
b=-1.097\pm 0.038, c=2.327\pm 0.030$. This fit gives the fitted 1.4-GHz total flux density of 143 mJy, and the fitted spectral index of $-1.24\pm 0.063$ at 1.4 GHz. The low- and high-frequency spectral indices between fitted flux densities at 408 and 1400 MHz, and at 1400 and 5000 MHz, are $-0.91$ and $-1.28$, respectively. Such a steep slope of the spectrum at frequencies above 1 GHz suggests that the radio source is related to a distant galaxy. However the spectrum evidently flattens at low frequencies, which allows to estimate the lifetime of relativistic electrons in the source. Therefore, we have fitted the straight lines to the spectral data at low and high frequencies, estimating a break of the spectrum at frequency of about 760 MHz.
The redshift estimate and size of the source
============================================
To determine linear (projected) extent of the source, a distance to the source must be known. The optical field suggests that the radio source may be associated with one of two objects, marked \#1 and \#2 in Fig. 1. Object \#1 is classified as a galaxy with R=19.35 mag in the DSS. Object \#2 is not classified there because it is not visible on the Palomar Observatory Sky Survey’s blue plates, however it is likely another red galaxy with R=19.5$\pm$0.4 mag. Because no radio core was detected, we can only assume that any counterpart optical galaxy is not brighter than R=19.35 mag. Taking into account the Hubble relation R(z), well established for 3CR, and other radio galaxies (e.g. Kristian, Sandage & Westphal 1978; Machalski 1988), the galaxy should be at redshift $z^{>}_{\sim}0.4$. This estimate is supported by the $R$-band Hubble diagram for the giant radio galaxies, recently published by Schoenmakers et al. (1998). On the basis of available spactroscopy of giants from the Westerbork sample and 7C sample of Cotter et al. (1996), they found
$$\log z=0.125 R-2.79$$
which, for R=19.35 mag, gives z=0.42. This redshift estimate corresponds to the source’s distance of about 2.6 Gpc (assuming $H_{o}=50, \Omega=1$), and linear size of 2.43 Mpc. If so, the source GB2 0909+353 is one of about the ten FRII-type radio sources extended over 2 Mpc. The radio source, investigated here, will be still over 1 Mpc (a conventional lower limit of size for ‘giant’ sources) at a redshift as low as 0.11.
The equipartition magnetic field and energy density
===================================================
The source is characterized by exceptionally low minimum energy density of relativistic particles and equipartition magnetic field. Recently, Ishwara- Chandra & Saikia (1999) have published very interesting statistics of the above parameters calculated for known radio sources extended more than 1 Mpc, and compared them with corresponding statistics of smaller 3CR sources. Following Ishwara-Chandra & Saikia, we calculate the minimum energy density $u_{min}$, and equipartition magnetic field $B_{eq}$ in GB2 0909+353 with the standard method (e.g. Miley 1980), assuming a cylindrical geometry of the source, a filling factor of unity, and equal energy distribution between relativistic electrons and protons. Integrating luminosity of the source between 10 MHz and 10 GHz, we found $B_{eq}=0.084^{+0.040}_{-0.024}$ nT and $u_{min}=(0.65^{+0.55}_{-0.33})\times 10^{-13}$ ergcm$^{-3}$. These values are weakly dependent of unknown distance $D(z)$ to the source; $B_{eq}\propto D^{-2/7}$, $u_{min}\propto D^{-4/7}$. This means that varying redshift by 2 (100 per cent), one can expect a change of $B_{eq}$ by 18 per cent, and $u_{min}$ by 33 per cent, only.
$B_{eq}$ and $u_{min}$ values, found for GB2 0909+353, are rather extremal among the corresponding values for known giant sources. Ishwara-Chandra & Saikia have showed that $B_{eq}$ in almost all radio sources larger than 1 Mpc is less than equivalent magnetic field of the microwave background radiation, i.e. $B_{eq}<B_{iC}\equiv0.324(1+z)^{2}$\[nT\]. They also have showed that oppositely, $B_{eq}>B_{iC}$ for more luminous and smaller 3CR radio sources, suggesting that the inverse-Compton losses dominate the synchrotron radiative losses in the evolution of the lobes of giant sources. Our calculation shows that the $B_{iC}/B_{eq}$ ratio for GB2 0909+353 may vary from 7.6$\pm$2.9 (if z=0.4) to 4.6$\pm$1.7 (if z=0.11), respectively, and $B^{2}_{eq}/(B^{2}_{iC}+B^{2}_{eq})$, which represents the ratio of the energy losses by synchrotron radiation to total energy losses due to both the processes, may vary from 0.017$\pm$0.013 (if z=0.4) to 0.044$\pm$0.035 (if z=0.11), respectively. In Figs. 3(a) and 3(b) we plot these values on the diagrams reproduced from the paper of Ishwara-Chandra & Saikia, and showing the above ratios as a function of the linear size of radio source. Similarly, the value of $u_{min}$ for GB2 0909+353 vs. linear size, and corresponding values for other giants from that paper, are plotted in Fig. 3(c). The solid line marks the expected relation $\log u_{min}=-(4/7)\log D+const$. The loci of the source GB2 0909+353 on these diagrams fully support our thesis that this source is one of the largest radio sources with extremely low energy density and equivalent magnetic field.
The age of relativistic electrons, derived from the value of $B_{eq}$ and $B_{iC}$ at the break frequency of 760 MHz may vary from $3.4\,10^{7}$ yr (if z=0.4) to $9.8\,10^{7}$ yr (if z=0.11), respectively. Assuming that (1) a redshift of the source is between the above values, and (2) the main axis of the source is close to the plane of the sky, and thus any physical distance from the center should not differ significantly from the projected one (this assumption is based on the high symmetry of the source) – a distance from the central galaxy to the brightest spots in the radio lobes should be within 640 kpc and 1540 kpc, the break frequency, characterizing the total radio spectrum, can be related to any distance between the above values, and the mean hotspot separation speed (resulting from this distance and the age of relativistic electrons) should be within 0.02$c$ and 0.15$c$. The range of these values is essentially the same as those found for much smaller, double 3CR radio sources (cf. Alexander & Leahy 1987; Liu, Pooley & Riley 1992), however the lower value seems to be much more likely in view of the strong positive correlation between the separation speed and 178-MHz luminosity, found by these authors. Detailed spectral observations and the ageing analysis of a sample of giant double sources are necessary to check whether this correlation holds for the largest radio sources.
Concluding, we argue that the source GB2 0909+353 is one of the largest, low-brightness, and distant classical double radio source.
The VLA is operated by the National Radio Astronomy Observatory (NRAO) for Associated Universities Inc. under a licence from the National Science Foundation of the USA. We acknowledge usage of the Digitized Sky Survey which was produced at the Space Telescope Science Institute based on photografic data obtained using the Oschin Schmidt Telescope on Palomar Mountain and the UK Schmidt Telescope. We thank the anonymous referee for the valuable comments.
, [99.]{}
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---
abstract: 'The physics of the metal-insulator coexistence region near the non-zero temperature Mott transition is investigated in presence of weak disorder. We demonstrate that disorder reduces the temperature extent and the general size of the coexistence region, consistent with recent experiments on several Mott systems. We also discuss the qualitative scenario for the disorder-modified Mott transition, and present simple scaling arguments that reveal the similarities to, and the differences from, the clean limit.'
author:
- 'M. C. O. Aguiar'
- 'V. Dobrosavljević'
- 'E. Abrahams'
- 'G. Kotliar'
title: 'Effects of disorder on the non-zero temperature Mott Transition'
---
Introduction and motivation
===========================
The physics of the metal-insulator transition has continued to attract considerable interest in recent years. Substantial progress has been achieved in understanding the behavior near the interaction-driven transition, where Dynamical Mean Field Theory [@georges] (DMFT) has been very successful in explaining the behavior of several classes of materials ranging from transition metal oxides such as V$_{2}$O$_{3}$ to organic Mott systems. This approach has been especially useful in describing the non-zero temperature behavior in the paramagnetic coexistence region between the metal and the insulator. In this regime, the two phases compete, and the resulting behavior emerges as a compromise between the energy gain to form coherent quasiparticles, and the larger entropy inherent to the incoherent insulating solution. There is not actual two-phase coexistence (as in conventional first order thermodynamic phase transitions) in this region. Rather, it is a region of parameters in which two local minima of the free energy coexist.
So far, most theoretical work has concentrated on clean systems, although several experimental studies indicate that effects of disorder are particularly important precisely in this coexistence regime. Measurements performed in compounds such as NiSSe mixtures [@sekine; @matsuura; @miyasaka] and $\kappa$-organics [@limelette; @strack] indicate that the presence of disorder pushes down the critical temperature end point of the metal and insulator coexistence region. In particular, experiments performed on a NiS$_{2}$ compound, which has much weaker disorder, show that the Mott transition occurs at 150K, [@sekine] with an external applied pressure of 3 GPa, while in the substituted NiS$_{2-x}$Se$_{x}$ compound it is seen only below 100K. [@miyasaka] It is important to notice that applying an external pressure to these compounds is equivalent to substituting S by Se, which might suggest that the results above would be in conflict. A speculation was made that the reduction in the transition temperature would be due to the local randomness introduced with Se substitution.[@matsuura]
We address the theoretical issues from the perspective of the Hubbard model. It is not a priori obvious what should be the effect of disorder on the size and the temperature range of the coexistence region. On the one hand, disorder tends to broaden the Hubbard bands and thus larger interaction is needed to open a Mott Hubbard gap. This may lead to a larger overall energy scale, which could stabilize the coexistence region. On the other hand, disorder generally leads to spatial fluctuations in all local quantities, an effect that could smear or decrease the jump at any first order phase transition, and thus reduce the coexistence energy scale. These considerations indicate that careful theoretical work is called for, which can address the interplay of interactions and disorder near the Mott metal-insulator transition.
A formalism that describes the effects of disorder within a DMFT approach was outlined some time ago,[@dmftdis] but a very limited number of calculations were explicitly carried out within this framework. More recently, the approach was reexamined to investigate strong correlation effects on disorder screening,[@screening] and the related temperature dependence of transport in the metallic phase.[@inelastic] These results shed light on several puzzling phenomena observed in experiments on two dimensional electron systems, but did not provide a description of the physics relevant to the coexistence region at non-zero temperature.
In this paper we examine the phase diagram for the Mott transition in the presence of moderate disorder at non-zero temperature within the DMFT approach.[@dmftdis] We present results describing the evolution of the coexistence region, showing that disorder generally reduces its size, in agreement with experiments. Our results give a physical picture that describes the gradual destruction of quasiparticles as the Mott insulator is approached, and establish the qualitative modification of the critical behavior resulting from the presence of disorder.
Our findings are valid in the regime of strong correlations but weak to moderate disorder, where Anderson localization effects, which are neglected in our theory, can be safely ignored. The latter have been included in earlier zero temperature DMFT-based strong correlation calculations. [@statdmft; @vollhardt] In particular, we mention that our lowest temperature results are consistent with the $T=0$ result at weak disorder of Byczuk et al,[@vollhardt] but give the temperature dependence of the metal-insulator coexistence region.
non-zero temperature DMFT for disordered electrons
==================================================
We consider a half-filled Hubbard model in the presence of random site energies, as given by the hamiltonian $$H=-t\sum_{<ij>\sigma}c_{i\sigma}^{\dagger}c_{j\sigma}+\sum_{i\sigma
}\varepsilon_{i}n_{i\sigma}+U\sum_{i}n_{i\uparrow}n_{i\downarrow}.$$
Here $c_{i\sigma}^{\dagger}$ ($c_{i\sigma}$) creates (destroys) a conduction electron with spin $\sigma$ on site $i$, $n_{i\sigma}=c_{i\sigma}^{\dagger
}c_{i\sigma}$ is the particle number operator, $t$ is the hopping amplitude, and $U$ is the on-site repulsion. The random site energies $\varepsilon_{i}$ are assumed to have a uniform distribution of width $W$.
Within DMFT for disordered electrons, [@dmftdis] a quasiparticle is characterized by a local but site-dependent [@zimanyi] self-energy function $\Sigma_{i}(\omega)=\Sigma(\omega,\varepsilon_{i})$. To calculate these self-energies, the problem is mapped onto an *ensemble* of Anderson impurity problems [@dmftdis] embedded in a self-consistently calculated conduction bath. In this approach, only quantitative details of the solution depend on the details of the electronic band structure; in the following we concentrate on a semi-circular model density of states. In this particular case, the hybridization function is given by $$\Delta(\omega)=t^{2}\bar{G}(\omega) \label{deltahd}%$$ and the average local Green’s function, $\bar{G}(\omega)$, is obtained by imposing the following self-consistent condition$$\bar{G}(\omega)=\left\langle \frac{1}{\omega-\varepsilon_{i}-\Delta
(\omega)-\Sigma_{i}(\omega)}\right\rangle , \label{autoconh}%$$ where $\left< ...\right> $ indicates the arithmetic average over the distribution of $\varepsilon_{i}$.
To solve the single-impurity problems at non-zero temperature for different site energies, we mostly used the iterated perturbation theory (IPT) method of Kajueter and Kotliar.[@kajueter; @nolting] However, to check the accuracy of the results, in several instances we also used the numerically exact quantum Monte Carlo method as an impurity solver, and generally found good qualitative and even quantitative agreement, supporting the validity of our IPT predictions in the relevant parameter ranges. Throughout the paper we express all energies in units of the bandwidth.
Phase diagram
=============
We first examine the evolution of the coexistence region as disorder is introduced. Within this region, both metallic and insulating solutions are found, depending on the initial guess used in the iterative scheme for solving the self-consistency condition. Typical results are presented in Fig. \[fig1\], showing the phase diagram obtained within DMFT-IPT at non-zero temperature, for varying levels of disorder $W$. For each level of disorder \[shown in panel (a)\] or temperature \[shown in panel (b)\], the first (from left) of the two lines, the so-called $U_{c1}$, indicates the stability boundary (i.e. the spinodal) of the insulating solution. Conversely, the second of the two lines, identified as $U_{c2}$, represents the boundary of the metallic solution. The coexistence region is found between these two lines, i.e. for $U_{c1}<U<U_{c2}$. Our results are in good quantitative agreement with previous results obtained in the $T=0$ limit in presence of disorder,[@screening] and also with non-zero temperature results in absence of disorder.[@georges]
As the disorder increases, the metal-insulator transition generally moves to larger $U$. Physically, this reflects the fact that disorder broadens the bands and smears the gap, making it harder for the Mott-Hubbard gap to open, so that a larger $U$ is necessary for the transition. At the same time, the temperature-dependent coexistence region is found to shrink \[Fig. \[fig1\](a)\], persisting only below a critical end-point temperature $T_{c}(W)$ . At any given temperature, the principal effects of introducing disorder \[Fig. \[fig1\](b)\] are as follows: (1) both the $U_{c1}$ and $U_{c2}$ lines move towards larger interaction potential; (2) the lines become closer to each other as disorder increases. In fact, they both approach the $W=U$ line as $W\rightarrow\infty$.
Having obtained these results in quantitative detail, we would like to understand the physical origin of this behavior. In the following we present simple analytical arguments relating the non-zero temperature aspects of the coexistence region to the evolution of its ground state properties. Our strategy is motivated by the following observations: (a) the shape of the non-zero temperature coexistence region \[Fig. \[fig1\](a)\] remains *very similar* at different values of disorder; (b) its size, both in terms of temperature and in terms of $U$-range, shrinks as disorder increases. This suggests that the physical mechanism for the destruction of the coexistence region as the temperature increases is similar to that of the clean limit, where it is governed by decoherence processes due to inelastic electron-electron scattering. Therefore, we begin our analysis by concentrating on the clean limit, where we show how simple estimates for the critical end-point temperature $T_{c}$ can be obtained.
Coexistence region in the clean limit
=====================================
The coexistence region at non-zero temperature is delimited by the two spinodal lines $U_{c1}(T)$ and $U_{c2}(T)$; the critical end-point temperature $T_{c}$ is reached when these two boundaries intersect. To estimate $T_{c}$ using the $T=0$ properties of the model, we need to understand the temperature dependence of each of these lines.
Insulating spinodal
-------------------
The insulating spinodal $U_{c1}(T)$ essentially corresponds to the closing of the gap separating the two Hubbard bands in the Mott insulator. Its temperature dependence should thus reflect that of the Hubbard bands. In contrast to the correlated metallic state close to the Mott transition, the insulating solution is not characterized by a small energy scale in the coexistence region. Accordingly, it is not expected to have strong temperature dependence; its weak temperature dependence reflects activated processes across the Mott-Hubbard gap. Such activations only lead to (exponentially) weak rounding/broadening of the Hubbard bands, which should very slowly reduce $U_{c1}(T)$ as temperature increases. Such behavior is indeed clearly seen in our results. This temperature dependence is, however, much weaker than that characterizing $U_{c2}(T)$. For purposes of roughly estimating $T_{c}$, to leading order we can ignore this weak temperature dependence, so that $$U_{c1}(T)\approx U_{c1}(T=0).$$
Metallic spinodal
-----------------
In the vicinity of the Mott transition, the metallic solution is characterized by a low energy scale corresponding to the coherence temperature $T^{\ast}$ of a low-temperature Fermi liquid.[@georges] Above $T^{\ast}$ the heavy quasiparticles are destroyed, and the metallic solution becomes unstable. To estimate $U_{c2}(T)$ we need to determine how this coherence temperature varies as the transition is approached. From detailed studies of the clean[@georges] and disordered[@inelastic] Hubbard models within DMFT, it is known that this coherence temperature can be estimated as$$T^{\ast}\approx AT_{F}Z$$ where $T_{F}$ is the Fermi temperature, $A$ is a constant of order one, and $Z$ is the quasiparticle (QP) weight defined as $$Z=\left[ \left. 1-\frac{\partial}{\partial\omega}\mbox{Im}\Sigma
(\omega)\right\vert _{\omega\rightarrow0}\right] ^{-1}. \label{e.2}%$$ The behavior of $Z$ is well known in the clean limit,[@georges] where it decreases linearly as $U$ increases toward the metallic spinodal, viz. $$Z=C[U_{c2}(0)-U].$$ From numerical studies,[@georges] the proportionality constant $C\approx0.45$. Therefore, the coherence temperature can be written as$$T^{\ast}(U)=ACT_{F}[U_{c2}(0)-U].$$
We can now estimate the temperature dependence of $U_{c2}(T)$ as that value of the interaction needed to set $T^{\ast}(U)=T$, i.e.$$T=ACT_{F}[U_{c2}(0)-U_{c2}(T)].$$ In other words$$U_{c2}(T)\approx U_{c2}(0)-BT,$$ where $B=1/ACT_{F}$. From our numerical results \[see Fig. \[fig1\](a)\] we find $B\approx22$, giving $A\approx0.2$, in reasonable agreement[@coeffA] with estimates[@inelastic] from the literature.
Using these expressions for $U_{c1}(T)$ and $U_{c2}(T)$, we arrive at the estimate for the critical end-point temperature$$T_{c}\approx\lbrack U_{c2}(0)-U_{c1}(0)]/B,$$ which agrees within $10\%$ with our numerical results (see Fig. \[fig7\]).
Critical behavior in presence of disorder
=========================================
Encouraged by the success of our analytical description of the coexistence regime in the clean limit, we now turn our attention to the effects of disorder. As in the clean limit, we would like to relate the finite temperature properties to the critical behavior of the quasiparticles at $T=0$. To do this, we therefore concentrate on describing the critical behavior in presence of disorder.
The principal new feature introduced by disorder within the DMFT scheme is the spatial variation of the spectral function, $\rho_i(\omega)$. This is shown in Fig. \[fig2\] at all energy scales: on the left we have the average spectral function and on the right the relative deviation of its distribution, in the metallic phase.
For each value of the interaction potential, the distribution of $\rho_i(\omega)$ presents a large dip at $\omega\approx0$ and becomes broader as the frequency increases. This comes from the fact that at small frequencies the system is in the Fermi liquid regime. At finite temperature we observe the reminiscence of the perfect disorder screening seen at $T=0$ close to the Mott transition.[@screening] For large frequencies, the quasiparticle regime is no more valid and the appropriate description is in terms of Hubbard bands, resulting in an increase of the fluctuation in $\rho_i(\omega)$.
In the disordered case, the self-energy function $\Sigma_{i}(\omega)$ presents site-to-site fluctuations, which lead to the spatial variations of the spectral function discussed above. The QP weights $Z_{i}=Z(\varepsilon_{i})$ now depend on the local site energy $\varepsilon_{i}$. To properly describe the approach to the Mott transition, we therefore must follow the evolution of the entire function $Z(\varepsilon_{i})$ as the transition is approached.[@sitener]
Behavior of local QP weights
----------------------------
Given the self-consistent solution of our ensemble of impurity models, we calculate the local QP weights as
$$Z_{i}=\left[ \left. 1-\frac{\partial}{\partial\omega}\mbox{Im}\Sigma
_{i}(\omega)\right\vert _{\omega\rightarrow0}\right] ^{-1}.$$
Typical results are shown in Fig. \[fig3\](a), where we plot $Z_{i}=Z(\varepsilon
_{i})$ at $T=0.005$, for disorder strength $W=1$, as the metallic spinodal is approached by increasing the interaction $U$ toward $U_{c2}\approx 1.9$.
We first observe that for small $U$, away from the transition, the QP weights $Z_{i}$ have strong $\varepsilon_{i}$ dependence, with the smallest $Z_{i}$ at $\varepsilon_{i}=0$. Physically, this reflects the tendency for correlation effects (suppression of $Z$) to be the strongest on sites which are locally close to half-filling (singly occupied). Nonzero site energies favor the local occupation departing from half-filling, thus reducing the correlation effect, and increasing $Z_{i}$.
As $U$ increases, all the $Z_{i}$’s decrease, as in the clean case. But how does this affect the distribution of QP weights $Z_{i}=Z(\varepsilon_{i})$? At first glance it seems that the $\varepsilon_{i}$ dependence becomes weaker, but a closer look reveals this not to be the case. As we shall now demonstrate, all the $Z_{i}$’s decrease linearly near the transition, i.e. they assume the form$$Z(U,\varepsilon_{i})=K(\varepsilon_{i})[U_{c2}-U],$$ where only the prefactor $K(\varepsilon_{i})$ depends on $\varepsilon_{i}$. If, to leading order, these prefactors remain independent of the distance to the spinodal, then the entire family of curves $Z(U,\varepsilon_{i})$ can all be collapsed on a single scaling function. To verify this hypothesis, we define reduced QP weights$$Z^{\ast}(\varepsilon_{i})=Z(U,\varepsilon_{i})/Z(U,0).$$ If our scaling ansatz is valid, then the $Z^{\ast
}(\varepsilon_{i})$ should approach a non-zero limit as $U\longrightarrow
U_{c2}$, i.e. they should all collapse onto a single scaling function. As shown in Fig. \[fig3\](b), this behavior is observed only for $U$ sufficiently close to $U_{c2}$ \[note that the data for $U=0.8$ (further from the transition) show deviations from leading scaling\]. This is precisely what we expect, since such simple scaling behavior typically occurs only within a critical region close to the metallic spinodal.
Distribution $P(Z_{i})$ of local QP weights
-------------------------------------------
Equivalently, we can characterize the QP weights by their probability distribution function $P(Z_{i})$. Typical results for $P(Z_{i})$ are shown in the inset of Fig. \[fig4\]. As the $Z_{i}$ decrease near the transition, the distribution function $P(Z_{i})$ changes its form and narrows down. However, if our scaling hypothesis is valid, then the *shape* of this distribution should approach a “fixed-point" form very close to the transition. More precisely, we expect the distribution for reduced QP weights $P(Z_{i}^{\ast})$ to collapse to a single scaling function close to $U_{c2}$. Results confirming precisely such behavior are presented in Fig. \[fig4\].
An interesting question relates to the precise form of the fixed-point distribution function $P(Z_{i}^{\ast})$, and how it may depend on disorder. In the clean limit, obviously, it reduces to $\delta(Z_{i}^{\ast}-1)$ indicating that spatial fluctuations are suppressed. As the disorder increases, $P(Z_{i}^{\ast})$ becomes very broad (as shown in Fig. \[fig5\]), reflecting large site-to-site fluctuations in the local QP weights. This behavior may be regarded as a precursor of electronic Griffiths phases,[@griffiths] which emerge for stronger disorder, as found within *stat*DMFT approaches.[@statdmft]
In essential contrast to the clean limit, the approach to the Mott transition in presence of disorder thus needs to be characterized by the entire *probability distribution function* of QP parameters. At first glance, this may appear to require a description considerably more complex than in the absence of disorder. However, we have demonstrated that in the critical region the distributions approach a fixed point form, allowing for single parameter scaling," in close analogy to the clean Mott transition. This finding immediately suggests that our arguments describing the finite temperature coexistence behavior in the clean limit may successfully be extended to the disordered case as well, allowing for a complete qualitative description, which we discuss in the following section.
Coexistence region in presence of disorder
==========================================
Within the DMFT formulation, the disorder is not expected to qualitatively affect the temperature dependence of the insulating spinodal, since the forms of the Hubbard bands remain qualitatively similar to that in the clean limit. The principal effect of disorder in the Mott insulating phase is to simply broaden the Hubbard bands, which retain well defined (sharp) band edges due to the CPA-like treatment of randomness in the DMFT limit. Indeed, our quantitative results \[see Fig. \[fig1\](a)\] confirm that $U_{c1}(T)\approx U_{c1}(0)$ retains very weak temperature dependence, as in the clean case. The only modification is that $U_{c1}(0)$ rapidly grows as disorder is increase, reflecting the disorder-induced broadening of the Hubbard bands.
The metallic solution is again found to be unstable above a certain coherence temperature $T^{\ast}(W,U)$, which defines the locus of the metallic spinodal $U_{c2}(T)$. An added subtlety is that different sites start to decohere at different temperatures, an effect that earlier work[@inelastic] found responsible for a nearly-linear temperature dependence of the resistivity in the disordered metallic phase. Nevertheless, sufficiently close to the Mott transition (within the coexistence region), a sharply defined temperature scale $T^{\ast}(W,U)$ emerges where the metallic solution suddenly disappears and where the qualitative form of the spectrum changes on *all* sites. This temperature scale defines the locus of the metallic spinodal, corresponding to the equation $$T=T^{\ast}(W,U_{c2}).$$
At first glance, it is anything but obvious how $T^{\ast}(W,U)$ should be estimated. As in the clean case, the reduction of this temperature scale as the transition is approached must reflect the behavior of the local quasi-particle weights $Z_{i}$, and presumably depend on the precise form of the distribution function $P(Z_{i})$. As we have seen, however, all the local QP weight scale in a similar fashion in the critical regime, which suggests that a reasonable estimate may be obtained simply from their average value $$\left\langle Z_{i}\right\rangle =\int d\varepsilon_{i}P(\varepsilon_{i})Z_{i}.$$ At least for sufficiently weak disorder, we may expect that \[cf. Eq. (5)\]$$T^{\ast}(W,U)\approx AT_{F}\left\langle Z_{i}\right\rangle ,$$ where $A\approx0.2$ as in the clean case. Using the fact that all $Z_{i}$’s decrease linearly near the transition, we expect$$\left\langle Z_{i}\right\rangle =C(W)[U_{c2}(0)-U].$$ To confirm this, we explicitly calculated $\left\langle Z_{i}\right\rangle $ as a function of $U$ for different levels of disorder; the results are shown in Fig. \[fig6\]. We conclude that $C(W)\approx C(0)\approx0.45$. These results suggest that the metallic spinodal should take the form, $$U_{c2}(W,T)\approx U_{c2}(W,0)-B(W)T,$$ where $B(W)\approx B(0)=22$. Our non-zero temperature results for $U_{c2}(W,T)$ \[see Fig. \[fig1\](a)\] fully confirm these expectations. Based on these results, we finally obtain the desired expression for $T_{c}(W)$ of the form $$T_{c}(W)\approx\lbrack U_{c2}(W,0)-U_{c1}(W,0)]/B(0).$$ To test the proposed procedure, we have used the values for $U_{c1}(W)$ and $U_{c2}(W)$ at the lowest temperature of our calculation ($T=0.005$) to estimate $T_{c}(W)$. As we can see from Fig. \[fig7\], our analytical estimates are found to be in excellent agreement with results of explicit non-zero temperature calculations. The decrease of $T_{c}(W)$ with disorder thus directly reflects the shrinking" of the coexistence region at low temperature, which in its turn reflects the decrease of the energy difference between the metallic and the insulating solution.
Conclusions
===========
In this paper we have used a DMFT approach to examine the effects of disorder on the critical behavior near the Mott metal-insulator transition, with special emphasis on non-zero temperature properties associated with the two spinodal lines $U_{c1}$ and $U_{c2}$. By using a combination of numerical results and analytical arguments we have demonstrated that simple scaling behavior emerges, providing a complete description of the critical regime.
In contrast to the clean case, the presence of disorder requires one to examine the entire distribution of local spectral functions, $\rho_i (\omega)$, describing how the local spectra varies with position in the sample. This can be probed with scanning tunneling microscopy (STM). Notice that the distribution function describing the site dependence of $\rho_i (\omega)$ will depend on the frequency of observation: it will be broader at higher energies \[as seen in Fig. \[fig2\](b)\], where a real space picture is appropriate to describe the Hubbard bands, and narrower at low frequencies, where a quasiparticle description in k space is appropriate. This is a manifestation of frequency dependence of the disorder screening discussed in an earlier paper by some of us.[@screening]
In the metallic regime, at low temperatures, the spectral function can be parametrized in terms of the distribution of quasiparticle parameters, which displays simple scaling properties. This allowed us to characterize the behavior near $U_{c2}$ using a single parameter scaling procedure. The approach to $U_{c2}$ thus retains a character qualitatively independent of the level of disorder, where the vanishing of quasiparticle weight signals the transmutation of itinerant electrons into localized magnetic moments.
Within the examined DMFT formulation, the region between the two spinodal lines $U_{c1}$ and $U_{c2}$ although reduced in size and extent cannot be completely eliminated no matter how large the disorder. Of course, these predictions are applicable only for weak enough disorder where Anderson localization effects can be ignored. Extensions of DMFT that incorporate Anderson localization mechanisms at zero temperature are available,[@statdmft; @vollhardt] but applying these approaches to examine the non-zero temperature behavior near Mott-Anderson transitions remains an interesting research direction. The behavior at the first order transition line and the actual nucleation of either the metallic or insulating phase, between $U_{c1}$ and $U_{c2}$, are also strongly modified by disorder, and this as well is left for future study.
The authors thank A. Georges and D. Tanasković for useful discussions. This work was supported by NSF grants DMR-9974311 and DMR-0234215 (VD) and DMR-0096462 (GK).
[99]{}
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|
---
author:
- 'Mariana B. Sánchez [^1]'
- 'Gonzalo C. de Elía'
- Juan José Downes
bibliography:
- 'Referencias.bib'
title: 'Tidal and general relativistic effects in rocky planet formation at the substellar mass limit using N-body simulations'
---
[Recent observational results show that very low mass stars and brown dwarfs are able to host close-in rocky planets. Low-mass stars are the most abundant stars in the Galaxy, and the formation efficiency of their planetary systems is relevant in the computation of a global probability of finding Earth-like planets inside habitable zones. Tidal forces and relativistic effects are relevant in the latest dynamical evolution of planets around low-mass stars, and their effect on the planetary formation efficiency still needs to be addressed.]{} [Our goal is to evaluate the impact of tidal forces and relativistic effects on the formation of rocky planets around a star close to the substellar mass limit in terms of the resulting planetary architectures and its distribution according to the corresponding evolving habitable zone.]{} [We performed a set of $N$-body simulations spanning the first 100 Myr of the evolution of two systems composed of 224 embryos with a total mass 0.25M$_\oplus$ and 74 embryos with a total mass 3 M$_\oplus$ around a central object of 0.08 M$_\odot$. For these two scenarios we compared the planetary architectures that result from simulations that are purely gravitational with those from simulations that include the early contraction and spin-up of the central object, the distortions and dissipation tidal terms, and general relativistic effects.]{} [We found that including these effects allows the formation and survival of a close-in ($r<0.07$ au) population of rocky planets with masses in the range $0.001<m/\mathrm{M_\oplus}<0.02$ in all the simulations of the less massive scenario, and a close-in population with masses $m \sim 0.35$ M$_\oplus$ in just a few of the simulations of the more massive scenario. The surviving close-in bodies suffered more collisions during the integration time of the simulations. These collisions play an important role in their final masses. However, all of these bodies conserved their initial amount of water in mass throughout the integration time.]{} [The incorporation of tidal and general relativistic effects allows the formation of an in situ close-in population located in the habitable zone of the system. This means that both effects are relevant during the formation of rocky planets and their early evolution around stars close to the substellar mass limit, in particular when low-mass planetary embryos are involved.]{}
Introduction
============
During the past decades several observational and theoretical results have suggested that the formation of rocky planets is a common process around stars of different masses [e.g., @Cumming2008; @Mordasini2009; @Howard2013; @Ronco2017]. In particular, observations and modeling have proven the existence and formation of rocky planets around very low mass stars (VLMS) and brown dwarfs (BDs) [e.g., @Payne2007; @Raymond2007; @Gillon2017]. These achievements are relevant because VLMS are the most abundant stars in the Galaxy and together with BDs are within the closest solar neighbors [e.g., @Padoan2004; @Henry2004; @Bastian2010]. This allows for surveys of rocky planets even in habitable zones, around numerous stellar samples, and through different observational techniques. This could be a crucial observational test of the processes driving the planet formation as suggested by theoretical modeling.
Although the detection of rocky planets around BDs is still challenging, some systems have been discovered [e.g., @Kubas2012; @Gillon2017; @Grimm2018; @Z2019]. Using photometry from *Spitzer*, @He2017 reported an occurrence rate of $\sim87\%$ of planets with radius $0.75<R/\mathrm{R_\oplus}<1.25$ and orbital periods $1.7<P/\mathrm{days}<1000$ around a sample of 44 BDs. From the M-type low-mass stars monitored by the *Kepler* mission, @Mulders2015b found that planets around VLMS are located close to their host stars, having an occurrence rate of small planets $(1<R/\mathrm{R_\oplus}<3),$ which is $\text{three}$ to $\text{four}$ times higher than for Sun-like stars, while @Hardegree2019 estimated a mean number of $1.19$ planets per mid-type M dwarf with radius $0.5<R/\mathrm{R_\oplus}<2.5$ and orbital period $0.5<P/\mathrm{days}<10$. The current *SPHERE* (Spectro-Polarimetric High-contrast Exoplanet REsearch) together with *ESPRESSO* (Echelle SPectrograph for Rocky Exoplanet and Stable Spectroscopic Observations) are already detecting Earth-sized planets around G, K, and M dwarfs [@Lovis2017; @Hojj2019], and *CARMENES* (Calar Alto High-Resolution Search for M Dwarfs with Exo-earths with a Near-infrared Echelle Spectrograph) is searching for exoplanets around M dwarfs and already found two Earth-mass planets around an M7 BD [@Z2019]. The ongoing and upcoming transit searches such as *TESS* (Transiting Exoplanet Survey Satellite) and *PLATO* (PLAnetary Transits and Oscillations of stars) are expected to find most of the nearest transiting systems in the next years [@Barclay2018; @Ragazzoni2016], opening the new era of atmospheric characterization of terrestrial-sized planets. Current observations suggest that there does not seem to be a discontinuity in the general properties of the circumstellar disks around VLMS and BDs [e.g., @luhman2012]. In particular, the dust growth to millimeter and centimeters sizes on the disk mid-plane of BDs is similar to the growth in VLMS disks, as has been inferred for a few BDs that have been observed with ALMA (Atacama Large Milimeter Array) and CARMA (Combined Array for Research in Milimeter-wave Astronomy) [@Ricci2012; @Ricci2013; @Ricci2014]. This suggests that similar processes in the evolution of the disk might also take place at either side of the substellar mass limit.
@Payne2007 investigated planet formation around low-mass objects using a standard core accretion model. They found that the formation of Earth-like planets is possible even around BDs, and that planets with masses up to 5 M$_\oplus$ can be formed. They reported that the mass-distribution of the resulting planets is strongly correlated with the disk masses. In particular, if the BD has a disk mass of about a few Jupiter masses, then only $10\%$ of the BDs might host planets with masses exceeding 0.3 M$_\oplus$. Through dynamical simulations of terrestrial planet formation from planetary embryos, @Raymond2007 found that the masses of planets located inside the habitable zone decrease while the mass of the host star decreases. The authors found small and dry planets around low-mass stars. Using $N$-body simulations with diverse water-mass fractions for objects beyond the snow line, @Ciesla2015 found both dry and water-rich planets close to low-mass stars. By studying planet formation around different host stars, their simulations predict that a greater number of compact and low-mass planets are located around low-mass stars, while higher mass stars will be hosting fewer and more massive planets. Recently, @Coleman2019 studied planet formation around low-mass stars similar to Trappist-1 and considered different accretion scenarios. They found water-rich rocky planets with periods $0.5<P/\mathrm{days}<1000$. Furthermore, @Miguel2019 studied planet formation around BDs and low-mass stars using a population synthesis code and found planets with a high ice-to-rock ratio.
It is still debated how the formation and evolution of rocky planetary systems around low-mass objects needs to be treated. Planets around these objects are thought to form close to them in a region where tides are very strong and lead to significant orbital changes [@Papaloizou2010; @Barnes2010; @Heller2010]. That BDs as well as VLMS collapse and spin up with time in their first 100 Myr allows the population of close-in tidally locked bodies around them to experience many different dynamical evolutions [@Bolmont2011]. @Bolmont2013 [@Bolmont2015] showed the importance of incorporating tidal effects and general relativity as well as the effect of rotation-induced flattering in the dynamical and tidal evolution of multi-planetary systems, particularly those with close-in bodies.
In this work we incorporate these effects in a set of $N$-body simulations to study rocky planet formation from a sample of embryos around an object with a mass of 0.08 M$_\odot$ close to the substellar mass limit in a period of 100 Myr in order to improve predictions for planetary system architectures. We evaluate how relevant tidal and relativistic effects are during rocky planet formation around an object at the substellar mass limit. Our aim is to estimate dynamical properties of the resulting population of close-in bodies and the efficiency of forming planets that remain in the habitable zone of the system.
In Section \[sec:vlmsbdcollapserotation\] we briefly describe the early gravitational collapse and rotation rates of VLMS and BDs. In Section \[sec:habzone\] we model the habitable zone around a star close to the substellar mass limit. In Section \[sec:disk\] we describe the protoplanetary disk model that is based on observations. In Section \[sec:numodel\] we explain the numerical method we used to include tidal forces and general relativity effects in the $N$-body simulations. In Section \[sec:results\] we describe the resulting planetary systems, and in Section \[sec:discussions\] we finally summarize our conclusions and future works.
Collapse and rotation of young VLMS and BD {#sec:vlmsbdcollapserotation}
==========================================
The BDs are stellar structures that are unable to reach the necessary core temperatures and densities to sustain stable hydrogen fusion. For solar metallicity, the substellar mass limit that separates BDs from VLMS is 0.072 M$_\odot$ [@Baraffe2015]. During the first $\sim100$ Myr of their evolution, BDs and VLMS share several properties such as circumstellar disks at different evolutionary stages [e.g., @luhman2012], the formation of planets around them, their gravitational collapse, and the subsequent increase in their rotation rates with time. These last two properties are particularly relevant in our modeling.
In the case of VLMS, the collapse stops when they reach the main sequence at ages of $\sim100$ Myr. Brown dwarfs continue to collapse slowly toward to a degenerate structure that is unable to sustain stable hydrogen fusion [e.g., @Kumar1963; @Hayashi1963].
Available observations show that the mean rotational periods of BD are shorter than those of VLMS. This has been interpreted as indicating that the magnetic braking on the early spin-up in the substellar mass domain is inefficient [@Mohanty2002; @Scholz2015]. The spin rate of the main object sets the position of the corotation radius $r_\textrm{corot}$ , which is the mid-plane orbital distance at which the mean orbital velocity $n$ of a planet is equal to the rotational velocity $\Omega_{\star}$ of the central object. In the case of a VLMS and BD, $r_\textrm{corot}$ shrinks while the object spins up. This behavior is essential for the treatment of close-in bodies because their orbital evolution depends on the initial eccentricity $e$ and semimajor axis $a$ with respect to the location of $r_{\textrm{corot}}$.
Modeling the habitable zone {#sec:habzone}
===========================
The classical habitable zone (CHZ) is the circular region around a single star or a multiple star system in which a rocky planet can retain liquid water on its surface [@Kasting1993]. The CHZ definition assumes that the most important greenhouse gases for habitable planets orbiting main-sequence stars are CO$_{2}$ and H$_{2}$O. This assumption extends the idea that the long-term ($\sim$1 Myr) carbonate-silicate cycle on Earth acts as a planetary thermostat that regulates the surface temperature [@Watson1981; @Walker1981; @Kasting1988] toward potentially habitable exoplanets.
![Evolution of the habitable zone around a star of 0.08 M$_\odot$ in a period of 1 Gyr.[]{data-label="fig:HZ_models"}](HZ_BD_80M.eps){width="9cm"}
To study potentially habitable exoplanets around a nonsolar-type star, it is necessary to take the relation between the albedo and the effective temperature of the central star into account. By extrapolating the cases studied by @Kasting1993, the inner and outer limits of the isolated habitable zone (IHZ) for stars with effective temperatures in between $3700 < T_{\textrm{eff}}/\mathrm{K} < 7200$ can be calculated as in @Selsis2007 by
$$l_{\mathrm{int}} = \left(l_{\mathrm{in},\sun}-a_{\mathrm{in}}T_{\star}-b_{\mathrm{in}}T_{\star}^{2}\right)\left(\frac{L_{\star}}{\mathrm{L_{\sun}}}\right)^{\frac{1}{2}},$$
$$l_{\mathrm{out}} = \left(l_{\mathrm{out},\sun}-a_{\mathrm{out}}T_{\star}-b_{\mathrm{out}}T_{\star}^{2}\right)\left(\frac{L_{\star}}{\mathrm{L_{\sun}}}\right)^{\frac{1}{2}},$$
where $l_{\textrm{in},\sun}=0.97$ au and $l_{\textrm{out},\sun}=1.67$ au are the inner and outer limits of a system with a Sun-like star as central object, considering runaway greenhouse and maximum greenhouse values, respectively [@Kopparapu2013; @Kopparapu2013erratum], and $a_{\textrm{in}} = 2.7619 \times 10^{-5}\ \ \mathrm{au K^{-1}}$, $b_{\mathrm{in}}= 3.8095 \times 10^{-9}\ \ \mathrm{auK^{-2}}$, $a_{\mathrm{out}}=1.3786 \times 10^{-4}\ \ \mathrm{auK^{-1}}$ , and $b_{\mathrm{out}}=1.4286 \times 10^{-9}\ \ \mathrm{auK^{-2}}$ are empirically determined constants; L$_\odot$ and $L_\star$ are the luminosity of the Sun and the considered star, and the temperature of the star is $T_\star=T_{\textrm{eff}}-5700K,$ where $T_{\textrm{eff}}$ can be expressed by
$$T_{\mathrm{eff}}=\left(\frac{L_{\star}}{4\pi\sigma R_{\star}^{2}}\right)^{\frac{1}{4}},$$
with R$_\star$ the radius of the central object and $\sigma$ the Stefan-Boltzmann constant. As in @Barnes2013, who studied habitable planets around BDs, we extended the calculation of the IHZ to the substellar mass limit. In this work $l_{\textrm{in}}$ and $l_{\textrm{out}}$ therefore represent the inner and outer limits of the IHZ around a 0.08 M$_\odot$ object. We used $R_\star$, $L_\star$ , and $T_{\textrm{eff}}$ as a function of time as predicted by the models of @Baraffe2015[^2]. In Fig. \[fig:HZ\_models\] we show the evolution of the IHZ of an object of 0.08 M$_\odot$. While the object is evolving, $L_\star$, $R_\star$ , and $T_{\textrm{eff}}$ decrease with time and the location of the IHZ becomes narrower and closer to the substellar object. When the central object is 1 Myr old, $l_{\textrm{in}} = 0.214$ au and $l_{\textrm{out}} = 0.369$ au, while at 1 Gyr, $l_{\textrm{in}} = 0.017$ au and $l_{\textrm{out}} = 0.029$ au. It is worth noting that the IHZ is located more than ten times closer to the central object after this time.
Protoplanetary disk {#sec:disk}
===================
In this section we describe the protoplanetary disk model that we adopted for the 0.08 M$_\odot$ central object. We calculate the dust species that might survive inside our region of study to determine the material that is avialable for the formation of larger structures such as protoplanetary embryos. We distinguish two scenarios, motivated by the diversity of disk masses and the observed distribution of the exoplanet mass around low-mass objects. Finally, we compare our model with others that have been used by different authors.
Region of study
---------------
Our region of study is defined between an inner radius $r_{\textrm{init}} = 0.015$ au and an outer radius of $r_{\textrm{final}} = 1$ au. The inner radius was selected by considering that tidal effects allow a planet located at this distance to survive without colliding with the central object for certain values of its eccentricity [@Bolmont2011]. The outer radius was defined in a way that the habitable zone and an outer region of water-rich embryos are contained in our region of study.
We evaluated the consistency of the selection of $r_{\textrm{init}}$ with the sublimation radius for different dust species by analyzing those species that could survive inside our region of study. We computed sublimation radii for a variety of species using the model of @Kobayashi2012. Sublimation temperatures were estimated according to @Pollack1994. In Fig. \[fig:sublimation\_radii\] we show the variation of the sublimation radii with time for different species, compared with the corotation radii $r_{\textrm{corot}}$, the inner radii $r_{\textrm{init}}$ of the region, and the radius of the stellar object $R_\star$. Components such as iron and volatile and refractory organics could survive during the first million years inside our region of study.
![Sublimation radii for different dust grain species and comparison with the corotation $r_{\textrm{corot}}$, the stellar radius $R_\star$ , and the inner radii of our region of study $r_{\textrm{init}}$.[]{data-label="fig:sublimation_radii"}](Radios_sublim_Pollack94_esc1_paper.eps){width="9cm"}
Protoplanetary disk model {#sec:protodisk}
-------------------------
The parameter that determines the distribution of the material within the disk is the surface density $\Sigma$. Based on physical models of viscous accretion disks [see @Lynden-Bell1974; @Hartmann1998], we adopted the dust surface density profile given by
$$\label{eq:sigmadust}
\Sigma_{\textrm{s}}(r)=\Sigma_{0\textrm{s}}\eta_{\textrm{ice}}\left(\frac{r}{r_{\textrm{c}}}\right)^{-\gamma}\textrm{e}^{{-(r/r_{\textrm{c}})}^{2-\gamma}}.$$
This profile is commonly used to interpret observational results in a wide range of stellar masses down to the substellar mass regime [e.g., @Andrews2009; @Andrews2010; @Guilloteau2011; @Testi2016]. The value $r$ represents the radial coordinate in the mid-plane of the disk, $r_\textrm{c}$ is the characteristic radius of the disk, $\gamma$ is the factor that determines the density gradient, and $\eta_\textrm{ice}$ represents the increase in the amount of solid material due to the condensation of water beyond the snow line $r=r_\textrm{ice}$. For large samples of stars, @Andrews2009 [@Andrews2010b] found that the factor $\gamma$ can take values between 1.1, and the mean value is 0.9. On the other hand, using a different technique, @Isella2009 found values for $\gamma$ between -0.8 and 0.8, with a mean value of 0.1. For BD and VLMS, the lower and upper bounds for $\gamma$ are -1.4 and 1.4, respectively, and the mean value is close to 1 [@Testi2016]. We took $r_\mathrm{c} = 15$ au and $\gamma=1,$ which are consistent with the latest observations of disks around BDs and VLMS [@Ricci2012; @Ricci2013; @Ricci2014; @Testi2016; @Hendler2017]. We fixed the location of the snow line at $r_\textrm{ice}=0.42$ au (see Appendix \[sec:Apendix\]). Following @Lodders2003, we propose that inside the snow line $\eta_\textrm{ice}=1$ and beyond the snow line $\eta_\textrm{ice}=2$. This jump of a factor 2 in the solid surface density profile is related to the water gradient distribution. Thus, we considered that bodies beyond $r_\textrm{ice}$ present $50\%$ of the water in mass, while bodies inside $r_\textrm{ice}$ have just $0.01\%$ water in mass. This small percentage of water for bodies inside the snow line is given considering that the inner region was affected by water-rich embryos from beyond the snow line during the gaseous phase that is related to the evolution of the disk. The water distribution was assigned to each body at the beginning of our simulations. The highest initial percentage of water in mass determines the value of the maximum percentage of water in mass that a resulting planet could have given the fact that the $N$-body code treats the collisions as perfectly inelastic ones, so that bodies conserve their mass and amount of water in mass in each collision.
By assuming an axial symmetric distribution for the solid material, we can express the dust mass of the disk $M_{\textrm{dust}}$ by
$$\label{eq:mdust}
M_{\textrm{dust}} = \int_{0}^{\infty} 2\pi r \Sigma_{\textrm{s}}(r) dr.$$
Solving Eq. means solving two integrals because of the jump in the content of water in the disk given by $\eta_{\textrm{ice}}$ at $r_{\textrm{ice}}$. Thus we can estimate the normalization constant for the solid component of the disk $\Sigma_{0s}$ for a given value of the solid mass in the disk.
Twofold parameterization of the disk density {#sec:twofolddensity}
--------------------------------------------
As discussed by @Manara2018, there is reliable observational evidence that protoplanetary disks are less massive than the known exoplanet populations. The authors suggested two mechanisms for this discrepancy in mass: an early formation of planetary cores at ages $<0.1-1$ Myr when disks may be more massive, and replenishment of disks by fresh material from the environment during their lifetimes. In order to consider the current uncertainties in estimating disk masses, we made the disk parameterization of the surface density profile from Eq. for two distinct values of $M_{\textrm{dust}}$. We refer to these two cases as the following **:
- **** is based on the latest observational results on the masses of dust in protoplanetary disks. We assumed $M_{\textrm{dust}} = 9 \times 10^{-6}$ M$_\odot$ ($\sim$ 3 M$_\oplus$) from the average of the dust masses obtained from observations of BDs and VLMS made with ALMA (see references in Section \[sec:protodisk\]). If we were to assume a gas-to-dust ratio of 100:1, this would be equivalent to taking $M_{\textrm{disk}} =1.1\%$ M$_\star$.
- **The planetary systems scenario (S2)** is based on the observational results on the masses of exoplanetary systems. We assumed $M_{\textrm{dust}} = 9 \times 10^{-5}$ M$_\odot$ ($\sim30$ M$_\oplus$) regarding the current terrestrial exoplanet detection around BDs [e.g., @Kubas2012; @Gillon2017; @Grimm2018]. If we were to assume a gas-to-dust radio of 100:1, this would be equivalent to taking $M_{\textrm{disk}} =11\%$ M$_{\star}$. In this case, we increased the percentage of the mass in order to extend the solid material in the disk that is available in our region of study to form rocky planets.
Contrasting $\Sigma$ parameterization
-------------------------------------
Many authors have also proposed a power-law surface density profile to model protoplanetary disks [e.g., @Ciesla2015; @Testi2016]. We therefore compared the model proposed in this work (Eq. \[eq:sigmadust\]) with a power-law density profile,
$$\label{eq:powlawdensprof}
\Sigma_{\textrm{sp}}(r) = \Sigma_{\textrm{sp}0}\eta_{\textrm{ice}}\left(\frac{r}{r_0}\right)^{-p}
,$$
where $r_0$ and $p$ are equivalent to $r_c$ and $\gamma$ in the exponentially tapered density profile. In Fig. \[fig:densprofiles\_varios\] we show the comparison of the two density profiles considering the same initial parameters as we chose to describe $\Sigma_\textrm{s}(r)$ (see Section \[sec:protodisk\]). As an example, we selected three different disk masses $M_{\textrm{disk}}$: $0.1\%$, $1\%,$ and $10\%$ of the mass of the central object, and we assumed a gas-to-dust ratio of $100:1$. The power-law profiles do not show significant differences with the exponentially tapered density profiles within our region of study.
![Power law (green line) and exponential tapered (blue line) surface density profiles in the protoplanetary-planetary disk for a total disk mass $M_{\textrm{disk}}$ of $0.1\%$, $1\%,$ and $10\%$ of the mass of the central object of 0.08 M$_\odot$. The region of study ($0.015<r/\mathrm{au}<1$) is indicated.[]{data-label="fig:densprofiles_varios"}](Density_profiles_varios.eps){width="9cm"}
Numerical model {#sec:numodel}
===============
In this section we describe the treatment of planet formation around an object of 0.08 M$_\odot$ by including a protoplanetary embryo distribution that interacts with the main object. We developed a set of $N$-body simulations with the well-known $\textsc{Mercury}$ code [@Chambers1999] by incorporating tidal and general relativistic acceleration corrections as external forces. Thus the dynamical evolution of protoplanetary embryos was affected not only by gravitational interactions between them and with the star, but also by tidal distortions and dissipation, as well as by general relativistic effects. The stellar contraction and rotational period evolution was included in the code as well as a fixed pseudo-synchronization period for protoplanetary embryos during the 100 Myr integration time of our simulations.
Tidal model {#sec:tidalmodel}
-----------
We followed the equilibrium tide model [@Hut1981; @Eggleton1998] that was rederivated by @Bolmont2011, which considers both the tide raised by the BD on the planet and by the planet on the BD in the orbital evolution of planetary systems. It also takes into account the spin-up and contraction of the BD. The authors followed the constant time-lag model and assumed constant internal dissipation for the BD and the planets involved.
Following the equilibrium tidal model, we incorporated tidal distortions and dissipation terms, considering the tide raised by the star on each protoplanetary embryo and by each protoplanetary embryo on the star and neglected the tide between embryos. Tidal interactions produce deformations on the bodies that in a heliocentric reference frame lead to precession of the argument of periastron $\omega$ and a decay in semimajor axis $a$ and eccentricity $e$ , which can be interpreted as distortions and dissipation terms, respectively.\
The correction in the acceleration of each protoplanetary embryo produced by the tidal distortion term was taken from @Hut1981 (for an explicit expression, see @Beauge2012) and is given by
$$\textbf{f}_\omega = -3\frac{\mu}{r^8}\left[k_{2,\star}\left(\frac{M_\mathrm{p}}{M_\star}\right)R_\star^5 + k_{2,\mathrm{p}}\left(\frac{M_\star}{M_\mathrm{p}}\right)R_\mathrm{p}^5\right] \textbf{r},$$
where **r** is the position vector of the embryo with respect to the central object, $k_{2,\star}=0.307$ and $k_{2,\textrm{p}}=0.305$ are the potential Love numbers of degree 2 of the star and the embryo, respectively [@Bolmont2015], $\mu=G(M_\star + M_\textrm{p})$, $G$ is the gravitational constant, and $M_\star$, $R_\star$, $M_\textrm{p}$ , and $R_\textrm{p}$ are the masses and radius of the star and the protoplanetary embryo under the approximation that these objects can instantaneously adjust their equilibrium shapes to the tidal force and considering only distoritions up to the second-order harmonic [@Darwin1908].
The evolution of $R_\star$ was taken from the models of @Baraffe2015, and the value of $R_\textrm{p}$ of each protoplanetary embryo was calculated by considering each of them as a spherical body with a fixed volume density $\rho=5~\mathrm{gr/cm^3}$.
The timescale associated with the tidal dissipation term was calculated based on the work of @Sterne1939 by considering the stellar and embryo tide, and it is given by
$$t_{\textrm{tide}} \sim \frac{2 \pi a^{5}}{7.5nf(e)}\left(\frac{M_{\star}M_{\mathrm{p}}}{k_{2,\star}M_{\mathrm{p}}^{2}R_{\star}^{5} + k_{2,\mathrm{p}}M_{\star}^{2}R_{\mathrm{p}}^{5}}\right),\\
\label{eq:jordantimescale}$$
with $f(e) = (1-e^2)^{-5}[1 + (3/2)e^2 + (1/8)e^4]$.\
\
The acceleration correction of each protoplanetary embryo induced by the tidal dissipation term, which produces $a$ and $e$ decay, was obtained from @Eggleton1998. After some algebra, this equals the expression from @Beauge2012,
$$\begin{aligned}
\textbf{f}_{\textrm{ae}} = & -3\frac{\mu}{r^{10}} \left[
\frac{M_{\textrm{p}}}{M_\star} k_{2,\star} \Delta \mathrm{t}_\star R_\star^{5}\left(2\textbf{r}(\textbf{r} \cdot \textbf{v}) + r^{2}(\textbf{r} \times \Omega_\star + \textbf{v})\right)\right]\end{aligned}$$
$$-3\frac{\mu}{r^{10}} \left[\frac{M_\star}{M_{\textrm{p}}}k_{2,\textrm{p}} \Delta \mathrm{t}_{\textrm{p}} R_\textrm{p}^{5}
\left(2\textbf{r}(\textbf{r}\cdot\textbf{v}) + r^{2}(\textbf{r} \times \Omega_\textrm{p} + \textbf{v})\right)\right],$$
where $\textbf{v}$ is the velocity vector of the embryo, and $\Delta \mathrm{t}_\star$ and $\Delta \mathrm{t}_\mathrm{p}$ are the time-lag model constants for the star and the protoplanetary embryo, respectively. The factors $k_{2,\star}\Delta \mathrm{t}_\star$ , and $k_{2,\mathrm{p}}\Delta \mathrm{t}_\mathrm{p}$ are related to the dissipation factors by
$$k_{2,\mathrm{p}} \Delta \mathrm{t} _\mathrm{p} = \frac{3R_\mathrm{p}^5\sigma_\mathrm{p}}{2G} \\
k_{2,\star} \Delta \mathrm{t}_{\star} = \frac{3R_{\star}^5\sigma_{\star}}{2G}
,$$
with the dissipation factor for each protoplanetary embryo $\sigma_\mathrm{p}=8.577\times10^{-50}\mathrm{g^{-1} cm^{-2} s^{-1}}$, the same dissipation factor as estimated for the Earth [@Neron1997], and the dissipation factor of the central object is $\sigma_\star=2.006\times10^{-60}\mathrm{g^{-1} cm^{-2} s^{-1}}$ [@Hansen2010].
In the constant time-lag model, in which the time-lag constant $\Delta \mathrm{t}_\star$ is independent of the tidal frequency, the rotation of the companions leads to pseudo-synchronization [@Hut1981; @Eggleton1998]. In preliminary simulations, @Leconte2010 [@Bolmont2011; @Bolmont2013] verified that a planet reaches the pseudo-synchronization very quickly in its evolution. For a planet, being at pseudo-synchronization means that its rotation tends to be synchronized with the orbital angular velocity at periastron, where the tidal interactions are stronger [@Hut1981]. As in @Bolmont2011, we fixed each protoplanetary embryo at pseudo-synchronization [@Hut1981] in each time-step of our simulations as
$$\label{eq:pseudosincro}
\Omega_\mathrm{p} = \frac{(1 + (15/2)e^2 + (45/8)e^4 + (5/16)e^6)}{(1 + 3e^2 + (3/8)e^4)(1 - e^2)^{3/2}} n,$$
where $\Omega_{\textrm{p}}$ is the rotational velocity of the protoplanetary embryo.
If $e=0$, then the embryo is in perfect synchronization, thus $\Omega_\textrm{p} = n$ and only the tide of the central object remain. When $e$ is small, the tide of the main object dominates and determines the evolution of the embryo: if the embryo is located beyond $r_{\textrm{corot}}$, then $\Omega_\textrm{p} < \Omega_\star,~\frac{da}{dt} > 0,$ so that the embryo is pushed outward, and if it is inside, $\Omega_\textrm{p} > \Omega_\star,~\frac{da}{dt} < 0,$ so that the embryo is pulled inward. On the other hand, when $e$ is high, the embryo tide will prevail. In this case, the embryo is pulled inward. This is always true because for a body in pseudo-synchronization, the body tide always acts to decrease the orbital distance [@Leconte2010].
The rotational velocity $\Omega_{\star}$ of the main object was calculated following the tidal model proposed by @Bolmont2011, who integrated its evolution as affected by its contraction and the influence of orbiting planets. They calibrated their results with a set of observationally determined $\Omega_{\star}$ for VLMS and BDs at different ages from @Herbst2007. Thus the evolution of $\Omega_\star$ can be expressed as
$$\Omega_\star(t)=\Omega_\star(t_0)\left[\frac{r_\mathrm{gyr}^2(t_0)}{r_\mathrm{gyr}^2(t)}\left(\frac{R_\star(t_0)}{R_\star(t)}\right)^{2}\times exp\left(\int_{t_0}^{t}f_{\textrm{t}} dt\right)\right]
\label{eq:omega}$$
[e.g., @Bolmont2011], where $r_\mathrm{gyr}^2$ is the square of the gyration radius, defined as $r_\mathrm{gyr}^2 = \frac{I_\star}{M_\star R_\star^{2}}$ , with $I_\star$ the moment of inertia of the main object [@Hut1981]. The function $f_{\textrm{t}}$ is given by
$$f_\mathrm{t}=\frac{1}{\Omega_\star}\frac{d\Omega_\star}{dt}.$$
If we were to consider $r_\mathrm{gyr}^2$ and $R_\star$ as constant values [@Bolmont2011], then
$$\label{eq:ftide}
f_\mathrm{t}=-\frac{\gamma_\star}{t_{\mathrm{dis},\star}}\left[No1(e) - \frac{\Omega_\star}{n}No2(e)\right]
,$$
with $\gamma_\star = \frac{h}{I_\star\Omega_\star}$, where $h$ is the orbital angular momentum, $t_\mathrm{dis,\star}$ is the dissipation timescale of the central object (see below), and the functions $No1(e)$ and $No2(e)$ are dependent on the eccentricity of the planetary companion, which is given by
$$\begin{aligned}
No1(e)&=\frac{1+(15/2)e^{2}+(45/8)e^{4}+(5/16)e^{6}}{(1-e^{2})^{13/2}}\\
\\
No2(e)&=\frac{1+3e^{2}+(3/8)e^{4}}{(1-e^{2})^{5}}.
\end{aligned}$$
When only terrestrial planets are considered to orbit the host object, then $f_\textrm{t}$ is small and the substellar object rotation period is mainly determined by the conservation of angular momentum, that is, by the initial rotation period [@Bolmont2011]. We therefore numerically integrated Eq. independently of the dynamics of the planetary system. We considered that the radius R$_\star$ evolves according to the structure and atmospheric models from @Baraffe2015, but we fixed its value for each time-step of our integration, which was small enough to be considered constant in order to simplify the integration and be able to use Eq. . We also assumed one Earth-like planet to orbit the main object with random initial values for $e$ and $a$ inside our region of study. From the different orbital elements initially given to the Earth-like planet, we verified that the evolution of $\Omega_\star$ was mainly determined by the evolution of the substellar object and was similar to the evolution achieved by @Bolmont2011. In Fig. \[fig:RandP\] we show the resulting evolution of the rotational period and the corresponding $R_{\star}$ in a period from 1 Myr to 100 Myr.\
![Rotation period evolution of an object of 0.08 M$_\odot$ based on @Bolmont2011 associated with its radius contraction. The radius evolution is taken from @Baraffe2015.[]{data-label="fig:RandP"}](Periodo_radioBD.eps){width="9cm"}
The dissipation timescales for eccentric orbits are determined from the secular tidal evolution of $a$ and $e$ [see @Hansen2010; @Bolmont2011; @Bolmont2013] as $t_\textrm{a}$ and $t_\textrm{e}$ , respectively, by $$\begin{split}
\frac{1}{t_\textrm{a}}=\frac{1}{a} \frac{da}{dt} = - \frac{1}{t_\textrm{dis,p}} \left[Na1(e) - \frac{\Omega_p}{n}Na2(e) \right]
\\
- \frac{1}{t_{\mathrm{dis},\star}} \left[Na1(e) - \frac{\Omega_\star}{n}Na2(e) \right]
\end{split}$$ $$\begin{split}
\frac{1}{t_\textrm{e}}=\frac{1}{e} \frac{de}{dt} = & - \frac{9}{2t_\textrm{dis,p}} \left[Ne1(e) - \frac{11}{18} \frac{\Omega_p}{n}Ne2(e)\right], \\
\\
&- \frac{9}{2t_{\mathrm{dis},\star}}\left[ Ne1(e) - \frac{11}{18} \frac{\Omega_\star}{n}Ne2(e) \right].
\end{split}$$ Here $t_\textrm{dis,p}$ and $t_{\textrm{dis},\star}$ are the dissipation timescales for circular orbits for the embryo and the main object, respectively, and $Na1$, $Na2$, $Ne1,$ and $Ne2$ are factors that take place in eccentric orbits and are defined by
$$\begin{aligned}
t_\textrm{dis,p} &= \frac{1}{9} \frac{M_\textrm{p}}{M_\star(M_\textrm{p} + M_\star)} \frac{a^8}{R_\textrm{p}^{10}} \frac{1}{\sigma_\textrm{p}},\\
\\
t_{\mathrm{dis},\star} &= \frac{1}{9} \frac{M_\star}{M_\textrm{p}(M_\textrm{p} + M_\star)} \frac{a^8}{R_\star^{10}} \frac{1}{\sigma_\star},\\
\\
Na1(e) &= \frac{1 + (31/2)e^{2} + (255/8)e^{4} + (185/16)e^{6} + (25/64)e^{8}}{(1-e^{2})^{15/2}},\\
\\
Na2(e) &= \frac{1 + (15/2)e^{2} + (45/8)e^{4} + (5/16)e^{6}}{(1-e^{2})^{6}},\\
\\
Ne1(e) &= \frac{1 + (15/4)e^{2} + (15/8)e^{4} + (5/64)e^{6}}{(1-e^{2})^{13/2}},\\
\\
Ne2(e) &= \frac{1 + (3/2)e^{2} + (1/8)e^{4}}{(1-e^{2})^{5}}.
\end{aligned}$$
General relativistic effect
---------------------------
The important effect derived from General Relativity theory (GRT) on the dynamic of planetary systems is the precession of $\omega$ [@Einstein1916]. In our case, we considered that only the main object contributes with relevant corrections. As we worked in the reference frame of the star, the associated correction in the acceleration of the embryo is
$$\textbf{f}_{\mathrm{GR}} = \frac{GM_\star}{r^3c^2}\left[\left(\frac{4GM_\star}{r} - \textbf{v}^2\right)\textbf{r}+4(\textbf{v}.\textbf{r})\textbf{v}\right],
\label{eq:grav}$$
with $c$ the speed of light. Eq. was proposed by @Anderson1975, who worked under the parameterize post-Newtonian theories. The authors obtained a relative correction associated with two parameters $\beta$ and $\gamma$, which are equal to unity in the GRT case. This expression has been used in several works that included relativistic corrections [e.g., @Quinn1991; @Shahid1994; @Varadi2003; @Benitez2008; @Zanardi2018]. The timescale associated with the precession of the longitude of periastron is given by
$$t_{\mathrm{GR}} \sim 2 \pi \frac{a^{\frac{5}{2}}c^2(1-e^2)}{3G^{\frac{3}{2}}(M_\star+M_{\textrm{p}})^{\frac{3}{2}}}.
\label{eq:GR}$$
Test simulations
----------------
We made a set of $N$-body simulations in order to test the agreement between the external forces that we incorporated in the <span style="font-variant:small-caps;">Mercury</span> code and the timescale associated with them. To test the precession of $\omega,$ we developed two simulations: one that included the tidal distortion term, and another that included the GR correction. In Fig. \[fig:perihelion\] we show the apsidal precession timescale of a planet with 1 M$_\oplus$ orbiting a 1 Myr substellar object of 0.08 M$_\odot$ with initial values $a = 0.01$ au and $e = 0.1$ for the two simulations we made. Our results show that the apsidal precession of $360^\circ$ is completed in 14750 yr and 3060 yr, respectively, which agrees with the time predicted by the timescales associated with each correction term in Eqs. and . These timescale values depend on the physical parameters of the protoplanetary embryos and the substellar object, as well as on the initial orbital elements. For instance, if the protoplanets are smaller than Earth in mass and radius, then the relativistic effect is more relevant than the tidal distortion regarding the precession of $\omega$.
![Apsidal precession due to tidal distortion (solid red line) and GR effects (dotted blue line) in a system composed of a 1 M$_\oplus$ planet around a 0.08 M$_\odot$ BD with initial $a = 0.01$ au and $e = 0.1$. In this example, the argument of periastron completes an orbit in $\sim$14750 yr when is affected by GR and in $\sim$3060 yr when is affected by tidal distortion.[]{data-label="fig:perihelion"}](Periastro_precesion.eps){width="45.00000%"}
To test the analytic expressions for the tidal dissipation with the timescales of $e$ and $a$ decay, we developed a $N$-body simulation that includes the dissipation term. Our aim was to compare our results with those obtained by @Bolmont2011, who used the analytic tidal model. In their work, they represent the evolution of $a$, $e,$ and the rotation period of a planet of 1 M$_\oplus$ evolving around a BD of 0.04 M$_\odot$. We chose as initial values $a=0.017242$ au and $e=0.744$. The semimajor axis we selected represents $a_{\textrm{switch}}$, that is, the one that determines the behavior of the protoplanet and separates inward migration and crash from inward migration but survival of the protoplanet or outward migration.
In Fig. \[fig:Bolmont\_example\] we show the evolution of $a$, $e,$ and the pseudo-synchronization period $P_\textrm{p}$ of a 1 M$_\oplus$ planet around an evolving 0.04 M$_\odot$ BD. In the middle panel, we also show the $r_{\textrm{corot}}$ evolution, while in the bottom panel we include the rotation period of the BD, $P_{\star}$ . First, as $P_\star > P_\textrm{p}$ , the planet moves inward of the central object. Under this condition, even tough $a = r_\textrm{corot}$, the orbit is not circular and the planet continues to move inward. When $P_\star = P_\textrm{p}$, the orbit is circular and then $\Omega_\textrm{p} = n,$ which means that this time when $a = r_{\textrm{corot}}$ , the planet starts to move outward because $P_\star < P_\textrm{p}$.
![Evolution of $e$, $a,$ and the pseudo-synchronization period of a 1 M$_\oplus$ around a 0.04 M$_\odot$ BD. Solid lines indicates the results considering the initial orbital elements from @Bolmont2011. The dashed lines indicate the evolution of $r_{\textrm{corot}}$ (middle panel) and the rotational period of the BD, $P_\star$ (bottom panel).[]{data-label="fig:Bolmont_example"}](Ejemplo_Bolmont.eps){width="48.00000%"}
$N$-body simulations
--------------------
We performed 20 $N$-body simulations using the modified version of the <span style="font-variant:small-caps;">Mercury</span> code. We developed 10 simulations for scenario *S1* and 10 simulations for scenario *S2,* as explained in Section \[sec:twofolddensity\]. We also ran 10 simulations for each scenario described above using the original version of the <span style="font-variant:small-caps;">Mercury</span> code, without external forces, in order to evaluate the relevance of tidal and general relativistic effects.
### Protoplanetary embryo distributions
For using $\textsc{Mercury}$ code, it is necessary to give physical and orbital parameters for the protoplanetary embryos. We modeled the initial mass distributions of embryos as a function of the radial distance from the central object, which are initial conditions for our numerical simulations. The initial spatial distribution of protoplanetary embryos was computed following Eq. , considering a distance range $0.015<r/\textrm{au}<1$ and defining 1 Myr as the initial time. We considered that at this age, the gas has already been dissipated from the disk.
Even though we are aware of the existence of a number of BDs that still accrete gas from their disks up to $\sim 10$ Myr [references, e.g., in @Pascucci2009; @Downes2015], incorporating the gas component is beyond the scope of this work, which reproduces the BDs that are not observed to show gas signatures at $\sim 1$ Myr, however. We calculated the mass of each protoplanetary embryo $M_\textrm{p}$ considering that at the initial time, they are at the end of the oligarchic growth stage, having accreted all the planetesimals in their feeding zones [@Kokubo2000] by
$$M_\textrm{p}=2\pi r\Delta r_\textrm{H}\Sigma_s(r),
\label{eq:masaemb}$$
where $\Delta r_\textrm{H}$ is the orbital separation between two consecutive embryos in terms of their mutual Hill radii $r_\textrm{H}$, with $\Delta$ an arbitrary integer number, given by
$$r_\textrm{H}=r\left(\frac{2M_\textrm{p}}{3 M_\star}\right)^{\frac{1}{3}}.
\label{eq:radiohill}$$
By replacing Eqs. and in Eq. , we obtain an expression for the mass of each protoplanetary embryo as a function of the radial distance in the disk mid-plane $r$, which is given by
$$M_{\mathrm{p}}=\left(2 \pi r^{2} \Delta \Sigma_{0\textrm{s}} \eta_{\textrm{ice}} \left(\frac{2}{3 M_{\star}}\right)^{\frac{1}{3}}\left(\frac{r}{r_{\textrm{c}}}\right)^{-\gamma}\textrm{e}^{{-\left(\frac{r}{r_{\textrm{c}}}\right)}^{2-\gamma}}\right)^{\frac{3}{2}}.
\label{eq:massemb}$$
We located our first embryo at the inner radii of our region $r_1 = 0.015$ au. Then we calculated its mass using Eq. . For the remaining embryos, we propose a separation of 10 r$_{\textrm{H}}$ by fixing $\Delta=10$ [@Kokubo1998].
Thus we calculated the initial positions $r_{i+1}$ and masses $M_{\textrm{p},i+1}$ for the embryos by
$$r_{i+1} = r_i + \Delta r_\textrm{i} \left(\frac{2M_i}{3 M_\star}\right)^{\frac{1}{3}},
\label{eq:distancias}$$
$$M_{\textrm{p},i+1}= \left(A\left(\frac{2}{3 M_\star}\right)^{\frac{1}{3}}\left(\frac{r_{i+1}}{r_c}\right)^{-\gamma}\textrm{e}^{{-\left(\frac{r_{i+1}}{r_c}\right)}^{2-\gamma}}\right)^{\frac{3}{2}},
\label{eq:masas}$$
for $i = 1,~2,~\text{etc.}$ with $A=2 \pi r_{i+1}^2\Delta\Sigma_{0\textrm{s}} \eta_\textrm{ice}$.
Using Eqs. and , we derived the initial distributions of masses of the protoplanetary embryos as a function of the radial distance, which represents the semimajor axis, from the central object for scenarios S1 and S2. In Fig. \[fig:Distribution\_embryos\] we illustrate the two distributions. S1 has a distribution of $224$ embryos with a total mass $M_{\textrm{pT}}\sim0.25$ M$_\oplus$ located in the region of study. $\text{Two hundred and ten}$ of them are distributed in the inner region up to the snow line, with a total mass $\sim0.06$ M$_\oplus$, while the remaining 14 embryos are distributed beyond the snow line and have a total mass $\sim0.19$ M$_\oplus$. S2 has a distribution of $74$ embryos that are located in the region of study with a total mass $M_{\textrm{pT}}\sim3$ M$_\oplus$. $\text{Sixty-nine}$ of them are distributed in the inner region up to the snow line, with a total mass $\sim0.72$ M$_\oplus$, while the remaining $5\text{}$ embryos have a total mass of $\sim2.25$ M$_\oplus$ and are placed beyond the snow line up to 1 au.
![Initial embryo distributions of masses as a function of their initial location given by their semimajor axis for S1 (circles) and S2 (stars). Blue represents the water-rich population ($50\%$ water in mass), and red represents the bodies with the lowest amount of water in mass ($0.01\%$).[]{data-label="fig:Distribution_embryos"}](Embriones_escenario_ambos.eps){width="8cm"}
We considered lower values than $0.02$ for the initial eccentricities and $0.5^\circ$ for the initial inclinations. The orbital elements, argument of periastron $\omega$, longitude of the ascending node $\Omega,$ and mean anomaly $M$ were determined randomly at the beginning of the simulations. They were between $0^\circ$ and $360^\circ$ from a uniform distribution for each protoplanetary embryo.
### $N$-body code: characterization {#sec:proto.planet.desc}
To develop our simulations, we chose the hybrid integrator, which uses a second-order symplectic algorithm to treat interactions between objects with separations greater than $3$ Hill radii, and we selected the Bulirsch-Stöer method to resolve closer encounters. The collisions were treated as perfectly inelastic, conserving the mass and the corresponding water content of protoplanetary embryos. We considered that a body is ejected from the system when it reaches a distance $a>100$ au.
We adopted a time step of 0.08 days, which corresponds to $1/30$ th of the orbital period of the innermost body in the simulations. In order to avoid any numerical error for small-perihelion orbits, we assumed $R_\star=0.004$ au, which corresponds to the maximum value of the radius of the central object.
All simulations were integrated over 100 Myr, which is a standard time for studying the dynamical evolution of planetary systems. Because of the stochastic nature of the accretion process between the protoplanets and eventually with the main object, we remark that it is necessary to carry out a set of $N$-body simulations. In this case, we performed ten simulations for each scenario, which required a mean CPU time of six months on 3.6 GHz processors.
Results {#sec:results}
=======
In this section we present the resulting planetary systems of the simulations in scenarios S1 and S2. We compare the resulting planets from simulations that included tidal and GR effects with those from simulations that neglected these effects to test their relevance in the formation of rocky planets. In particular, we focus our analysis on the population that is located close to the central object.
Planetary architectures {#sec:planet_arch}
-----------------------
Our simulations predict a diversity of final planetary system architectures at 100 Myr regarding all the simulations made in both scenarios. Fig. \[fig:Scenario1\_architectures\] shows the final location of the resulting planetary systems of each simulation in scenario S1 for the final masses and fraction of water in mass. The planetary masses are between 0.01 M$_\oplus$ and 0.12 M$_\oplus$ (this is approximately the mass of the ** and ** respectively) and the fraction of water in mass is between $0.01\%$ and $50\%$. The left panel shows the planetary architectures from simulations that included tidal and GR effects, and the right panel presents their counterparts in the simulations that neglect these effects. In both panels, the IHZ of the system at 100 Myr and at 1 Gyr overlap. In Fig. \[fig:Scenario2\_architectures\] we show the resulting architectures of the simulations for scenario S2. The resulting planets have a range of masses between 0.2 M$_\oplus$ and 1.8 M$_\oplus$ and a percentage of water in mass between $0.01\%$ and $50\%$.
The main difference we found is the close-in planet population that survived in the simulations that included tidal and GR effects, which did not survive in the simulations that neglected these effects. This becomes more relevant in S1, where the protoplanetary embryos involved were an order of magnitude less massive than in S2.
In simulations S2, the embryos involved suffered stronger gravitational interactions between them than those from S1 because they are more massive bodies. Therefore they generate more excitation in the system, allowing some embryos to collide with the central object and to be ejected from the system. These interactions became more relevant than tidal and GR effects for the population of very close-in bodies in S2. This is supported by the percentage of embryos that collided with the central object or were ejected from the system (reached $a > 100$ au), as shown in Fig. \[fig:colisiones\]. The percentage is given over the initial amount of embryos in each scenario of work: 224 embryos in S1, and 74 embryos in S2. The number of bodies that either collided with the central object or were ejected from the system in S2 is much higher than in S1 because the system is more highly excited. Moreover, the simulations that included tidal and GR effects reduced the collisions of embryos with the central object and had almost no effect on the ejection of embryos because at long distances from the central object, tidal effects became irrelevant.
{width="40.00000%"} {width="40.00000%"}
{width="40.00000%"} {width="40.00000%"}
![Percentage of embryos that collided or were ejected from the system during the integration time of each scenario. Blue bars correspond to embryos from simulations that included tidal and GR effects, and red bars represent embryos from simulations that neglected these effects.[]{data-label="fig:colisiones"}](colis_eyec.eps){width="8cm"}
By compacting all the resulting planetary systems in Fig. \[fig:masas\_vs\_a\], we represent their distribution in $a$ for their masses for S1 and S2. The IHZ at 100 Myr and at 1 Gyr overlap as well. The surviving population of close-in bodies has low masses and mainly results from simulations with tidal and GR effects from S1. This shows that the relevance of tidal and GR effects also depends on the mass of the bodies that are involved in the simulations.
The Fig. \[fig:avse\] shows the eccentricities of the resulting planets as a function of their semimajor axis for S1 and S2. Low-mass and close-in planets that survived while external effects were included appear to have eccentricities values greater than zero. This is because the many gravitational interactions between embryos produce excitation in their orbital parameters, and the timescale for eccentricity damping is far longer ($\text{approximately some billion years}$) than the integration time of our simulations. As we discussed in previous sections, the tidal effects added in our simulations affect the distribution of eccentricities and the semimajor axis of the resulting planets, but the long decay timescales prevent us from seeing the damping in $e$ at this point. Nevertheless, the $e$ damping will be efficient by the time the central object reaches 1 Gyr for the planets that remain located close in to the central object, where the IHZ will be located by that time.
Our results strongly suggest that a formation scenario that includes tidal and GR effects is more realistic for planet formation at the substellar mass limit. Although GR corrections are relevant during planet formation, the tidal effects are mor important to map more realistic orbits and therefore more realistic encounters between embryos. These effects play a primary role in the survival of an in situ population. However, when the masses of the bodies involved increase (like in S2), tidal effects became less relevant than the gravitational interactions between them (see Section \[sec:discussions\]).
![Distribution in mass of the resulting planets for their semimajor axis at 100 Myr in S1 (top panel) and S2 (bottom panel). Blue dots represent the planets from simulations that included tidal and GR effects, while red dots represent those from simulations that neglected these effects. The pink band represents the location of the IHZ at 1 Gyr, while the cream band represents its location at 100 Myr.[]{data-label="fig:masas_vs_a"}](masavsa_final_ambosescenarios.eps){width="9cm" height="6cm"}
![Orbital distribution of the resulting planets regarding their location in the system and eccentricity in S1 (top panel) and S2 (bottom panel). Colors are as in Figure \[fig:masas\_vs\_a\].[]{data-label="fig:avse"}](evsa_final_ambosescenarios.eps){width="9cm" height="6cm"}
Water mass fraction {#sec:water_planets}
-------------------
The two scenarios show a diversity in the fraction of water in mass, but the resulting planets inside $a<0.1$ au always conserved this initial fraction of water in mass. We assumed this to be $0.01\%$ of the mass of the embryos that are located inside the snow line at the beginning of the simulations.
For S1, planets at $a>0.1$ au present a range in percentage of water in mass that is between $10\%$ and $35\%$ for planets with a semimajor axis $0.1<a/\textrm{au}<0.42$, and it is between $35\%$ and $50\%$ for planets with $0.42<a/\textrm{au}<1$. The outer water-rich planets maintain their high content of water in mass, and an intermediate population of water-rich resulting planets appears close to the location of the snow line.
For S2, planets at $a>0.1$ au present a high mass percentage of water, between $35\%$ and $50\%$. Embryos located outside the snow line either have suffered impacts of other water-rich bodies or have been ejected from the system. No intermediate water-rich population as in S1 evolved. In order to explain the origin of the resulting distribution of water of the surviving planets in our region of study, in the next section we analyze their whole collisional history.
{width="65.00000%"}
{width="65.00000%"}
Collisional history {#sec:sollisions}
-------------------
During the first 100 Myr of planet formation that we studied in our simulations, embryos had gravitational interactions in the form of encounters and collisions among them. From the initial location in the system, each embryo can interact with others that have different orbital and physical parameters in its gravitational influence zone. The $N$-body code treats all collisions as perfectly inelastic. After two bodies collide, the resulting body therefore is a merger of the initial two.
From the previous analysis of the final distributions of the resulting planets and their final fraction of water in mass, we can distinguish different subregions in which the collisional history of the resulting planets can be studied: an inner region of planets that are finally located at $a < 0.1$ au, an intermediate region, between $0.1<a/\mathrm{au}<0.42,$ and an outer region beyond the snow line, between $0.42<a/\mathrm{au}<1$.
Fig. \[fig:historia\_colisional\] shows the collisional history of all the resulting planets of S1 when tidal and GR effects are included and when these effects are neglected. Each peak in the lines represents the initial location of the embryo that collided with the resulting planet and the percentage of mass that it added to the planet after the perfect collision. In the inner region the resulting planets collided with embryos that were initially located at $a < 0.35$ au, all located inside the snow line, which means that they preserve the initial fraction of water in mass. In the intermediate region, planets accreted embryos that were initially located between $0.05 < a/\mathrm{au} < 0.88$ and between $0.05 < a/\mathrm{au} < 0.95,$ which explains the intermediate range of water in mass after collision with embryos inside and outside the snow line. Finally, in the outer region, planets collided with embryos that were initially located between $0.2 < a/\mathrm{au} < 0.95$ and $0.22 < a/\mathrm{au} < 0.95$ . In this outer region, even though the resulting planets suffered collisions with embryos that were distributed in a wide range of the system, only a few planets that were located inside the snow line collided with these planets and thus maintained a huge fraction of water in mass. It is important to remark that the close-in population located at $a < 0.1$ au did not collide with other embryos beyond this semimajor axis when tidal effects were included in the simulations.
On the other hand, Fig. \[fig:historia\_colisional2\] shows the collisional history of the resulting planets of S2 when tidal and GR effects are included and when these effects are neglected. In this case, in the inner-region planets collided with embryos that were initially located at $a < 0.36$ au and $a < 0.22$ au. In the intermediate region, planets accreted embryos that were initially located between $0.06 < a/\mathrm{au} < 0.72$ and $0.07 < a/\mathrm{au} < 0.91$ . Finally, in the outer region, planets collided with embryos that were initially located at $0.17 < a/\mathrm{au} < 0.72$ and $0.097 < a/\mathrm{au} < 0.72$. In S2, planets in the intermediate and outer region accreted material from a similar region, with a few embryos initially located inside the snow line. Thus all these planets retained a huge amount of water in mass. In this case, the three planets that survived with at $a < 0.1$ au did not collide with other embryos beyond this semimajor axis as happened in S1 when tidal effects were included in the simulations.
The collisional history of the resulting planets explains their final masses and the fraction of water in mass. Moreover, it allows us to conclude that the close-in surviving population that was most affected by tidal effects only suffered collisions with the embryos that were initially located close in.
Close-in population: potentially habitable planets {#sec:HZ_planets}
--------------------------------------------------
We focus our analysis on the close-in bodies that survived in the simulated planetary systems. We gave physical and orbital parameters of those bodies candidates to be potentially habitable planets.
### Characterization
Inside the location of the IHZ at 100 Myr (final time of simulations), two planets in S1 remained when tidal and GR effects were included in the simulations and only one planet remained when these effects were excluded from the simulations regarding their semimajor axis. On the other hand, in S2 two planets remained in the IHZ when external effects were included and no candidate survived when these effects were excluded from the simulations. Moreover, when we consider the value of the eccentricity and calculate the periastron distance (q) and apastro distance (Q) of these planets, only one planet remained inside the IHZ in a, q, and Q in S1 and 1 in S2 when tidal and general relativistic effects were considered in the simulations.
In Section \[sec:habzone\] we discussed the behavior of the IHZ around very low mass stars that are located close to the star and evolve toward a smaller radius as the star evolves with time. We therefore extended our analysis to bodies that ended up closer in to the central object, in particular, inside the location of the IHZ at 1 Gyr. The S1 alone generated such a close-in body population: nine planets when tidal and GR effects were incldued, and only one planet when these effects were neglected in the simulations regarding their semimajor axis. Even though we do not have the orbital parameters at 1 Gyr, it is expected that the eccentricities of these planets are small enough for them to remain in the IHZ at 1 Gyr in S1 (see Section \[sec:numodel\]). In Table \[tab:ZH\_info\] we present some physical parameters of the planets that remained in the IHZ at 100 Myr or at 1 Gyr in both scenarios. When tidal and GR effects are included in the simulations, S1 is the most favorable scenario to allow these candidates of potentially habitable planets (see Section \[sec:discussions\] for further discussion.)
---------- -------- ---------- ------- ---------------- -------- -----
Scenario Embryo a $e$ M $H_2O$ IHZ
\[$au$\] \[M$_\oplus$\] \[%\]
S1t 39 0.020 0.23 0.004 0.01 b
S1t 132 0.025 0.18 0.010 0.01 b
S1t 113 0.026 0.23 0.008 0.01 b
S1t 113 0.027 0.27 0.013 0.01 b
S1t 132 0.024 0.43 0.007 0.01 b
S1t 33 0.018 0.07 0.001 0.01 b
S1t 126 0.046 0.19 0.010 0.01 a
S1t 115 0.022 0.23 0.013 0.01 b
S1t 63 0.022 0.30 0.006 0.01 b
S1t 142 0.052 0.08 0.013 0.01 a
S1t 62 0.019 0.28 0.007 0.01 b
S1wt 163 0.066 0.195 0.017 0.01 a
S1wt 90 0.022 0.18 0.007 0.01 b
S2t 59 0.051 0.45 0.337 0.01 a
S2t 39 0.053 0.20 0.370 0.01 a
---------- -------- ---------- ------- ---------------- -------- -----
: Potentially habitable planets from both scenarios when tidal and GR effects are included in the simulations (S1t and S2t) and when these effects are neglected in the case of S1 (S1wt). The initial numbers of embryos that become the resulting planet are listed in Col. 2 with their respective final semimajor axes, eccentricity, mass, and percentage of water in mass. In their final locations, the planets are located in the IHZ at 100 Myr (IHZ = a) or in the IHZ at 1 Gyr (IHZ = b).[]{data-label="tab:ZH_info"}
### Mass accretion history {#sec:colid_history_IHZ}
![Cumulative collisions between the resulting planets that survived inside the IHZ at 100 Myr and at 1 Gyr for S1 with tide and GR (orange line), S1 without considering tide or GR (cyan dots), and S2 with tide and GR.[]{data-label="fig:Scenario1_colissions"}](embriones_colisiones.eps){width="8cm"}
All the close-in population that we consider as candidates for potentially habitable planets had many collisions during the integration time of our simulations. All of them were targets of many impacts. We show in Fig. \[fig:Scenario1\_colissions\] the number of collisions of all the IHZ candidates at 100 Myr and 1 Gyr. In S1, all the resulting planets received more than 50 impacts and a maximum of 160 impacts, and $50\%$ of the total received more than 100 impacts when tidal and GR effects were included, while the planets received 70 and 127 impacts when these effects were neglected. In S2, one of the planets received 29 impacts and the other almost 70 impacts when tidal and GR effects were included. Each impact corresponds to a collision with another embryo of the simulation. In S1 more impacts are allowed because the total number of embryos is much higher ($224$ in total) than in S2 ($74$ in total). In any case, all the candidate planets collided with between 25$\%$ and $50\%$ of the total number of embryos during the first 100 Myr of their evolution.
![Initial location of each embryo that collided with planets that survived inside the IHZ at 100 Myr or 1 Gyr related to the percentage of mass that the candidate planet obtained after each collision in S1 with tide and GR (orange curves), in S2 without tide and GR (cyan curves), and in S2 with tide and GR (red curves).[]{data-label="fig:planets_impactors"}](impactores_IHZ_total.eps){width="8cm"}
Table \[tab:ZH\_info\] shows that all the planets inside the IHZ conserved their initial fraction of water in mass because all the impacts that they suffered came from embryos that were located inside the snow line of the system. Fig. \[fig:planets\_impactors\] shows the location on the semimajor axis of each embryo that collided with one IHZ candidate, related to the percentage of mass that the candidate obtained after each collision in S1 with tide and GR, in S2 without tide and GR, and in S2 with tide and GR. This figure is a zoom of Fig. \[fig:historia\_colisional\] and Fig. \[fig:historia\_colisional2\] for the planets located in the two determined IHZ.
![Evolution of the mass (top panel) and its fraction with respect to the final mass (bottom panel) of the resulting planets that survived inside the IHZ at 100 Myr (solid lines) and 1 Gyr (dotted lines) in S1 with tide and GR (orange curves), in S2 without tide and GR (cyan curves), and in S2 with tide and GR (red curves).[]{data-label="fig:planets_masses"}](embriones_masas.eps){width="45.00000%"}
The impacts that each body suffered during the integration time always produced an increase in mass because the N-body code we used to develop the simulations considers all collisions as completely inelastic, so that every time a collision between embryos occurs, it ends in a perfect merger (see Section \[sec:discussions\]). In Fig \[fig:planets\_masses\] we show the evolution in mass of each IHZ candidate planet and its fraction of mass. Each step in the curves represents the mass or fraction of mass, respectively, that is gained by the IHZ candidate planet after each collision. There is no difference between candidate planets that were finally located in the IHZ at 100 Myr and 1 Gyr in each scenario with respect to the mass accretion history.\
Conclusions and discussions {#sec:discussions}
===========================
We studied the rocky planet formation around a star close to the substellar mass limit using $N$-body simulations that included tidal and GR effects and did not include the effect of gas in the disk. Our aim was to evaluate the relevance of tidal effects following the equilibrium tidal model during the formation and evolution of the system and to improve the accuracy in the calculation of the orbit of the protoplanetary embryos by considering GR effects.
The equilibrium tide model we used is based on the assumption that when a star suffers tidal disturbance from a companion body, it instantly adjusts to hydrostatic equilibrium [@Darwin1879]. A more general approach must include the dynamical tide model for a more reliable description of the very high eccentricity orbits, as shown by @Ivanov2011 and references therein. As shown in Figure \[fig:colisiones\], in simulations that do not include tides, 7% and 20% of the embryos collide with the central star in S1 and S2, respectively, because of their high eccentricity. When tides are included through the equilibrium model, these fractions change to 1% and 16%, respectively. New simulations that include the dynamical tide model are needed in order to explore the possible change in these fractions, which we expect to occur toward lower values because the tidal damping produced by this model is stronger.
Using a different model for close-in bodies, @Makarov2013 found non-pseudo-synchronization in the rotational periods for terrestrial planets and moons. In our case, we adopted the pseudo-synchronization to maintain consistency with the [@Hut1981] model that we adopted for the $N$-body simulations because the hypothesis of pseudo-synchronization is a direct consequence of the constant time-lag model [@Darwin1879; @Hut1981; @Eggleton1998]. A comparative analysis of our results with those from other treatments as well as the self-consistent inclusion of the rotational evolution of embryos is beyond the goals of this work.
Because of the current uncertainties in the determination of disk masses, the simulations were performed for two different scenarios S1 and S2, which basically differ in the initial mass of solid material in the disk, $\sim 3$ M$_\oplus$ and $\sim 30$ M$_\oplus$ , respectively. These values roughly represent the corresponding upper and lower limits of the disk mass for stars close to the substellar mass limit.
The resulting planets have masses between $0.01 < m/\mathrm{M_\oplus} < 0.12$ in scenario S1 and $0.2 < m/\mathrm{M_\oplus} < 1.8$ in scenario S2. Even though we used lower values for the disk mass than those from @Payne2007, we found the same correlation between the resulting planet masses and the initial amount of solid material in the disk. When the disk mass increases, more massive planets could be formed.
When tidal and general relativistic effects are included, a close-in planetary population located at $a < 0.07$ au in scenario S1 survived in all the simulations, while in the more massive scenario S2, embryos suffered stronger gravitational interaction and the formation of this close-in population occurred only in two of the ten simulations. S1 therefore is the most favorable scenario for generating close-in planets. Then, we found that tidal and general relativistic effects are relevant during the formation and evolution of rocky planets around an object at the substellar mass limit, in particular when the protoplanetary embryos involved are low-mass bodies. Our work together with the model developed by @Bolmont2011 [@Bolmont2013; @Bolmont2015] shows that tidal effects in both the formation and later evolution of such systems are required.
The close-in population resulted from a large number of collisions among the protoplanetary embryos, which was treated in our simulations as perfectly inelastic collisions. This gives upper limits on the final mass value and water content for the resulting planets. This shows why it is necessary to reproduce the simulations using an $N$-body code that includes fragmentation during collisions, which can decrease the final masses of the resulting planets considerably, as shown by @Chambers2013 [@Dugaro2019].
The close-in population we found is of particular interest because it is located inside the evolving IHZ of the system. We classified a set of 15 close-in bodies as candidates to potentially habitable planets based on their semimajor axes and eccentricities. A complete analysis of their probability of being habitable planets considering additional constraints [e.g., @Martin2006] is beyond the scope of this work.
Our model presents an important improvement in the scenario of rocky planet formation at the substellar mass limit by including tidal and general relativistic effects. We stress that even tough @Coleman2019 did not incorporate these effects in their simulations of planet formation around Trappist 1, they showed the relevance of considering the gas component of the disk during the first 1 Myr. A more realistic simulation of this scenario of planet formation must therefore clearly include all these effects. A new set of such $N$-body simulations considering low-mass stars and BDs as central objects with different masses is currently ongoing.
We conclude that tidal and GR effects are relevant during rocky planet formation at the substellar mass limit because they allow the survival of close-in bodies that are located inside the IHZ. This supports the hypothesis that these systems are important candidates for future searches of life in the solar neighborhood.
This work was partially financed by Agencia Nacional de Promoción Científica y Tecnológica (ANPCyT) through PICT 201-0505, and by Universidad Nacional de La Plata (UNLP) through PID G144. We acknowledge the financial support by Facultad de Ciencias Astronómicas y Geofísicas de La Plata (FCAGLP) and Instituto de Astrofísica de La Plata (IALP) for extensive use of their computing facilities. We specially appreciate the kind support and advice from Bolivia Cuevas-Otahola (INAOE, México) during the numerical simulations and Adrián Rodríguez Colucci (UFRJ, Brazil) for his valuable comments. We thank the editor Benoît Noyelles and the anonymous referee for very valuable suggestions which helped improve the presentation of our results.
Snow-line location in the substellar regime {#sec:Apendix}
===========================================
The snow-line location corresponds to the radial distance to the central object where water condenses. This occurs when the partial pressure of the protoplanetary disk exceeds the saturation pressure. The exact temperature where this occurs depends on the physical structure of the disk and on the relative abundances of elements, but is expected to be in a range between $140<T/\textrm{K}<170$. To determine the snow-line location in a system with a substellar object of a mass 0.08 M$_\odot$ as a primary object, we took the same temperature profile of a disk that radiates as a blackbody as @Chiang1997,
$$T(r)=T_\star\left(\frac{\alpha}{2}\right)^{\frac{1}{4}}\left(\frac{r}{R_{\star}}\right)^{-\frac{1}{2}}.
\label{eq:temp}$$
The parameter $\alpha$ represents the glazing angle at which the starlight strikes the disk. @Chiang1997 considered vertical hidrostatic equilibrium and derived the expression
$$\alpha = \frac{0.4R_\star}{a} + a \frac{d}{da}\left(\frac{H}{a}\right),
\label{eq:alpha}$$
where the parameter $H$ represents the height of the visible photosphere above the disk mid-plane, and the factor $(H/a)$ can be expressed by
$$\frac{H}{a} = 4T_\star^{\frac{4}{7}}R_{\star}^{\frac{2}{7}}\left(\frac{k}{GM_\star\mu_\textrm{g}}\right)^\frac{4}{7}a^{\frac{2}{7}},
\label{eq:Ha}$$
where $k$ is the Boltzman constant and $\mu_{\textrm{g}}$ is the mass of the molecular hydrogen.
By replacing Eq. in Eq. , we estimated a value of $\alpha$. We considered then that the snow line is located where the disk reaches a temperature of $T(r = r_{\mathrm{ice}}) = T_{\mathrm{ice}} = 140 K$. Thus we could estimated the location of the snow line by
$$r_{\textrm{ice}} = R_\star \left(\frac{T_\star}{T_{\textrm{ice}}}
\right)^2\left(\frac{\alpha}{2}\right)^{\frac{1}{2}}.
\label{eq:snow}$$
We estimated the location of the snow line at $r_{\mathrm{ice}}=0.42$ au, when the BD is 1 Myr old, that is, the initial time of our simulations. We considered this value fixed in all the simulations. Because we know that $T_\star$ and $R_\star$ evolve with time, the location of the snow line would also evolve and continuously approach the star. When we consider that the location of the snow line evolves with time, it might have important consequences for the final amount of water in mass of the resulting planets of the simulations. However, this treatment is beyond the scope of this work.
[^1]: e-mail:msanchez@fcaglp.unlp.edu.ar
[^2]: http://perso.ens-lyon.fr/isabelle.baraffe/BHAC15dir/
|
---
abstract: 'The kaons decays to the pairs of charged and neutral pions are considered in the framework of the non-relativistic quantum mechanics. The general expressions for the decay amplitudes to the two different channels accounting for the strong interaction between pions are obtained. The developed approach allows one to estimate the contribution of terms of any order in strong interaction and correctly takes into account the electromagnetic interaction between the pions in the final state.'
author:
- '**S.R.Gevorkyan [^1] [^2], A.V.Tarasov, O.O.Voskresenskaya[^3]**'
title: '**Final state interaction in kaons decays.**'
---
Joint Institute for Nuclear Research, 141980 Dubna, Russia
PACS:11.30.Rd;13.20.Eb\
Keywords:Decays of K-mesons; two channels decays, relative momenta, pions scattering lengths\
Introduction
============
It has long been known [@gribov58; @gribov61] that the K-mesons decays with a pions in the final state can give unique information on the pions s-wave scattering lengths $a_0,a_2$ , whose values are predicted by Chiral Perturbation Theory (ChPT) with high accuracy [@colangelo01].\
Recently the high quality data on $K^\pm\to \pi^\pm\pi^0\pi^0$ decays have been obtained by NA48/2 collaboration at CERN [@batley06]. The dependence of the decay rate on the invariant mass of neutral pions $M^2=(p_1+p_2)^2$ reveals a prominent anomaly (cusp) at the charged pions production threshold $M_c^2=4m_c^2$.\
As was explained in [@cabibbo04; @cabibbo05] this anomaly is due to the possibility for the kaon to decay to three charged pions, which after charge exchange reaction $\pi^+\pi^-\to \pi^0\pi^0$ gives the observed neutral pions. This possibility is provided by mass difference of charged and neutral pions. The detail consideration of this decay using the technique of non-relativistic field theory [@colangelo06] or ChPT [@gamiz07] supports the proposed picture.\
Nevertheless there are two challenges crucial in scattering lengths extraction from kaons decays. One needs a reliable way to estimate the contribution of higher order terms in strong interaction and calculates the electromagnetic interaction among the charged pions.\
These issues are very close connected with each other. Calculation of the electromagnetic interaction in every order of strong interaction [@bisseger09] doesn’t solve the problem of bound states (pionium atoms), as to take into account electromagnetic interaction leading to unstable bound states one needs expressions for decay amplitudes including the strong interaction between pions in all orders [@gevorkyan07]. The problem of correct accounting of the electromagnetic effects are also necessary in a wide class of decays with two pions in the final state as for instance $K_{e4}$ decay [@gevorkyan1; @gevorkyan2].\
The phenomenon of cusp in elastic scattering at the threshold relevant to inelastic channel is known for many years and was widely discussed in the framework of non-relativistic quantum mechanics [@wigner48; @breit57; @baz57].For the elastic process $\pi^0\pi^0\to\pi^0\pi^0$ this anomaly at the $\pi^+\pi^-$ threshold was firstly discussed in the framework of ChPT in [@meisner97]. In the present work we consider the kaon decay to pion pairs with pions of different masses. Using the well known results of quantum mechanics we obtain the matrix elements for decay $K\to \pi\pi$ where the final pions consist from pions of different masses ($\pi^+\pi^-,\pi^0\pi^0$).
Two channel decay
=================
We are interested in two channel decay of kaon to the pion pairs in the final state, where the pions in the pair can be neutral or charged . The well examples are $K_L\to \pi\pi$ as well as $K^\pm\to\pi^+\pi^-e^\pm\mu$ ($ K_{e4}$ decay). In what follows all quantities relevant to the neutral pions pair ($\pi^0\pi^0$) are labeled by index “n”, whereas the charged pions pairs ($\pi^+\pi^-$) are labeled by index “c”.\
We do not consider here the electromagnetic interaction in the pair, the effect discussed in our previous work [@gevorkyan2]. Our main goal is to obtain the matrix elements of the kaon decay to the pion pair accounting for different masses of neutral and charged pions and the possibility of charged exchange reaction $\pi^+\pi^-\to \pi^0\pi^0$ and the elastic scattering of pions in the final state.\
The general form of matrix element for kaon decay to the final state with two charged or neutral pions can be written in the operator form: M\_c=\_c\^+(r)M\_0(r)d\^3r; M\_n=\^+\_n(r)M\_0(r)d\^3r The two component operator $M_0=\left (\begin{array}{c}M_c^{(0)}(r)\\M_n^{(0)}(r)\end{array}\right)$, where $M_c^{(0)}(r),M_n^{(0)}(r)$ are the matrix elements of kaon decay to noninteracting charged and neutral pions pairs, while $\Psi_c(r), \Psi_n(r)$ are the appropriate two component wave functions.\
These wave functions would satisfy to couple Shrődinger equations [^4] -\_c(r)+U\_[cc]{}\_c(r)+U\_[cn]{}\_n(r)&=&k\_c\^2\_c(r)-\_n(r)+U\_[nn]{}\_n(r)+U\_[nc]{}\_c(r)&=&k\_n\^2\_n(r) where $U_{ij}$ are the strong potentials describing elastic $cc\to cc; nn\to nn$ scattering and charge exchange reaction $cn\to cn$ . $k_c,k_n$ are the charge and neutral pions momenta in the appropriate center of mass system.\
According to the general principles of scattering theory the asymptotic behavior of the wave functions $\Psi_c(r) ,\Psi_n(r)$ can be written through the s-wave amplitudes $f_{cc}, f_{nn}, f_{cn}, f_{nc}$ in the following form: \_c(r)&=&(
[c]{}\
0
)+ (
[c]{}f\^\*\_[cc]{}\
f\^\*\_[nc]{}
)\_n(r)&=&(
[c]{}0\
)+ (
[c]{}f\^\*\_[cn]{}\
f\^\*\_[nn]{}
) The first columns in these expressions describe the noninteracting s-waves pions pairs, whereas the second columns correspond to the interacting charge and neutral pions pair in the far asymptotic of corresponding wave function.\
One can rewritten these equations through the elements of appropriate S-matrix [@landau63]: S\_[cc]{}=1+2ik\_cf\_[cc]{}; S\_[nn]{}=1+2ik\_nf\_[nn]{}; S\_x=2if\_x Substituting these relations in the expressions (3) one immediately obtains: \^\*\_c(r)=(
[c]{}i\
-i
) \^\*\_n(r)=(
[c]{} -i\
i
) From the other hand the wave functions $\Psi_c(r)$ and $\Psi_n(r)$ can be constructed as the linear combination of two real solutions of equations (2) \^[(1)]{}=(
[c]{}\_c\^[(1)]{}\
\_n\^[(1)]{}
)\^[(2)]{}=(
[c]{}\_c\^[(2)]{}\
\_n\^[(2)]{}
) with the standard boundary conditions[^5] $ \Psi^{(1)}(0)=\Psi^{(2)}(0)=0.$ Keeping this in mind we will look for the desired wave functions in the form: \^\*\_c(r)=A\_c\^[(1)]{}\^[(1)]{} +A\_c\^[(2)]{}\^[(2)]{}; \^\*\_n(r)=A\_n\^[(1)]{}\^[(1)]{} +A\_n\^[(2)]{}\^[(2)]{} where $A_c^{(1)},A_c^{(2)},A_n^{(1)},A_n^{(2)}$ are arbitrary complex numbers.\
Substituting the expressions (6),(7) in (1) one gets: M\_c&=&A\_c\^[(1)]{}(\_c\^[(1)]{}M\_c\^[(0)]{} +\_n\^[(1)]{}M\_n\^[(0)]{})d\^3r&+&A\_c\^[(2)]{}(\_c\^[(2)]{}M\_c\^[(0)]{} +\_n\^[(2)]{}M\_n\^[(0)]{})d\^3r =A\_c\^[(1)]{}I\_1+A\_c\^[(2)]{}I\_2M\_n&=&A\_n\^[(1)]{}(\_c\^[(1)]{}M\_c\^[(0)]{} +\_n\^[(1)]{}M\_n\^[(0)]{})d\^3r&+&A\_n\^[(2)]{}(\_c\^[(2)]{}M\_c\^[(0)]{} +\_n\^[(2)]{}M\_n\^[(0)]{})d\^3r =A\_n\^[(1)]{}I\_1+A\_n\^[(2)]{}I\_2 Making use that any real solution of equations (2) out of the potential range $(U_{ij}=0)$ can be taken in the form: (r)==(e\^[ikr+i(k)]{}-e\^[-ikr-i(k)]{}) we will look for the real solutions out of potential range as [^6]: \_c\^[(1)]{}(r)&=&; \_c\^[(2)]{}(r)= \_n\^[(1)]{}(r)&=&; \_n\^[(2)]{}(r)= In order to obtain the relations between the unknown coefficients[^7] in the above expressions let us at first compare the asymptotic behavior of the initial wave functions $\Psi_c(r)$ in (5) with the first raw in the parametrization (10): (g\_1e\^[ik\_cr]{}-g\_1\^\*e\^[-ik\_cr]{})&+&(g\_2e\^[ik\_cr]{}-g\_2\^\*e\^[-ik\_cr]{})= (e\^[-ik\_cr]{}-S\_[cc]{}e\^[ik\_cr]{})(h\_1e\^[ik\_nr]{}-h\_1\^\*e\^[-ik\_nr]{})&+&(h\_2e\^[ik\_nr]{}-h\_2\^\*e\^[-ik\_nr]{})= -S\_[cn]{}e\^[ik\_nr]{}Gathering the structures in front of the appropriate exponents and solving the system of obtained equations after a bit algebra we get: A\_c\^[(1)]{}&=&; A\_c\^[(2)]{}=-;H=g\_1\^\*h\_2\^\*-h\_1\^\*g\_2\^\*;S\_[cc]{}&=&; S\_[cn]{}= Carry out the same procedure for $\Psi(r)$ we obtain the relevant relations for the case of kaon decay to pair of neutral pions: A\_n\^[(1)]{}&=&-; A\_n\^[(2)]{}=-;S\_[nn]{}&=&-; S\_[nc]{}= In respect that due to T-invariance $S_{cn}=S_{nc}$ it can be checked that obtained relations satisfied the unitarity constraints: |S\_[nn]{}|\^2+|S\_[nc]{}|\^2=1; |S\_[cc]{}|\^2+|S\_[cn]{}|\^2=1 As has been seen from expression (9) the imaginary parts of functions $g_{1(2)}, h_{1(2)}$ are determined by appropriate phases.[^8] For instance, from first equation in (10): $$g_1=ge^{i\delta(k_c)}=g\cos{\delta(k_c)}+ig\sin{\delta(k_c)}=
g\cos{\delta(k_c)}+ik_cg\frac{\sin{\delta(k_c)}}{k_c}$$ At considered low energy one can safely confined by linear term in phases dependence on momenta : g\_1&=&d\_c\^[(1)]{}+ik\_ca\_c\^[(1)]{};g\_2=d\_c\^[(2)]{}+ik\_ca\_c\^[(2)]{}h\_1&=&d\_n\^[(1)]{}+ik\_na\_n\^[(1)]{};h\_2=d\_n\^[(2)]{}+ik\_na\_n\^[(2)]{} Substituting these relations in expressions (12), (13) after cumbersome, but simple algebra we obtain the energy dependence of S-matrix elements in the two channel case[^9]: S\_[cc]{}&=&S\_[nn]{}&=&S\_[cn]{}&=&S\_[nc]{}= where : a\_[nn]{}&=&; a\_[cc]{}=;a\_x&=&= Now we are in the position to get the dependence of matrix elements (1) on pairs momenta $k_c,k_n$. Introducing the real combinations[^10]: M\_[0c]{}= M\_[0n]{}= and making use the expressions (8),(12), (13),(15) we obtain our final result : M\_c&=&M\_[0c]{}+ik\_nM\_[0n]{} M\_n=M\_[0n]{}+ik\_cM\_[0c]{}D&=&(1-ik\_ca\_[cc]{})(1-ik\_na\_[nn]{})+k\_nk\_ca\_x\^2 For applications it is more convenient to rewritten these relations through the amplitudes of elastic pion-pion scattering $f_{cc},f_{nn}$ and charge exchange $f_x$: M\_c&=&M\_[0c]{}(1+ik\_cf\_[cc]{})+ik\_nM\_[0n]{}f\_x; M\_n=M\_[0n]{}(1+ik\_nf\_[nn]{})+ik\_cM\_[0c]{}f\_xf\_[cc]{}&=&; f\_[nn]{}=;f\_x=;These relations expressing the decay matrix elements (1) through the amplitudes of pion-pion scattering are the main result of present work. Their application to $K\to 3\pi$ and $K^\pm\to \pi^+\pi^-e^{\pm}\nu$ decays allow us [@gevorkyan07; @gevorkyan2] to take into account the electromagnetic interaction among the charged pions in the final state for any invariant mass of the pion pair.\
The first terms in the expansion (20) coincide with appropriate expressions in [@cabibbo04; @cabibbo05], i.e. the $M_{0c}, M_{0n}$ introduced above (see eq. (18)) can be interpreted as so called “unperturbed” amplitudes introduced in [@cabibbo04].\
The two channel task considered in the present work permits to estimate the accuracy of the scattering lengths values extracting from experimental data on kaons decays. Moreover obtained expressions allows one to correctly take into account the electromagnetic effects in the final state not only above the charged pions production threshold, but also for bound states. [@gevorkyan07; @gevorkyan2] We are grateful to V.D. Kekelidze who initiated and support this work and D.T. Madigozhin for many stimulating and useful discussions.
[99]{} V.N. Gribov, Nucl. Phys. 5 (1958) 653. V.N. Gribov, ZhETF 41 (1961) 1221. G.Colangelo, J.Gasser, H.Leutwyler, Nucl. Phys. B 603 (2001) 125. J.R.Batley et al., Phys. Lett. B 633 (2006) 173. N. Cabibbo, Phys. Rev. Lett. 93 (2004) 121801. N. Cabibbo, G. Isidori, JHEP 0503 (2005) 021. G.Colangelo, J.Gasser, B.Kubis, A.Rusetsky, Phys. Lett. B 638 (2006) 187. E.Gamiz, J. Prades, I. Shiemi, Eur.Phys. J. C50 (2007) 405 M. Bisseger, A. Fuhrer , J. Gasser, B. Kubis, A. Rusetsky, Nucl. Phys. B 806 (2009) 178 S.Gevorkyan, A.Tarasov, O.Voskresenskaya, Phys.Lett. B649 (2007) 159 S.Gevorkyan, A. Sissakian, A.Tarasov, H.Torosyan, O.Voskresenskaya, arXiv: 0704.2675 \[hep-ph\]; To be published in Yad. Phys. 73 (2010) S.Gevorkyan, A. Sissakian, A.Tarasov, H.Torosyan, O.Voskresenskaya, arXiv: 0711.4618 \[hep-ph\]; To be published in Yad.Phys. 73 (2010) E.Wigner, Phys. Rev. 73 (1948) 1002. G. Breit, Phys. Rev. 107 (1957) 1612. A.Baz, ZhETF 33 (1957) 923. Ulf-G.Mei$\beta$ner, G.M$\ddot u$ller, S.Steininger, Phys. Lett. B 406 (1997) 154. L.Landau and I.Lifshitz, Quantum mechanics, FM,Moscow (1963) A.I. Baz, Ya.B. Zeldovich, A.M. Perelomov, “Scattering, reactions and decays in nonrelativistic quantum mechanics”, Nauka, Moscow, 1971.
[^1]: Corresponding author: gevs@jinr.ru (S.Gevorkyan)
[^2]: On leave of absence from Yerevan Physics Institute
[^3]: On leave of absence from Siberian Physical Technical Institute
[^4]: Throughout this paper we restricted by s-wave $\pi\pi$ scattering in the final state.
[^5]: In terms of the wave function $\Phi(r)=\frac{\Psi(r)}{r}$ this condition requires the regularity at r=0.
[^6]: We consider only the class of strong potentials with the sharp boundary.
[^7]: These factors are functions of pions momenta $k_c,k_n$
[^8]: The phases are odd functions of relevant momenta $\delta(-k)=-\delta(k)$ .
[^9]: The similar expressions are cited in the textbook [@baz71], but with wrong numerator in the inelastic case.
[^10]: The integrals $I_{1,2}$ are real quantities.
|
---
author:
- |
Mayank Bansal$^{1,2}$\
$^1$Center for Vision Technologies, SRI International\
Princeton, NJ, USA\
[mayank.bansal@sri.com]{}
- |
Kostas Daniilidis$^2$[^1]\
$^2$GRASP Lab, University of Pennsylvania\
Philadelphia, PA, USA\
[kostas@cis.upenn.edu]{}
bibliography:
- 'egbib.bib'
title: 'Geometric Polynomial Constraints in Higher-Order Graph Matching'
---
[^1]: The authors are grateful for support through the following grants: NSF-IIP-0742304, NSF-OIA-1028009, ARL MAST-CTA W911NF-08-2-0004, ARL RCTA W911NF-10-2-0016, NSF-DGE-0966142, and NSF-IIS-1317788.
|
---
abstract: 'Previously unrecognized weak emission lines originating from high excitation states of Si [ii]{} (12.84 eV) and Al [ii]{} (13.08 eV) are detected in the red region spectra of slowly rotating early B-type stars. We surveyed high resolution spectra of 35 B-type stars covering spectral sub-types between B1 and B7 near the main sequence and found the emission line of Si [ii]{} at 6239.6 Å in all 13 stars having spectral sub-types B2 and B2.5. There are 17 stars belonging to sub-type B3 and seven stars among them are found to show the emission line of Si [ii]{}. The emission line of Al [ii]{} at 6243.4 Å is detected in a narrower temperature range ([*$T_{\rm eff}$*]{} between 19000K and 23000 K) in nine stars. Both of these emission lines are not detected in cooler ([*$T_{\rm eff}$*]{} $ < $ 16000 K) stars in our sample. The emission line of Si [ii]{} at 6239.6 Å shows a single-peaked and symmetric profile and the line center has no shift in wavelength with respect to those of low excitation absorption lines of Si II. Measured half width of the emission line is the same as those of rotationally broadened low excitation absorption lines of Si [ii]{}. These observations imply that the emitting gas is not circumstellar origin, but is located at the outermost layer of the atmosphere, covering the whole stellar surface and co-rotates with the star.'
author:
- 'Kozo <span style="font-variant:small-caps;">Sadakane</span>, and Masayoshi <span style="font-variant:small-caps;">Nishimura</span>'
title: 'Weak Metallic Emission Lines in Early B-Type Stars'
---
Introduction
============
The phenomenon of sharp and weak emission lines (WELs) in optical spectra of middle to late B-type stars has been reported mainly in chemically peculiar (CP) stars. Detections of these weak features have been made possible by achieving both high spectral resolution and high signal-to-noise (SN) ratio observations. @sigut2000 reported detections of emission lines of Mn [ii]{}, P [ii]{}, and Hg [ii]{} in the red spectral region of the helium-weak star 3 Cen A (B5 III-IVp). They also found very weak emission lines of Mn [ii]{} in a mild HgMn star 46 Aql (B9 III). Sigut (2001a, b) carried out detailed analyses of emission lines of Mn [ii]{} observed in 3 Cen A and 46 Aql and concluded that observations can be naturally explained by interlocked non-local thermodynamic equilibrium (NLTE) effect combined with the vertical stratification of the manganese abundance, with manganese concentrated high in the photosphere.
@wahlgren2000 reported detections of weak emission lines originating from high excitation states of Ti [ii]{}, Cr [ii]{} and Mn [ii]{} in sharp lined late B-type stars including HgMn stars and suggested that the these emission lines arise from a selective excitation process involving H Ly$\alpha$ photon energies. @sadakane2001 noted that all of the Ti [ii]{} lines in 46 Aql with high excitation potential ($\chi$ $ > $ 8.05 eV) and large transition probabilities (log $\it gf$ $ > $ 0.1) are observed in emission near 6000 Å. @wahlgren2004 published an extensive list of emission lines of 3 Cen A. Their list includes emission line spectra of Si [ii]{}, P [ii]{}, Ca [ii]{}, Mn [ii]{}, Fe [ii]{}, Ni [ii]{}, Cu [ii]{}, and Hg [ii]{}. Abundances of 11 elements (from C to Hg) have been determined using a synthetic spectrum fitting technique.
Numerous weak emission lines of Cr [ii]{} and Ti [ii]{} have been reported in a cool HgMn star HD 175640 (B9 V) by @castelli2004. They noted that emission lines are selectively found for high excitation lines having large transition probabilities (log $\it gf$ $ > $ –1.0) and that these emission lines are found in the red part ($\lambda$ $ > $ 5850 Å) of the spectrum. @castelli2007 observed emission lines of Cr [ii]{}, Mn [ii]{} and Fe [ii]{} and found isotopic anomalies for Ca and Hg in the Bp star HR 6000. They noted that there are disagreements between observed and calculated line strengths for some Fe [ii]{} lines and suggest these discrepancies are due either to incorrect log $\it gf$ values or to the emission component filling the absorptions (filling-in effects).
@hubrig2007 observed high resolution spectra of the sharp-lined magnetic helium-variable star a Cen (HD 125823) over the rotation period of 8.82 d and found variable high excitation S [ii]{}, Mn [ii]{} and Fe [ii]{} emission lines. They found a correlation between the probable location of surface spots of Mn and Fe and the strength of the emission lines. It is interesting to notice that they note that an emission line detected in this star at 6239.80 Å as an unidentified line (their table 1). Recently, @alexeeva2016 constructed a comprehensive model atom for C [i]{} and C [ii]{} and computed the NLTE line formation for C [i]{} and C [ii]{}. They analysed the lines of C [i]{} and C [ii]{} in seven B to early A-type stars, including $\iota$ Her, Sirius, and Vega, and found that the C [i]{} emission lines were detected in the four hottest stars, and these lines were well reproduced by their NLTE calculations.
Summarizing these published results, we notice that stars showing WELs are mainly middle B to early A-type stars including CP stars. Almost all WELs are arising from high excitation states of singly ionized ions. At the same time, WELs are found in the red and near-IR spectral regions preferentially and no WELs have been reported in the blue or in the near UV regions so far.
@nieva2011 analysed high resolution spectra of 13 early B-type main-sequence stars of spectral classes B0 V to B2 V and low projected rotational velocities ($\it v$ sin $\it i$ $ < $ 60 km s${}^{-1}$) in the Ori OB1 association. They published a graphical presentation of the spectrum of HD 35299 (B1.5 V) in the appendix. @nieva2012 carried out a comprehensive spectral analyses of 27 B-type stars and published graphical presentations of spectra of four B-type stars including a B2 IV star $\gamma$ Peg. Examining graphical data of HD 35299 [@nieva2011] and $\gamma$ Peg [@nieva2012], we noticed an emission like spike near 6239.7 Å in both stars. Curiously, however, no identification has been given to this feature on graphs of both stars.
Interested in the nature of this unidentified feature, we examined high resolution spectral data of $\gamma$ Peg obtained with the HIgh-Dispersion Echelle Spectrograph (HIDES, $\it R$ $\sim$ 70000) at the coudé focus of the 188-cm reflector of the Okayama Astrophysical Observatory (OAO) and data downloaded from the Elodie archive ($\it R$ $\sim$ 42000, [@moultaka2004]). We find that the emission like feature is definitely present on both data and conclude that the feature is not an observational artifact. We surveyed for candidate lines by simulating the spectrum and found two highly excited (12.84 eV) lines of Si [ii]{} at 6239.61 and 6239.66 Å are expected to appear as weak absorption lines in early B-type stars. Measured wavelength of the emission line just coincides with those of the two Si [ii]{} lines. Furthermore, we found two additional weak emission features at 6231.8 Å and at 6243.4 Å in $\gamma$ Peg. Observed wavelengths of these two features coincide with those of two highly excited (13.08 eV) lines of Al [ii]{} at 6231.75 Å and at 6243.37 Å.
Because we can find no alternative identification for these features, we conclude that WELs of Al [ii]{} and Si [ii]{} are present in the optical region spectrum of $\gamma$ Peg. This finding is the first case of WELs found not only in $\gamma$ Peg but also in hot ([*$T_{\rm eff}$*]{} $ > $ 20000 K) B-type stars.
We collected as many high resolution spectral data of early B-type stars as possible from various data archives, then tried a systematic survey for the Al [ii]{} and Si [ii]{} emission lines near 6240 Å in these stars to find that the Si [ii]{} emission line appears frequently in B2 - B3 sharp-lined main sequence stars while the Al II emission line appears in a narrower spectral range of B2-type stars. Details of our survey for WELs in early B-type stars and results are described in the following sections.
Observational data
==================
We collected optical spectral data of 35 B-type stars of low rotational velocities corresponding to spectral types from B1 to B7 and luminosity classes from III to V from various data archives. Spectral data of 26 targets were observed using the HIDES spectrograph at OAO. Data of 11 targets were obtained with the Elodie spectrograph of the 193-cm telescope at the Observatoire de Haute Provence. Data were obtained by the UVES spectrograph of the ESO VLT (three targets) and by the High Dispersion Spectrograph (HDS) of the Subaru telescope (three targets). Raw data obtained by the HIDES and HDS spectrographs were downloaded from the SMOKA database, which is operated by the Astronomy Data Center, National Astronomical Observatory of Japan. Processed data of the Elodie and UVES spectrographs were obtained from the Elodie archive and the UVES-POP archive [@bagnulo2003], respectively.
The reduction of two-dimensional echelle spectral data obtained by the HIDES and HDS spectrographs (bias subtraction, flat-fielding, scattered-light subtraction, extraction of spectral data, and wavelength calibration) was performed in a standard manner using the IRAF software package. The wavelength calibration was done using the Th-Ar comparison spectra obtained during the observations. The observed wavelengths of all stars observed by all four spectrographs have been converted into the laboratory scale using measured wavelengths of He [i]{} and Si [ii]{} absorption lines. Finally, the continuum levels of all spectral data have been normalized to unity by a polynomial fitting technique.
Relevant data of our target objects are summarized in table 1. Spectral types are taken from the Bright Star Catalog 5-th edition [@hoffleit1991], except for three stars (HD 89587, HD 133518 and HD 181858). Spectral types of two southern stars (HD 89587 and HD 133518) are taken from @houk1978 and data of HD 181858 is taken from @houk1999. Data of rotational velocities ($\it v$ sin $\it i$) are taken from @abt2002. Data of $\it v$ sin $\it i$ for five stars (HD 3360, HD 29248, HD 35039, HD35468, and HD36591) have been replaced with new data taken from @simon2014 and that of HD 133518 is taken from @alecian2014. We could find no published data of $\it v$ sin $\it i$ for HD 89587.
Effective temperatures ([*$T_{\rm eff}$*]{}) and surface gravities (log $\it g$) are taken from various sources. We adopt data given in @takeda2010 and @nieva2013. When no data can be found in these two papers, we use [*$T_{\rm eff}$*]{} and log $\it g$ values given in @lefever2010 (HD 35468 and HD 214993), @aerts2014 (HD 163472), @morel2008 (HD 170580), and @alecian2014 (HD 133518). We can find neither published data of [*$T_{\rm eff}$*]{} and log $\it g$ nor $\it uvby$ and $\beta$ photometric data for the southern star HD 89587. We estimated its effective temperature by comparing the line intensity ratio of the S [ii]{} line at 5664.8 Å to the N [ii]{} line at 5666.6 Å with those measured in 15 B2 and B3 type stars in table 1. The ratio is found to be sensitive to a change in temperature for stars belonging to spectral types B2 and B3. Measured equivalent widths (in mÅ) and central relative intensities (C. I.) of the Si [ii]{} line at 6239.6 Å and the Al [ii]{} line at 6243.4 Å are given in columns from 12 to 15. Measurements of equivalent widths are carried out using the direct integration method.
Figures 1 and 2 show spectral data near 6240 Å for five representative examples of hot B-type stars and five cooler stars, respectively. We plot observed and simulated spectra for each star. Simulated spectra were computed using the tabulated atmospheric parameters and the rotational velocity for each star, interpolating the ATLAS9 model atmospheres [@kurucz1993]. The solar abundances, the LTE line formation and the microturbulent velocity, $\xi$${}_{\rm t}$ = 4 km s${}^{-1}$, have been assumed in these simulations. We use log $\it gf$ values of the Si [ii]{} lines and the Al [ii]{} lines taken from the NIST Atomic Database [@kramida2015], while log $\it gf$ values of other lines are taken from @kurucz1995.
We surveyed for other emission features of high excitation Al [ii]{} and Si [ii]{} ions on a high SN ratio ($\sim$ 900) spectrum of $\gamma$ Peg and found two and six additional emission features of these two ions, respectively, as listed in table 2. Figure 3 displays two small sections of the spectra of two sharp lined stars $\gamma$ Peg and $\iota$ Her near the Si [ii]{} line at 5688.81 Å (upper panel) and the Al [ii]{} line at 5593.30 Å (lower panel). We can see that the Si [ii]{} line is seen in emission in both stars. The Al [ii]{} line appears as an emission feature in $\gamma$ Peg, while the line is seen as a weak absorption feature in the cooler star $\iota$ Her.
In figure 4, we compare observed profiles of the Si [ii]{} emission line at 6239.6 Å and those of a low excitation (8.12 eV) absorption line of Si [ii]{} at 6371.37 Å of three B2 type stars ($\gamma$ Peg, $\zeta$ Cas, and $\gamma$ Ori) on the velocity scale.
Discussion
==========
We have presented observations of weak emission lines (WELs) of high excitation Si [ii]{} and Al [ii]{} ions in the red region spectra of early B-type stars. WELs in hot B-type stars and those of Al [ii]{} ions have not been reported in previous publications. We find that the emission line of Si [ii]{} at 6239.6 Å is observed in all 13 stars in our sample contained in the [*$T_{\rm eff}$*]{} range between 23000 K and 17500 K. The line is observed less frequently in middle B-type stars (between 17500 K and 16500 K) and the line is not observed in emission among cooler ([*$T_{\rm eff}$*]{} $ < $ 16000 K) stars. The observed high frequency of detection of the Si [ii]{} line at 6239.6 Å in emission among early B-type stars strongly suggests that the occurrence of this emission line is not a rare case but a common phenomenon among these stars. The line of Al [ii]{} at 6243.4 Å is observed in emission only in a narrow temperature range (between 23000 K and 19000 K) less frequently than the Si [ii]{} line. The line is observed not in emission but as a weak absorption feature in stars with [*$T_{\rm eff}$*]{} $ < $ 17500 K. Data of equivalent widths of emission lines (table 1, columns 12 and 14) show that maxima of both emission lines are observed between the range in [*$T_{\rm eff}$*]{} between 22000 K and 19500 K.
We have data of multi-epoch observations for two objects $\gamma$ Peg and $\iota$ Her, both of which show a clear emission line of Si [ii]{} at 6239.6 Å. Data of six epochs (from 1998 November to 2006 October) and seven epochs (from 1994 April to 2013 June) are available for $\gamma$ Peg and $\iota$ Her, respectively. These data are obtained by the Elodie, HIDES, and HDS spectrographs. We examined averaged spectral data of each epoch of both stars and found no significant time variation in the line peak intensity, central wavelength, and width of the Si [ii]{} emission line at 6239.6 Å. These data strongly suggest that the appearances of the emission line in these two stars are not temporal phenomena but they are stationary features lasting for at least a few decades.
Figure 4 clearly demonstrates that very faint WELs can be detected in a relatively fast rotating star such as $\gamma$ Ori when using high SN and high resolution data. We can see from this figure that the emission lines show single-peaked and symmetric profiles in all three stars and that their line centers have no shift in wavelength with respect to those of low excitation absorption lines of Si [ii]{}. Measured half widths of the emission lines are the same as those of low excitation absorption lines of Si [ii]{} in all three stars. These observations imply that the faint WELs are formed nearly at the same location as the absorption lines and their origins are not circumstellar. Furthermore, the observed widths of WELs, which are the same as those of the rotationally broadened absorption lines, imply that the emitting gas is covering the whole stellar surface and is co-rotating with the star.
The mechanism for populating the highly excited states is still now under investigation. In a review article, @wahlgren2008 noted that NLTE can be a significant mechanism for the production of WELs. @sigut1996 carried out a theoretical work on near IR region WELs of Mg [ii]{} in A and B-type stars and published predictions of line profiles of four Mg [ii]{} lines between 1.01 $\mu$m and 4.76 $\mu$m. No observational confirmation of these Mg [ii]{} emission lines has been published yet. Sigut (2001a, b) carried out analyses of emission lines of Mn [ii]{} observed in 3 Cen A and 46 Aql and concluded that these emission lines can be explained by interlocked NLTE effect combined with the vertical stratification. Wahlgren and Hubrig (2000, 2004) proposed an alternative mechanism that the population of highly excited states might be due to excitation from the far-UV continuum radiation.
Our present sample covers a different temperature domain from that containing the previously known WELs stars on the HR diagram, and contains only one CP star (HD 133518: B2 IVp He-strong) and none of the remaining sample stars are known to show spectroscopic anomalies. Thus, it seems difficult to postulate a stratification of some specific elements, such as Si and Al, in their atmospheres. Thus, our results may have a significant bearing upon our understanding of the outermost atmospheric regions of B-type stars and the formation of the abundance anomalies observed in CP stars, which is generally attributed to the diffusion processes. Finally, we notice in table 1 that 12 stars belong to the $\beta$ Cep type, the SPB type, or the hybrid type pulsational variable stars. The observed frequency of these variable stars among B-type stars is steadily increasing in recent years. Furthermore, weak magnetic fields have recently been detected in some of these variable stars [@wade2016].
We plan to carry out spectroscopic observations of early B-type stars in order to increase the number of sample stars. One of our interest is observing hot stars with spectral sub-types between B1 and B2, where we have only a few objects in the present sample. It might be possible to determine the hot end of the region in the [*$T_{\rm eff}$*]{} range between 23000 K and 27000 K, in which the Si [ii]{} line at 6239.6 Å appears as an emission line. We also plan to observe wavelength regions not included in the present study in order to survey for new emission features. These observations will provide useful boundary conditions or constraints to be incorporated in theoretical interpretations.
This research has made use of the SIMBAD database, operated by CDS, Strasbourg, France. We thank Dr. E. Kambe for comments and Mr. Y. Notsu for his help in preparation of figure materials.
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-------- ---------------- ----------- ------------------- --------------------- ------------ ----------------- ----- --------------------- --------------- ------ ---------------- ------- ---------------- -------
HD Name MK type Variable type $\it v$ sin $\it i$ Instrument Obs. Date SN [*$T_{\rm eff}$*]{} log $\it g$ Ref. Si [ii]{} 6239 Al [ii]{} 6243
km s${}^{-1}$ J. D. 2450000 + K cm s${}^{-2}$ E.W. C.I. E.W. C.I.
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)
205021 $\beta$ Cep B1 IV $\beta$ Cep$^{a}$ 20 1,3 3257 800 27000 4.05 1 \* 1.000 \* 0.999
36591 HR 1861 B1 IV 9 3 3322 350 27000 4.12 1 \* 1.005 \* 0.997
216916 16 Lac B2 IV $\beta$ Cep$^{a}$ 10 2,3 2194 520 23000 3.95 1 –15 1.040 –3 1.008
214993 12 Lac B2 III Hybrid$^{b}$ 30 3 3015 440 23000 3.6 3 –11 1.012 \* 0.999
163472 V2052 Oph B2 IV-V $\beta$ Cep$^{a}$ 75 3 2902 360 22490 3.95 4 –14 1.007 \* 0.998
35468 $\gamma$ Ori B2 III 52 3,4 2538 410 22000 3.6 3 –22 1.011 –4 1.002
29248 $\nu$ Eri B2 III Hybrid$^{b}$ 32 1 4030 380 22000 3.85 1 –12 1.006 \* 1.001
886 $\gamma$ Peg B2 IV Hybrid$^{b}$ 0 1,3 4026 940 22000 3.95 1 –19 1.057 –6 1.020
16582 $\delta$ Cet B2 IV $\beta$ Cep$^{a}$ 5 3 1712 430 21250 3.8 1 –20 1.042 –6 1.013
3360 $\zeta$ Cas B2 IV SPB$^{c}$ 23 1,3 3301 650 20750 3.8 1 –21 1.027 –8 1.008
35708 o Tau B2.5 IV 10 1 4030 460 20700 4.15 1 –15 1.020 –8 1.008
35039 o Ori B2 IV-V 11 1 4030 520 19600 3.56 1 –20 1.048 –9 1.020
42690 HR 2205 B2 V 0 1 4030 410 19299 3.81 2 –17 1.030 –7 1.011
170580 HR 6941 B2V Hybrid$^{b}$ 0 2 3184 970 19175 4.02 5 –13 1.039 \* 1.001
133518 HIP 73966 B2 IVpHe 0 4 1983 270 19000 4.0 6 –10 1.053 –5 1.024
32249 $\psi$ Eri B3 V 30 1 4027 510 18890 4.13 2 –4: 1.005 \* 1.000
34447 HR 1731 B3 IV 10 1 4027 270 18480 4.10 2 –8 1.021 \* 1.000
160762 $\iota$ Her B3 IV SPB$^{d}$ 0 1,2,3 4026 610 17500 3.8 1 –9 1.030 +3 0.990
196035 HR 7862 B3 IV 20 1 4028 360 17499 4.36 2 +3: 0.998 +3: 0.996
43157 HR 2224 B5 V 30 1 4027 410 17486 4.12 2 +9: 0.983 +5: 0.993
223229 HR 9011 B3 IV 30 1 4026 370 17327 4.20 2 –8: 1.005 \* 0.999
89587 HIP 50519 B3 III no data 4 2217 540 17000 no data 7 –6 1.016 +3 0.988
176502 V543 Lyr B3 V 0 1 4027 390 16821 3.89 2 +6 0.986 +9 0.972
41753 $\nu$ Ori B3 V 30 1 4029 410 16761 3.9 2 –4 1.005 \* 0.994
25558 40 Tau B3 V SPB$^{d}$ 30 1 4028 390 16707 4.29 2 +9 0.982 +5: 0.992
44700 HR 2292 B3 V 0 1 4029 340 16551 4.29 2 +6 0.982 +6 0.972
186660 HR 7516 B3 III 0 1 4027 310 16494 3.57 2 –6 1.025 +6 0.976
181858 HR 7347 B3 II-III 0 1 4027 270 16384 4.19 2 +9: 0.979 +8: 0.984
184171 8 Cyg B3 IV 15 1 4027 430 15858 3.54 2 +6 0.988 +6 0.989
198820 HR 7996 B3 III 15 1 4027 280 15852 3.86 2 +6 0.989 +7: 0.989
209008 18 Peg B3 III SPB$^{e}$ 5 1 4028 360 15800 3.75 1 +9 0.980 +12 0.982
28375 HR 1415 B3 V 0 1 4027 380 15278 4.30 2 +30 0.950 +20 0.971
11415 $\epsilon$ Cas B3 III 30 1,3 4027 650 15174 3.54 2 +10 0.984 +14 0.986
147394 $\tau$ Her B5 IV 30 1 4026 610 14898 4.01 2 +27 0.970 +18 0.980
17081 $\pi$ Cet B7 V 25 1 4029 710 13063 3.72 2 +37 0.944 +20 0.966
-------- ---------------- ----------- ------------------- --------------------- ------------ ----------------- ----- --------------------- --------------- ------ ---------------- ------- ---------------- -------
\(1) HD number. (2) Star name. (3) Spectral type. (4) References: a: @stankov2005, b: @moravveji2016, c: @briquet2016, d: @szewczuk2015, and e: @Irrgang2016. (5) Rotational velocity ($\it v$ sin $\it i$). (6) Used spectrographs. 1: HIDES, 2: HDS, 3: Elodie, 4: UVES. (7) Date of observation in J. D. When multiple observations are available, date of observations used in measurements (12) – (15) are given. (8) SN ratio measured at the continuum level near 6250 Å. (9) Effective temperature . (10) Logarithm of surface gravity. (11) Sources of [*$T_{\rm eff}$*]{} and log $\it g$. 1: @nieva2013, 2: @takeda2010, 3: @lefever2010, 4: @aerts2014, 5: @morel2008, 6: @alecian2014, 7: Estimated using the S [ii]{} and N [ii]{} line intensity ratio. (12) and (14) Equivalent widths (E. W.) of the Si [ii]{} 6239.6 and the Al [ii]{} 6243.4 in mÅ. Data of emission and absorption features are given in negative and positive numbers, respectively. Entries marked with a colon(:) are less accurate. An asterisk (\*) indicates that no emission or absorption feature can be recognized. (13) and (15) Central intensities (C. I.) of the Si [ii]{} 6239.6 and the Al [ii]{} 6243.4 lines relative to the local continuum level.
----------- ----------- ----------- ---------------------- --------------------------------------------- --------------------------------------------- -------------- ----------------
Ion Mult. No. $\lambda$ Excitation potential Configuration log $\it gf$ Peak intensity
(Å) (eV) Upper Lower
Al [ii]{} – 5593.30 13.26 3$\it s$4$\it p$ 3$\it s$4$\it d$ 0.337 1.003
10 6226.18 13.07 3$\it s$4$\it p$ 3$\it s$4$\it d$ 0.037 1.004
10 6231.75 13.07 3$\it s$4$\it p$ 3$\it s$4$\it d$ 0.389 1.014
10 6243.36 13.08 3$\it s$4$\it p$ 3$\it s$4$\it d$ 0.659 1.020
Si [ii]{} – 5185.52 12.84 3$\it s$$^{2}$4$\it f$ 3$\it s$$^{2}$7$\it g$ -0.302 1.013
– 5185.56 12.84 3$\it s$$^{2}$4$\it f$ 3$\it s$$^{2}$7$\it g$ -0.456 1.013
– 5466.43 12.53 3$\it s$$^{2}$4$\it d$ 3$\it s$$^{2}$6$\it f$ -0.237 1.016
– 5466.89 12.53 3$\it s$$^{2}$4$\it d$ 3$\it s$$^{2}$6$\it f$ -0.082 1.012
– 5688.81 14.19 3$\it s$3$\it p$($^{3}$P$^{\circ}$)3$\it d$ 3$\it s$3$\it p$($^{3}$P$^{\circ}$)4$\it p$ 0.126 1.009
– 5701.37 14.17 3$\it s$3$\it p$($^{3}$P$^{\circ}$)3$\it d$ 3$\it s$3$\it p$($^{3}$P$^{\circ}$)4$\it p$ -0.057 1.004
– 6239.61 12.84 3$\it s$$^{2}$4$\it f$ 3$\it s$$^{2}$6$\it g$ 0.177 1.057
– 6239.66 12.84 3$\it s$$^{2}$4$\it f$ 3$\it s$$^{2}$6$\it g$ 0.021 1.057
----------- ----------- ----------- ---------------------- --------------------------------------------- --------------------------------------------- -------------- ----------------
: Emission lines of Al [ii]{} and Si [ii]{} observed in $\gamma$ Peg[]{data-label="first"}
Multiplet numbers are taken from @moore1959 and atomic data (wavelengths $\lambda$, excitation potentials, electron configurations and log $\it gf$ values) are taken from the NIST atomic database.
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---
abstract: 'We prove several results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are quite short.'
author:
- 'David Conlon[^1]'
- 'Jacob Fox[^2]'
- 'Benny Sudakov[^3]'
title: Short proofs of some extremal results II
---
Introduction
============
We study several questions from extremal combinatorics, a broad area of discrete mathematics which deals with the problem of maximizing or minimizing the cardinality of a collection of finite objects satisfying a certain property. The problems we consider come mainly from the areas of extremal graph theory and Ramsey theory. In many cases, we give complete or partial solutions to open problems posed by researchers in the area.
While each of the results in this paper is interesting in its own right, the proofs are all quite short. Accordingly, in the spirit of Alon’s ‘Problems and results in extremal combinatorics’ papers [@Al1; @Al2; @Al3] and our own earlier paper [@CFS14], we have chosen to combine them. We describe the results in brief below. For full details on each topic we refer the reader to the relevant section, each of which is self-contained and may be read separately from all others.
In Section \[sec:setmap\], we address old questions of Erdős and Hajnal [@EH58] and Caro [@C87] concerning extremal problems for set mappings. In Section \[sec:FKP\], we answer a question of Foucaud, Krivelevich and Perarnau [@FKP], which extends an old problem of Bollobás and Erdős [@E68], about finding large $K_{r,r}$-free subgraphs in graphs with a given number of edges. In Section \[sec:cube\], we show how to use the Lovász local lemma to embed sparse hypergraphs in large dense hypergraphs and apply this technique to improve the Ramsey number of the cube and other bipartite graphs. In Section \[sec:minors\], we address a problem of Erdős and Hajnal [@EH89] on estimating the bounds on a variant of the classical Ramsey problem and extend this to estimate the extremal number for small and shallow clique minors. We find a connection between induced Ramsey numbers and the Ruzsa–Szemerédi induced matching problem in Section \[sec:matchings\]. Finally, in Section \[sec:coloredremoval\], we prove a colored variant of the famous triangle removal lemma with reasonable bounds.
All logarithms are base $2$ unless otherwise stated. For the sake of clarity of presentation, we systematically omit floor and ceiling signs whenever they are not crucial. We also do not make any serious attempt to optimize absolute constants in our statements and proofs.
Extremal problems for set mappings {#setmappings}
==================================
\[sec:setmap\]
Consider all mappings $f:{M \choose k} \to {M \choose l}$ with $|M|=m$ such that $X$ is disjoint from $f(X)$ for all $X \in {M \choose k}$. Let $p(m,k,l)$ be the maximum $p$ such that for every such mapping $f$ there is a subset $P \subset M$ with $|P| = p$ where $f(X)$ and $P$ are disjoint for all $X \in {P \choose k}$. In 1958, Erdős and Hajnal [@EH58] proved that, for all $k$ and $l$, $$cm^{1/(k+1)} \leq p(m,k,l) \leq c'(m \log m)^{1/k},$$ where $c$ and $c'$ depend only on $k$ and $l$, and asked for a more exact determination of the dependence of $p(m,k,l)$ on $m$.
In 1972, in a textbook application of the probabilistic method, Spencer [@S72] proved an extension of Turán’s theorem to $k$-uniform hypergraphs and used this to prove that $p(m,k,l) \geq cm^{1/k}$, where $c$ depends only on $k$ and $l$. More precisely, Spencer showed that every $r$-uniform hypergraph with $n$ vertices and $t$ edges contains an independent set of order at least $c_rn^{1+1/(r-1)}/t^{1/(r-1)}$. Now consider the $(k+1)$-uniform hypergraph with $m$ vertices and the $t=l{m \choose k}$ edges given by $X \cup \{y\}$ for all $X \in {[m] \choose k}$ and $y \in f(X)$. An independent set $P$ in this hypergraph is a set for which $f(X)$ and $P$ are disjoint for all $X \in {P \choose k}$. Since there is an independent set of order at least $\Omega(m^{1+1/k}/t^{1/k})=\Omega(m^{1/k})$, we get the desired lower bound on $p(m,k,l)$.
Despite the attention the set mapping problem has received over the years, the upper bound has not been improved. Here we solve the Erdős–Hajnal problem when $l$ is sufficiently large as a function of $k$, showing that $p(m, k, (k-1)!) = \Theta(m^{1/k})$, where the implied constants depend only on $k$. For simplicity, we first describe a construction for $l=k!$ and then show how to modify it to get $l=(k-1)!$.
For $m= n^k$, there is a function $f:{M \choose k} \to {M \choose k!}$ such that $|M|=m$, $X$ and $f(X)$ are disjoint for all $X \in {M \choose k}$, and every $P \subset M$ of order greater than $k^2 n$ contains a $k$-set $X$ such that $f(X)$ is not disjoint from $P$.
Let $M=[n]^k$ and consider the function $f:{M \choose k} \to {M \choose k!}$ such that if $X$ consists of the $k$-tuples $(x_{1i},x_{2i},\ldots,x_{ki})$ for $i \in [k]$, then $f(X)$ consists of all $k$-tuples $(x_{1\pi(1)},x_{2\pi(2)},\ldots,x_{k\pi(k)})$, where $\pi$ is a permutation of $[k]$, with the caveat that if $(x_{1\pi(1)},x_{2\pi(2)},\ldots,x_{k\pi(k)})$ is equal to an element of $X$ or a previously chosen element of $f(X)$, we instead choose an arbitrary element of $M$ which is not equal to any of these elements. With this choice, $f(X)$ is well defined and $X$ and $f(X)$ are disjoint.
Let $P \subset [n]^k$ with $|P|>k^2 n$. As long as one of the hyperplanes with one fixed coordinate has at most $k$ elements of $P$, delete those elements from $P$. As there are at most $kn$ such hyperplanes, there are at most $k^2 n$ deleted points. Therefore, there is a remaining point $p \in P$ such that there are at least $k$ other points remaining on each of the $k$ hyperplanes containing $p$ with one coordinate fixed. Picking distinct points $p_1,\ldots,p_k \in P$ which are distinct from $p$ with $p_i$ having the same $i^{\textrm{th}}$ coordinate as $p$ and letting $X=\{p_1,\ldots,p_k\}$, we have $p \in f(X)$, which completes the proof.
To improve the bound on $l$ in this theorem from $k!$ to $(k-1)!$, we need a minor modification. Indeed, suppose that the elements of $M$ are ordered lexicographically (that is, order is determined by first comparing the first coordinates, then the second coordinates and so on) and $f(X)$ is that subset of the $k$-tuples $(x_{1\pi(1)},x_{2\pi(2)},\ldots,x_{k\pi(k)})$ for which $\pi(1)$ is fixed so that the first coordinate $x_{1\pi(1)}$ is equal to the minimum of the first coordinates of the elements of $X$. The proof then proceeds as above, but we choose $p$ and $p_1$ to be the smallest, in the lexicographic ordering, among the remaining vertices after all deletions. This guarantees that $p_1$ is the first element in the set $\{p_1, \dots, p_k\}$, though the remaining elements may be ordered arbitrarily. We note that $l=(k-1)!=1$ is best possible when $k = 2$. After this paper was written, we learned that this particular case was independently solved much earlier by Füredi [@F91]. It would be very interesting to determine whether $l$ can be decreased to $1$ for all $k$.
A related question of Caro [@C87] (see also [@AC86]) asks for an estimate on $q(m, k, d)$, the maximum $q$ such that for every mapping $f:{M \choose k} \to {M \choose k}$ with $|M|=m$ such that $|X \cap f(X)| \leq d$ for all $X \in {M \choose k}$, there is a subset $Q \subset M$ with $|Q| = q$ such that $f(X)$ is not a subset of $Q$ for any $X \in {Q \choose k}$. This differs from the Erdős–Hajnal question on three counts: we take $l = k$; we allow $X$ to overlap with $f(X)$ by a certain controlled amount $d$; and we only require a subset where the image of each element is not contained within the subset rather than being entirely disjoint from it. If we let $t = (k-d)/(2k - d - 1)$, Caro [@C87] proved that $$c m^t \leq q(m, k, d) \leq c' (m \log m)^t,$$ where $c$ and $c'$ depend only on $k$ and $d$. Here we partially answer a question of Caro [@C87] by removing the log factor from the upper bound when $k = 2$.
There exist constants $c_1$ and $c_2$ such that
- $q(m, 2, 1) \leq c_1 m^{1/2}$,
- $q(m, 2, 0) \leq c_2 m^{2/3}$.
\(i) We define a mapping of the complete graph with vertex set $[m]^2$. If $x < x'$ and $y \neq y'$, we map the edge $((x,y),(x',y'))$ to $((x,y),(x,y'))$ and otherwise we map arbitrarily while ensuring that $((x,y), (x',y'))$ doesn’t map to itself.
Suppose now that $Q$ is a subset of order at least $2m + 1$. On every horizontal or vertical line, we delete the highest point which is in $Q$. Since we delete at most $2m$ points, some point $q \in Q$ must remain. If $q = (x, y')$, we see that there are points $(x, y)$ and $(x',y')$ with $x < x'$ and $y' < y$ which are also in $Q$. But, if $e = ((x,y), (x', y'))$, its image is $((x,y), (x, y'))$, which is also in $Q$.
\(ii) We define a mapping of the complete graph with vertex set $[m]^3$. If $x < x'$, $y \neq y'$ and $z \neq z'$, we map the edge $((x,y,z), (x', y', z'))$ to $((x',y,z),(x',y,z'))$ and otherwise we map arbitrarily while ensuring that $((x,y,z)), (x', y', z'))$ is disjoint from its image.
Suppose now that $Q$ is a subset of order at least $3m^2 + 1$. On every line $\{(a,y,z): 1 \leq a \leq m\}$, we remove the lowest point, while on every line $\{(x,b,z): 1 \leq b \leq m\}$, we remove the highest point. The remaining set still has $m^2 + 1$ points. Therefore, there are two points which have the same $x$ and $y$ coordinate, say, $(x', y, z)$ and $(x', y, z')$, where $z < z'$. Since we removed the highest point on the line $\{(x', b, z'): 1 \leq b \leq m\}$, there exists $y' > y$ such that $(x', y', z')$ is in $Q$. Similarly, since we removed the lowest point on the line $\{(a, y, z): 1\leq a \leq m\}$, there exists $x < x'$ such that $(x, y, z)$ is in $Q$. But, if $e = ((x,y,z), (x', y', z'))$, its image is $((x',y,z),(x',y,z'))$, which is also in $Q$.
Large subgraphs without complete bipartite graphs {#sec:FKP}
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Given a family of graphs $\mathcal{H}$, we let $f(m, \mathcal{H})$ be the size of the largest $\mathcal{H}$-free subgraph that can be found in any graph with $m$ edges, where a graph is $\mathcal{H}$-free if it contains no graph from the family $\mathcal{H}$ as a subgraph. The problem of estimating $f(m, \mathcal{H})$ was first raised by Bollobás and Erdős [@E68] at a workshop in 1966, where they asked whether every graph with $m$ edges contains a $C_4$-free subgraph with $\Omega(m^{3/4})$ edges. Erdős [@E71] later remarked that the answer in this case is likely $\Theta(m^{2/3})$, based on an example due to Folkman and private communication from Szemerédi.
Recently, the function $f(m, \mathcal{H})$ was rediscovered by Foucaud, Krivelevich and Perarnau [@FKP], who considered the case where $\mathcal{H}$ is the family of even cycles of length at most $2k$, obtaining estimates that are tight up to a logarithmic factor. In this section, we address a question asked by these same authors and extend the Folkman–Szemerédi result by determining $f(m, H)$ up to a constant factor when $H$ is a complete bipartite graph.
Let $K_{r,s}$ be the complete bipartite graph with parts of order $r$ and $s$, where $2 \leq r \leq s$. The following theorem gives a lower bound on $f(m,K_{r,s})$.
\[th1\] Every graph $G$ with $m$ edges contains a $K_{r,r}$-free subgraph of size at least $\frac{1}{4}m^{\frac{r}{r+1}} $.
To prove this theorem, we need an upper bound on the number of copies of $K_{r,r}$ which can be found in a graph with $m$ edges. The problem of maximizing the number of copies of a fixed graph $H$ over all graphs with a given number of edges was solved by Alon [@Al] (and the corresponding problem for hypergraphs was solved by Friedgut and Kahn [@FK]). For our purposes, the following simpler estimate will suffice.
\[l1\] Every graph $G$ with $m$ edges contains at most $2m^r$ copies of $K_{r,r}$.
Note that every copy of $K_{r,r}$ in $G$ contains a matching of size $r$. Since the number of such matchings is at most ${m \choose r}$ and every matching of size $r$ can appear in at most $2^r$ copies of $K_{r,r}$, the total number of such copies is at most $2^r{m \choose r} \leq 2m^r$.
Using this lemma, together with a simple probabilistic argument, one can prove the required lower bound on $f(m,K_{r,s})$.
[**Proof of Theorem \[th1\]:**]{} Let $G$ be a graph with $m$ edges. Consider a subgraph $G'$ of $G$ obtained by choosing every edge independently at random with probability $p=\frac{1}{2}m^{-1/(r+1)}$. Then the expected number of edges in $G'$ is $mp$. Also, by Lemma \[l1\], the expected number of copies of $K_{r,r}$ in $G'$ is at most $2p^{r^2}m^{r}$. Delete one edge from every copy of $K_{r,r}$ contained in $G'$. This gives a $K_{r,r}$-free subgraph of $G$, which, by linearity of expectation, has at least $$pm-2p^{r^2}m^{r} \geq \frac{1}{2} m^{\frac{r}{r+1}}-\frac{1}{8} m^{\frac{r}{r+1}} \geq \frac{1}{4} m^{\frac{r}{r+1}}$$ edges on average. Hence, there exists a choice of $G'$ which produces a $K_{r,r}$-free subgraph of $G$ of size at least $\frac{1}{4}m^{\frac{r}{r+1}} $. $\Box$
We will now show that this estimate is tight when $G$ is an appropriately chosen complete bipartite graph with $m$ edges. Since the Turán number for $K_{r,s}$ is not known in general, it is somewhat surprising that one can prove a tight bound on the size of the largest $K_{r,s}$-free subgraph in graphs with $m$ edges.
\[th2\] Let $2 \leq r \leq s$ and let $G$ be a complete bipartite graph with parts $U$ and $V$, where $|U|=m^{1/(r+1)}$ and $|V|=m^{r/(r+1)}$. Then $G$ has $m$ edges and the largest $K_{r,s}$-free subgraph of $G$ has at most $s m^{r/(r+1)}$ edges.
The proof is a simple application of the counting argument of Kővári, Sós and Turán [@KST]. Let $G'$ be a $K_{r,s}$-free subgraph of $G$ and let $d=e(G')/|V|$ be the average degree of vertices in $V$ within $G'$. If $d \geq s$, then, by convexity, $$\sum_{v \in V} { d_{G'}(v) \choose r} \geq |V| {d \choose r} \geq {s \choose r}m^{r/(r+1)}\geq s m^{r/(r+1)}/r!\,.$$ On the other hand, since $G'$ is $K_{r,s}$-free we have that $$\sum_{v \in V} {d_{G'}(v) \choose r} < s {|U| \choose r} \leq s |U|^r/r!=s m^{r/(r+1)}/r!\,.$$ This contradiction completes the proof of the theorem.
These results can also be extended to $k$-uniform hypergraphs, which, for brevity, we call $k$-graphs. Let $K^{(k)}_{r,\ldots,r}$ denote the complete $k$-partite $k$-graph with parts of order $r$.
\[th3\] Every $k$-graph $G$ with $m$ edges contains a $K^{(k)}_{r,\ldots,r}$-free subgraph of size at least $\frac{1}{2k!}m^{\frac{q-1}{q}} $, where $q=\frac{r^k-1}{r-1}$.
Let $G$ be a $k$-graph with $m$ edges. Every copy of $K^{(k)}_{r,\ldots,r}$ in $G$ contains a matching of size $r$ and the number of such matchings is at most ${m \choose r}$. On the other hand, every matching in $G$ of size $r$ can appear in at most $(k!)^r$ copies of $K^{(k)}_{r,\ldots,r}$. This implies that the total number of such copies is at most $(k!)^r{m \choose r}$.
Consider a subgraph $G'$ of $G$ obtained by choosing every edge independently at random with probability $p=\frac{1}{k!}m^{-1/q}$. Then the expected number of edges in $G'$ is $mp$ and the expected number of copies of $K^{(k)}_{r,\ldots,r}$ in $G'$ is at most $(k!)^rp^{r^k}{m \choose r}$. Delete one edge from every copy of $K^{(k)}_{r,\ldots,r}$ contained in $G'$. This gives a $K^{(k)}_{r,\ldots,r}$-free subgraph of $G$ with at least $$pm- (k!)^rp^{r^k}{m \choose r} \geq \frac{1}{2k!}m^{\frac{q-1}{q}}$$ expected edges. Hence, there exists a choice of $G'$ which produces a $K^{(k)}_{r,\ldots,r}$-free subgraph of $G$ of this size.
We can again show that this estimate is tight when $G$ is an appropriately chosen $k$-partite $k$-graph.
\[th4\] Let $k, r \geq 2$, $q=\frac{r^k-1}{r-1}$ and let $G$ be a complete $k$-partite $k$-graph with parts $U_i, 1\leq i \leq k$, such that $|U_i|=m^{r^{i-1}/q}$. Then $G$ has $m$ edges and the largest $K^{(k)}_{r,\ldots,r}$-free subgraph of $G$ has $O(m^{(q-1)/q})$ edges.
This result follows from the next statement, which is proved using a somewhat involved extension of the counting argument used in the graph case. This technique has its origins in a paper of Erdős [@E64]. Throughout the proof, we use the notation $\binom{t}{r}$ as a shorthand for $\frac{t(t-1)\dots(t-r+1)}{r!} 1_{t \geq r-1}$, thus extending the definition of the binomial coefficient to a convex function on all of $\mathbb{R}$.
Let $G$ be a $k$-partite $k$-graph with parts $U_i, 1\leq i \leq k$, such that $|U_i|=n^{r^{i-1}}$ and with $a \prod_{i \geq 2}|U_i|$ edges. Then $G$ contains at least ${a - k + 1\choose r}\prod_{i \leq k-1} {|U_i| \choose r}$ copies of $K^{(k)}_{r,\ldots,r}$.
We prove the result by induction on $k$. We may always assume that $a > r + k - 2$, since otherwise $\binom{a - k + 1}{r} = 0$ and the result is trivial. First, suppose that $k=2$ and we have a bipartite graph with parts $U_1$ of order $n$, $U_2$ of order $n^r$ and $an^r$ edges. Let $d(S)$ denote the number of common neighbors of a subset $S$ in $G$ and let $D=\sum_{S\subset U_1, |S|=r}d(S)/{n \choose r}$ be the average number of common neighbors taken over all subsets of order $r$ in $U_1$. Note that $$\sum_{S\subset U_1, |S|=r}d(S)=\sum_{x \in U_2} { d(x) \choose r} \geq {a \choose r} |U_2|={a \choose r}n^r.$$ Therefore, since $a > r$, $D \geq {a \choose r}n^r/{n \choose r} \geq a$ and the number of copies of $K_{r,r}$ in $G$ is $$\sum_{S\subset U_1, |S|=r}{d(S) \choose r} \geq {D \choose r} {n \choose r} \geq {a \choose r} {n \choose r},$$ completing the proof in this case.
Now suppose we know the statement for $k-1$. For every vertex $x \in U_k$, let $G_x$ be the $(k-1)$-partite $(k-1)$-graph which is the link of vertex $x$ (i.e., the collection of all subsets of order $k-1$ which together with $x$ form an edge of $G$). Let $a_x \prod_{i=2}^{k-1}|U_i|$ be the number of edges in $G_x$. By definition, $\sum_x a_x=a|U_k|=an^{r^{k-1}}$. By the induction hypothesis, each $G_x$ contains at least ${a_x - k + 2 \choose r}\prod_{i \leq k-2} {|U_i| \choose r}$ copies of $K^{(k-1)}_{r,\ldots,r}$. By convexity, the total number of such copies added over all $G_x$ is at least $$\begin{aligned}
{a - k + 2 \choose r}n^{r^{k-1}}\prod_{i \leq k-2} {|U_i| \choose r} & = {a - k + 2 \choose r}|U_{k-1}|^r\prod_{i \leq k-2} {|U_i| \choose r}\\
& \geq r!{a - k + 2 \choose r} \prod_{i \leq k-1} {|U_i| \choose r} \geq (a - k + 2)\prod_{i \leq k-1} {|U_i| \choose r},\end{aligned}$$ where in the final inequality we use that $a > r + k - 2$.
For every subset $S$ which intersects every $U_i$, $i\leq k-1$, in exactly $r$ vertices, let $d(S)$ be the number of vertices $x \in U_k$ such that $x$ forms an edge of $G$ together with every subset of $S$ of order $k-1$ which contains one vertex from every $U_i$. By the above discussion, we have that $$\sum_S d(S) \geq (a - k + 2)\prod_{i \leq k-1} {|U_i| \choose r},$$ that is, at least the number of copies of $K^{(k-1)}_{r,\ldots,r}$ counted over all $G_x$. On the other hand, by the definition of $d(S)$, the number of copies of $K^{(k)}_{r,\ldots,r}$ in $G$ equals $\sum_S {d(S) \choose r}$. Since the total number of sets $S$ is $\prod_{i \leq k-1} {|U_i| \choose r}$, the average value of $d(S)$ is at least $a-k+2$ and the result now follows by convexity.
Ramsey numbers and embedding large sparse hypergraphs into dense hypergraphs {#sec:cube}
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For a graph $H$, the [*Ramsey number*]{} $r(H)$ is the least positive integer $N$ such that every two-coloring of the edges of the complete graph $K_N$ contains a monochromatic copy of $H$. One of the most important results in graph Ramsey theory is a theorem of Chvátal, Rödl, Szemerédi and Trotter [@CRST83] which says that for every positive integer $\Delta$ there is a constant $C(\Delta)$ such that every graph $H$ with $n$ vertices and maximum degree $\Delta$ satisfies $r(H) \leq C(\Delta) n$. That is, the Ramsey number of bounded-degree graphs grows linearly in the number of vertices.
The original proof of this theorem used the regularity lemma and gives a very poor tower-type bound for $C(\Delta)$. Following improvements by Eaton [@E98] and Graham, Rödl and Ruciński [@GRR00], the bound $C(\Delta) \leq 2^{c \Delta \log \Delta}$ was given by the authors [@CFS12]. This is close to optimal, since Graham, Rödl and Ruciński [@GRR00; @GRR01] showed that there exist graphs $H$ (even bipartite graphs) with $n$ vertices and maximum degree $\Delta$ for which $r(H) \geq 2^{c' \Delta} n$.
If we assume that $H$ is bipartite, work of Conlon [@C09] and Fox and Sudakov [@FS09] shows that $r(H) \leq 2^{c \Delta} n$ for any $H$ with $n$ vertices and maximum degree $\Delta$. By the results of Graham, Rödl and Ruciński mentioned above, this is optimal up to the constant $c$. The bound proved by Fox and Sudakov [@FS09], $r(H) \leq \Delta 2^{\Delta + 5} n$, remains the best known. Here we remove the $\Delta$ factor from this bound.
\[thm:bip\] For every bipartite graph $H$ on $n$ vertices with maximum degree $\Delta$, $r(H) \leq 2^{\Delta + 6}n$.
This theorem allows us to give a slight improvement on the Ramsey number of cubes. The [*$d$-cube*]{} $Q_d$ is the $d$-regular graph on $2^d$ vertices whose vertex set is $\{0,1\}^d$ and where two vertices are adjacent if they differ in exactly one coordinate. Burr and Erdős [@BE75] conjectured that $r(Q_d)$ is linear in the number of vertices $|Q_d|$. After several improvements over the trivial bound $r(Q_d) \leq r(|Q_d|) \leq 4^{|Q_d|} = 2^{2^{d+1}}$ by Beck [@B83], Graham, Rödl and Ruciński [@GRR01] and Shi [@S01; @S07], Fox and Sudakov [@FS09] obtained the bound $r(Q_d) \leq d 2^{2d + 5}$, which is nearly quadratic in the number of vertices. This follows immediately from their general upper bound on the Ramsey numbers of bipartite graphs. Theorem \[thm:bip\] improves this to a true quadratic bound.
\[cor:cube\] For every positive integer $d$, $r(Q_d) \leq 2^{2d + 6}$.
We note that results similar to Theorem \[thm:bip\] and Corollary \[cor:cube\] were proved by Lee [@L15] in his recent breakthrough work on the Ramsey numbers of degenerate graphs. However, the method he uses is very different to ours. To understand our approach, it will be useful to first describe the method used in [@FS09] to prove the bound $r(H) \leq \Delta 2^{\Delta+5}n$.
Suppose that $H$ is a bipartite graph with $n$ vertices and parts $V_1$ and $V_2$, where every vertex in $V_1$ has degree at most $\Delta$ and every vertex in $V_2$ has degree at most $k$. The proof from [@FS09] has two main ingredients. The first ingredient is a powerful probabilistic technique known as dependent random choice (see, for example, the survey [@FS11] for a discussion of its many variants and applications) which allows one to find a large vertex subset $U$ in a dense graph $G$ such that almost all subsets of at most $k$ vertices from $U$ have many common neighbors.
To prove an upper bound on the Ramsey number of $H$, we take $G$ to be the denser of the two monochromatic graphs which edge-partition the complete graph $K_N$, so that $G$ has edge density at least $1/2$. We use the dependent random choice lemma to find a subset $U$ with the property that almost every subset with at most $k$ vertices has at least $n$ common neighbors. We then form an auxiliary hypergraph $\mathcal{G}$ on $U$ by letting a subset $S$ with at most $k$ vertices be an edge of $\mathcal{G}$ if the vertices of $S$ have at least $n$ common neighbors in $G$. We also define a hypergraph $\mathcal{H}$ on $V_1$ by saying that a subset $T$ with at most $k$ vertices is an edge if there is a vertex of $H$ (which will necessarily be in $V_2$) whose neighborhood is $T$. It is easy to show that if $\mathcal{H}$ is a subhypergraph of $\mathcal{G}$, then $H$ is a subgraph of $G$. Thus, to prove an upper bound on Ramsey numbers, it suffices to show that every sparse hypergraph is a subhypergraph of every not much larger but very dense hypergraph. An embedding lemma of this form is the second ingredient used in [@FS09].
To state the appropriate lemma from [@FS09], we say that a hypergraph is down-closed if $e_1 \subset e_2$ and $e_2 \in E$ implies $e_1 \in E$.
[@FS09] \[lem:FSembed\] Let $\mathcal{H}$ be an $n$-vertex hypergraph with maximum degree $\Delta$ such that each edge of $\mathcal{H}$ has size at most $k$ and suppose that $\delta \leq (4\Delta)^{-k}$. If $\mathcal{G}$ is a down-closed hypergraph on vertex set $U$ with $N \geq 4n$ vertices and more than $\left(1-\delta\right){N \choose k}$ edges of size $k$, then there is a copy of $\mathcal{H}$ in $\mathcal{G}$.
The proof of this lemma uses a greedy embedding process. However, we may improve it by a simple application of the Lovász local lemma, which we now recall.
\[lem:LLL\] Let $A_1,\ldots,A_n$ be events in an arbitrary probability space. A directed graph $D=(V,E)$ on the set of vertices $V=\{1,\ldots,n\}$ is called a dependency digraph for the events $A_1,\ldots,A_n$ if for each $i$, $1 \leq i \leq n$, the event $A_i$ is mutually independent of all the events $\{A_j:(i,j) \not \in E\}$. Suppose $D=(V,E)$ is a dependency digraph for the above events and suppose there are real numbers $x_1,\ldots,x_n$ such that $0 \leq x_i < 1$ and $\textrm{Pr}[A_i] \leq x_i \prod_{(i,j) \in E} (1-x_j)$ for all $1 \leq i \leq n$. Then $$\textrm{Pr}\left[ \bigwedge_{i=1}^n \bar A_i \right] \geq \prod_{i=1}^n \left(1-x_i\right).$$ In particular, with positive probability no event $A_i$ holds.
Using this result, we now improve Lemma \[lem:FSembed\] as follows.
\[lem:embed\] Let $\mathcal{H}$ be an $n$-vertex hypergraph with maximum degree $\Delta$ such that each edge of $\mathcal{H}$ has size at most $k$ and suppose that $\delta \leq \frac{1}{4k\Delta}2^{-8kn/N}$. If $\mathcal{G}$ is a down-closed hypergraph on vertex set $U$ with $N \geq 16n$ vertices and more than $\left(1-\delta\right){N \choose k}$ edges of size $k$, then there is a copy of $\mathcal{H}$ in $\mathcal{G}$.
When $k = 1$, the result follows from Lemma \[lem:FSembed\]. We may therefore assume that $k \geq 2$.
Consider a uniform random mapping $f:V(\mathcal{H}) \rightarrow V(\mathcal{G})$. For two vertices $uv$ of $\mathcal{H}$, consider the bad event $A_{uv}$ that $f(u)=f(v)$. For an edge $e$ of $\mathcal{H}$, we also consider the bad event $B_e$ that $|f(e)|=|e|$ (the vertices of $e$ map to distinct vertices), but $f(e)$ is not an edge of $\mathcal{G}$.
Clearly, $\textrm{Pr}[A_{uv}]=1/N$. We also have $\textrm{Pr}[B_e] \leq\delta$. Indeed, suppose $|e|=\ell$. If $f(e)$ is not an edge in $\mathcal{G}$, then, since $\mathcal{G}$ is down-closed, none of the ${N-\ell \choose k-\ell}$ $k$-sets containing it are in $\mathcal{G}$ either. However, the number of pairs consisting of an edge of size $k$ which is not in $\mathcal{G}$ and a subset of size $\ell$ is at most $\delta \binom{N}{k} \binom{k}{\ell}$. It follows that the number of $\ell$-sets which are not edges of $\mathcal{G}$ is at most $$\delta {N \choose k} \cdot \frac{{k \choose \ell}}{{N-\ell \choose k-\ell}}=\delta{N \choose \ell},$$ which implies that $\textrm{Pr}[B_e] \leq\delta$.
Provided $\{u',v'\}$ and $e$ are disjoint from $\{u,v\}$, the event $A_{uv}$ is independent of $A_{u'v'}$ and $B_e$. Therefore, $A_{uv}$ is dependent on at most $2(n-2)$ of the events $A_{u'v'}$ and at most $2 \Delta$ of the events $B_e$. Similarly, provided $\{u,v\}$ and $e'$ are disjoint from $e$, the event $B_e$ is independent of $A_{uv}$ and $B_{e'}$. Therefore, $B_e$ is dependent on at most ${n \choose 2}-{n-|e| \choose 2} < kn$ of the events $A_{uv}$ and at most $k \Delta$ of the events $B_{e'}$.
We now apply the local lemma. For each $A_{uv}$, we let the corresponding $x_i$ be $x$ and, for each $B_e$, we let the corresponding $x_i$ be $y$. Let $x=\frac{4}{N}$ and $y=\frac{1}{2k\Delta}$. As $x,y \leq 1/2$, we have $1-x \geq 4^{-x}$ and $1-y \geq 4^{-y}$. Therefore, since $N \geq 16 n$ and $k \geq 2$, $$x(1-x)^{2n}(1-y)^{2\Delta} \geq \frac{4}{N}4^{-8n/N}4^{-1/k} \geq \frac{1}{N}=\textrm{Pr}[A_{uv}]$$ and $$y(1-y)^{k\Delta}(1-x)^{kn} \geq \frac{1}{2k\Delta}4^{-1/2}4^{-4kn/N} \geq \delta \geq \textrm{Pr}[B_e].$$ By Lemma \[lem:LLL\], the probability that none of the bad events $A_{uv}$ and $B_e$ occur is positive and, therefore, $\mathcal{H}$ is a subhypergraph of $\mathcal{G}$.
To prove Theorem \[thm:bip\], we need an appropriate variant of the dependent random choice lemma. The version we use follows easily from Lemma 2.1 of [@FS09].
\[lem:DRC\] Let $G$ be a bipartite graph with parts $V_1$ and $V_2$ of order $N$ and at least $\epsilon N^2$ edges, where $N \geq \epsilon^{-k} \max(bn, 4k)$. Then there is a subset $U \subset V_1$ with $|U| \geq 2^{-1/k}\epsilon^k N$ such that the number of $k$-sets $S \subset U$ with $|N(S)|< n$ is less than $2^{k+1} b^{-k}{|U| \choose k}$.
Combining Lemmas \[lem:embed\] and \[lem:DRC\], we arrive at the following theorem.
Let $H$ be a bipartite graph with $n$ vertices such that one part has maximum degree $k$ and the other part has maximum degree $\Delta$. If $G$ is a bipartite graph with edge density $\epsilon$ and at least $16\Delta^{1/k}\epsilon^{-k}n$ vertices in each part, then $H$ is a subgraph of $G$.
Let $N = 16\Delta^{1/k}\epsilon^{-k}n$. Applying Lemma \[lem:DRC\] with $b = 16 \Delta^{1/k} \geq 2(8 k \Delta)^{1/k} 2^{8n/N}$, we find a set $|U|$ with $|U| \geq 2^{-1/k} \epsilon^k N \geq 16n$ vertices such that the number of $k$-sets $S \subset U$ with $N(S) < n$ is less than $2^{k+1} b^{-k}{|U| \choose k}$. Since $2^{k+1} b^{-k} \leq \frac{1}{4k\Delta} 2^{-8kn/N}$, we may apply Lemma \[lem:embed\] to embed the auxiliary hypergraph $\mathcal{H}$ in the hypergraph $\mathcal{G}$ (as described before Lemma \[lem:FSembed\]). This in turn implies that $H$ is a subgraph of $G$.
By considering the denser color in any two-coloring, this result has the following immediate corollary. Theorem \[thm:bip\] follows as a special case.
If $H$ is a bipartite graph with $n$ vertices such that one part has maximum degree $k$ and the other part has maximum degree $\Delta$, then $r(H) \leq \Delta^{1/k}2^{k+5}n.$
Weakly homogeneous sequences and small minors {#sec:minors}
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In 1989, Erdős and Hajnal [@EH89] studied an extension of the fundamental problem of estimating Ramsey numbers. A sequence $S_1,\ldots,S_t$ of disjoint vertex subsets of a graph is called a [*weakly complete $r$-sequence of order $t$*]{} if each subset $S_i$ has cardinality $r$ and, for each pair $1 \leq i < j \leq t$, there is an edge from a vertex in $S_i$ to a vertex in $S_j$. Let $g(r,n)$ be the largest $t$ for which every graph on $n$ vertices or its complement contains a weakly complete $r$-sequence of order $t$. Note that determining $g(1,n)$ is simply the classical Ramsey problem, since the vertices making up $S_1,\ldots,S_t$ form either a clique or an independent set.
For $r$ fixed and $n$ sufficiently large, Erdős and Hajnal [@EH89] proved that $$(1/2 - o_r(1))(3/2)^{r}\log n \leq g(r,n) \leq 2^{r^2+1}r\log n,$$ where the upper bound comes from considering a random $2$-coloring of the edges of the complete graph on $n$ vertices. These estimates naturally lead one to ask whether the power of $r$ in the exponent of the constant factor should be $1$ or $2$. Improving the lower bound of Erdős and Hajnal, we answer this question by showing that the upper bound is much closer to the truth. Moreover, for $r \geq 2$, we will show that a density theorem holds, that is, every dense graph contains a large weakly complete $r$-sequence.
A sequence $S_1,\ldots,S_t,T_1,\ldots,T_t$ of disjoint vertex subsets of a graph is called a [*weakly bi-complete $r$-sequence of order $t$*]{} if $|S_i|=|T_i|=r$ for $1 \leq i \leq t$ and, for each pair $1 \leq i, j \leq t$, there is an edge from a vertex in $S_i$ to a vertex in $T_j$. Given such a weakly bi-complete $r$-sequence, $S_1 \cup T_1,\ldots,S_t \cup T_t$ is clearly a weakly complete $2r$-sequence.
\[thm1\] Let $n$ be sufficiently large and $G$ be a graph with $n$ vertices and edge density $p$. The graph $G$ contains a weakly bi-complete $r$-sequence of order $t$ if
- $p \geq n^{-1/3}$, $r \leq 2p^{-1/2}$ and $t \leq \frac{\log n}{4\log (32/pr^2)}$;
- $p \geq n^{-1/5}$, $4p^{-1/2} \leq r \leq \sqrt{p^{-1}\log n}$ and $t \leq \frac{1}{16}e^{pr^2/8}\log n$;
- $r \geq 4\sqrt{p^{-1}\log n}$ and $t \leq \min(pn/64\sqrt{\log n}, n/2r)$.
In particular, $G$ also contains a weakly complete $2r$-sequence of order $t$.
Note that if a graph has a weakly bi-complete $r$-sequence of order $t$, then, by arbitrarily adding additional vertices of the graph to the $r$-sets to obtain $r'$-sets, the graph also has a weakly bi-complete $r'$-sequence of order $t$ for any $r'$ satisfying $r \leq r' \leq n/2t$. This is useful for the third bound and for interpolating between the bounds. In particular, for the third bound, it will suffice to prove it for the case $r= 4\sqrt{p^{-1}\log n}$.
All three bounds on $t$ will follow from reducing the problem to a special case of the Zarankiewicz problem in which we want to guarantee a $K_{t,t}$ in a bipartite graph with parts of order at least $pn/16r$ and edge density at least $1-e^{-pr^2/8}$. Although it is not difficult to improve our bounds by being a little more careful at a few points in the argument, we have chosen to present proofs which determine the correct behavior while remaining as simple as possible.
By considering a random graph with edge density $p$, we see that the bounds in Theorem \[thm1\] are close to being tight. Noting that every graph or its complement has edge density at least $1/2$, we have the following immediate corollary of the second bound.
\[cor1\] For $r$ fixed and $n$ sufficiently large, $$g(r,n) \geq \frac{1}{16}e^{2^{-6}r^2}\log n.$$
[*Clique minors*]{} are a strengthening of weakly complete sequences, with the added constraint that the sets $S_i$ are required to be connected. A classical result of Mader [@Ma67] guarantees that for each $t$ there is $c(t)$ such that every graph on $n$ vertices with at least $c(t)n$ edges contains a clique minor of order $t$. Kostochka [@Ko82; @Ko84] and Thomason [@Th84] independently determined the order of $c(t)$, proving that $c(t)=\Theta(t\sqrt{\log t})$. Almost two decades later, Thomason [@Th01] determined an asymptotic formula: $c(t)=(\alpha+o(1))t \sqrt{\ln t}$, where $\alpha = 0.319...$ is a computable constant.
In recent years, there has been a push towards extending these classical results on the extremal problem for graph minors to small graph minors, that is, where few vertices are used in making the minor. A result of Fiorini, Joret, Theis and Wood [@FJRW] says that, for each $t$, there are $h(t)$ and $f(t)$ such that every graph with at least $f(t)n$ edges contains a $K_t$-minor with at most $h(t)\log n$ vertices. The $\log n$ factor here is necessary. Indeed, for each $C$ there is $c>0$ and an $n$-vertex graph with at least $Cn$ edges and girth (which is defined as the length of the shortest cycle, but is also the minimum number of vertices in a $K_3$-minor) at least $c\log n$. Fiorini et al. also conjectured that, for each $\epsilon>0$, one may take $f(t)=c(t)+\epsilon$ and $h(t)=C(\epsilon,t)$. Shapira and Sudakov [@ShSu] came close to proving this conjecture, showing that every $n$-vertex graph with at least $(c(t)+\epsilon)n$ edges contains a $K_t$-minor of order at most $C(\epsilon,t)\log n \log \log n$. Building upon their approach, Montgomery [@Mont] then solved the conjecture by removing the $\log \log n$ factor.
These results are all about finding clique minors in sparse graphs. Here, we study the dense case and find conditions on $t$ and $r$ such that every dense graph contains a $K_t$-minor where each connected set corresponding to a vertex of the minor contains at most $r$ vertices. In particular, this shows that we may prove an analogue of Corollary \[cor1\] where the required subgraph is a clique minor rather than just a weakly complete $r$-sequence.
\[minortheorem\] Let $n$ be sufficiently large and $G$ be a graph with $n$ vertices and edge density $p \geq n^{-1/8}$. If $24p^{-1/2} \leq r \leq \frac{1}{2}\sqrt{p^{-1}\log n}$ and $t \leq \frac{1}{32}e^{pr^2/256}\log n$, then $G$ contains a $K_t$-minor such that the connected sets corresponding to its vertices have size at most $8r$.
Minors in which the connected sets corresponding to vertices have small diameter are known as [*shallow minors*]{}. This concept was introduced in a paper by Plotkin, Rao and Smith [@PRS], though they attribute the idea to Leiserson and Toledo. Shallow minors also play a fundamental role in the work of Nešetřil and Ossona de Mendez on the theory of nowhere dense graphs (see their book [@NM]).
We mention this concept because the proof of Theorem \[minortheorem\] also gives that the connected subset corresponding to each vertex has diameter at most $9$. A variant of this argument (using a different version of dependent random choice) can be used to reduce the diameter of the sets to $3$, but with a slightly weaker bound on $t$. We also note that there are analogues of Theorem \[thm1\] when $r$ is larger or smaller than the assumed range. However, the proof is the same, so we omit the details.
We begin by proving Theorem \[thm1\] and then deduce Theorem \[minortheorem\]. We will make use of the following three lemmas.
\[firsta1\] Let $H=(V_1,V_2,E)$ be a bipartite graph with edge density $p$. There is a subset $B \subset V_2$ with $|B| \geq p |V_2|/2$ such that every vertex in $B$ has more than $p|V_1|/2$ neighbors in $V_1$.
Delete all vertices in $V_2$ of degree at most $p|V_1|/2$ and let $B$ be the remaining subset of $V_2$. The number of deleted edges is at most $p|V_1||V_2|/2$ and hence there are at least $p|V_1||V_2|-p|V_1||V_2|/2=p |V_1||V_2|/2$ remaining edges from $B$ to $V_1$. As each vertex in $B$ is in at most $|V_1|$ edges, $|B| \geq p |V_1||V_2|/2|V_1|=p |V_2|/2$.
\[firstbd\] Let $H=(V_1,V_2,E)$ be a bipartite graph with edge density $1-q$. Then there is a subset $B \subset V_2$ with $|B| \geq |V_2|/2$ such that every vertex in $B$ has more than $(1-2q)|V_1|$ neighbors in $V_1$.
Delete all vertices in $V_2$ of degree at most $(1-2q)|V_1|$ and let $B$ be the remaining subset of $V_2$. The number of nonedges touching the deleted vertices is at least $2q |V_1||V_2 \setminus B|$ and at most $q|V_1||V_2|$. Hence, $|V_2 \setminus B| \leq |V_2|/2$ and $|B| \geq |V_2|/2$.
\[secondb\] If $H=(V_1,V_2,E)$ is a bipartite graph in which every vertex in $V_2$ has at least $p|V_1|$ neighbors in $V_1$, then there is a partition $V_1=A_1 \cup \ldots \cup A_{d}$ into subsets of order $r$ (so $d=|V_1|/r$) such that the fraction of pairs $(A_i,b)$ with $b \in V_2$ for which $b$ does not have a neighbor in $A_i$ is at most $(1-p)^r \leq e^{-pr}$.
Partition $V_1$ uniformly at random into subsets $A_i$ of size $r$. The probability that $b$ has no neighbor in a subset $A_i$ chosen uniformly at random is precisely the same as $A_i$ not containing any of the at least $p|V_1|$ neighbors of $b$ in $V_1$, which is at most ${(1-p)|A| \choose r}/{|A| \choose r} \leq (1-p)^r$. Therefore, the expected fraction of pairs $(A_i,b)$ for which $b$ does not have a neighbor in $V_1$ is at most $(1-p)^r$. Hence, there is such a partition of $V_1$ where the fraction of pairs $(A_i,b)$ is at most this expected value.
[**Proof of Theorem \[thm1\]:**]{} Let $G=(V,E)$ be a graph on $n$ vertices with edge density $p$. By considering a random equitable vertex partition of $G$, there is a vertex partition $V=V_1 \cup V_2$ into parts of order $n/2$ such that the bipartite graph induced by this partition has edge density at least $p$. By Lemma \[firsta1\], there is $B \subset V_2$ with $|B| \geq p|V_2|/2 \geq p n/4$ such that every vertex in $B$ has at least $p|V_1|/2$ neighbors in $V_1$. By Lemma \[secondb\], there is a partition $V_1=A_1 \cup \ldots \cup A_{d}$ into subsets of order $r$ (so $d=|V_1|/r=n/2r$) such that the fraction of pairs $(A_i,b)$ with $b \in B$ for which $b$ does not have a neighbor in $V_1$ is at most $\rho:=(1-p/2)^r$.
Consider the auxiliary bipartite graph $X$ with parts $\{1,\ldots,d\}$ and $B$, where $i$ is adjacent to $b \in B$ if there is at least one edge from $b$ to $A_i$. The density of $X$ between its parts is at least $1-\rho$.
[*Case 1*]{}: $p \leq 3/r$. In this case, we have $\rho \leq e^{-pr/2} \leq 1-pr/4$ and hence the density of $X$ between its parts is at least $1-\rho \geq pr/4$.
Let $S \subset \{1,\ldots,d\}$ consist of those vertices with at least $pr|B|/8$ neighbors in $B$. By Lemma \[firsta1\], we have $|S| \geq prd/8 = pr(n/2r)/8=pn/16$. By Lemma \[secondb\], there is a partition $B=B_1 \cup \ldots \cup B_{h}$ into subsets of order $r$ (so $h=|B|/r \geq p n /4r$) such that the fraction of pairs $(i,j)$ with $i \in S$ and $j \in [h]$ for which $i$ does not have a neighbor in $B_j$ in $X$ (and hence $A_i$ does not have an edge to $B_j$ in $G$) is at most $(1-pr/8)^r \leq e^{-pr^2/8}$.
[*Case 2*]{}: $p>3/r$. In this case, we have $\rho = (1-p/2)^r \leq e^{-pr/2}$ and hence the density of $X$ between its parts is at least $1-\rho \geq 1-e^{-pr/2}$.
Let $S \subset \{1,\ldots,d\}$ consist of those vertices with at least $(1-2e^{-pr/2}) |B| \geq (1-e^{-pr/4})|B|$ neighbors in $B$. By Lemma \[firstbd\], we have $|S| \geq d/2 =n/4r$. By Lemma \[secondb\], there is a partition $B=B_1 \cup \ldots \cup B_{h}$ into subsets of order $r$ (so $h=|B|/r \geq p n /4r$) such that the fraction of pairs $(i,j)$ with $i \in S$ and $j \in [h]$ for which $i$ does not have a neighbor in $B_j$ in $X$ (and hence $A_i$ does not have an edge to $B_j$ in $G$) is at most $\left(e^{-pr/4}\right)^r =e^{-pr^2/4}$.
In either case, we obtain a bipartite graph $T$ with parts $S$ and $[h]$ where $(i,j) \in S \times [h]$ is an edge if $A_i$ has at least one edge to $B_j$, the parts are of order at least $N := pn/16r$ and the edge density is $1 - \delta$ for some $\delta \leq e^{-pr^2/8}$. Note that any $K_{t,t}$ in $T$ forms a weakly bi-complete $r$-sequence of order $t$ in $G$.
If $r \geq 4\sqrt{p^{-1}\log n}$, then $pr^2 \geq 16 \log n$ and this edge density is at least $1-n^{-2}$, so $T$ is a complete bipartite graph with parts of order at least $pn/16r$. This gives the third desired bound.
A classical result of Kővári, Sós and Turán [@KST] on the Zarankiewicz problem shows that if a bipartite graph $T$ with parts of order at least $N$ has density at least $1-\delta$ and $N{(1-\delta)N \choose t} > (t-1){N \choose t}$, then the bipartite graph contains $K_{t,t}$. Notice that this inequality holds if $(1-\delta-\frac{t}{N})^t \geq t/N$.
If $r \leq 2p^{-1/2}$, $p \geq n^{-1/3}$ and $t \leq \frac{\log n}{4\log (32/pr^2)}$, then, letting $x=pr^2/8 \leq 1/2$, we see that $T$ has edge density at least $1-e^{-x} \geq x/2$. However, $x \geq 4t/N$, so $T$ contains a $K_{t,t}$ if $(x/4)^t \geq t/N$. But $$(x/4)^t \geq n^{-1/4} \geq 32p^{-3/2}tn^{-1} \geq t16r/pn = t/N,$$ and we have shown the first desired bound.
Suppose now that we are trying to obtain the second desired bound. Since $4p^{-1/2} \leq r < \sqrt{p^{-1}\log n}$, $p \geq n^{-1/5}$ and $t \leq \frac{1}{16}e^{pr^2/8}\log n$, we have $\delta \geq e^{-pr^2/8} \geq n^{-1/8}$ and $t \leq n^{1/4}$. Therefore, we have $\delta N = \delta pn/16r \geq pn^{7/8}/16r \geq n^{1/2} \geq t$ and $t/N \leq n^{-1/4}$, so the desired inequality for the Kővári–Sós–Turán result holds if $(1-2\delta)^t \geq n^{-1/4}$. Taking the logarithm of both sides and noting that $\log(1-2\delta) \geq -4\delta$ as $\delta \leq e^{-2}$, we see that this holds as long as $t \leq \frac{\log n}{16\delta}\leq\frac{1}{16}e^{pr^2/8}\log n$, which is a given assumption. This completes the proof of Theorem \[thm1\].
To prove Theorem \[minortheorem\], we combine the previous embedding technique with the following consequence of dependent random choice (discussed in the previous section) taken from [@FLS].
\[drclemma\] Let $H=(U,V,E)$ be a bipartite graph with $|U|=|V|=n/2$ and at least $pn^2/4$ edges. Then, if $p^2 n \geq 1600$, there is a subset $X \subset U$ with $|X| \geq pn/50$ such that for every pair of vertices $x,y \in X$, there are at least $10^{-9}p^5n$ internally vertex-disjoint paths with four edges between $x$ and $y$ with internal vertices not in $X$.
[**Proof of Theorem \[minortheorem\]:**]{} Let $G$ be a graph with edge density $p$ on $n$ vertices, so it has $p{n \choose 2} \geq pn^2/4$ edges. By deleting vertices of degree less than $pn/8$ one at a time, we arrive at a subgraph $G'$ with minimum degree at least $pn/8$ and at least $pn^2/4-n(pn/8)=pn^2/8$ edges. Let $v$ denote the number of vertices in $G'$, so $p^{1/2}n/4 \leq v \leq n$.
Let $H=(U,V,E)$ be a bipartite subgraph of $G'$ with parts of order $v/2$ and at least $pn^2/32$ edges such that the minimum degree of $H$ is at least $pn/32$. Such a bipartite subgraph exists by considering a random bipartition. Note that the number of edges in $H$ is at least $pn^2/32=(pn^2/8v^2)v^2/4$. By Lemma \[drclemma\], there is a subset $X \subset U$ with $|X| \geq (pn^2/8v^2)v/50 \geq pn/400$ such that for every pair of vertices $x,y \in X$, there are at least $10^{-9}(pn^2/8v^2)^5v \geq 10^{-14}p^5n$ internally vertex-disjoint paths with four edges between $x$ and $y$ with internal vertices not in $X$. Let $X'$ be an arbitrary subset of $X$ of size exactly $pn/400$.
As every vertex in $X$ (and hence $X'$) has degree at least $pn/32$ in $H$, there are at least $(pn/32)|X'|$ edges between $X'$ and $V$. Delete all vertices in $V$ with fewer than $(pn/32v)|X'|$ neighbors in $X'$ and let $Z$ be the remaining subset of $V$. The number of edges between $X'$ and $Z$ is at least $(pn/32)|X'|-(pn/32v)|X'|(v/2)=pn|X'|/64$. Note that $|Z| \geq pn/64>|X'|$. Let $Z' \subset Z$ be a subset with $|Z'|=|X'|$ such that the number of edges between $X'$ and $Z'$ at least $p|X'||Z'|/64$. Such a subset $Z'$ exists by considering a random subset of $Z$ of order $|Z'|$. Consider the bipartite graph $H'$ between $X'$ and $Z'$. It has $2|Z'|$ vertices and at least $p|X'||Z'|/64=(p|X'|/64|Z'|)(2|Z'|)^2/4$ edges. Applying Lemma \[drclemma\] to $H'$, there is a subset $Y \subset Z'$ with $|Y| \geq (p|X'|/64|Z'|)(2|Z'|)/50=p|X'|/1600$ such that for every pair of vertices $x,y \in Y$, there are at least $10^{-9} (p|X'|/64|Z'|)^5(2|Z'|) \geq 10^{-18}p^5 |X'|$ internally vertex-disjoint paths with four edges between $x$ and $y$ with internal vertices not in $Y$. Since every vertex in $Z$ has at least $(pn/32v)|X'|$ neighbors in $X'$, the density between $X'$ and $Y$ is at least $pn/32v$. Let $W \subset X'$ be a subset of order $|Y|$ such that the edge density between $W$ and $Y$ is at least $pn/32v \geq p/32$. Such a subset exists by considering a random subset of $X'$ of order $|Y|$.
By Theorem \[thm1\] (or rather its proof, as we pass to a balanced bipartite subgraph), the bipartite graph between $W$ and $Y$ contains a weakly bi-complete $r$-sequence of order $t$ with $t = \frac{1}{16}e^{pr^2/256}\log |Y| \geq \frac{1}{32}e^{pr^2/256}\log n$. For each part $A$ used to make this weakly bi-complete $r$-sequence, fix a vertex $a \in A$ and consider any other vertex $b \in A$. There are at least $10^{-18}p^5|X'|>10^{-21}p^6 n>8rt$ internally vertex disjoint paths, so we can find one of these internal paths so that the vertices have not already been used and add the three internal vertices of the path to connect $a$ and $b$. Doing this for every vertex $b \in A \setminus a$, we get a connected set $A'$ with at most $4r$ vertices. We can do this for each of the $2t$ sets $A$ making up the weakly bi-complete $r$-sequence of order $t$. We thus obtain a $K_{t,t}$-minor with each part corresponding to a vertex of order at most $4r$. From a matching in the $K_{t,t}$, we get a $K_t$-minor with each part corresponding to a vertex of order at most $8r$.
To prove the claim that each set in the $K_t$-minor may be chosen to have diameter at most $9$, suppose that $A$ and $B$ are two sets in the weakly bi-complete $r$-sequence and the union of the sets $A'$ and $B'$ formed from $A$ and $B$ corresponds to a vertex in the $K_t$-minor. If we let $ab$ be an edge with $a \in A$ and $b \in B$, we see that we could have chosen the sets $A'$ and $B'$ so that every vertex in $A'$ is connected to $a$ by a path of length $4$ and every vertex in $B'$ is connected to $b$ by a path of length $4$. Since $a$ and $b$ are joined, this clearly implies that $A' \cup B'$ has diameter at most $9$.
Induced Ramsey numbers and Ruzsa–Szemerédi graphs {#sec:matchings}
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A graph $H$ is said to be an [*induced subgraph*]{} of $G$ if $V(H) \subset V(G)$ and two vertices of $H$ are adjacent if and only if they are adjacent in $G$. We write $G \rightarrow_{ind} (H_1,H_2)$ if every red/blue-coloring of the edges of $G$ contains an induced copy of $H_1$ all of whose edges are red or an induced copy of $H_2$ all of whose edges are blue. The [*induced Ramsey number*]{} $r_{\textrm{ind}}(H_1,H_2)$ is the minimum $N$ for which there exists a graph $G$ on $N$ vertices with $G \rightarrow_{ind} (H_1,H_2)$. When $H_1 = H_2 = H$, we simply write $r_{\textrm{ind}}(H)$.
We will be interested in the induced Ramsey number of trees. It is easy to show that the usual Ramsey number for trees is linear in the number of vertices. For some trees, such as paths, it is even possible to show [@HKL95] that the induced Ramsey number grows linearly in the number of vertices. However, Fox and Sudakov [@FS08] showed that there exist trees $T$ for which the induced Ramsey number grows superlinearly. More precisely, they showed that $r_{\textrm{ind}}(K_{1, t}, M_t)$ is superlinear in $t$, where $M_t$ is the matching with $t$ edges. It is then sufficient if $T$ contains both $K_{1,t}$ and $M_t$ as induced subgraphs.
Fox and Sudakov [@FS08] proved their result by an appeal to the regularity lemma. In this section, we prove a strengthening of this result by showing that there is a close connection between $r_{\textrm{ind}}(K_{1, t}, M_n)$ and the celebrated Ruzsa–Szemerédi induced matching problem [@RuSz]. We say that a graph $G=(V,E)$ is an [*$(n,t)$-Ruzsa–Szemerédi graph*]{} (or an $(n,t)$-RS graph, for short) if its edge set is the union of $t$ pairwise disjoint induced matchings, each of size $n$.
\[thm:toRS\] If $G \rightarrow_{\textrm{ind}} (K_{1,t},M_n)$, then $G$ contains a subgraph which is an $(n,t)$-RS graph.
Pick out disjoint induced matchings of size $n$ from $G$ until there are no more induced matchings of this size. If at least $t$ induced matchings are picked out, then the union of these $t$ induced matchings makes the subgraph of $G$ which is an $(n,t)$-RS graph. Otherwise, we color the edges of the (fewer than $t$) induced matchings in red and the remaining edges in blue. The red graph won’t contain a red $K_{1,t}$ as each of the (fewer than $t$) induced matchings contributes at most one to the degree of each vertex. Moreover, since we cannot pick out another disjoint induced matching, the blue graph does not contain an induced $M_n$, contradicting our assumption that $G \rightarrow_{\textrm{ind}} (K_{1,t},M_n)$.
To recover the quantitative statement that $r_{\textrm{ind}}(K_{1, t}, M_t)$ is superlinear in $t$, we use Theorem \[thm:toRS\] to deduce that if $G \rightarrow_{ind} (K_{1,t},M_t)$ then $G$ contains a $(t,t)$-RS graph. However, the work of Ruzsa and Szemerédi [@RuSz] shows that any such graph must have $t (\log^*t)^c$ vertices for some positive constant $c$, where $\log^*t$ is the slowly-growing function defined by $\log^* t = 0$ if $t \leq 1$ and $\log^*t = 1 + \log^*(\log t)$ if $t > 1$. Fox’s bound for the removal lemma [@Fo], which we will discuss further in the next section, improves this estimate to $t e^{c \log^*t}$ for some positive constant $c$. This yields the following corollary.
There exists a positive constant $c$ such that $r_{\textrm{ind}}(K_{1, t}, M_t) \geq t e^{c \log^* t}$.
To show that Ruzsa–Szemerédi graphs also give rise to graphs $G$ for which $G \rightarrow_{\textrm{ind}} (K_{1,t},M_n)$, we first show that we may assume our Ruzsa–Szemerédi graph is bipartite.
\[lem:tobip\] If there is an $(n,t)$-RS graph $G=(V,E)$ on $N$ vertices, then there is a bipartite $(2n,t)$-RS graph $B$ on $2N$ vertices.
Let $B$ be the bipartite graph with parts $V_1$ and $V_2$, each a copy of $V$, where $(u,v) \in V_1 \times V_2$ is an edge if and only if $(u,v)$ is an edge of $G$. Each of the $t$ induced matchings of size $n$ in $G$ corresponds to an induced matching of size $2n$ in $B$ and these induced matchings make up $B$.
It is now straightforward to show that bipartite Ruzsa–Szemerédi graphs $G$ satisfy $G \rightarrow_{\textrm{ind}} (K_{1,n},M_n)$.
\[thm:toind\] Suppose that $c \geq 2$ and $G$ is a bipartite $(cn,N/c)$-RS graph on $N$ vertices. Then $G \rightarrow_{ind} (K_{1,n},M_n)$.
In any red/blue-coloring of the edges of $G$, at least half of the edges are blue or half of them are red. In the former case, at least one of the induced matchings of size $cn$ has at least half of its edges in color blue and, since $cn/2 \geq n$, these edges form a blue induced matching of size $n$. In the latter case, as there are $nN$ edges and so at least $nN/2$ red edges, there is a vertex of red degree at least $n$. Since $G$ is bipartite, this induces a red star $K_{1,n}$.
As observed by Ruzsa and Szemerédi [@RuSz], a construction due to Behrend [@Be46] allows one to show that there are $(N/e^{c \sqrt{\log N}}, N/5)$-RS graphs on $N$ vertices. Applying Lemma \[lem:tobip\] and Theorem \[thm:toind\], we get the following corollary.
There exists a constant $c$ such that $r_{\textrm{ind}}(K_{1, t}, M_t) \leq t e^{c \sqrt{\log t}}$.
Colored triangle removal {#sec:coloredremoval}
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The triangle removal lemma of Ruzsa and Szemerédi [@RuSz] states that for each $\epsilon>0$ there is $\delta>0$ such that every graph on $n$ vertices with at most $\delta n^3$ triangles can be made triangle-free by removing at most $\epsilon n^2$ edges. That is, a graph with a subcubic number of triangles can be made triangle-free by removing a subquadratic number of edges. The triangle removal lemma has many applications in graph theory, additive combinatorics, discrete geometry and theoretical computer science.
Until recently, the only known proof of the triangle removal lemma used Szemerédi’s regularity lemma [@Sz76] and gave a weak quantitative bound for $\delta^{-1}$, namely, a tower of $2$s of height polynomial in $\epsilon^{-1}$. Recently, Fox [@Fo] found a new proof which avoids the regularity lemma and improves the bound on $\delta^{-1}$ to a tower of $2$s of height logarithmic in $\epsilon^{-1}$. It remains a major open problem to find a bound of constant tower height.
It is easy to show that the triangle removal lemma is equivalent to the following statement: for each $\epsilon>0$ there is $\delta > 0$ such that every tripartite graph on $n$ vertices whose edge set can be partitioned into $\epsilon n^2$ triangles contains at least $\delta n^3$ triangles. We present a simple proof of the following Ramsey-type weakening of this statement with a much better (only double-exponential) bound.
\[main\] Every $r$-edge-coloring of a tripartite graph $G$ with parts $V_0,V_1,V_2$, where $|V_1|=|V_2|=n$, $|V_0| = cn$ and $V_1$ is complete to $V_2$, such that there is a collection of $n^2$ edge-disjoint monochromatic triangles which cover the edges between $V_1$ and $V_2$ contains at least $(4cr)^{-2^{r+3}}n^3$ monochromatic triangles.
When $(4cr)^{-2^{r+3}}n^3>n^2$, two of the monochromatic triangles must share an edge between $V_1$ and $V_2$ and so we have the following corollary.
\[cor:col\] Suppose $n>(4cr)^{2^{r+3}}$. Every $r$-edge-coloring of a tripartite graph $G$ with parts $V_0,V_1,V_2$, where $|V_1|=|V_2|=n$, $|V_0| = cn$ and $V_1$ is complete to $V_2$, such that there is a collection of $n^2$ edge-disjoint monochromatic triangles which cover the edges between $V_1$ and $V_2$ contains a monochromatic diamond, i.e., an edge in two monochromatic triangles.
One application of the triangle removal lemma, noted by Solymosi [@So], is a short proof for the corners theorem of Ajtai and Szemerédi [@AjSz]. This theorem states that for each $\epsilon>0$ there is $N(\epsilon)$ such that, for $N \geq N(\epsilon)$, any subset $S$ of the $N \times N$ grid with $|S| \geq \epsilon N^2$ contains a corner, i.e., the vertices $(x,y)$, $(x+d,y)$, $(x,y+d)$ of an isosceles right triangle. This in turn gives a simple proof of Roth’s theorem [@Ro] that every subset of the integers of positive upper density contains a three-term arithmetic progression.
The best known upper bound for the corners theorem, due to Shkredov [@Sh], states that any subset of the $N \times N$ grid with no corner has cardinality at most $N^2/(\log \log N)^{c}$, where $c>0$ is an absolute constant. Graham and Solymosi [@GrSo] proved a better bound for the Ramsey-type analogue of the corners theorem. They showed that there is $c>0$ such that any coloring of the $N \times N$ grid with fewer than $c\log \log N$ colors contains a monochromatic corner. This in turn implies a double-exponential bound for the van der Waerden number $W(3; r)$ (the smallest $N$ such that any $r$-coloring of the set $\{1, 2, \dots, N\}$ contains a monochromatic three-term arithmetic progression), but this is weaker than the exponential bound that follows from the best quantitative estimate for Roth’s theorem [@B15; @San11].
Just as the triangle removal lemma implies the corners theorem, Corollary \[cor:col\] implies the Graham–Solymosi bound on monochromatic corners in colorings of the grid. Indeed, consider an $r$-coloring of the $N \times N$ grid with $N>(8r)^{2^{r+3}}$. Let $V_0$ denote the set of $2N-1$ lines with slope $-1$ that each contain at least one of the grid points, $V_1$ denote the set of $N$ vertical lines that each contain $N$ of the grid points and $V_2$ denote the set of $N$ horizontal lines that each contain $N$ of the grid points. Consider the tripartite graph $G$ with parts $V_0,V_1,V_2$, where two lines are adjacent if and only if they intersect in one of the points of the $N \times N$ grid, and color the edge between them the color of their intersection point. Note that every line in $V_1$ intersects every line in $V_2$ in the grid, so $V_1$ is complete to $V_2$ in the graph. Moreover, the three lines passing through any grid point form a monochromatic triangle and this collection of $N^2$ monochromatic triangles gives an edge-partition of $G$. Therefore, by Corollary \[cor:col\], $G$ contains a monochromatic diamond. This, in turn, implies that the coloring of the grid must contain a monochromatic corner.
The proof of Theorem \[main\] follows from iterating the following simple lemma.
\[mainlemma\] In every $r$-edge coloring of a tripartite graph $G$ with parts $V_0,V_1,V_2$, where $|V_1|=|V_2|=n$, $|V_0|=cn$ and the number $m$ of edges between $V_1$ and $V_2$ is at least $n^2/2$, such that there are fewer than $\delta n^3$ monochromatic triangles and there is a collection of edge-disjoint monochromatic triangles which cover the edges between $V_1$ and $V_2$, there are subsets $V_1' \subset V_1$ and $V_2' \subset V_2$ with $|V_1'|=|V_2'| \geq \frac{n}{4cr}$ and a color such that the number of edges of that color between $V_1'$ and $V_2'$ is at most $4\delta n^2$.
Let $\mathcal{T}$ denote a collection of edge-disjoint monochromatic triangles which cover the edges of $G$ between $V_1$ and $V_2$, so $|\mathcal{T}|=m$. Partition $V_0=A \cup B$, where $v \in V_0$ is in $A$ if the number of triangles in $\mathcal{T}$ containing $v$ is at least $\frac{m}{2|V_0|}$. The number of triangles in $\mathcal{T}$ containing a vertex in $B$ is less than $|B|\frac{m}{2|V_0|} \leq \frac{m}{2}$. Hence, at least half of the triangles in $\mathcal{T}$ contain a vertex from $A$.
Since each vertex in $A$ is in at most $n$ triangles from $\mathcal{T}$, we have $|A| \geq (m/2)/n \geq n/4$. If every vertex in $A$ is in at least $4\delta n^2$ monochromatic triangles, the total number of monochromatic triangles is at least $|A| 4\delta n^2 \geq \delta n^3$. Hence, there is a vertex $v \in A$ in fewer than $4\delta n^2$ monochromatic triangles. Since $v \in A$, there is a color, say red, such that $v$ is in at least $\frac{m}{2|V_0|r} \geq \frac{n^2}{4cnr} = \frac{n}{4cr}$ monochromatic red triangles from $\mathcal{T}$. For $i=1,2$, let $V_i'$ denote those vertices in $V_i$ which are in a monochromatic red triangle from $\mathcal{T}$ with vertex $v$, so $|V_1'|=|V_2'| \geq \frac{n}{4cr}$. Since $v$ is in fewer than $4\delta n^2$ monochromatic triangles and $v$ is complete in red to $V_1'$ and $V_2'$, the number of red edges between $V_1'$ and $V_2'$ is at most $4\delta n^2$, which completes the proof.
[**Proof of Theorem \[main\]:**]{} Let $f(n,r,q,s)$ be the minimum number of monochromatic triangles in an $r$-edge coloring of a tripartite graph $G$ with parts $V_0,V_1,V_2$, where $|V_1|=|V_2| \geq n$ and $|V_0| \leq q$, such that the number of edges between $V_1$ and $V_2$ is at least $|V_1|^2-s$ and the edges between $V_1$ and $V_2$ can be covered by edge-disjoint monochromatic triangles. In Theorem \[main\], $V_1$ is complete to $V_2$ and $|V_0|=cn$, so we are trying to prove a lower bound on $f(n,r,cn,0)$, namely $$f(n,r,cn,0) \geq (4cr)^{-2^{r+3}}n^3.$$
When $r=1$, Lemma \[mainlemma\] implies that if $G$ is a tripartite graph $G$ with parts $V_0,V_1,V_2$, where $|V_1|=|V_2|=n$, $|V_0|=cn$ and $e(V_1,V_2) \geq n^2/2$, such that $G$ contains at most $\delta n^3$ triangles and there is a collection of edge-disjoint triangles in $G$ which cover the edges between $V_1$ and $V_2$, then there are subsets $V_1' \subset V_1$ and $V_2' \subset V_2$ with $|V_1'|=|V_2'| \geq \frac{n}{4c}$ for which $e(V_1',V_2') \leq 4\delta n^2$. In particular, if $e(V_1,V_2)=n^2-s$, then $$4 \delta n^2 \geq e(V_1',V_2') \geq |V_1'||V_2'|-s \geq \left(\frac{n}{4c}\right)^2-s,$$ and hence $\delta n^3 \geq \frac{n^3}{64c^2}-\frac{ns}{4}$. Substituting $q=cn$, we get the bound $$f(n,1,q,s) \geq \frac{n^5}{64q^2}-\frac{ns}{4}.$$
When $s<n^2/2$, Lemma \[mainlemma\] implies that by deleting the edges of the sparsest color between $V_1'$ and $V_2'$ and letting $f_0=f(n,r,q,s)$, we have $$f_0> f\left(\frac{n^2}{4qr},r-1,q,s+4f_0/n\right).$$ Let $n_i=\frac{n^{2^i}}{(4qr)^{2^i-1}}$, $s_0=s$ and, for $i\geq 1$, $s_i=s_{i-1}+4f_0/n_{i-1}$, so $s_i<s+4f_0\sum_{j=0}^{i-1}\frac{1}{n_{j}} <s+5f_0/n_{i-1}$. After $i$ iterations, if $s_{i-1}<n_{i-1}^2/2$, we get $$f_0> f\left(n_i,r-i,q,s_i\right).$$
We set $s=0$ and $q=cn$. We use the above inequalities to compute a lower bound on $f_0=f(n,r,cn,0)$. Either we have $s_{i-1} \geq n_{i-1}^2/2$ for some $i<r$ or $f_0 \geq f\left(n_{r-1},1,cn,s_{r-1}\right)$. In the first case, we have $5f_0/n_{i-2} \geq s_{i-1} \geq n_{i-1}^2/2$ and, therefore, $$f_0 \geq \frac{n_{i-2}n_{i-1}^2}{10} \geq \frac{n_{r-3}n_{r-2}^2}{10} \geq (4cr)^{-2^r}n^3.$$ In the second case, we have $$f_0 \geq f\left(n_{r-1},1,cn,s_{r-1}\right) \geq \frac{n_{r-1}^5}{64q^2}-\frac{ns_{r-1}}{4} \geq \frac{n^3}{(4cr)^{5(2^{r-1}-1)}64c^2}-\frac{5}{4}f_0\frac{n}{n_{r-2}},$$ in which case we get $$f_0 \geq \frac{1}{2}\frac{n_{r-2}}{n} \frac{n^3}{(4cr)^{5(2^{r-1}-1)}64c^2} \geq (4cr)^{-2^{r+3}}n^3.$$ In either case, we have the desired inequality, which completes the proof of Theorem \[main\].
[**Acknowledgements.**]{} We would like to thank the anonymous referees for their helpful remarks and Zoltan Füredi for bringing the reference [@F91] to our attention.
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[^1]: Mathematical Institute, Oxford OX2 6GG, United Kingdom. Email: [david.conlon@maths.ox.ac.uk]{}. Research supported by a Royal Society University Research Fellowship.
[^2]: Department of Mathematics, Stanford University, Stanford, CA 94305. Email: [fox@math.mit.edu]{}. Research supported by a Packard Fellowship, by NSF Career Award DMS-1352121 and by an Alfred P. Sloan Fellowship.
[^3]: Department of Mathematics, ETH, 8092 Zurich, Switzerland. Email: [benjamin.sudakov@math.ethz.ch]{}. Research supported by SNSF grant 200021-149111.
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abstract: 'We study transport of interacting electrons in a low-dimensional disordered system at low temperature $T$. In view of localization by disorder, the conductivity $\sigma(T)$ may only be non-zero due to electron-electron scattering. For weak interactions, the weak-localization regime crosses over with lowering $T$ into a dephasing-induced “power-law hopping". As $T$ is further decreased, the Anderson localization in Fock space crucially affects $\sigma(T)$, inducing a transition at $T=T_c$, so that $\sigma(T<T_c)=0$. The critical behavior of $\sigma(T)$ above $T_c$ is $\ln\sigma(T)\propto - (T-T_c)^{-1/2}$. The mechanism of transport in the critical regime is many-particle transitions between distant states in Fock space.'
author:
- 'I.V. Gornyi$^{1,*}$'
- 'A.D. Mirlin$^{1,2,\dagger}$'
- 'D.G. Polyakov$^{1,*}$'
date: 'May 20, 2005'
title: 'Interacting electrons in disordered wires: Anderson localization and low-$T$ transport'
---
In a pathbreaking paper [@anderson58] Anderson demonstrated that a quantum particle may become localized by a random potential. In particular, in non-interacting systems of one-dimensional (1D) or two-dimensional (2D) geometry even weak disorder localizes all electronic states [@abrahams79], thus leading to the exactly zero conductivity, $\sigma(T)=0$, whatever temperature $T$. A non-zero $\sigma(T)$ in such systems may only occur due to inelastic scattering processes leading to dephasing of electrons. Two qualitatively different sources of dephasing are possible: (i) scattering of electrons by external excitations (in practice, phonons) and (ii) electron-electron (e-e) scattering. In either case, at sufficiently high temperatures, the dephasing rate $\tau_\phi^{-1}$ is high, so that the localization effects are reduced to a weak-localization (WL) correction to the Drude conductivity. This correction behaves as $\ln\tau_{\phi}$ in 2D and as $\tau_{\phi}^{1/2}$ in quasi-1D (many-channel wire) systems [@aa], and thus diverges with lowering $T$, signaling the occurrence of the strong localization (SL) regime. This prompts a question as to how the system conducts at low $T$.
For the case of electron-phonon scattering the answer is well known. The conductivity is then governed by Mott’s variable-range hopping (VRH) [@mott], yielding $\sigma(T)\propto \exp\{-(T_0/T)^{\mu}\}$ with $\mu = 1/(d+1)$, where $d$ is the spatial dimensionality. In the presence of a long-range Coulomb interaction, the Coulomb gap in the tunneling density of states modifies the VRH exponent, $\mu={1\over 2}$ [@efros-shklovskii].
But what is the low-$T$ behavior of $\sigma(T)$ if the electron-phonon coupling is negligibly weak and the only source of the inelastic scattering is the e-e interaction? Our purpose here is to solve this long-standing fundamental problem, which is also of direct experimental relevance; see, e.g., Refs. [@gershenson] and [@hsu95; @khondaker99], where the crossover from WL to SL with lowering $T$ was studied for 1D and 2D systems, respectively. For definiteness, we concentrate on the case of a many-channel 1D system with a short-range interaction. Our results are, however, more general (including single-channel wires, 2D systems, Coulomb interaction), as we discuss in the end of the paper.
It was proposed in [@fleishman78] that the e-e interaction by itself is sufficient to induce VRH at low $T$. This idea was widely used for interpretation of experimental [@khondaker99; @shlimak99] and numerical [@berkovits99] results on 2D systems. Further, Ref. used bosonization to study the problem in 1D and concluded that transport is of VRH character. These results are, however, in conflict with the argument [@fleishman80] – supported by our analysis – that elementary hops in the low-$T$ limit are forbidden for $d<3$ even for the case of long-range ($1/r$) Coulomb interaction, since energy conservation cannot be respected when an electron attempts a real transition by exciting an electron-hole pair [@malinin04]. The situation is particularly interesting in 1D and 2D, where no mobility edge exists, activation to which otherwise might give $\sigma (T)\neq 0$. If neither VRH nor activation, then what?
Let us now specify the model. We consider a many-channel weakly disordered wire, so that the relevant length scales satisfy $k_F^{-1}
\ll l \ll \xi$, where $k_F$ is the Fermi momentum, $l$ the mean free path, and $\xi\sim \pi\nu D$ the localization length ($\nu$ is the density of states per unit length and $D$ the diffusion constant) [@efetov; @adm-review]. The corresponding energy scales are the Fermi energy $E_F$, the elastic scattering rate $\tau^{-1}$, and the level spacing in the localization volume, $\Delta_\xi=1/\nu\xi$, with $E_F \gg \tau^{-1} \gg
\Delta_\xi$. We will assume a short-range interaction $U(\bf r-\bf r')$ between electrons, characterized by a dimensionless coupling $\alpha=\nu\tilde{U}(0)$, where $\tilde{U}({\bf q})$ is the Fourier transform of $U({\bf r})$. We assume that $\alpha\ll 1$, which yields a richer behavior of $\sigma(T)$ and allows better understanding of underlying physics; the case $\alpha\sim 1$ (as well as Coulomb interaction) will be discussed in the end.
At sufficiently high $T$, the conductivity $\sigma(T)\simeq\sigma_{\rm D}+\Delta\sigma_{\rm WL}+\Delta\sigma_{\rm AA}$ is close to its Drude value $\sigma_{\rm D}$, with quantum corrections related to the weak localization ($\Delta\sigma_{\rm WL}$) and to interplay of interaction and disorder (Altshuler-Aronov contribution $\Delta\sigma_{\rm AA}$) [@aa], && [|\_[WL]{}|\_[D]{}]{} \~\_[l\_\^[-1]{}]{}[dqDq\^2]{} \~[l\_]{} \~([\_\^2 T]{})\^[1/3]{}. \[e1\] Here we used the result for the dephasing rate length $l_\phi=(D\tau_\phi)^{1/2}$ due to e-e interaction [@aa], \[e3\] \_\^[-1]{} \~\^2 T \_[l\_\^[-1]{}]{} [dq D q\^2]{} \~\^2 T [l\_]{}. The WL correction grows with lowering $T$ and finally becomes strong ($\Delta\sigma_{\rm WL}/\sigma_0\sim 1$) when $l_\phi$ reaches $\xi$, or, equivalently, when $\tau_\phi^{-1} \sim \Delta_\xi$. This happens at $T\sim T_1 = \alpha^{-2}\Delta_\xi$, marking the beginning of the SL regime. The interaction-induced correction $\Delta\sigma_{\rm AA}/\sigma_{\rm D}\sim
(\alpha^2\Delta_\xi/T)^{1/2}$ remains small at $T\sim T_1$ and thus is of no relevance in the present context. (For $\alpha\sim 1$, $\Delta\sigma_{\rm AA}$ is of order $\sigma_{\rm D}$ at $T\sim T_1$ and does not lead to any qualitative changes either.) The subject of our interest is $\sigma(T)$ for $T<T_1$.
In fact, SL does not necessarily mean $\sigma(T)$ is exponentially small. Specifically, in the high-$T$ part of the SL regime the transport mechanism – we will call it power-law hopping (PLH) – is analogous to the one identified in [@gogolin75] for the case of inelastic electron-phonon scattering, i.e., hopping over length $\sim\xi$ in time $\sim\tau_\phi$. In other words, the dephasing time $\tau_\phi$ serves in this regime as a lifetime of localized states, which adds an imaginary part $i/2\tau_\phi$ to the single-particle energies $\epsilon_\alpha$. This yields \[e4\] (T) \~\_[ac]{}()|\_[= i/\_]{}\~e\^2\^2/\_, where $\sigma_{\rm ac}(\Omega)$ is the zero-$T$ conductivity of noninteracting electrons at frequency $\Omega$. The crucial point here is that $\tau_\phi$ in this SL regime can still be calculated via Fermi’s golden rule, as we are going to show. The lowest-order decay process of a localized state $|\alpha\rangle$ is the transition to a three-particle state – two electrons $|\beta\rangle$, $|\gamma\rangle$ and a hole $|\delta\rangle$, all located within a distance $\sim \xi$, Fig. \[fig1\]a. The corresponding matrix element of the interaction is \[e5\] V\_ V \~\_; |\_-\_|, |\_-\_|\_and decays fast for larger energy differences (cf. results for a metallic sample [@agkl; @adm-review] with the dimensionless conductance set to $g\sim 1$). Since energies of all the relevant single-particle states are within the window of width $\sim T$, the level spacing of three-particle states to which the original state $|\alpha\rangle$ is coupled according to (\[e5\]), reads \[e6\] \_\^[(3)]{} \~\_\^2/T. Using (\[e5\]), (\[e6\]) and the golden rule, we find \[e7\] \_\^[-1]{} \~|V|\^2/\_\^[(3)]{} \~\^2 T. Note that this result could also be obtained from (\[e3\]) if $\xi^{-1}$ is used as the infrared cutoff, as appropriate for the SL regime. The condition of validity of the golden-rule calculation is $V \gg \Delta_\xi^{(3)}$, or, equivalently, $\tau_\phi^{-1} \gg \Delta_\xi^{(3)}$. This introduces a new temperature scale $T_3=\alpha^{-1}\Delta_\xi$, so that the PLH regime is restricted to the range $T_3 \ll T \ll T_1$. Combining (\[e4\]) and (\[e7\]), we get the conductivity in this regime, \[e8\] (T) \~e\^2\^2 \^2 T \~e\^2T/T\_1.
![Diagrams for the golden rule (a) and higher-order decay amplitude (b).[]{data-label="fig1"}](GR.eps){width="8cm"}
What happens below $T_3$? Simple hops on a distance $\sim\xi$ are then not sufficient to delocalize electrons. Increasing the distance $r$ of a hop does not help: the matrix element vanishes exponentially with $r/\xi$, while the level spacing $\Delta^{(3)}$ only as a power law. We thus have to analyze higher-order processes by exploring the structure of the theory in the many-body Fock space, similarly to the ideas developed for the problem of a quasiparticle decay in quantum dots [@agkl; @mf97; @silvestrov]. The process of $n$-th order represents a transition with excitation of $n$ electron-hole pairs, $|\alpha\rangle
\longrightarrow |B^{(n)}\rangle\equiv |\beta_0\beta_1\bar{\beta}_1
\ldots\beta_n\bar{\beta}_n\rangle$ \[with the energy difference $\lesssim
\Delta_\xi$ for each pair in view of (\[e5\])\], see Fig. \[fig2\]b. Let us estimate the dimensionless coupling of $n$-th order (ratio of the matrix element to the level spacing of final states), $V^{(n)}/\Delta^{(2n+1)}$, which is the $n>1$ generalization of the ratio $V/\Delta_\xi^{(3)}$ considered above. In this estimate, it will be sufficient for us to keep track of factors of the type $n^n$ (or $n!$) and $(T/T_3)^n$. Factors of the type $c^n$, where $c\sim 1$, will be unessential and thus neglected.
Let us assume that the set of states $B^{(n)}$ is spread over the length $m\xi$, with $\sim n/m$ pairs in each box of length $\xi$; later we optimize with respect to $m$ (for $1\lesssim m
\lesssim n$). The corresponding level spacing can be estimated as \[e9\] \_[m]{}\^[(2n+1)]{}/\_ \~\[(n/m) \_/T \]\^n. The matrix element $V_{\alpha;\beta_1,\ldots,\beta_{n+1}}\equiv V^{(n)}$ is given by \[e10\] V\^[(n)]{} = \_[diagrams]{} \_[\_1,…\_[n-1]{}]{} V\_1 \_[i=1]{}\^[n-1]{} [V\_[i+1]{}E\_i-\_[\_i]{}]{}, where the matrix element $V_i$ corresponds to the $i$-th interaction line, $\gamma_i$ are the virtual states corresponding to internal lines, and $E_i$ the corresponding energy variable which can be expressed as a linear superposition of $\epsilon_{\alpha_i}$ using the energy conservation. We need to take into account only those contributions where all states forming each matrix element are within a distance $\sim\xi$ from each other, so that $V_i\sim\alpha\Delta_\xi$. Therefore, the summation over each $\gamma_i$ is effectively taken over a single localization domain, and we can replace it by taking the “optimal” $\gamma_i$, with $|E_i-\epsilon_{\gamma_i}|\sim\Delta_\xi$. We thus get \[e11\] V\^[(n)]{}/\_\~\[M\_m\^[(n)]{}\]\^[1/2]{} (\_)\^n, where $M_m^{(n)}$ is the number of diagrams contributing to the amplitude of the transition $|\alpha\rangle \to
|B^{(n)}\rangle$. These contributions have random signs, hence the factor $[M_m^{(n)}]^{1/2}$.
To find $M_m^{(n)}$, we first calculate the number of topologically different diagrams, $D^{(n)}$, which satisfies the recursion relation \[e12\] D\^[(n)]{}=\_[n\_1+n\_2+n\_3=n-1; n\_[1,2,3]{}0]{} D\^[(n\_1)]{}D\^[(n\_2)]{}D\^[(n\_3)]{}, with the initial condition $D^{(0)}=1$. It is easy to show that its solution increases only as $D^{(n)}\sim a^n$, with $a\sim 1$, so that with the required accuracy it can be replaced by unity. Thus $M_m^{(n)} \sim A_m^{(n)}$, the number of allowed permutations of the set $B^{(n)}$ over the final-state lines. To estimate $A_m^{(n)}$, we notice that only electron-hole pairs within the same (or nearby) localization volume can be interchanged, which yields $A_m^{(n)}\sim [(n/m)!]^{m} \sim (n/m)^{n}$. Combining this with (\[e9\]), (\[e11\]), we finally get \[e14\] [V\^[(n)]{} / \^[(2n+1)]{}]{}\~ \^n.
Let us now analyze the result. First of all, for given $n$ the most favorable case is $m\sim n$, which corresponds to “ballistic” paths. In such a process, an electron makes a many-body transition over the distance $n\xi$, leaving behind a string of $n$ particle-hole pairs, as illustrated in Fig. \[fig2\]. Second, $V^{(n)} / \Delta^{(2n+1)}$ increases with $n$ at sufficiently high $T$ and decreases with $n$ (thus remaining small for all $n$) at low $T$. Therefore, at low $T$ the higher-order processes do not help a localized single-particle state to decay, so that $\sigma(T)$ is exactly zero. In contrast, at high $T$, the increase of the coupling (\[e14\]) with $n$ guarantees that the golden-rule calculation performed for the WL and PLH regimes is not spoiled by the higher-order effects. The temperature $T_c$ of the transition into the zero-conductivity regime can be immediately estimated from (\[e14\]), $T_c\sim \Delta_\xi/\alpha$. In fact, (\[e14\]) misses a $\ln\alpha^{-1}$ factor recovered in a more accurate treatment below, Eq. (\[e16\]).
![Scheme of a high-order ballistic process: the string of particle-hole pairs (black and white circles). The initial state and intermediate virtual states ($\gamma_i$) are shown in grey.[]{data-label="fig2"}](quarks.eps){width="8.2cm"}
What helps us to analyze the critical behavior at the transition is that the structure of the theory, when restricted to the optimal (“ballistic”) paths, reduces essentially to that of the Anderson model on the Bethe lattice – a tree with a fixed branching number. Indeed, consider the process shown in Fig. \[fig2\]: an electron hops to an adjacent localization volume, creating an electron-hole pair, then to the next one and so forth. Clearly, the density of final states increases at each step by the same factor \[e15\] K\~\_/ \_\^[(3)]{} \~T/\_, which is the branching number of the Bethe lattice. The Bethe-lattice character of the problem can also be inferred from the exponential dependence of the coupling (\[e14\]) on $n$ at $m=n$. This should be contrasted with the opposite limit, $m=1$, corresponding to the case of a quantum dot, when (\[e14\]) contains an additional $n^{-n/2}$ factor. The latter is related to a decrease of the effective branching number with $n$ in this case, as was noticed in [@mf97; @silvestrov]. In other words, the mapping on the Bethe lattice model, oversimplified for a quantum dot (for which it was originally proposed in [@agkl]), turns out to work perfectly in the case of localized states in a wire (or, more generally, in a non-restricted geometry), where going to higher generations in Fock space can be accompanied by the exploration of new regions in real space.
We can now use the results for the Anderson transition on the Bethe lattice that has been studied extensively [@abou73; @efetov; @zirnbauer86; @mf94; @mf97]. For a large branching number $K$ the equation for the transition point reads \[e16\] /V = 4 K, where $V$ is the hopping matrix element, and $\Delta$ is the mean level spacing of states of generation $n+1$ coupled to a given state of generation $n$. Using (\[e5\]), (\[e15\]), and $\Delta=\Delta_\xi^{(3)}$, we find the transition temperature, \[e17\] T\_c \~\_/\^[-1]{}, so that $T_c\sim T_3/\ln \alpha^{-1}$. The critical behavior of $\sigma(T)$ above $T_c$ is governed by that of the decay rate $\tau_\phi^{-1}$, which translates into the imaginary part of the self-energy ${\rm Im}\:\Sigma$ for the Bethe-lattice problem. The critical behavior of the latter was found in [@mf94; @mf97], yielding \[e18\] (T) {- c\_\^[-1/2]{} }, with $c_\alpha \sim \ln\alpha^{-1}$ for $\alpha\ll 1$. Near the transition, the local density of states on the Bethe lattice acquires an increasingly more sparse “spatial” structure [@mf94; @mf97], so that the transport is governed by processes connecting remote states in Fock space, Fig. \[fig2\], which implies a glassy character of the system. When $T\to T_c$, the length of the particle-hole strings diverges. We term the low-$T$ phase “Anderson-Fock glass" (AFG), since its physics is governed by the Anderson localization in Fock space.
{width="8cm"}
\[fig3\]
The found behavior of $\sigma(T)$ is illustrated in Fig. \[fig3\]. If $\alpha\sim 1$, all scales become of the same order, $T_1\sim T_3\sim
T_c\sim \Delta_\xi$, and the range of PLH disappears. In realistic systems, weak coupling to phonons will lead to PLH (for $\alpha\ll 1$) and VRH below $T_c$, but with a small prefactor, so that the transition at $T_c$ should be well observable. Note the peculiar character of the transition: not only the exponential critical behavior (\[e18\]) but also that the ordered (metallic) phase corresponds to $T>T_c$. An apparent conflict with the Mermin-Wagner theorem is related to the unconventional (functional) nature of the order parameter for Anderson localization [@efetov; @zirnbauer86; @mf94].
Before closing the paper, let us briefly mention a few extensions of our results [@tobe].
\(i) [*1D: Single channel.*]{} We have recently shown [@luttinger-dephasing] that the notion of WL and dephasing are also applicable to a disordered Luttinger liquid and calculated $\sigma(T)$ in the WL regime. The low-$T$ results presented above can then be easily generalized to the single-channel case. An important difference is that $\tau$ (and thus, $\xi$) is strongly renormalized by interaction, $\tau(T)/\tau\sim
(T/E_F)^{\alpha}$.
\(ii) $d>1$. Generalization to 2D systems is straightforward, with $\xi$ depending exponentially on disorder, $\xi\propto \exp(2\pi^2\nu D)$. Our consideration is also applicable to 3D Anderson insulators; however, in this case $\sigma(T<T_c)$ will not be exactly zero but rather determined by the activation above the single-particle mobility edge.
\(iii) [*Coulomb interaction.*]{} For 1D and 2D geometry, the transition survives also for $1/r$ Coulomb interaction, since correlated hops of two electrons separated by a large distance $r\gg\xi$ will not delocalize them. Indeed, the corresponding matrix elements decrease with $r$ as $V_{\alpha\beta\gamma\delta}\propto 1/r^3$, which is not compensated by the increase $\propto r^d$ of the density of final states for $d<3$ and thus does not help. On the other hand, in 3D such processes will lead to delocalization for any $T$ [@fleishman80]; the behavior of $\sigma(T)$ in this case requires a separate study.
\(iv) [*Creep.*]{} Our approach can be used to analyze the non-linear conductivity $\sigma(E)$ at weak electric field $E$.
In conclusion, we have studied the conductivity of interacting electrons in a disordered quantum wire; very similar results hold for a 2D system. In contrast to a popular belief, the e-e interaction is not sufficient to support the VRH transport in the low-$T$ limit. Instead, the system undergoes a localization transition at temperature $T_c$, below which $\sigma(T)=0$ (assuming vanishing coupling to phonons). We have shown that the conductivity vanishes as $\ln\sigma(T)\propto - (T-T_c)^{-1/2}$ as $T\to T_c$. Transport in the critical regime is governed by many-particle transitions between distant states in Fock space, corresponding to the formation of long strings of electron-hole pairs. For weak interactions, this Anderson-Fock glass phase is separated from the WL regime by an intermediate temperature regime of power-law hopping.
We thank V. Cheianov, D. Maslov, G. Minkov, T. Nattermann, and B. Shklovskii for valuable discussions. We are particularly grateful to I. Aleiner, B. Altshuler, and D. Basko for very useful discussions and criticism of the earlier version of this work (cond-mat/0407305v1). This helped us to correct an error in counting of the number of diagrams $M_m^{(n)}$ \[which yielded a spurious double exponential tail of $\sigma(T)$ in the AFG phase\], whose elimination gave a true phase transition at $T=T_c$, as also found in [@aleiner]. The work was supported by SPP “Quanten-Hall-Systeme” and CFN of the DFG and by RFBR.
[1]{}
Also at A.F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia.
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|
---
abstract: 'We consider a linear power flow model with interval-bounded nodal power injections and limited line power flows. We determine the minimal number of power injections to control based on a minimal set of measurements, such that the overall system is feasible for all assignments of the non-controlled power injections. For the important case where the possible measurements are the nodal power injections, we show that the problem can be solved efficiently as a mixed-integer linear program (MILP). When also line power flows are considered as potential measurements, we derive an iterative, greedy algorithm that provides a feasible, but potentially conservative solution. We apply the developed algorithms to both a small microgrid and a modified version of the IEEE 118 bus test power system. We show that in both cases a sparse solution in terms of the number of required controllers and measurements can be obtained. Moreover, the number of required measurements can be reduced significantly if line flow measurements are considered additionally to nodal power injections.'
author:
-
bibliography:
- 'IEEEabrv.bib'
- 'root.bib'
title: |
On the minimal set of controllers and sensors\
for linear power flow
---
=1
Controllability, observability, power flow, resilience
|
**Comment on “Modified Coulomb Law in a Strongly Magnetized Vacuum”**\
In a recent Letter [@Shabad:2007xu], Shabad and Usov studied the electric potential of a charge placed in a strong magnetic field $B\gg B_0\equiv m^2/e$ ($m$ is the electron mass and $e$ is its charge), as modified by the vacuum polarization. According to these authors, in the limit $B\to\infty$ the modified potential becomes a Dirac delta function plus a regular background. With this potential they showed that the unboundedness from below of the nonrelativistic hydrogen spectrum is removed. In this Comment, we point out that Shabad and Usov used an incorrect calculation to arrive at their, otherwise, correct result and conclusion.
It is firmly established that the magnetic catalysis of chiral symmetry breaking [@Gusynin:1994re] is a universal phenomenon. The general result states that a strong magnetic field leads to the generation of a dynamical fermion mass even at the weakest attractive interaction between fermions. The realization of this phenomenon in the chiral limit in QED has been studied in detail in the literature [@Gusynin:1995gt; @Gusynin:1998zq; @Leung:2005yq]. In particular, in Ref. [@Leung:2005yq], we obtained an asymptotic expression for the dynamical fermion mass, reliable in the weak coupling regime and the strong field approximation.
The dynamical fermion mass in massless QED in a strong magnetic field, $m_\mathrm{dyn}$, is given by [@Leung:2005yq] $$m_\mathrm{dyn}\simeq\alpha\,
\exp\left(-\frac{\pi}{\alpha\log\alpha^{-1}}\right)\,\sqrt{eB},$$ where $\alpha=e^2/4\pi\simeq 1/137$ is the fine structure constant. In a moderately strong magnetic field $B\gtrsim B_0$, the ratio $m_\mathrm{dyn}/m$ is vanishingly small hence the calculation of Ref. [@Shabad:2007xu] with the nonperturbative effect of magnetic catalysis completely ignored is justified. In an infinitely strong magnetic field, however, the dynamical fermion mass $m_\mathrm{dyn}$ approaches infinity linearly with $\sqrt{eB}$. The universal nature of the magnetic catalysis of chiral symmetry breaking implies that in realistic QED with massive electrons, the electron in an infinitely strong magnetic field will acquire an effective dynamical mass that is much larger than $m$. As a result, the calculation of Ref. [@Shabad:2007xu] is not correct in the limit $B\to\infty$.
Taking into consideration the nonperturbative effect of dynamical mass generation in realistic QED in a strong magnetic field, we find the vacuum polarization that is quoted in Ref. [@Shabad:2007xu] to be modified by $$\kappa_2(k_3^2,k_\perp^2)=-\frac{2\alpha}{\pi}\,eB\,\exp\left(-
\frac{k_\bot^2}{2eB}\right)\,T\left(\frac{k_3^2}{4m_\ast^2}\right),$$ where the explicit dependence on $eB$ has been restored. In Eq. (2), $m_\ast$ is the effective electron mass in a strong magnetic field, yet to be determined by a gap equation that couples to $\kappa_2$ through the full photon propagator. At this point it might appear that the only way to proceed is to first solve for $m_\ast$, which is a numerically intensive task. Fortunately, this is not the case. Since, as can be seen from Eq. (1), the effect of magnetic catalysis increases with increase of $B$, it is conceivable that $m_\ast$ has the asymptotic behavior $m_\ast\approx m_\mathrm{dyn}\to\infty$ as $B\to\infty$. The key point is to note that while both $m_\ast$ and $B$ approach infinity in the limit $B\to\infty$, there is still a wide separation of scales in this problem, namely, $m_\ast\ll\sqrt{eB}$. Furthermore, in the limit $B\to\infty$ the ratio of these two scales, $m_\ast/\sqrt{eB}\ll 1$, is a constant independent of the magnetic field. Indeed, as shown in Ref. [@Leung:2005yq], the wide separation of scales $m_\mathrm{dyn}\ll\sqrt{eB}$ is the underlying physics behind the fact that Eq. (1) is reliable for weak couplings and strong fields.
Based on the above analysis, the arguments of Ref. [@Shabad:2007xu] that lead to an isotropic Yukawa law and, consequently, to a Dirac delta function plus a regular background in the limit $B\to\infty$ can be rendered valid, provided that we make the two corrections: (i) the electron Compton length, $m^{-1}$, is changed to the effective electron Compton length in a strong magnetic field, $m_\ast^{-1}$; (ii) the separation of scales $L_B\ll m^{-1}$ is changed to $L_B\ll m_\ast^{-1}$, where $L_B=1/\sqrt{eB}$ is the electron Larmor length. The reason that Shabad and Usov obtained the correct result and conclusion is because the separation of scales $L_B\ll m_\ast^{-1}\ll m^{-1}$ holds in the limit $B\to\infty$.
*Note added in proof*. The asymptotic behavior of $m_\ast$ has been numerically verified in a recent study [@Wang:2007sn].
Shang-Yung Wang\
\
PACS numbers: 12.20.-m
[9]{}
A.E. Shabad and V.V. Usov, Phys. Rev. Lett. **98**, 180403 (2007). V.P. Gusynin, V.A. Miransky, and I.A. Shovkovy, Phys. Rev. Lett. **73**, 3499 (1994); Phys. Lett. B **349**, 477 (1995). V.P. Gusynin, V.A. Miransky, and I.A. Shovkovy, Phys. Rev. D **52**, 4747 (1995); Nucl. Phys. **B462**, 249 (1996); D.-S. Lee, C.N. Leung, and Y.J. Ng, Phys. Rev. D **55**, 6504 (1997); E. J. Ferrer and V. de la Incera, Phys. Rev. D **58**, 065008 (1998). V.P. Gusynin, V.A. Miransky, and I.A. Shovkovy, Phys. Rev. Lett. **83**, 1291 (1999); Nucl. Phys. **B563**, 361 (1999); Phys. Rev. D **67**, 107703 (2003); A.V. Kuznetsov and N.V. Mikheev, Phys. Rev. Lett. **89**, 011601 (2002). C.N. Leung and S.-Y. Wang, Nucl. Phys. **B747**, 266 (2006); Ann. Phys. (N.Y.) **322**, 701 (2007). S.-Y. Wang, arXiv:0709.4427 \[hep-ph\].
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abstract: 'In this paper, we introduce a flexible notion of safety verification for nonlinear autonomous systems by measuring how much time the system spends in given unsafe regions. We consider this problem in the particular case of nonlinear systems with a polynomial dynamics and unsafe regions described by a collection of polynomial inequalities. In this context, we can quantify the amount of time spent in the unsafe regions as the solution to an infinite-dimensional linear program (LP). This LP measures the volume of the unsafe region with respect to the occupation measure of the system trajectories. Using Lasserre hierarchy, we approximate the solution to the infinite-dimensional LP using a sequence of finite-dimensional semidefinite programs (SDPs). The solutions to the SDPs in this hierarchy provide monotonically converging upper bounds on the optimal solution to the infinite-dimensional LP. Finally, we validate the performance of our framework using numerical simulations.'
author:
- 'Ximing Chen, Shaoru Chen, Victor M. Preciado[^1]'
title: Safety Verification of Nonlinear Autonomous System via Occupation Measures
---
Introduction {#sec:introduction}
============
Our ability to provide safety certificates about the behavior of complex systems is critical in many engineering applications, such as air traffic control [@hu2003probabilistic], life support devices [@glavaski2005safety], motion planning in robotics manipulations [@ziegler2010fast], and connected autonomous vehicles [@althoff2007safety; @althoff2009model]. Although safety verification is a mature area with many success stories [@bemporad2000optimization; @chutinan2003computational], the verification of nonlinear dynamical systems over nonconvex unsafe regions remains a challenging problem [@bollobas1997volume; @dyer1988complexity]. In the past decades, various solutions have been proposed to verify the safety of dynamical systems. The solution approaches often fall into the following two categories: (i) reachable set methods [@anai2001reach; @tomlin2003computational; @asarin2003reachability], and (ii) Lyapunov function methods [@prajna2004safety; @prajna2006barrier; @prajna2007framework; @sloth2012compositional]. Essentially, reachable set methods aim to find a set containing all possible states at a given time, for a given set of initial conditions. Subsequently, if the reachable set does not intersect with the pre-specified unsafe regions, the system is considered to be safe. For example, in [@anai2001reach] the reachable set is found for continuous-time linear systems, whereas in [@tomlin2003computational] and [@asarin2003reachability] the reachable sets are computed via approximations for nonlinear dynamical systems. In [@kousik2017safe], the authors applied a reachable set method to plan safe trajectories for autonomous vehicles.
While reachable set methods can be used to obtain quantitative guarantees for safety, the reliability of the result largely depends on the assumptions made about the system, as well as the form of the unsafe regions. For instance, calculating the volume of the intersection of two sets, such as the reachable set and the unsafe regions, can become computationally challenging [@dyer1988complexity], jeopardizing the practical application of reachable set methods. An alternative approach to safety verification is based on using Lyapunov-like functions. In [@prajna2006barrier], the authors proposed the use of barrier certificates for safety verification of nonlinear systems. In contrast with the reachable set method, this line of work does not require to solve differential equations and is computationally more tractable. Furthermore, it also allows to provide safety certificates for various types of hybrid [@prajna2004safety] and stochastic systems [@prajna2007framework].
Despite a tremendous amount of solutions proposed to solve the safety verification problem, the majority of existing methods only provide binary safety certificates. More specifically, these certificates concern only *whether the system is safe* rather than *how safe the system is*. Lacking a detailed analysis of how unsafe a system is may result in a restricted and conservative design space. To illustrate this point, let us consider the operation of a solar-powered autonomous vehicle. Naturally, regions without solar exposure are considered to be unsafe, since the battery of the vehicle could be drained after a period of time. However, it would be inefficient to plan a path for the vehicle completely avoiding all these shaded regions. Instead, a more suitable requirement would be that the amount of time the vehicle spends in the shaded regions is bounded. More generally, this framework can be useful in those situations where the system is able to tolerate the exposure to a deteriorating agent, such as excessive heat or radiation, for a limited amount of time.
In this paper, we consider this alternative, more flexible notion of safety. More precisely, we aim to compute the time that a (nonlinear) system spends in the unsafe regions. In particular, we focus our analysis on the case of systems described by a polynomial dynamics and unsafe regions described by a collection of polynomial inequalities. To calculate the amount of time spent in the unsafe regions, we use *occupation measures* to quantify how much time the system trajectory spends in a particular set [@vinter1993convex]. Using this alternative viewpoint of the system dynamics, the safety quantity of interest can be calculated by finding the volume of the unsafe region with respect to the occupation measure [@henrion2009approximate]. The usage of occupation measures allows us to leverage powerful numerical procedures developed in the context of control of polynomial systems [@lasserre2008nonlinear; @henrion2014convex; @majumdar2014convex].
The contribution of this paper is threefold. First, we formulate a flexible notion of safety allowing a trade-off between safety and performance. Second, we provide an *exact* formulation of the problem under consideration in terms of an infinite-dimensional LP. Furthermore, we provide a hierarchy of relaxations that can be efficiently solved using semidefinite programming. Finally, we provide numerical examples to demonstrate the applicability of our method.
The rest of the paper is structured as follows. The safety verification problem formulation is stated in Section II. In Section III, we introduce concepts from measure theory that are necessary for developing our framework. Based on those notions, we show that the problem under consideration can be stated as an infinite-dimensional linear program, and in Section IV, we provide approximate solutions to this LP using a sequence of semidefinite programs. The performance of our framework is illustrated through numerical experiments in Section V and we conclude the paper in Section VI.
**Notations:** We use bold symbols to represent real-valued vectors. Given $n\in \mathbb{N},$ we use the shorthand notation $[n]$ to denote the set of integers $\{1,\ldots, n\}.$ The indicator function of a given set $\mathcal{S}$ is defined by $\mathbf{1}_\mathcal{S}(\cdot).$ We use $\delta_\mathbf{x}$ to denote the Dirac measure centered on a fixed point $\mathbf{x} \in \mathbb{R}^n$ and we use $\otimes$ to denote the product between two measures. The ring of polynomials in $\mathbf{x}$ with real coefficients is denoted by $\mathbb{R}[\mathbf{x}]$, and $\mathbb{R}[\mathbf{x}]_r$ denotes the subset of polynomials of degree $\leq r$. Given $\mathbf{x}\in\mathbb{R}^n$ and ${\ensuremath{\boldsymbol{\alpha}}}\in \mathbb{N}^n,$ we let $\mathbf{x}^{\boldsymbol{\alpha}}$ denote the quantity $\mathbf{x}^{{\ensuremath{\boldsymbol{\alpha}}}} = \prod_{i = 1}^n x_i^{\alpha_i}$. Let $|\boldsymbol{\alpha}| = \sum_{i = 1}^n \alpha_i$ and $\mathbb{N}_r^n = \lbrace {\ensuremath{\boldsymbol{\alpha}}}\in \mathbb{N}^n \mid |{\ensuremath{\boldsymbol{\alpha}}}| \leq r \rbrace$.
Problem Statement {#sec:statement}
=================
Occupation Measure-based Reformulation {#sec:prelim}
======================================
Semidefinite and Sum-of-Squares Relaxation {#sec:relaxation}
==========================================
Numerical Examples
==================
Conclusion
==========
In this paper, we have proposed a flexible safety verification notion for nonlinear autonomous systems described via polynomial dynamics and unsafe regions described via polynomial inequalities. Instead of verifying safety by checking whether the dynamics completely avoids the unsafe regions, we consider the system to be safe if it spends less than a certain amount of time in these regions. This more flexible notion can be of relevance in, for example, solar-powered vehicles where the vehicle should avoid spending too much time is dark areas. More generally, this framework can be useful in those situations where the system is able to tolerate the exposure to a deteriorating agent, such as excessive heat or radiation, for a limited amount of time. In this paper, we first propose an infinite-dimensional LP over the space of measures whose solution is equal to the (expected) time our (nonlinear) system spends in the (possibly nonconvex) unsafe regions. We then approximate the solution of the LP through a monotonically converging sequence of upper bounds by solving a hierarchy of SDPs. We have validated our approach via a simple example involving a nonlinear Van der Pol oscillator. As future work, we are working on the problem of path planning using the flexible safety notion herein proposed.
{#appen:proof}
[^1]: The authors are with the Department of Electrical and Systems Engineering at the University of Pennsylvania, Philadelphia PA 19104. e-mail:{ximingch, srchen, preciado}@seas.upenn.edu.
|
---
abstract: 'Let ${{\mathcal M}}$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $\tau$. We show that the symmetrically $\Delta$-normed operator space $E({{\mathcal M}},\tau)$ corresponding to an arbitrary symmetrically $\Delta$-normed function space $E(0,\infty)$ is an interpolation space between $L_0({{\mathcal M}},\tau)$ and ${{\mathcal M}}$, which is in contrast with the classical result that there exist symmetric operator spaces $E({{\mathcal M}},\tau)$ which are not interpolation spaces between $L_1({{\mathcal M}},\tau)$ and ${{\mathcal M}}$. Besides, we show that the ${{\mathcal K}}$-functional of every $X\in L_0({{\mathcal M}},\tau)+{{\mathcal M}}$ coincides with the ${{\mathcal K}}$-functional of its generalized singular value function $\mu(X)$. Several applications are given, e.g., it is shown that the pair $(L_0({{\mathcal M}},\tau),{{\mathcal M}})$ is ${{\mathcal K}}$-monotone when ${{\mathcal M}}$ is a non-atomic finite factor.'
address:
- 'School of Mathematics and Statistics, University of New South Wales, Kensington, 2052, NSW, Australia *E-mail :* [jinghao.huang@unsw.edu.au]{}'
- 'School of Mathematics and Statistics, University of New South Wales, Kensington, 2052, NSW, Australia *E-mail :* [f.sukochev@unsw.edu.au]{} '
author:
- 'J. Huang'
- 'F. Sukochev'
title: 'Interpolation between $L_0({{\mathcal M}},\tau)$ and $L_\infty({{\mathcal M}},\tau)$'
---
Introduction
============
Recall the Calkin correspondence between symmetrically $\Delta$-normed operator spaces and symmetrically $\Delta$-normed function spaces introduced in [@HLS]. Let $E(0,\infty)$ be an arbitrary symmetrically $\Delta$-normed function space equipped with a $\Delta$-norm $\|\cdot\|_E$ (see Section \[prel\]) and let $\mathcal{M}$ be an arbitrary semifinite von Neumann algebra equipped with a faithful normal semifinite trace $\tau$. Then, $$E(\mathcal{M},\tau):=\{X\in S(\mathcal{M},\tau):\ \mu(X)\in E(0,\infty)\},
\ \|X\|_{E(\mathcal{M},\tau)}:=\|\mu(X)\|_E$$ is a symmetrically $\Delta$-normed operator space, where $S({{\mathcal M}},\tau)$ is the space of all $\tau$-measurable operators affiliated with ${{\mathcal M}}$. Moreover, if $E(0,\infty)$ is complete, then $E({{\mathcal M}},\tau)$ is also complete (see [@HLS Theorem 3.8], see also [@Kalton_S; @Sukochev] for the Banach case and the quasi-Banach case). For brevity, we omit below the term “operator” and refer just to symmetrically $\Delta$-normed spaces.
Let $A({{\mathcal M}},\tau)$ and $B({{\mathcal M}},\tau)$ be two symmetrically $\Delta$-normed spaces. A symmetrically $\Delta$-normed space $E({{\mathcal M}},\tau)$ is said to be *intermediate* for $A({{\mathcal M}},\tau)$ and $B({{\mathcal M}},\tau)$ if the continuous embeddings $$A({{\mathcal M}},\tau)\cap B({{\mathcal M}},\tau) \subset E({{\mathcal M}},\tau) \subset A({{\mathcal M}},\tau)+ B({{\mathcal M}},\tau)$$ hold. Let $E({{\mathcal M}},\tau)$ be a symmetrically $\Delta$-normed space intermediate between $A({{\mathcal M}},\tau)$ and $B({{\mathcal M}},\tau)$. If every linear operator on $A({{\mathcal M}},\tau)+ B({{\mathcal M}},\tau)$ which is bounded from $A({{\mathcal M}},\tau)$ to $A({{\mathcal M}},\tau)$ and $B({{\mathcal M}},\tau)$ to $B({{\mathcal M}},\tau)$ is also a bounded operator from $E({{\mathcal M}},\tau)$ to $E({{\mathcal M}},\tau)$, then $E({{\mathcal M}},\tau)$ is called an *interpolation* space between the spaces $A({{\mathcal M}},\tau) $ and $ B({{\mathcal M}},\tau)$.
Interpolation function spaces have been widely investigated (see e.g. [@AK; @Astashkin; @HM; @KPS; @Rotfeld; @Bergh_L; @AM; @Mali1992]) since Mityagin [@Mityagin] and Calderón [@Calderon] gave characterizations of the class of all interpolation spaces with respect to $(L_1(0,\infty),L_\infty(0,\infty))$ (see also [@DDS2014; @DDP] for results in the noncommutative setting). Among several real interpolation methods, the K-method of interpolation linked to the so-called ${{\mathcal K}}$-functional is very important (we refer [@MP1991; @Mali1992; @Astashkin; @STZ] for applications of ${{\mathcal K}}$-functionals in different areas). Calculating the ${{\mathcal K}}$-functionals for a given couple of spaces is very in the K-method [@Mali]. In [@HM], the ${{\mathcal K}}$-functionals for the couple $L_0(0,\infty)$ and $L_\infty(0,\infty)$ are obtained. We give a description of the ${{\mathcal K}}$-functional of every element $X\in (L_0+L_\infty)({{\mathcal M}},\tau)$ in terms of singular value function as well as in terms of its distribution function, showing that the ${{\mathcal K}}$-functional of $X$ coincides with the ${{\mathcal K}}$-functional of its generalized singular value function $\mu(X)$ (see Section \[prel\]), which extends [@HM Proposition 3].
It is well-known (see e.g. [@Mityagin; @Calderon], see also [@KPS; @DPS; @DDP]) that the operator space $E({{\mathcal M}},\tau)$ corresponding to a fully symmetric (see e.g Section \[imbedding\]) function space $E(0,\infty)$ is an interpolation space between $L_1({{\mathcal M}},\tau)$ and ${{\mathcal M}}$. In particular, there exist symmetric normed spaces which are not interpolation spaces between $L_1({{\mathcal M}},\tau)$ and $L_\infty({{\mathcal M}},\tau)$ (see [@KPS Chapter II, $\S$ 4.2 and $\S$ 5.7]). However, in this paper, it is shown that if we consider $L_0({{\mathcal M}},\tau)$ (the set of all $\tau$-measurable operators having finite-trace support) instead of $L_1({{\mathcal M}},\tau)$, then the operator space $E({{\mathcal M}},\tau)$ corresponding to an arbitrary symmetrically $\Delta$-normed function space $E(0,\infty)$ is necessarily an interpolation space between $L_0({{\mathcal M}},\tau)$ and ${{\mathcal M}}$, which is a noncommutative version of results in [@HM] (see also [@Astashkin]).
As an application of the previous result, we describe the orbits and ${{\mathcal K}}$-orbits for an arbitrary $A \in S({{\mathcal M}},\tau)$ in the last section. It is shown that the unit balls of ${{\mathcal K}}$-orbits do not coincide with the unit balls of orbits in the pair $(L_0({{\mathcal M}},\tau),{{\mathcal M}})$, which generalises [@Astashkin Theorem 4]. In [@Astashkin], it is asserted that the commutative pair $(L_0(0,\infty),L_\infty(0,\infty))$ is not ${{\mathcal K}}$-monotone, that is, ${{\mathcal K}}$-orbits do not necessarily coincide with orbits in the pair $(L_0(0,\infty),L_\infty(0,\infty))$. However, it is known that this assertion is incorrect and $(L_0(0,\infty),L_\infty(0,\infty))$ is indeed ${{\mathcal K}}$-monotone (see e.g. Section \[O\]). A non-commutative version of this result is established, that is, the pair $(L_0({{\mathcal M}},\tau),{{\mathcal M}})$ is ${{\mathcal K}}$-monotone in the setting when ${{\mathcal M}}$ is a non-atomic finite factor. We would like to thank Professor Astashkin for providing us with the proof for the commutative pair $(L_0(0,\infty),L_\infty(0,\infty))$.
Preliminaries {#prel}
=============
Generalized singular value functions
------------------------------------
$\\$ In what follows, ${{\mathcal H}}$ is a Hilbert space and $B({{\mathcal H}})$ is the $*$-algebra of all bounded linear operators on ${{\mathcal H}}$, and $\mathbf{1}$ is the identity operator on ${{\mathcal H}}$. Let $\mathcal{M}$ be a von Neumann algebra on ${{\mathcal H}}$. For details on von Neumann algebra theory, the reader is referred to e.g. [@Dixmier], [@KR1; @KR2] or [@Tak]. General facts concerning measurable operators may be found in [@Nelson], [@Se] (see also [@Ta2 Chapter IX] and the forthcoming book [@DPS]). For convenience of the reader, some of the basic definitions are recalled.
A linear operator $X:\mathfrak{D}\left( X\right) \rightarrow {{\mathcal H}}$, where the domain $\mathfrak{D}\left( X\right) $ of $X$ is a linear subspace of ${{\mathcal H}}$, is said to be [*affiliated*]{} with $\mathcal{M}$ if $YX\subseteq XY$ for all $Y\in \mathcal{M}^{\prime }$, where $\mathcal{M}^{\prime }$ is the commutant of $\mathcal{M}$. A linear operator $X:\mathfrak{D}\left( X\right) \rightarrow {{\mathcal H}}$ is termed [*measurable*]{} with respect to $\mathcal{M}$ if $X$ is closed, densely defined, affiliated with $\mathcal{M}$ and there exists a sequence $\left\{ P_n\right\}_{n=1}^{\infty}$ in the logic of all projections of $\mathcal{M}$, ${{\mathcal P}}\left(\mathcal{M}\right)$, such that $P_n\uparrow \mathbf{1}$, $P_n({{\mathcal H}})\subseteq\mathfrak{D}\left(X\right) $ and $\mathbf{1}-P_n$ is a finite projection (with respect to $\mathcal{M}$) for all $n$. It should be noted that the condition $P_{n}\left(
{{\mathcal H}}\right) \subseteq \mathfrak{D}\left( X\right) $ implies that $XP_{n}\in \mathcal{M}$. The collection of all measurable operators with respect to $\mathcal{M}$ is denoted by $S\left(
\mathcal{M} \right) $, which is a unital $\ast $-algebra with respect to strong sums and products (denoted simply by $X+Y$ and $XY$ for all $X,Y\in S\left( \mathcal{M}\right) $).
Let $X$ be a self-adjoint operator affiliated with $\mathcal{M}$. We denote its spectral measure by $\{E^X\}$. It is well known that if $X$ is a closed operator affiliated with $\mathcal{M}$ with the polar decomposition $X = U|X|$, then $U\in\mathcal{M}$ and $E\in
\mathcal{M}$ for all projections $E\in \{E^{|X|}\}$. Moreover, $X\in S(\mathcal{M})$ if and only if $X$ is closed, densely defined, affiliated with $\mathcal{M}$ and $E^{|X|}(\lambda,
\infty)$ is a finite projection for some $\lambda> 0$. It follows immediately that in the case when $\mathcal{M}$ is a von Neumann algebra of type $III$ or a type $I$ factor, we have $S(\mathcal{M})= \mathcal{M}$. For type $II$ von Neumann algebras, this is no longer true. From now on, let $\mathcal{M}$ be a semifinite von Neumann algebra equipped with a faithful normal semifinite trace $\tau$.
An operator $X\in S\left( \mathcal{M}\right) $ is called $\tau$-measurable if there exists a sequence $\left\{P_n\right\}_{n=1}^{\infty}$ in $P\left(\mathcal{M}\right)$ such that $P_n\uparrow \mathbf{1},$ $P_n\left({{\mathcal H}}\right)\subseteq \mathfrak{D}\left(X\right)$ and $\tau(\mathbf{1}-P_n)<\infty $ for all $n$. The collection of all $\tau $-measurable operators is a unital $\ast $-subalgebra of $S\left(
\mathcal{M}\right) $ denoted by $S\left( \mathcal{M}, \tau\right)
$. It is well known that a linear operator $X$ belongs to $S\left(
\mathcal{M}, \tau\right) $ if and only if $X\in S(\mathcal{M})$ and there exists $\lambda>0$ such that $\tau(E^{|X|}(\lambda,
\infty))<\infty$. Alternatively, an unbounded operator $X$ affiliated with $\mathcal{M}$ is $\tau$-measurable (see [@FK]) if and only if $$\tau\left(E^{|X|}\bigl(\frac1n,\infty\bigr)\right)=o(1),\quad n\to\infty.$$
\[mu\] Let a semifinite von Neumann algebra $\mathcal M$ be equipped with a faithful normal semi-finite trace $\tau$ and let $X\in
S(\mathcal{M},\tau)$. The generalized singular value function $\mu(X):t\rightarrow \mu(t;X)$ of the operator $X$ is defined by setting $$\mu(s;X)
=
\inf\{\|XP\|_\infty:\ P=P^*\in\mathcal{M}\mbox{ is a projection,}\ \tau(\mathbf{1}-P)\leq s\}.$$
An equivalent definition in terms of the distribution function of the operator $X$ is the following. For every self-adjoint operator $X\in S(\mathcal{M},\tau),$ setting $$d_X(t)=\tau(E^{X}(t,\infty)),\quad t>0,$$ we have (see e.g. [@FK]) $$\begin{aligned}
\label{dis}
\mu(t; X)=\inf\{s\geq0:\ d_{|X|}(s)\leq t\}.\end{aligned}$$
Consider the algebra $\mathcal{M}=L^\infty(0,\infty)$ of all Lebesgue measurable essentially bounded functions on $(0,\infty)$. Algebra $\mathcal{M}$ can be seen as an abelian von Neumann algebra acting via multiplication on the Hilbert space $\mathcal{H}=L^2(0,\infty)$, with the trace given by integration with respect to Lebesgue measure $m.$ It is easy to see that the algebra of all $\tau$-measurable operators affiliated with $\mathcal{M}$ can be identified with the subalgebra $S(0,\infty)$ of the algebra of Lebesgue measurable functions which consists of all functions $x$ such that $m(\{|x|>s\})$ is finite for some $s>0$. It should also be pointed out that the generalized singular value function $\mu(x)$ is precisely the decreasing rearrangement $\mu(x)$ of the function $x$ (see e.g. [@KPS]) defined by $$\mu(t;x)=\inf\{s\geq0:\ m(\{|x|\geq s\})\leq t\}.$$
The two-sided ideal ${{\mathcal F}}({{\mathcal M}},\tau)$ in ${{\mathcal M}}$ consisting of all elements of $\tau$-finite rank is defined by setting $${{\mathcal F}}({{\mathcal M}},\tau) =\{X \in {{\mathcal M}}: \tau(r(X)) <\infty \} =\{X \in {{\mathcal M}}: \tau(s(X)) <\infty \}.$$
For convenience of the reader we also recall the definition of the measure topology $t_\tau$ on the algebra $S({{\mathcal M}},\tau)$. For every $\varepsilon,\delta>0,$ we define the set $$V(\varepsilon,\delta)=\{X\in S(\mathcal{M},\tau):\ \exists P\in P\left(\mathcal{M}\right)\mbox{ such that } \|X(\mathbf{1}-P)\|\leq\varepsilon,\ \tau(P)\leq\delta\}.$$ The topology generated by the sets $V(\varepsilon,\delta)$, $\varepsilon,\delta>0,$ is called the measure topology $t_\tau$ on $S({{\mathcal M}},\tau)$ [@DPS; @FK; @Nelson]. It is well known that the algebra $S({{\mathcal M}},\tau)$ equipped with the measure topology is a complete metrizable topological algebra [@Nelson] (see also [@Muratov]). A sequence $\{X_n\}_{n=1}^\infty\subset S({{\mathcal M}},\tau)$ converges to zero with respect to measure topology $t_\tau$ if and only if $\tau\big(E^{|X_n|}(\varepsilon,\infty)\big)\to 0$ as $n\to \infty$ for all $\varepsilon>0$ [@DPS; @DP2].
$\Delta$-normed spaces
----------------------
$\\$ For convenience of the reader, we recall the definition of $\Delta$-norm. Let $\Omega$ be a linear space over the field $\mathbb{C}$. A function $\|\cdot\|$ from $\Omega$ to $\mathbb{R}$ is a $\Delta$-norm, if for all $x,y \in \Omega$ the following properties hold:
- $\|x\| \geqslant 0$, $\|x\| = 0 \Leftrightarrow x=0$;
- $\|\alpha x\| \leqslant \|x\|$ for all $|\alpha| \le1$;
- $\lim _{\alpha \rightarrow 0}\|\alpha x\| = 0$;
- $\|x+y\| \le C_\Omega \cdot (\|x\|+\|y\|)$ for a constant $C_\Omega\geq 1$ independent of $x,y$.
The couple $(\Omega, \left\|\cdot \right\|)$ is called a $\Delta$-normed space. We note that the definition of a $\Delta$-norm given above is the same with that given in [@KPR]. It is well-known that every $\Delta$-normed space $(\Omega, \left\| \cdot \right\|)$ is metrizable and conversely every metrizable space can be equipped with a $\Delta$-norm [@KPR]. Note that properties $(2)$ and $(4)$ of a $\Delta$-norm imply that for any $\alpha\in\mathbb{C}$, there exists a constant $M$ such that $\|\alpha x\|\leq M\|x\|,\, x\in \Omega$, in particular, if $\|x_n\|\to 0, \{x_n\}_{n=1}^\infty\subset \Omega$, then $\|\alpha x_n\|\to 0$.
Let $E(0,\infty)$ be a space of real-valued Lebesgue measurable functions on $(0,\infty)$ (with identification $m$-a.e.), equipped with a $\Delta$-norm $\left\| \cdot \right\|_E$. The space $E(0,\infty)$ is said to be [*absolutely solid*]{} if $x\in E(0,\infty)$ and $|y|\leq |x|$, $y\in S(0,\infty)$ implies that $y\in E(0,\infty)$ and $\|y\|_E\leq\|x\|_E.$ An absolutely solid space $E(0,\infty)\subseteq S(0,\infty)$ is said to be [*symmetric*]{} if for every $x\in E(0,\infty)$ and every $y\in S(0,\infty)$, the assumption $\mu(y)=\mu(x)$ implies that $y\in E(0,\infty)$ and $\|y\|_E=\|x\|_E$ (see e.g. [@KPS]).
We now come to the definition of the main object of this paper.
\[opspace\] Let a semifinite von Neumann algebra $\mathcal M$ be equipped with a faithful normal semi-finite trace $\tau$. Let $\mathcal{E}$ be a linear subset in $S({\mathcal{M}, \tau})$ equipped with a $\Delta$-norm $\left\| \cdot \right\|_{\mathcal{E}}$. We say that $\mathcal{E}$ is a *symmetrically $\Delta$-normed operator space* if $X\in
\mathcal{E}$ and every $Y\in S({\mathcal{M}, \tau})$ the assumption $\mu(Y)\leq \mu(X)$ implies that $Y\in \mathcal{E}$ and $\|Y\|_\mathcal{E}\leq \|X\|_\mathcal{E}$.
One should note that a symmetrically $\Delta$-normed space $E({{\mathcal M}},\tau)$ does not necessarily satisfy $$\|AXB\|_E \le \|A\|_\infty \|B\|_\infty \|X\|_E, ~A, B\in {{\mathcal M}},~ X\in E({{\mathcal M}},\tau).$$
It is clear that in the special case, when ${{\mathcal M}}=L_\infty(0,1)$, or ${{\mathcal M}}=L_\infty(0,\infty)$, or ${{\mathcal M}}=l_\infty$, the definition of symmetrically $\Delta$-normed operator spaces coincides with definition of the symmetric function (or sequence) spaces. In the case, when $\mathcal{M}=B(H)$ and $\tau$ is a standard trace ${\rm Tr}$, we shall call a symmetrically $\Delta$-normed operator space introduced in Definition \[opspace\] a symmetrically $\Delta$-normed operator ideal (for the symmetrically normed ideals we refer to [@GK1; @GK2; @Simon]).
As mentioned before, the operator space $E({{\mathcal M}},\tau)$ defined by $$E(\mathcal{M},\tau):=\{X\in S(\mathcal{M},\tau):\ \mu(X)\in E(0,\infty)\},
\ \|X\|_{E(\mathcal{M},\tau)}:=\|\mu(X)\|_E$$ is a complete symmetrically $\Delta$-normed operator space whenever the symmetrically $\Delta$-normed function space $E(0,\infty)$ equipped with a $\Delta$-norm $\|\cdot\|_E$ is complete [@HLS].
$(L_0+L_\infty)({{\mathcal M}},\tau) = S({{\mathcal M}},\tau)$
==============================================================
By $L_0(0,\infty)$ we denote the space of all measurable functions on $(0,\infty)$ whose support has finite measure. This space is endowed with the group-norm [@HM] $$\|f\|_{L_0} = m({{\mbox{supp}}}(f)),$$ where ${{\mbox{supp}}}(f) = \{t\in (0,\infty):f(t)\ne 0\}$. It is clearly that the corresponding operator space $L_0({{\mathcal M}},\tau)$ is the subspace of $S({{\mathcal M}},\tau)$ which consists of all operators $X$ such that $\tau(s(X))<\infty$. It is easy to see that $(L_0+L_\infty) ({{\mathcal M}},\tau)$ coincides with $S({{\mathcal M}},\tau)$. For the sake of completeness, we present a brief proof below.
\[prop:0infty\] $(L_0+L_\infty) ({{\mathcal M}},\tau) =S({{\mathcal M}},\tau)$.
Since $ (L_0+L_\infty) ({{\mathcal M}},\tau) \subset S({{\mathcal M}},\tau)$, it suffices to prove $ S({{\mathcal M}},\tau)\subset (L_0+L_\infty) ({{\mathcal M}},\tau) $. For any operator $X\in S({{\mathcal M}},\tau)$, there exists $\lambda >0$ such that $t:= \tau(E^{|X|}(\lambda,\infty))<\infty$. By [@DPS Chapter III, Eq. (4)], we have $$\begin{aligned}
d_{\mu(X)}(\lambda)= \tau(E^{|X|}(\lambda,\infty)) =t .\end{aligned}$$ This together with [@HM Proposition 3] implies that $\mu(X)\in (L_0+L_\infty) (0,\infty)$. Thus, $X\in (L_0+L_\infty) ({{\mathcal M}},\tau)$.
In [@HM Proposition 3], the description of the ${{\mathcal K}}$-functional (which is a $\Delta$-norm on $S(0,\infty)$) $K_u(f)=\inf\{\|g\|_0 +u\|h\|_\infty : f=g+h, g\in L_0(0,\infty), h\in L_\infty(0,\infty)\}$, $u>0$, of any $f\in S(0,\infty)$ is given in terms of its distribution function and its singular value function. That is, for every $f\in S(0,\infty)$, we have $$\begin{aligned}
\label{ineq:K}
K_u(f)= \inf_{s>0} [su + d_{\mu(f)}(s)]= \inf_{t>0} [ t+ u \mu(t;f)].\end{aligned}$$ Similarly, for every $X\in S({{\mathcal M}},\tau)$, the ${{\mathcal K}}$-functional is defined by $$K_u(X) : =\inf\{\|G\|_0 +u\|H\|_\infty : X=G+H,G\in L_0({{\mathcal M}},\tau), H\in L_\infty({{\mathcal M}},\tau)\}, u>0.$$ In particular, we define a $\Delta$-norm (see e.g. Remark \[remark:tri\]) on $S({{\mathcal M}},\tau)$ by $$\|X\|_S:= K_1 (X)= \inf\{\|G\|_0 +\|H\|_\infty : X=G+H,G\in L_0({{\mathcal M}},\tau), H\in L_\infty({{\mathcal M}},\tau)\}$$ for any $X\in S({{\mathcal M}},\tau)$. The following result complements an earlier result from [@DDP] for the pair $(L_1({{\mathcal M}},\tau),L_\infty({{\mathcal M}},\tau))$.
\[prop:3.3\] For every $X\in S({{\mathcal M}},\tau)$, we have $$K_u(X) =K_u(\mu(X)), ~u>0.$$ In particular, $\|X\|_S= \inf_{s>0} [s + d_{\mu(X)}(s)]= \inf_{t>0} [ t+ \mu(t;X)]$. Moreover, $S({{\mathcal M}},\tau)$ is a complete $\Delta$-normed space with respect to the $\Delta$-norm $K_u$ for every $u>0$.
Firstly, for every $X\in S({{\mathcal M}},\tau)$, we have $$\begin{aligned}
K_u(X)&= \inf\{\|G\|_0 +u\|H\|_\infty : X=G+H, G\in L_0({{\mathcal M}},\tau), H\in L_\infty({{\mathcal M}},\tau)\}\\
&\le \inf_{ s>0 } \{\| X E^{|X|}(s,\infty)\|_0 +u\|X E^{|X|}(0,s]\|_\infty \} \\
&\le \inf_{ s>0 } \{ d_{ |X E^{|X|}(s,\infty)|}(0 ) +us \} \\
&= \inf_{ s>0 } \{ d_{\mu( X E^{|X|}(s,\infty))}(0 ) +us \} ~\quad \mbox{(by \cite[Chapter III, Eq. (4)]{DPS})} \\
&= \inf_{ s>0 } \{ d_{\mu( X E^{|X|}(s,\infty))}(s ) +us \} \\
&\le \inf_{ s>0 }\{d_{\mu(X)}(s) +us \} \\
& \stackrel{\eqref{ineq:K} }{=} K_u(\mu(X)).\end{aligned}$$ Conversely, for every $G\in L_0({{\mathcal M}},\tau) $ with $t:= \|G \|_0 $ and $X-G\in L_\infty({{\mathcal M}},\tau)$, by [@DPS Chapter III, Proposition 2.20], we have $$\begin{aligned}
&~\quad t+ u \mu(t;X)\\
& = t+ u\cdot \inf\{\|X-Y\|_\infty : Y\in L_0({{\mathcal M}},\tau), \|Y\|_0\le t, X-Y\in {{\mathcal M}}\} \\
& \le \|G\|_0 + u \| X - G \|_\infty .\end{aligned}$$ Hence, we obtain $$\inf_{t>0}[t+ u \mu(t;X) ]\le \inf _{\substack{G\in L_0({{\mathcal M}},\tau)\\ X- G\in L_\infty({{\mathcal M}},\tau)}}( \|G\|_0 + u \| X - G \|_\infty )=K_u(X) .$$ The fact that $S(0,\infty)$ is complete with respect with $K_u$ together with [@HLS Theorem 3.8] implies the completeness of $S({{\mathcal M}},\tau)$ with respect with $K_u$.
Recall that $d_{\mu(X)}(s)=d_{|X|}(s)$ (see e.g. [@DPS Chapter III, Eq. (4)]). For every $X\in S({{\mathcal M}},\tau)$, the ${{\mathcal K}}$-functional can be also defined by the formula $$\begin{aligned}
K_u(X) = \inf_{s>0} [su + d_{|X|}(s)],~ u>0.\end{aligned}$$
\[remark:tri\] It is still unknown whether the Calkin correspondence preserves the constant $C_E$ for an arbitrary symmetrically $\Delta$-normed function space $E(0,\infty)$ (see [@HLS; @Sukochev]). However, it is well-known that $(S(0,\infty),K_u(\cdot))$, $u>0$, is an $F$-space (i.e., a complete $\Delta$-normed space with $C_E=1$) and Proposition \[prop:3.3\] implies that for every $u>0$, $K_u(\mu( \cdot))$ is not only a $\Delta$-norm but also an $F$-norm on $S({{\mathcal M}},\tau)$. Indeed, for every $X,Y\in S({{\mathcal M}},\tau)$, by Proposition \[prop:3.3\], we have $$\begin{aligned}
&~\quad K_u(\mu(X)) +K_u(\mu(Y))\\
&= K_u(X) +K_u(Y)\\
& =\inf\{\|X_1\|_0 +u\|X_2\|_\infty : X=X_1+X_2,X_1\in L_0({{\mathcal M}},\tau), X_2\in L_\infty({{\mathcal M}},\tau)\} \\
&~\quad +\inf\{\|Y_1\|_0 +u\|Y_2\|_\infty : Y=Y_1+Y_2,Y_1\in L_0({{\mathcal M}},\tau), Y_2\in L_\infty({{\mathcal M}},\tau)\}\\
& =\inf\{\|X_1\|_0 +\|Y_1\|_0 +u\|X_2\|_\infty +u\|Y_2\|_\infty : \\
&~\quad X=X_1+X_2, Y=Y_1+Y_2, X_1,Y_1\in L_0({{\mathcal M}},\tau), X_2,Y_2\in L_\infty({{\mathcal M}},\tau)\}\\
& \ge \inf\{\|X_1+Y_1\|_0 +u\|X_2+ Y_2\|_\infty : \\
&~\quad X=X_1+X_2, Y=Y_1+Y_2, X_1,Y_1\in L_0({{\mathcal M}},\tau), X_2,Y_2\in L_\infty({{\mathcal M}},\tau)\}\\
& \ge \inf\{\|Z_1\|_0 +u\|Z_2\|_\infty : X+Y=Z_1+Z_2, Z_1\in L_0({{\mathcal M}},\tau), Z_2\in L_\infty({{\mathcal M}},\tau)\}\\
&= K_u(X+Y)= K_u(\mu(X+Y)),\end{aligned}$$ where we used the fact that $\|X\|_0+ \|Y\|_0 = \tau({{\mbox{supp}}}(X)) +\tau({{\mbox{supp}}}(Y)) \ge \tau({{\mbox{supp}}}(X+Y))=\|X+Y \|_0$ for every $X,Y\in S({{\mathcal M}},\tau)$.
An embedding theorem
====================
It is well-known that for every symmetrically normed function space $E(0,\infty)$, the corresponding operator space $E({{\mathcal M}},\tau)$ is symmetrically normed [@Kalton_S] and is an intermediate space for the noncommutative pair $(L_1({{\mathcal M}},\tau),{{\mathcal M}})$ [@DP2; @DPS]. In this section, we prove an analogue for the $\Delta$-normed case, that is, every operator space $E({{\mathcal M}},\tau)$ corresponding to a $\Delta$-normed function space $E(0,\infty)$ is an intermediate space for the noncommutative pair $(L_0({{\mathcal M}},\tau),{{\mathcal M}})$.
Before we proceed to the proof of the embedding theorem, we show that the topology given by $\|\cdot\|_S$ is equivalent with the measure topology.
\[prop:SandM\] Let $\{X_n\}$ be a sequence in $S({{\mathcal M}},\tau)$. Then, $\|X_n\|_S\rightarrow 0$ if and only if $X_n \rightarrow _{t_\tau} 0$.
By [@HLS Lemma 2.4], it suffices to show that $\|X_n\|_S\rightarrow 0$ whenever $X_n \rightarrow _{t_\tau} 0$. By [@DPS Chapter II, Proposition 5.7], we have $\tau(E^{|X_n|} (\varepsilon,\infty))\rightarrow_n 0$ for every $\varepsilon>0$. By Proposition \[prop:3.3\], we have $\|X_n\|_S \le \varepsilon +\tau(E^{|X_n|} (\varepsilon,\infty))$, which completes the proof.
Notice that the two-sided ideal ${{\mathcal F}}(\tau)$ in ${{\mathcal M}}$ coincides with $(L_0\cap L_\infty) ({{\mathcal M}},\tau)$. For every $X\in {{\mathcal F}}(\tau)$, we define the group-norm $\|X\|_{{\mathcal F}}$ by $$\|X\|_{{\mathcal F}}:= \max \{\|X\|_0, \|X\|_\infty\}.$$ The following embedding theorem is the main result of this section, which extends [@HM Theorem 1] to the non-commutative case.
\[th:embedding\] If $E(0,\infty)$ is a nontrivial symmetrically $\Delta$-normed function space, then $${{\mathcal F}}(\tau) \subset E({{\mathcal M}},\tau)\subset S({{\mathcal M}},\tau).$$ Moreover, the embeddings are continuous. That is, $E({{\mathcal M}},\tau)$ is an intermediate space between $L_0({{\mathcal M}},\tau)$ and ${{\mathcal M}}$.
Since $E(0,\infty)$ is not empty, there is a non-zero element $ x_0 \in E(0,\infty)$. Then, there is a scalar $t>0$ such that $\mu(t;x_0)>0$. It is clear that $\mu(t;x_0)\chi_{(0,t]} \le \mu(x_0)$, which implies that $\chi_{(0,t]} \in E(0,\infty)$. Since $ m(\chi_{(0,t]})= m(\chi_{(t,2t]})= \cdots = m(\chi_{((n-1)t,nt]})$ and $E(0,\infty)$ is a linear space, it follows that $\chi_{(0,nt]}\in E(0,\infty)$. Hence, ${{\mathcal F}}(0,\infty )\subset E(0,\infty)$.
Let $\{X_n\}_n \subset {{\mathcal F}}(\tau)$ be a sequence such that $\|X_n\|_{{\mathcal F}}\rightarrow 0$. For every $0<\varepsilon<1$, we can find an $N$ such that for every $n\ge N$, we have $\|X_n \|_{{{\mathcal F}}}\le \varepsilon$, that is, $${{\mbox{supp}}}(\mu(X_n))\le \varepsilon ~ \mbox{ and } ~ \|X_n\|_\infty \le \varepsilon.$$ Hence, $\mu(X_n) \le \varepsilon \chi_{(0,\varepsilon]}\le \varepsilon \chi_{(0,1]}$ and therefore $\|X_n \|_E \le \|\varepsilon \chi_{(0,1]}\|_E $. By the continuity of $\Delta$-norm $\|\cdot\|_E$, we obtain that $\|X_n\|_E\rightarrow_n 0$.
Lemma 2.4 in [@HLS] together with Proposition \[prop:SandM\] implies that $E({{\mathcal M}},\tau)$ is continuously embedded into $ S({{\mathcal M}},\tau)$.
The set of all self-adjoint elements in $E({{\mathcal M}},\tau)$ is denoted by $E_h({{\mathcal M}},\tau)$. Then, [@DPS Chapter II, Proposition 6.1] together with Proposition \[prop:SandM\] and Theorem \[th:embedding\] implies the following results immediately.
\[cor:cone\] Let $E(0,\infty)$ is a symmetrically $\Delta$-normed function space. The following statements hold.
1. The positive cone $E({{\mathcal M}},\tau)^+$ is closed in $E({{\mathcal M}},\tau)$ with respect to $\|\cdot\|_E$.
2. If $\{X_n\}_{n=1}^\infty$ is a sequence in $E({{\mathcal M}},\tau)$ and $X,Y\in E_h({{\mathcal M}},\tau)$ are such that $\|X_n -X\|_E\rightarrow_n 0$ and $X_n\le Y$ for all $n$, then $X\le Y$.
3. If $\{X_n\}_{n=1}^\infty$ is an increasing sequence in $E_h({{\mathcal M}},\tau)$ and $X\in E_h({{\mathcal M}},\tau)$ with $\|X_n-X\|_E \rightarrow_n 0$, then $X_n\uparrow_n X$.
Interpolation in the pair $(L_0({{\mathcal M}},\tau),{{\mathcal M}})$ {#imbedding}
=====================================================================
Introduce the dilation operator $\sigma_s$ on $S(0,\infty)$, $s>0$, by setting $$\begin{aligned}
(\sigma_s(x))(t) = x \left( \frac{t}{s} \right), ~t >0.\end{aligned}$$ It is well-known that $\mu(X+Y)\leq \sigma_2(\mu(X)+\mu(Y))$, $X,Y\in S({{\mathcal M}},\tau)$ [@LSZ]. We note also that $\sigma_{2^k}x \in E(0,\infty)$ with $$\label{norm_sigma}
\|\sigma_{2^k}x\|_E\leq (2C_E)^k\|x\|_E$$ for all $x\in E(0,\infty)$ and $k\in\mathbb{N}$ (see e.g. [@KPS], see also [@HM]).
Recall that $(L_0+L_\infty)({{\mathcal M}},\tau)=S({{\mathcal M}},\tau)$ (see Proposition \[prop:0infty\]). Let $T: S ({{\mathcal M}},\tau)\rightarrow S({{\mathcal M}},\tau)$ be a homomorphism, i.e., $$T(X+Y) =TX+TY \mbox{ and }T(-X)=-TX$$ for any $X,Y \in S({{\mathcal M}},\tau)$. Let $E(0,\infty)$ be a $\Delta$-normed function space. A homomorphism $T: E({{\mathcal M}},\tau) \rightarrow E({{\mathcal M}},\tau)$ is called **continuous** if for any given $\varepsilon>0$, there exists $\delta(\varepsilon)>0$ such that $\|X\|_E <\delta(\varepsilon)$ implies that $\|TX\|_E <\varepsilon$ [@KPR Chapter I, Section 4]. A homomorphism is called **bounded** if $$\|T\|_{E\rightarrow E} =\sup _{x\ne 0} \frac{\|TX\|_E}{\|X\|_E}<\infty.$$ The homomorphism $T$ is said to be bounded on the pair $(L_0({{\mathcal M}},\tau),{{\mathcal M}})$ if $T$ is a bounded mapping from $L_0({{\mathcal M}},\tau)$ into $L_0({{\mathcal M}},\tau)$ and from ${{\mathcal M}}$ into ${{\mathcal M}}$.
\[th:bounded\] Let $E(0,\infty)$ be a symmetrically $\Delta$-normed function space and $T: S ({{\mathcal M}},\tau)\rightarrow S({{\mathcal M}},\tau)$ be a homomorphism which is bounded on $(L_0({{\mathcal M}},\tau),{{\mathcal M}})$ with $$\| TX\|_0 \le M_0 \|X\|_0, \ \forall X\in L_0({{\mathcal M}},\tau),$$ $$\|TX\|_\infty \le M_1 \|X\|_\infty, \ \forall X\in {{\mathcal M}}$$ for some constants $M_0,M_1>0$. Then, $T$ maps $E({{\mathcal M}},\tau)$ into itself and $$\begin{aligned}
\label{ineq:M1}
\mu(M_0t; TX) \le \mu(t;M_1X),~ X\in E({{\mathcal M}},\tau).
\end{aligned}$$
For any $P\in {{\mathcal P}}({{\mathcal M}})$ with $\tau(\mathbf{1}-P)<t$, we have $ \|T X({\bf 1}-P)\|_0 \le M_0\|X( \mathbf{1}-P)\|_0 \le M_0 t$. By [@LSZ Theorem 2.3.13], for every $t>0$, we have $$\begin{aligned}
\mu(M_0t;TX)
&=
\inf\{ \|TX-B\|_\infty : B\in S({{\mathcal M}},\tau), \ \|B\|_0 \le M_0t\}\\
&\le \|TX - TX({\bf 1}-P)\|_\infty = \| TX P\|_\infty \le M_1 \|XP\|_\infty.\end{aligned}$$ By Definition \[mu\], we have $$\begin{aligned}
\mu(M_0t; TX) \le M_1 \mu(t; X).\end{aligned}$$ This implies that $\sigma_{1/M_0}\mu( TX) \in E(0,\infty)$ and therefore, by appealing to , we conclude that $\mu( TX) =\sigma_{M_0}(\sigma_{1/M_0}\mu( TX) )\in E(0,\infty)$.
If $X,Y\in S({{\mathcal M}},\tau)$, then $X$ is said to be submajorized by $Y$, denoted by $X\prec\prec Y$, if $$\begin{aligned}
\int_{0}^{t} \mu(s;X) ds \le \int_{0}^{t} \mu(s;Y) ds , ~t\ge 0.\end{aligned}$$ A linear subspace $E$ of $S({{\mathcal M}},\tau)$ equipped with a complete norm $\|\cdot\|_E$, is called *fully symmetric space* (of $\tau$-measurable operators) if $X\in S({{\mathcal M}},\tau)$, $Y \in E$ and $X\prec\prec Y$ imply that $X\in E$ and $\|X\|_E \le \|Y\|_E$ [@DP2; @DPS; @LSZ].
For a symmetric normed function space $E(0,\infty)$, by [@DPS Theorem 10.13] (see also [@DDP] and [@KPS]), $E({{\mathcal M}},\tau)$ is an interpolation space between $L_1({{\mathcal M}},\tau)$ and ${{\mathcal M}}$ if $ E(0,\infty)$ is fully symmetric. In particular, one can find symmetric normed spaces $E({{\mathcal M}},\tau)$ which are not interpolation spaces between $L_1({{\mathcal M}},\tau)$ and $L_\infty({{\mathcal M}},\tau)$ [@KPS Chapter II, $\S$ 5.7]. However, for an arbitrary symmetrically $\Delta$-normed function space $E(0,\infty)$, $E({{\mathcal M}},\tau)$ is, in fact, an interpolation space between $L_0({{\mathcal M}},\tau)$ and ${{\mathcal M}}$.
\[cor:inter\] Let $E(0,\infty)$ be a symmetrically $\Delta$-normed function space and $T: S ({{\mathcal M}},\tau)\rightarrow S({{\mathcal M}},\tau)$ be a homomorphism which is bounded on $(L_0({{\mathcal M}},\tau),{{\mathcal M}})$ with $$\| TX\|_0 \le M_0 \|X\|_0, \ \forall X\in L_0({{\mathcal M}},\tau),$$ $$\|TX\|_\infty \le M_1 \|X\|_\infty, \ \forall X\in {{\mathcal M}}$$ for some constants $M_0,M_1>0$. Then, $T$ is a bounded homomorphism from $E({{\mathcal M}},\tau)$ into itself. In particular, $\sup_{X\in S({{\mathcal M}},\tau)}\frac{\|TX\|_{S}}{\|X\|_{S}} <\infty$.
By Theorem \[th:bounded\], for any $X\in E({{\mathcal M}},\tau)$, we have $TX\in E({{\mathcal M}},\tau)$ with $$\begin{aligned}
\label{ineq:M}
\mu(M_0t; TX) \le M_1 \mu(t; X).\end{aligned}$$ Let $k\ge 0$ be an integer such that $2^k \ge M_0$. Noticing that $\sigma_{2^k}(\mu(X) ) \ge \sigma_{M_0} (\mu(X))$. By (\[norm\_sigma\]), we have $$\label{ineq:CE}
\|\sigma_{M_0} (\mu(X))\|_E \le \|\sigma_{2^k}\mu(X)\|_E\le (2C_E)^k\|\mu(X)\|_E$$ for any $X\in E({{\mathcal M}},\tau)$.
Then, we get $$\begin{aligned}
\|TX\|_E &= \|\mu(TX)\|_E \stackrel{\eqref{ineq:CE}}{\le} (2C_E)^k \|\sigma_{1/M_0}(\mu(TX))\|_E \stackrel{\eqref{ineq:M}}{\le} (2C_E)^k\|M_1 \mu(X)\|_E\\
&\le (2C_E)^k \|([M_1]+1) \mu(X)\|_E \le (2C_E)^k \sum_{i=1}^{[M_1]+1}C_E^i \| \mu(X)\|_E,
\end{aligned}$$ where $[M_1]$ is the integer part of $ M_1$. The proof is complete.
The results in this section are applied to the study of orbits and ${{\mathcal K}}$-orbits in the pair of $(L_0({{\mathcal M}},\tau),{{\mathcal M}})$ in the next section.
Orbits and ${{\mathcal K}}$-orbits {#O}
==================================
For an element $X\in S ({{\mathcal M}},\tau)$, the orbit ${{\mbox{Orb}}}(X;L_0({{\mathcal M}},\tau),{{\mathcal M}})$ of $X$ is the set of all $Y\in S ({{\mathcal M}},\tau)$ such that $Y = TX$ for some homomorphism $T$ which is *bounded* on the pair $(L_0 ({{\mathcal M}},\tau), {{\mathcal M}})$. Furthermore, we define $$\|Y\|_{{{\mbox{Orb}}}} :=\inf_{Y=TX} \|T\|_{S({{\mathcal M}},\tau)},$$ where the infimum is taken over all bounded homomorphisms $T$ such that $TX=Y$ and $\|T\|_{S({{\mathcal M}},\tau)} =\max \{ \|T\|_{L_0\rightarrow L_0}, \|T \|_{L_\infty \rightarrow L_\infty}\}$.
By Theorem \[th:bounded\], we have the following proposition, which is an analogue of [@Astashkin Theorem 1].
\[prop:6.1\] Let $X\in S({{\mathcal M}},\tau)$. Then, for every $Y\in {{\mbox{Orb}}}(X; L_0({{\mathcal M}},\tau),{{\mathcal M}})$, we have $$\begin{aligned}
\label{ineq:C}
\mu( t; Y) \le \|Y\|_{{\mbox{Orb}}}\mu(\frac{t}{\|Y\|_{{\mbox{Orb}}}}; X),~ t>0.\end{aligned}$$
Notice for every $\varepsilon>0$, we can find a $T$ such that $Y=TX$ with $\|T\|_{S({{\mathcal M}},\tau)} \le \|Y\|_{{{\mbox{Orb}}}} +\varepsilon $. Then, by (\[ineq:M1\]), we have $$\begin{aligned}
\mu(\|T\|_{S({{\mathcal M}},\tau)} t; Y)=\mu(\|T\|_{S({{\mathcal M}},\tau)} t; TX) \le \|T\|_{S({{\mathcal M}},\tau)} \mu(t; X) .\end{aligned}$$ Hence, $\mu( (\|Y\|_{{{\mbox{Orb}}}} +\varepsilon) t; Y) \le (\|Y\|_{{{\mbox{Orb}}}} +\varepsilon)\mu(t; X) $ for every $t>0$. By the right-continuity of singular value functions, we have $$\mu(\|Y\|_{{{\mbox{Orb}}}} t; Y) \le \|Y\|_{{{\mbox{Orb}}}} \mu(t; X)$$ for every $t>0$, which completes the proof.
Let $(X_0,X_1)$ be a pair of symmetrically $\Delta$-normed spaces. The ${{\mathcal K}}$-orbit of $A\in X_0+X_1$ is defined by the set ${{\mathcal K}}{{\mathcal O}}(A;X_0,X_1)$ of all $X \in X_0+X_1$ such that $$\|X\|_{{{\mathcal K}}{{\mathcal O}}} :=\sup_{t>0} \frac{{{\mathcal K}}(t,X; X_0,X_1)}{{{\mathcal K}}(t, A; X_0,X_1)}<\infty,$$ where ${{\mathcal K}}(t,Z; X_0,X_1) := \inf\{\|Z_0\|_{X_0}+t\|Z_1\|_{X_1}: Z= Z_0 +Z_1 , Z_0\in X_0, Z_1\in X_1\}$.
A pair $(X_0,X_1)$ is called *${{\mathcal K}}$-monotone* if ${{\mathcal K}}{{\mathcal O}}(A;X_0,X_1) = {{\mbox{Orb}}}(A;X_0,X_1)$ for all $A\in X_0+X_1$. The pair $(L_1(0,\infty), L_\infty(0,\infty))$ is a classical example of a ${{\mathcal K}}$-monotone pair (see e.g. [@Calderon]). Moreover, the noncommutative pair $(L_1({{\mathcal M}},\tau),L_\infty({{\mathcal M}},\tau))$ is ${{\mathcal K}}$-monotone (see [@DDP Proposition 2.5, Theorem 4.7]).
It follows from the definition that the unit ball of ${{\mbox{Orb}}}(A;X_0,X_1)$ is a subset of the unit of ${{{\mathcal K}}{{\mathcal O}}}(A;X_0,X_1)$. Moreover, [@DDP Proposition 2.5 and Theorem 4.7] imply that the the closed unit ball of ${{\mbox{Orb}}}(A;L_1({{\mathcal M}},\tau),{{\mathcal M}})$ coincides with the unit ball of ${{{\mathcal K}}{{\mathcal O}}}(A;L_1({{\mathcal M}},\tau),{{\mathcal M}})$. However, it is known that the reverse inclusion may fail for certain element in the pair $L_0(0,\infty)+ L_\infty(0,\infty)$ [@Astashkin]. One of the main results of this section is a non-commutative version of [@Astashkin Theorem 4].
By Proposition \[prop:3.3\], the ${{\mathcal K}}$-orbit ${{\mathcal K}}{{\mathcal O}}(A;L_0({{\mathcal M}},\tau),{{\mathcal M}})$ of every ${{\mathcal A}}\in S({{\mathcal M}},\tau)$ is the set of all $X \in S({{\mathcal M}},\tau)$ such that $$\|X\|_{{{\mathcal K}}{{\mathcal O}}} :=\sup_{t>0} \frac{K_t(\mu(X))}{K_t(\mu(A))}<\infty.$$
If ${{\mathcal M}}$ is a non-trivial von Neumann algebra (${{\mathcal M}}\ne \mathbb{C}{\bf 1}$ and ${{\mathcal M}}\ne 0$), then there exist $A,X\in S({{\mathcal M}},\tau)$ such that $$K_t(A) = K_t(X)$$ whereas $\mu(t;A)<\mu(t;X)$, $t\in E$, for some measurable set $E$, $m(E)>0$. In particular, the unit ball of the ${{{\mathcal K}}{{\mathcal O}}}(A;L_0({{\mathcal M}},\tau),{{\mathcal M}})$ does not coincide with the unit ball of ${{\mbox{Orb}}}(A;L_0({{\mathcal M}},\tau),{{\mathcal M}})$.
Since ${{\mathcal M}}\ne \mathbb{C}\bf 1$, there exist two $\tau$-finite projections $P_1,P_2\in {{\mathcal P}}({{\mathcal M}})$ such that $P_1\perp P_2 $. Let $\tau_1:= \tau(P_1)>0$ and $\tau_2:=\tau(P_2)>0$.
Let $ k_1, k_2>0$ be such that $$\begin{aligned}
\label{ineq:tau_k}
k_1>k_2 >\frac{\tau_2 k_1}{\tau_1+\tau_2}.\end{aligned}$$
Define $$X:= k_1( P_1 +P_2).$$ Then, $\mu(X) = k_1 \chi_{(0, \tau_1+\tau_2)}$. By (\[ineq:K\]), we have $$K_t(X) = \min\{tk_1, \tau_1+\tau_2\}.$$ Then, we have $$K_t(X)=\left\{
\begin{aligned}
t k_1, & \qquad\qquad t< \frac{\tau_1+\tau_2}{k_1 }, \\
\tau_1+\tau_2 , & \qquad\qquad t \ge \frac{\tau_1+\tau_2}{k_1 }.
\end{aligned}
\right.$$ Define $$A:= k_1 P_1 + k_2 P_2.$$ Then, $\mu(A) = k_1 \chi_{(0,\tau_1)} + k_2 \chi_{[\tau_1,\tau_1+\tau_2)}$. By (\[ineq:K\]), we have $$K_t(A) = \min\{tk_1, \tau_1 + t k_2, \tau_1+\tau_2\}.$$ However, (\[ineq:tau\_k\]) implies that there is no such a $t$ such that $ \tau_1 + t k_2 \le \min\{tk_1, \tau_1+\tau_2\}$. Hence, we have $$K_t(A)=\left\{
\begin{aligned}
t k_1, & \qquad\qquad t< \frac{\tau_1+\tau_2}{k_1 } ,\\
\tau_1+\tau_2 , & \qquad\qquad t \ge \frac{\tau_1+\tau_2}{k_1 }.
\end{aligned}
\right.$$ That is, $K_t(A)=K_t(X)$. However, it is clear that $$\begin{aligned}
\label{9}
\mu(t;A) < \mu(t;X), ~\quad \tau_1\le t <\tau_1+\tau_2.\end{aligned}$$
Assume that $X$ lies in the unit ball of $ {{\mbox{Orb}}}(A;L_0({{\mathcal M}},\tau),{{\mathcal M}})$. By (\[ineq:C\]) $$\begin{aligned}
\mu( t; X) \le \|X\|_{{\mbox{Orb}}}\mu(\frac{t}{\|X\|_{{\mbox{Orb}}}}; A) \le \mu(t ; A), ~t>0,\end{aligned}$$ which is a contradiction with . Hence, $X$ lies in the unit ball of ${{{\mathcal K}}{{\mathcal O}}}(A;L_0({{\mathcal M}},\tau),{{\mathcal M}})$ but not in the unit ball of ${{\mbox{Orb}}}(A;L_0({{\mathcal M}},\tau),{{\mathcal M}})$.
In [@Astashkin], it is asserted incorrectly that the ${{\mbox{Orb}}}(A; L_0(0,\infty), L_\infty(0,\infty)) \ne {{{\mathcal K}}{{\mathcal O}}}(A;L_0(0,\infty), L_\infty(0,\infty))$. The following proposition together [@Astashkin Theorem 1] explains why ${{\mbox{Orb}}}(A; L_0(0,\infty), L_\infty(0,\infty)) = {{{\mathcal K}}{{\mathcal O}}}(A;L_0(0,\infty), L_\infty(0,\infty))$, i.e., the pair $(L_0(0,\infty),L_\infty(0,\infty))$ is ${{\mathcal K}}$-monotone. We would like to thank Professor Astashkin for providing the proof for the special case when ${{\mathcal M}}=L_\infty(0,\infty)$.
\[prop:6.5\] Let $A,X\in S({{\mathcal M}},\tau)$. Then, the following statements are equivalent.
1. There exists $C >0$ such that $\mu(s;X)\le C \mu(\frac{s}{C};A)$ for every $s>0$.
2. $\sup_{t>0} \frac{K_t(X)}{K_t(A)}<\infty$.
\(i) For every $A,X$ satisfying condition (1), we have $$\begin{aligned}
\sup_{t>0} \frac{K_t(X)}{K_t(A)} &= \sup_{t>0} \frac{\inf_s\{s + t \mu(s;X) \}}{\inf_s\{s + t \mu(s;A) \}}\le \sup_{t>0} \frac{\inf_s\{s + t C \mu(s/C ;A) \}}{\inf_s\{s + t \mu(s;A) \}}\\
&= \sup_{t>0} \frac{\inf_s\{C s + t C \mu(s ;A) \}}{\inf_s\{s + t \mu(s;A) \}}= C <\infty,\end{aligned}$$ which proves the validity of condition (2).
\(ii) Conversely, assume that $$\sup_{t>0} \frac{\inf_s\{s + t \mu(s;X) \}}{\inf_s\{s + t \mu(s;A) \}} \le C$$ for some $C>0$. For every $Z\in S({{\mathcal M}},\tau)$, we define $$M_t(Z):= \inf_s\{\max\{s,t \mu(s;Z)\}\}, ~ t>0 .$$ Clearly, we have $$M_t(Z)\le K_t(Z)\le 2M_t(Z),\;\;Z\in S({{\mathcal M}},\tau), ~t>0 \mbox{\cite{MP1991}}.$$ Therefore, $$\begin{aligned}
\label{eq1}
M_t(X)\le 2CM_t(A),\;\;t>0.\end{aligned}$$
Let $s\in(0,\infty)$ and let $t:=\frac{s}{\mu(s;X)}$ (without loss of generality, we may assume that $\mu(s;X)>0$). Notice that $s=t\mu(s;X)\le t\mu(s-\Delta_1 ;X)$ and $s=t\mu(s;X)\le s+\Delta_2 $ for any $\Delta_1 ,\Delta_2 >0$ with $s-\Delta_1 > 0$. We have, $$\begin{aligned}
\label{s=}
M_t(X)=s=t\mu(s;X).\end{aligned}$$ Let $s_1:=M_t(A)$. Then, we have $$\begin{aligned}
\label{s-}
t \mu(s_1^-;A )(:=t\lim_{k \uparrow s_1} \mu(k;A))\ge s_1\end{aligned}$$ (otherwise, we have $\max\{t\mu(s_1-\varepsilon;A ),s_1-\varepsilon \}<s_1 = M_t(A)$ for some $\varepsilon>0$, which is a contradiction to the definition of $M_t (A)$.).
Since $s \stackrel{\eqref{s=}}{=} M_t(X) \stackrel{\eqref{eq1}}{\le} 2C M_t(A) = 2Cs_1 $, it follows that $$\begin{aligned}
\label{esi1}
\mu(s_1^-;A)\le \mu((\frac{s}{2C})^-;A).
\end{aligned}$$ Then, we obtain $$t\mu(s;X) \stackrel{\eqref{s=}}{=} M_t(X) \stackrel{\eqref{eq1}}{\le} 2CM_t(A) = 2Cs_1 \stackrel{\eqref{s-}}{\le} 2C t \mu(s_1^-;A)\stackrel{\eqref{esi1}}{\le} 2Ct \mu((\frac{s}{2C})^-;A),$$ that is, $$\begin{aligned}
\mu(s;X)\le 2C \mu((\frac{s}{2C})^-;A) .\end{aligned}$$ Since $s>0$ is arbitrary taken, it follows that $$\mu(s;X)\le 3C \mu(\frac{s}{3C};A)$$ for every $s>0$, which completes the proof.
By the above proposition and [@Astashkin Theorem 1], we obtain the following result immediately, which implies that the commutative pair $(L_0(0,\infty), L_\infty(0,\infty))$ is indeed ${{\mathcal K}}$-monotone.
For every $a\in S(0,\infty)$, we have $${{\mbox{Orb}}}(a;L_0(0,\infty),L_\infty(0,\infty)) = {{{\mathcal K}}{{\mathcal O}}}(a;L_0(0,\infty),L_\infty(0,\infty)).$$
It is known that there cannot in general be a conditional expectation from $S({{\mathcal M}},\tau)$ onto a subalgebra of $S({{\mathcal M}},\tau)$ (see e.g. [@DSZ Appendix B]), which is the main obstacle in extending [@Astashkin Theorem 1] to the non-commutative case. The following theorem is the last result of this section, giving a non-commutative version of [@Astashkin Theorem 1] in the setting of non-atomic finite factors by approaches which are completely different from those used in [@Astashkin].
For the sake of convenience, we denote $L_\infty(0,\tau(s(X)))$ by $M_{\mu(X)}$. If ${{\mathcal M}}$ is a non-atomic semifinite von Neumann algebra, then for every $X\in S_0({{\mathcal M}},\tau)$, there exists a non-atomic commutative von Neumann subalgebra ${{\mathcal M}}_{|X|}$ in $s(|X| ){{\mathcal M}}s(|X|)$ and a trace-preserving $*$-isomorphism $J$ from $S({{\mathcal M}}_{|X|},\tau)$ onto the algebra $S(M_{\mu(X)},m)$ [@CKS; @DDP1992].
\[th:6.5\] If ${{\mathcal M}}$ is a non-atomic finite von Neumann factor with a faithful normal finite trace $\tau$, then for every $0 \ne A\in S({{\mathcal M}},\tau)$, ${{\mbox{Orb}}}(A;L_0({{\mathcal M}},\tau),L_\infty({{\mathcal M}},\tau))$ is the set of all $X\in S({{\mathcal M}},\tau)$ for which there exists $C>0$ such that $$\begin{aligned}
\label{CCC}
\mu(t;X) \le C \mu(\frac{t}{C};A),~t>0.\end{aligned}$$ In particular, ${{\mbox{Orb}}}(A;L_0({{\mathcal M}},\tau),L_\infty({{\mathcal M}},\tau)) = {{{\mathcal K}}{{\mathcal O}}}(A;L_0({{\mathcal M}},\tau),L_\infty({{\mathcal M}},\tau))$.
\(i) It follows from Proposition \[prop:6.1\] that for every $X\in {{\mbox{Orb}}}(A;L_0({{\mathcal M}},\tau),L_\infty({{\mathcal M}},\tau))$, there exists such a $C>0$ satisfying .
\(ii) Conversely, assume that $A,X\in S({{\mathcal M}},\tau)$ satisfies .
Then, by [@CKS Lemma 1.3], there are isomorphisms $J_A$ between $S(M_{\mu(A)},m)$ and $S({{\mathcal M}}_{|A|},\tau)$ such that $J_A \mu(A)=|A|$ and $J_X$ between $S(M_{\mu(X)},m)$ and $S({{\mathcal M}}_{|X|},\tau)$ such that $J_X\mu(X)=|X|$.
Then, by , we have $$2C \mu(\frac{t}{2C};A )-\mu(t;x) \ge C \mu(\frac{t}{2C};A ) + (C \mu(\frac{t}{2C};A )-\mu(t;x)) \ge C \mu(\frac{\tau(s(X))}{2C};A )$$ for every $t\in (0, \tau(s(X)))$. Without loss of generality, we can assume that $C$ is an integer which is large enough such that $$\begin{aligned}
\label{ineq2C}
2C \mu(\frac{t}{2C};A )-\mu(t;X)\ge C \mu(\frac{\tau(s(X))}{2C};A ) \ge 1\end{aligned}$$ for every $t\in (0, \tau(s(X)))$.
Let $\varepsilon<\frac{1}{2C}$. Since $J_A \mu(A) = |A|$, we can define $A_n:= [a_n,b_n)$, $n\ge 0$, by $$\chi_{A_n}= \chi_{ [a_n,b_n)}=J_A^{-1} (E^{|A|} (n\varepsilon,(n+1)\varepsilon]).$$ For every $ 0\le j\le 2C-1$, we set $$[a_{nj},b_{nj}) := [2C a_n + j(b_n -a_n ), 2C a_n + (j+1)(b_n -a_n )).$$
By [@CKS Lemma 1.3], for every $ [a_{nj},b_{nj})\subset [0,\tau(s(X)))$, we have $$\begin{aligned}
\label{16}
\tau(E^{|A|} (n\varepsilon,(n+1)\varepsilon])=\tau(J_A \chi_{[a_n,b_n)}) = \tau(J_X \chi_{[a_{nj},b_{nj})}).\end{aligned}$$ Since ${{\mathcal M}}$ is a finite factor, due to , there exist partial isometries $U_{nj} \in {{\mathcal M}}$ such that $U_{nj}U_{nj}^* = E^{|A|}(n\varepsilon,(n+1)\varepsilon]=J_A \chi_{[a_n,b_n)} $ and $U_{nj}^*U_{nj} = J_X \chi_{[a_{nj},b_{nj})} $. If $ [a_{nj},b_{nj}) \cap [0,\tau(s(X)))=\varnothing$, we define $U_{nj}:=0$. In the case when $ [a_{nj},b_{nj})\nsubseteq [0,\tau(s(X)))$ but $ [a_{nj},b_{nj}) \cap [0,\tau(s(X)))\ne \varnothing $, we define $U_{nj}$ as the partial isometry such that $U_{nj}U_{nj}^* =J_A \chi_{[a_n, a_n -a_{nj} + \tau(s(X))) }\le E^{|A|}(n\varepsilon,(n+1)\varepsilon] $ and $U_{nj}^*U_{nj} = J_X \chi_{[a_{nj},\tau(s(X)))} $.
Denote $U_j :=\sum_{n=0}^\infty U_{nj}$, $0\le j\le 2C-1$ (note that $U_{kj}^* U_{lj}=0$ and $U_{lj} U_{kj} ^* =0$ whenever $k\ne l$). Note that every $U_j$ is a partial isometry. Let $$B_1:=\sum_{n=1}^{\infty} n\varepsilon E^{|A|}(n\varepsilon,(n+1)\varepsilon]=\sum_{n=1}^{\infty} n\varepsilon J_A \chi_{[a_n,b_n)} \in S({{\mathcal M}}_{|A|},\tau)$$ and $$B_2:= \sum_{n,j}n\varepsilon J_X ( \chi_{[a_{nj},b_{nj})}\chi_{[0,\tau(s(X)))} )\in S({{\mathcal M}}_{|X|},\tau).$$ Then, noting that $U_{n,j}^* E^{|A|}(n\varepsilon,(n+1)\varepsilon]U_{n,j}= U_{n,j}^* J_A \chi_{[a_n,b_n)} U_{n,j} = J_X ( \chi_{[a_{nj},b_{nj})}\chi_{[0,\tau(s(X)))} )$ for every $n,j$, we have $$B_ 2 = \sum_{n,j} U_{nj} ^* n\varepsilon E^{|A|}(n\varepsilon,(n+1)\varepsilon] U _{nj}= \sum_{j=0}^{2C-1} U_{j} ^* B_1 U _{j}.$$ Since $\varepsilon<\frac{1}{2C}$, it follows that $0 \le \mu(A) - \mu(B_1 ) <\frac{1}{2C}$ and therefore, by , we have $$\begin{aligned}
\label{17}
2C \mu(t;B_2 ) -\mu(t;X) &=2C \mu(\frac{t}{2C};B_1 ) -\mu(t;X) \nonumber\\
&> 2C (\mu(\frac{t}{2C};A ) -\frac{1}{2C})-\mu(t;X) \ge 0\end{aligned}$$ for every $t\in [ 0, \tau(s(X)))$ (note that the $ \mu(t;B_2 ) =\mu(\frac{t}{2C};B_1 )$ follows immediately from the definitions of $B_1$ and $B_2$).
Let $A_\Delta:= \int \frac{[\lambda/\varepsilon] \varepsilon }{\lambda} dE ^{|A|}(\lambda)$ and $B_\Delta:= J_X \frac{\mu(X)}{\mu(B_2)}$. Here, $[\lambda/\varepsilon]$ is the integer part of $\lambda/\varepsilon$. Note that implies that $B_\Delta$ is a bounded operator. Clearly, we have $A_\Delta |A| = B_1$ and $|X| =J_X \frac{\mu(X)}{\mu(B_2)} J_X \mu(B_2)= B_\Delta B_2 $ (notice that $\sum_{n,j}n\varepsilon \chi_{[a_{nj},b_{nj})}\chi_{[0,\tau(s(X)))} =\mu(B_2)$). Let $U_A |A| = A$ and $U_X|X| =X$ be the polar decompositions. Define a homomorphism $T :S({{\mathcal M}},\tau) \rightarrow S({{\mathcal M}},\tau) $ by setting $$T Z =U_X B_\Delta \Big( \sum_{j=0}^{2C-1}U_j^*( A_\Delta U_A^* Z )U_j \Big) , ~ Z\in S({{\mathcal M}},\tau).$$ It is easy to verify that $TA=X$ and $T$ is a bounded homomorphism on the pair $(L_0({{\mathcal M}},\tau),{{\mathcal M}})$ (one should note that operators of multiplication by $U_j$, $0\le j\le 2C-1$, $A_\Delta$ and $B_\Delta$ are bounded homomorphisms on the pair $(L_0({{\mathcal M}},\tau),{{\mathcal M}})$ by Corollary \[cor:inter\]).
The last statement follows immediately from Proposition \[prop:6.5\].
The assumption that ${{\mathcal M}}$ is a finite factor plays a crucial role in the above proof. The authors did not succeed in extending the result to the case for general semifinite von Neumann algebras.
[**Acknowledgements**]{} The authors would like to thank Sergei Astashkin and Dima Zanin for helpful discussions.
The first author acknowledges the support of University International Postgraduate Award (UIPA). The second author was supported by the Australian Research Council.
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abstract: 'Evidence for the accretion of material in spiral galaxies has grown over the past years and clear signatures can be found in H0.1em[I]{} observations of galaxies. We describe here new detailed and sensitive H0.1em[I]{} synthesis observations of a few nearby galaxies (NGC 3359, NGC 4565 and NGC 6946) which show that indeed accretion of small amounts of gas is taking place. These should be regarded as examples illustrating a general phenomenon of gas infall in galaxies. Such accretion may also be at the origin of the gaseous halos which are being found around spirals. Probably it is the same kind of phenomenon of material infall as observed in the stellar streams in the halo and outer parts of our galaxy and M 31'
author:
- Thijs van der Hulst
- Renzo Sancisi
title: Gas accretion in galactic disks
---
Introduction
============
In this paper we present new evidence bearing on the formation of disks and halos of spiral galaxies through the accretion of small companions. This process is generally indicated as the nurture of galaxies. Evidence in support of it comes from various directions. The asymmetric shape of stellar disks (Zaritsky 1995 and Zaritsky & Rix 1997) and the morphological and kinematic lopsidedness observed in the H0.1em[I]{} density distributions and velocity fields of spirals (Verheijen 1997, Swaters et al. 1999) may have originated from recent minor mergers. Furthermore, there is an increasingly large number of galaxies which in H0.1em[I]{} show either peculiar features or clear signs of interactions with small companions (Sancisi 1999a and b). This indicates that galaxies often are in an environment where material for accretion is available.
Cold extra-planar gas has been found in several spiral galaxies. The best examples are those of NGC 891, NGC 2403 and UGC 7321 (cf. Oosterloo et al., Fraternali et al., Matthews, this volume). The origin of this gas is not known. It has been suggested that it may, at least partly, be the product of galactic fountains. But some of its structural properties suggest that it may have originated from minor mergers.
Recently, clear evidence that accretion events play an important role has come from studies of the distribution and kinematics of stars in the Milky Way halo. The discovery of the Sagittarius dwarf galaxy (Ibata et al. 1994) is regarded as proof that accretion is still taking place at the present time. Since such minor merger remnants retain information about their origin for a long time (Helmi & White 2000) studies of the distribution and kinematics of “stellar streams” can in principle be used to trace the merger history of the Milky Way (Helmi & de Zeeuw 2001). Such “stellar streams” are not only seen in the Milky Way, but have also been discovered in the Local Group galaxy M 31 (Ibata et al. 2001, Ferguson et al. 2002, McConnachie et al. 2003). The substructure in the halo of M 31 is another piece of clear evidence that minor mergers still take place.
Such events are difficult to trace in more distant galaxies, where we can not observe individual stars. Other means are needed for detecting the signature of accretion. In this respect, the use of H0.1em[I]{} is very powerful as it can image interactions very effectively by studying the H0.1em[I]{} distributions and kinematics. The latter is particularly useful for modelling. Examples can be found in Sancisi (1999a). The improved sensitivity of modern synthesis radio telescopes brings within reach the detection of faint H0.1em[I]{} signatures of accretion events and we expect that new observations of nearby and also more distant galaxies will reveal these in the coming decade. To further illustrate this point we here present a few examples of such signatures: NGC 3359, NGC 4565 and NGC 6946.
The observations
================
All three galaxies have been observed recently with the Westerbork Synthesis Radio Telescope (WSRT) using the new front-end and correlator providing a much improved sensitivity. We will discuss each case individually below. Details of the observations and some of these results have already been reported by van der Hulst & Sancisi (2004).
NGC 3359
--------
NGC 3359 is a nearby barred spiral galaxy (Hubble type SBc) which has been observed in H0.1em[I]{} by Broeils (1992) as part of a study of the mass distribution of a sample of nearby spiral galaxies. It has a total mass of $1.2 \times 10^{11}$ M$_{\odot}$ and an H0.1em[I]{} mass of $7.5 \times 10^{9}$ M$_{\odot}$ (Broeils & Rhee, 1997, adjusted for a Hubble constant of 72 km/s/Mpc). It has well developed spiral structure both in the optical and in H0.1em[I]{}. Kamphuis & Sancisi (1994, see also Sancisi 1999a) pointed out the presence of an H0.1em[I]{} companion which appears distorted and has a long tail which may connect to the H0.1em[I]{} disk of NGC 3359. This observation already indicated the possible accretion of gas by a large galaxy. Our new, more sensitive observations (rms noise of 0.85 mJy/beam for velocity and spatial resolutions of 10 km s$^{-1}$ and 30$^{\prime\prime}$ respectively) are shown in Figure 1 (left panel) and convincingly display an H0.1em[I]{} connection between the distorted H0.1em[I]{} companion and the main galaxy. The mass of the H0.1em[I]{} companion is $1.8
\times 10^{8}$ M$_{\odot}$ or 2.4% of the H0.1em[I]{} mass of NGC 3359. The H0.1em[I]{} distribution of the companion is clearly distorted and shows a tail pointing towards and connecting with the outer spiral structure of NGC 3359. No clear optical counterpart of the H0.1em[I]{} companion has yet been found.
The velocity structure of the H0.1em[I]{} companion and the connecting H0.1em[I]{} fits in very well with the regular velocity field of NGC 3359. This is shown in the right panel of Figure 1 where we display the emission in the individual channels superposed on the total H0.1em[I]{} image of NGC 3359. Contours of different shades of grey (low velocities are dark, high velocities are light) denote the outer edge of the H0.1em[I]{} emission in each of the velocity channels and thus display the basic kinematics of the H0.1em[I]{} without any further analysis of individual velocity profiles. The regularity of the velocities suggests that the process has been going on slowly for at least one rotational period which is of the order of 1.7 Gy.
NGC 4565
--------
NGC 4565 is a large edge-on galaxy of Hubble type Sb which was first observed in H0.1em[I]{} by Sancisi (1976) in an early search for galaxies with warped H0.1em[I]{} disks. Rupen (1991) observed NGC 4565 with much higher resolution and presented a detailed study of the kinematics and the warp. NGC 4565 has a small optical companion 6$^{\prime}$ to the north of the center, F378-0021557, which has $7.4 \times 10^{7}$ M$_{\odot}$ of H0.1em[I]{} compared to an H0.1em[I]{} mass of $2.0 \times 10^{10}$ M$_{\odot}$ for NGC 4565 (using a distance of 17 Mpc). An H0.1em[I]{} detection of this companion called NGC 4565A has also been reported by Rupen (1991). Another H0.1em[I]{} companion, NGC 4562, somewhat larger in H0.1em[I]{} ($2.5 \times 10^{8}$ M$_{\odot}$) and brighter optically can be found 15$^{\prime}$ to the south-west of the center of NGC 4565. The H0.1em[I]{} distribution, derived from a new sensitive WSRT observation by Dahlem (priv. comm.), is shown in Figure 2 superposed on the DSS. The asymmetric warp is clearly visible.
Inspection of individual channel maps brings to light that in addition to the warp the H0.1em[I]{} distribution shows additional, low surface brightness emission to the north of the center, in the direction of the faint companion F378-0021557. The H0.1em[I]{} emission in the velocity range from 1250 to 1290 km s$^{-1}$ (close to the velocity of F378-0021557 and to the systemic velocity, 1230 km s$^{-1}$, of NGC 4565) clearly shows distortions above the plane pointing towards the companion. This is best seen in Figure 3 which shows four channel maps chosen at velocities in this range. In these maps one can clearly see the H0.1em[I]{} layer bending towards F378-0021557, indicating a connection between F378-0021557 and a strong disturbance in the H0.1em[I]{} disk of NGC 4565. This disturbance is not associated with the warp. However, this bending of the H0.1em[I]{} layer, undoubtedly caused by the companion, is remarkably similar tot the outer H0.1em[I]{} warping. One wonders whether the type of interaction we are witnessing here is the mechanism also responsible for the creation of a warp. This interaction between NGC 4565 and its companion will eventually lead to a merger of the latter with NGC 4565.
To further elucidate the connection between F378-0021557 and NGC 4565 we show a position-velocity diagram along the H0.1em[I]{} connection in figure 4. This position-velocity cut has been taken parallel to the minor axis of NGC 4565 through F378-0021557 and clearly shows the velocity continuity of the H0.1em[I]{} disturbance in the disk of NGC 4565 and F378-0021557.
NGC 6946
--------
NGC 6946 is a bright, nearby spiral galaxy of Hubble type Scd which has been studied in H0.1em[I]{} numerous times (Rogstad et al. 1973, Tacconi & Young 1986, Kamphuis 1993). It was in this galaxy that Kamphuis and Sancisi (1993) found the first evidence for an anomalous velocity H0.1em[I]{} component which they associated with outflow of gas from the disk into the halo as a result of stellar winds and supernova explosions. Evidence for such a component is now being found in more galaxies as discussed by Fraternali et al. (2001, 2002, and also this volume). A detailed study of the anomalous H0.1em[I]{} and the structure in the H0.1em[I]{} disk is being carried out by Boomsma et al. (this volume) on the basis of very sensitive observations with the WSRT (rms of 0.5 mJy/beam for spatial and velocity resolutions of 60$^{\prime\prime}$ and 5 km s$^{-1}$ respectively).
Here we use a low resolution (60$^{\prime\prime}$) version of these data. Figure 6 shows a total H0.1em[I]{} image of NGC 6946 down to column density levels of $1.3 \times 10^{19}$ cm$^{-2}$. To the west two small companion galaxies can be seen. The most intruiging feature is the faint whisp to the north-west of the H0.1em[I]{} disk of NGC 6946. This faint H0.1em[I]{} extension can only be brought out at this resolution and appears to form a faint H0.1em[I]{} filament which blends smoothly (also kinematically) with the H0.1em[I]{} disk of NGC 6946 at a position some 11$^{\prime}$ (or 19 kpc) south of the tip of the filament. There is no detected connection with the two companion galaxies farther to the west. The spatial and velocity structure of the object are so regular, yet only connected to the main H0.1em[I]{} disk at one side that we prefer an explanation in terms of a tidally stretched, infalling H0.1em[I]{} object. So this looks like yet another example of accretion of small amounts of gas onto a large H0.1em[I]{} disk.
Similar examples are perhaps the filament discovered in NGC 2403 (Fraternali et al. 2001, 2002 and also this volume), a long H0.1em[I]{} filament in M 33 (van der Hulst, unpublished) and the extra-planar filaments in the northern part of the H0.1em[I]{} halo of NGC 891 (Fraternali et al., this volume).
Concluding remarks
==================
We have shown three cases with strong evidence for the accretion of small amounts of H0.1em[I]{}. These are certainly not unique. There are several more cases known (Sancisi 1999a and b). Furthermore, there are cases with such faint features that can only be seen in sensitive H0.1em[I]{} observations as the H0.1em[I]{} masses involved are rather modest. We therefore expect that with the increased sensitivity of modern synthesis radio telescopes, more examples will be discovered in the coming decade. There probably is a range of H0.1em[I]{} masses for these accretion events as is already apparent from the six cases mentioned here: NGC 891, NGC 2403, NGC 3359, NGC 4565, NGC 6946 and M 33.
What is the effect of accretion on the main galaxy? This can influence local star formation in the disks and starbursts. For instance, there may very well be a connection with the star formation activity in the disks of galaxies such as NGC 6946 and NGC 2403. The infall of gas may be at the origin of the extra-planar gas and gaseous halos recently discovered in spiral galaxies. Also, it may affect the structure of galactic disks in the outer parts and possibly also contribute to the formation of the outer layers and of warps in particular.
It is quite clear that future sensitive and detailed studies of the H0.1em[I]{} in nearby galaxies will provide a more complete census of the phenomena discussed in this paper and enable us to address these issues further and obtain more definitive answers. In particular, it should be possible to obtain estimates of the mean gas accretion rate in galaxies.
Broeils, A. H., 1992, PhD thesis, University of Groningen Broeils, A. H., Rhee, M. -H., 1997, , 324, 877 Ferguson, A. M. N., Irwin, M. J., Ibata, R. A., Lewis, G. F., & Tanvir, N. R. 2002, , 124, 1452 Fraternali, F., Oosterloo, T., Sancisi, R., & van Moorsel, G. 2001, , 562, L47 Fraternali, F., van Moorsel, G., Sancisi, R., & Oosterloo, T. 2002, , 123, 3124 Helmi, A. & Tim de Zeeuw, P. 2000, , 319, 657 Helmi, A. & White, S. D. M. 2001, , 323, 529 Ibata, R., Irwin, M., Lewis, G., Ferguson, A. M. N., & Tanvir, N. 2001, Nature, 412, 49 Ibata, R. A., Gilmore, G., & Irwin, M. J. 1994, Nature, 370, 194 Kamphuis, J. 1993, PhD thesis, University of Groningen Kamphuis, J. & Sancisi, R. 1993, , 273, L31 Kamphuis, J. & Sancisi, R. 1994, Panchromatic View of Galaxies. Their Evolutionary Puzzle, eds. G. Hensler, C. Theis and J.S. Gallagher, Editions Frontiers, p. 317 McConnachie, A. W., Irwin, M. J., Ibata, R. A., Ferguson, A. M. N., Lewis, G. F., & Tanvir, N. 2003, , 343, 1335 Rogstad, D. H., Shostak, G. S., & Rots, A. H. 1973, , 22, 111 Rupen, M. P. 1991, , 102, 48 Sancisi, R. 1976, , 53, 159 Sancisi, R. 1999a, IAU Symp. 186: Galaxy Interactions at Low and High Redshift, eds. J.E. Barnes and D. B. Sanders, p. 71 Sancisi, R. 1999b, , 269, 59 Tacconi, L. J. & Young, J. S. 1986, , 308, 600 van der Hulst, J. M. & Sancisi, S. 2004, IAU Symp. 217: Recycling intergalactic and interstellar matter, eds. P.-A. Duc, J. Braine and E. Brinks, p. 122 Zaritsky, D. 1995, , 448, L17 Zaritsky, D. & Rix, H. 1997, , 477, 118
|
---
abstract: 'In this talk – based on the results of a forthcoming paper (Coletti, Scozzafava and Vantaggi 2002), presented also by one of us at the Conference on “Non Classical Logic, Approximate Reasoning and Soft-Computing” (Anacapri, Italy, 2001) – we discuss the problem of representing default rules by means of a suitable coherent conditional probability, defined on a family of conditional events. An event is singled-out (in our approach) by a [*proposition*]{}, that is a statement that can be either [*true*]{} or [*false*]{}; a conditional event is consequently defined by means of two propositions and is a 3–valued entity, the third value being (in this context) a conditional probability.'
author:
- |
[**Giulianella Coletti**]{}\
Dipartimento Matematica e Informatica\
Università di Perugia, 06100 Perugia (Italy)\
[**Romano Scozzafava**]{}\
Dipartimento Metodi e Modelli Matematici\
Università La Sapienza, 00161 Roma (Italy)\
[**Barbara Vantaggi**]{}\
Dipartimento Metodi e Modelli Matematici\
Università La Sapienza, 00161 Roma (Italy)\
title: Default Logic in a Coherent Setting
---
INTRODUCTION
============
The concept of conditional event (as dealt with in this paper) plays a central role for the probabilistic reasoning. We give up (or better, in a sense, we generalize) the idea of de Finetti of looking at a conditional event $E|H$, with $H\neq\emptyset$ (the [*impossible*]{} event), as a $3$–valued logical entity looked on as “undetermined” when $H$ is false: it is [*true*]{} when both $E$ and $H$ are true, [*false*]{} when $H$ is true and $E$ is false, while we let the third value [*suitably depend on the given ordered pair*]{} $(E,H)$ and not being just an undetermined [*common value*]{} for all pairs. It turns out (as explained in detail in Coletti and Scozzafava 1999) that this function can be seen as a measure of the degree of belief in the conditional event $E|H$, which under “natural” conditions reduces to the conditional probability $P(E|H)$, in its most general sense related to the concept of [*coherence*]{}, and satisfying the classic axioms as given by de Finetti (1949), Rényi (1956), Krauss (1968), Dubins (1975): see Section 2. Notice that our concept of conditional event differs from that adopted, for example, by Adams (1975), Benferhat, Dubois and Prade (1997), Goodman and Nguyen (1988), Schay (1968).
Among the peculiarities (which entail a large flexibility in the management of any kind of uncertainty) of this concept of [*coherent*]{} conditional probability versus the usual one, we recall the following ones:
- due to its [*direct*]{} assignment as a whole, the knowledge (or the assessment) of the “joint” and “marginal” unconditional probabilities $P(E
\wedge H)$ and $P(H)$ is not required;
- the [*conditioning*]{} event $H$ (which [*must*]{} be a [*possible*]{} one) may have [*zero probability*]{}, but in the assignment of $P(E|H)$ we are driven by [*coherence*]{}, contrary to what is done in those treatments where the relevant conditional probability is given an [*arbitrary*]{} value in the case of a conditioning event of zero probability;
- a suitable interpretation of its extreme values $0$ and $1$ for situations which are different, respectively, from the trivial ones $E\wedge H =\emptyset$ and $H\subseteq E$, leads to a “natural” treatment of the [*default reasoning*]{}.
In this talk we deal with the latter aspect.
COHERENT CONDITIONAL\
PROBABILITY
=====================
The classic [*axioms for a conditional probability*]{} read as follows (given a set $\C=\G \times \B^o$ of conditional events $E|H$ such that $\G$ is a Boolean algebra and $\B \subseteq \G$ is closed with respect to (finite) logical sums, with $\B^o=\B\setminus\{\emptyset\}\,$):
- $P(H|H) = 1$, for every $H\in \B^o\,$,
- $P(\cdot|H)$ is a (finitely additive) probability on $\G$ for any given $H \in \B^o\,$,
- $P(E \wedge A|H)=P(E|H)P(A|E \wedge H)$,\
for any $A, E\in \G$, $H, E \wedge H \in \B^o$.
Conditional probability $P$ has been defined on $\G\times\B^o$; however it is possible, through the concept of [*coherence*]{}, to handle also those situations where we need to assess $P$ on an [*arbitrary*]{} set ${\cal C}$ of conditional events.
[**Definition 1**]{} - The assessment $P(\cdot|\cdot)$ on ${\cal C}$ is [*coherent*]{} if there exists $\C'\supset \C$, with $\C'= \G \times \B^o$, such that $P(\cdot|\cdot)$ can be extended from $\C$ to $\C'$ as a [*conditional probability*]{}.
A characterization of coherence is given (see, e.g., Coletti and Scozzafava 1996) by the following
[**Theorem 1**]{} - Let $\C$ be an arbitrary finite family of conditional events $E_1|H_1, \ldots, E_n|H_n$ and $\A_o$ denote the set of atoms $A_r$ generated by the (unconditional) events $E_1,H_1,\ldots,E_n, H_n$. For a real function $P$ on $\C$ the following two statements are equivalent:
\(i) $P$ is a [*coherent*]{} conditional probability on $\C$;
\(ii) there exists (at least) a [*class*]{} of probabilities $\{ P_0 ,P_1
,\ldots\,P_k\}$, each probability $P_\alpha$ being defined on a suitable subset $\A_\alpha \subseteq\A_o$, such that for any $E_i |H_i \in \C$ there is a unique $P_\alpha$ with $$\Sumer_{A_r \subseteq H_i} P_\alpha (A_r) >0 \,,\,$$ $$P(E_i|H_i) = \frac{\Sumer_{A_r \subseteq E_i \wedge H_i} P_\alpha
(A_r)}{\Sumer_{A_r \subseteq H_i} P_\alpha (A_r)}\,\,;\leqno(1)$$ moreover $\A_{\alpha '} \subset \A_{\alpha''}$ for $\alpha'
>\alpha''$ and $P_{\alpha''} (A_r) = 0$ if $A_r \in \A_{\alpha'}$.
According to Theorem 1, a coherent conditional probability gives rise to a suitable class $\{ P_o ,P_1 ,\ldots\,P_k\}$ of “unconditional” probabilities.
Where do the above classes of probabilities come from? Since $P$ is coherent on $\C$, there exists an extension $P^*$ on $\G \times \B^o$, where $\G$ is the algebra generated by the set $\A_o$ of atoms and $\B$ the additive class generated by $H_1,
\ldots, H_n$: then, putting $\F=\{\Omega, \emptyset\}$, the restriction of $P^*$ to $\A_o \times \F^o$ satisfies (1) with $\alpha = 0$ for any $E_i|H_i$ such that $P_o(H_i)>0\,$. The subset $\A_1\subset\A_o$ contains only the atoms $A_r \subseteq
H_o^1$, the union of $H_i$’s with $P_o(H_i)=0$ (and so on): we proved (see, e.g., Coletti and Scozzafava 1996, 1999) that, starting from a coherent assessment $P(E_i|H_i)$ on $\C$, a relevant family $\P = \{P_\al\}$ can be suitably defined that allows a representation such as (1). Every value $P(E_i|H_i)$ constitutes a constraint in the construction of the probabilities $P_\al$ $(\al = 0, 1,...)$; in fact, given the set $\A_o$ of atoms generated by $E_1,...,E_n,H_1,...,H_n$, and its subsets $\A_{\alpha}$ (such that $P_\b(A_r )=0$ for any $\b < \al$, with $A_r \in
\A_{\alpha}$), each $P_\al$ must satisfy the following system $(S_\alpha)$ with unknowns $P_\alpha (A_r ) \ge 0$, $A_r \in \A_{\alpha}$, $$(S_\alpha) \cases{
\displaystyle \Sumer_{A_r \subseteq E_i H_i} P_\alpha (A_r) = P (E_i|H_i) \Sumer_{A_r
\subseteq H_i} P_\alpha (A_r)\,,\vspace{1mm} \,\, \cr
\smallskip \big[{\rm if}\ P_{\alpha-1} (H_i) =0\big]\cr \noalign{\medskip}
\displaystyle \Sumer_{A_r\subseteq H_0^\alpha} P_\alpha (A_r) =1}$$
where $P_{-1} (H_i) =0$ for all $H_i$’s, and $H_o^\alpha$ denotes, for $\alpha \ge 0$, the union of the $H_i$’s such that $P_{\alpha-1}(H_i ) = 0$; so, in particular, $H_o^o = H_o = H_1 \vee \ldots \vee H_n\,.$
Any class $\{ P_\alpha\}$ singled-out by the condition $(ii)$ is said [*to agree*]{} with the conditional probability $P$. Notice that in general there are infinite classes of probabilities $\{P_\alpha\}$; in particular we have [*only one agreeing class*]{} in the case that $\C$ is a product of Boolean algebras.
A coherent assessment $P$, defined on a set $\C$ of conditional events, can be extended in a natural way to all the conditional events $E|H$ such that $E \wedge H$ is an element of the algebra $\G$ spanned by the (unconditional) events $E_i, H_i\,,\,
i=1,2,...,n$ taken from the elements of $\C$, and $H$ is an element of the additive class spanned by the $H_i$’s. Obviously, this extension is not unique, since there is no uniqueness in the choice of the class $\{ P_\alpha\}$ related to condition [*(ii)*]{} of Theorem 1.
In general, we have the following result (see, e.g., Coletti and Scozzafava 1996):
[**Theorem 2**]{} - If $\C$ is a given family of conditional events and $P$ a corresponding assessment, then there exists a (possibly not unique) coherent extension of $P$ to an arbitrary family $\K$ of conditional events, with $\K \supseteq
\C$, [*if and only if*]{} $P$ is coherent on $\C$.
Notice that if $P$ is coherent on a family $\C$, it is coherent also on $\E \subseteq
\C$.
ZERO-LAYERS
===========
Given a class $\P = \{P_\alpha\}_{\alpha=0,1,\dots,k}$, agreeing with a conditional probability on $\C$, it [*naturally induces*]{} the [*zero-layer*]{} ${\circ (H)}$ of an event $H$, defined as $${\circ (H)}=\beta\;\;\mbox{ if }
P_\beta(H)>0\,;\vspace{-2mm}$$ if $P_\alpha(H)=0$ for every $\alpha=0,1,\dots,k$ (obviously, we necessarily have $H \neq H_i$ for every $i=1,2,\dots,n$), then $\,{\circ (H)}=k+1$.
The zero-layer of a conditional event $E|H$ is defined as $${\circ (E|H)}={\circ (E\wedge H)}-{\circ (H)}.\vspace{-1mm}$$ Obviously, for the certain event $\Omega$ and for any event $E$ with positive probability, we have ${\circ (\Omega)}={\circ (E)}=0$ (so that, if the class contains only an [*everywhere positive*]{} probability $P_o$, there is only one (trivial) zero-layer, [*i.e.*]{} $\al=0$), while we put ${\circ (\emptyset)}=+\infty $. Clearly, $${\circ (A\vee
B)}=\min\{{\circ (A)}, {\circ (B)}\}. \vspace{-.1cm}$$ Moreover, notice that $P(E|H)>0$ if and only if ${\circ (E H)}= {\circ (H)}$, [*i.e.*]{} ${\circ (E|H)}=0$.
On the other hand, Spohn (see, for example, Spohn 1994, 1999) considers degrees of plausibility defined via a [*ranking*]{} function, that is a map $\kappa$ that assigns to each [*possible*]{} proposition a natural number (its [*rank*]{}) such that
1. either $\kappa(A)=0$ or $\kappa(A^c)=0$, or both;
2. $\kappa(A\vee B)=\min\{\kappa(A), \kappa(B)\}$;
3. for all $A\wedge B\neq \emptyset$, the conditional rank of $B$ given $A$ is $\kappa(B|A)=\kappa(A\wedge B)-\kappa(A)$.
Ranks represent degrees of “disbelief”. For example, $A$ is [*not*]{} disbelieved iff $\kappa(A)=0$, and it is disbelieved iff $\,\kappa(A)>0$.
[**Remark 1**]{} - [*Ranking functions are seen by Spohn as a tool to manage [*plain belief*]{} and [*belief revision*]{}, since he maintains that probability is inadequate for this purpose. In our framework this claim can be challenged (see [*Coletti, Scozzafava and Vantaggi 2001*]{}), since our tools for belief revision are [*coherent conditional probabilities*]{} and the [**ensuing**]{} concept of [*zero-layers*]{}: it is easy to check that zero-layers have the same formal properties of ranking functions*]{}.
COHERENT PROBABILITY AND DEFAULT LOGIC
======================================
We recall that in Coletti, Scozzafava and Vantaggi (2001) we showed that a sensible use of events whose probability is $0$ (or $1$) can be a more general tool in revising beliefs when new information comes to the fore, so that we have been able to challenge the claim contained in Shenoy (1991) that probability is inadequate for revising plain belief. Moreover, as recalled in Section 1, we may deal with the extreme value $P(E|H)=1$ also for situations which are different from the trivial one $H\subseteq
E$.
The aim of this Section is to handle, by means of a [*coherent*]{} conditional probability, some aspects of [*default reasoning*]{} (see, e.g., Reiter 1980, Russel and Norvig 1995): as it is well-known, a default rule is a sort of weak implication.
First of all, we discuss briefly some aspects of the classic example of Tweety.
The usual [*logical implication*]{} (denoted by $\subseteq$) can be anyway useful to express that a penguin ($\pi$) is [*certainly*]{} a bird ($\beta$), i.e. $$\pi
\subseteq \beta\,,$$ so that $$P(\beta|\pi)=1\,;$$ moreover we know that Tweety ($\tau$) is a penguin (that is, $\tau \subseteq \pi$), and so also this fact can be represented by a conditional probability equal to $1$, that is $$P(\pi|\tau)=1\,.$$
But we can express as well the statement “a penguin [*usually*]{} does not fly” (we denote by $\varphi^c$ the contrary of $\varphi$, the latter symbol denoting “flying”) by writing $$P(\varphi^c|\pi)=1\,.$$
(For simplicity, we have avoided to write down explicit a [*proposition*]{} – that is, an event – such as “a given animal is a penguin”, using the short-cut “penguin” and the symbol $\pi$ to denote this event; similar considerations apply to $\beta$, $\tau$ and $\varphi$).
The question “can Tweety fly?” can be faced through an assessment of the conditional probability $P(\varphi|\tau)$, which must be coherent with the already assessed ones: by Theorem 1, it can be shown that [*any value*]{} $p\in [0,1]$ is a coherent value for $P(\varphi|\tau)$, so that no conclusion can be reached – [*from the given premises*]{} – on Tweety’s ability of flying.
In other words, interpreting an equality such as $P(E|H)=1$ like a [*default rule*]{} (denoted by $\longmapsto$), which in particular (when $H \subseteq E$) reduces to the usual implication, we have shown its [*nontransitivity*]{}: in fact we have $$\tau\longmapsto\pi \mbox{\,\, and \,\,} \pi\longmapsto
\varphi^c\,,$$ but it [*does not*]{} necessarily follow the further default rule $\tau\longmapsto \varphi^c$ (even if we [*might*]{} have that $P(\varphi^c|\tau)=1$, [*i.e.*]{} that “Tweety usually does not fly”).
[**Definition 2**]{} - Given a [*coherent*]{} conditional probability $P$ on a family $\C$ of conditional events, a [*default rule*]{}, denoted by $H \longmapsto E$, is any conditional event $E|H \in \C$ such that $P(E|H)=1$.
Clearly, any logical implication $A\subseteq B$ (and so also any equality $A=B$) between events can be seen as a (trivial) default rule.
[**Remark 2**]{} - [*By resorting to the systems $(S_\alpha)$ to check the coherence of the assessment $P(E|H)=1$ (which implies, for the relevant zero-layer, ${\circ (E|H)}=0$), a simple computation gives $P_o(E^c \wedge H)=0$ (notice that the class $\{P_\alpha\}$ has in this case only one element $P_o$). It follows $\,{\circ (E^c|H)}=1$, so that $${\circ (E^c|H)} > {\circ (E|H)}\,.$$ In terms of Spohn’s ranking functions (we recall – [*and underline*]{} – that our zero-layers are – so to say – “incorporated” into a coherent conditional probability, so that [**we do not need**]{} an “autonomous” definition of ranking!) we could say, when $P(E|H)=1$, that the disbelief in $E^c|H$ is greater than that in $E|H$. This conclusion [**must not**]{} be read as $P(E|H)> P(E^c|H)$!*]{}
Given a [*set $\Delta \subseteq \C$ of default rules*]{} $H_i \longmapsto E_i\,$, with $i=1,...,n\,,$ we need to check its [*consistency*]{}, that is the coherence of the “global” assessment $P$ on $\C$ such that $P(E_i|H_i)=1\,$, $i=1,...,n\,$.
We stress that, even if our definition involves a conditional probability, the condition given in the following theorem refers [*only*]{} to logical (in the sense of [*Boolean*]{} logic) relations.
[**Theorem 3**]{} - Given a [*coherent*]{} conditional probability $P$ on a family $\C$ of conditional events, the following two statements are equivalent:
\(i) the set $\Delta \subseteq \C$ of default rules $$H_i \longmapsto E_i\,,\,\,
i=1,2,...,n\,,$$ represented by the assessment $$P(E_i|H_i)=1\,,\,\, i=1,2,...,n\,,$$ is consistent;
\(ii) for every subset $$\{H_{i_1} \longmapsto E_{i_1},\ldots, H_{i_s} \longmapsto
E_{i_s} \}\,$$ of $\Delta\,$, with $s=1,2,...,n$, we have $$\bigvee_{k=1}^s (E_{i_k}
\wedge H_{i_k}) \not\subseteq \bigvee_{k=1}^s (E_{i_k}^c \wedge H_{i_k}). \leqno(2)$$
[*Proof*]{} - We prove that, assuming the above logical relations (2), coherence of $P$ is compatible with the assessment $P(E_i|H_i)=1\;(i=1,2,...,n)\,,$ on $\Delta$.
We resort to the characterization Theorem 1: to begin with, put $P(E_i|H_i)=1\;(i=1,2,...,n)\,,$ in the system $(S_o)$. The unconditional probability $P_o$ can be obtained by putting $P_o(A_r)=0$ for all atoms $A_r \subseteq
\bigvee_{j=1}^n\Big(E_j^c \wedge H_j\Big)$, so for any atom $A_k \subseteq E_i \wedge
H_i $ which is not contained in $\bigvee_{j=1}^n\Big(E_j^c \wedge H_j\Big)\,$ – notice that condition (2) ensures that there is such an atom $A_k$, since $\bigvee_{j=1}^n (E_j\wedge H_j)\not\subseteq \bigvee_{j=1}^n (E_{j}^c \wedge H_{j})$ – we may put $P_o(A_k)> 0$ in such a way that these numbers sum up to 1, and we put $P_o(A_r)=0$ for all remaining atoms.
This clearly gives a solution of the first system $(S_o)$. If, for some $i$, $E_i
\wedge H_i \subseteq \bigvee_{j=1 }^n\Big(E_j^c \wedge H_j\Big)$, then $P_o(E_i\wedge
H_i)=0$. So we consider the second system (which refers to all $H_i$ such that $P_o(H_i)=0$), proceeding as above to construct the probability $P_1$; and so on. Condition (2) ensures that at each step we can give positive probability $P_\alpha$ to (at least) one of the remaining atoms.
Conversely, consider the (coherent) assignment $P(E_i|H_i)=1$ (for $i=1,...,n$). Then, for any index $j\in \{1,2,\dots,n\}$ there exists a probability $P_\alpha$ such that $P_\alpha(E_{j} \wedge H_{j})>0$ and $P_\alpha(E_{j}^c \wedge H_j)=0$. Notice that the restriction of $P$ to some conditional events $E_{i_1}|H_{i_1},...,E_{i_s}|H_{i_s}$ of $\Delta$ is coherent as well.
Let $P_o$ be the first element of an agreeing class, and $i_k$ an index such that $P_o(H_{i_k})>0\,$: then we have $P_o(E_{i_k} \wedge H_{i_k})>0$ and $P_o(E_{i_k}^c
\wedge H_{i_k})=0$. Suppose that $E_{i_k} \wedge H_{i_k} \subseteq \bigvee_{k=1}^s
(E_{i_k}^c \wedge H_{i_k})\,$: then $P_o(E_i{_k}\wedge H_{i_k})=0$. This contradiction shows that condition (2) holds.
[**Definition 3**]{} - A set $\Delta$ of default rules [*entails*]{} the default rule $H\longmapsto E$ if the [*only*]{} coherent value for $P(E|H)$ is 1. In other words, the rule $H\longmapsto E$ is entailed by $\Delta$ (or by a subset of $\Delta$) if every possible extension (cf. Theorem 2) of the probability assessment $P(E_{i_r}|H_{i_r})=1\,$, $r=1 \dots s\,,$ assigns the value $1$ also to $P(E|H)$.
Going back to the previous example of Tweety, its possible ability (or inability) of flying can be expressed by saying that the default rule $\tau \longmapsto \varphi$ (or $\tau \longmapsto
\varphi^c$) [*is not entailed*]{} by the premises (the given set $\Delta$).
INFERENCE
=========
Several formalisms for default logic have been studied in the relevant literature with the aim of discussing the minimal conditions that an entailment should satisfy. In our framework this “inferential” process is ruled by the following
[**Theorem 4**]{} - Given a set $\Delta$ of consistent default rules, we have:
[**(Reflexivity)**]{}
$\Delta$ entails $A\longmapsto A\;\; \mbox{ for any }
A\neq\emptyset\,$;
[**(Left Logical Equivalence)**]{}
$(A=B) \,,\, (A\longmapsto C)\in \Delta$ entails $ B\longmapsto
C\,$;
[**(Right Weakening)**]{}
$(A\subseteq B) \,,\, (C\longmapsto A)\in \Delta$ entails $
C\longmapsto B\,$;
[**(Cut)**]{}
$(A\wedge B\longmapsto C) \,,\, (A\longmapsto B)\in \Delta$ entails $A\longmapsto C\,$;
[**(Cautious Monotonicity)**]{}
$(A\longmapsto B) \,,\, (A\longmapsto C)\in \Delta$ entails $A\wedge B\longmapsto C\,$;
[**(Equivalence)**]{}
$(A\longmapsto B) \,,\, (B\longmapsto A) \,,\, (A\longmapsto C)\in \Delta$ entails $ B\longmapsto C\,$;
[**(And)**]{}
$(A\longmapsto B) \,,\, (A\longmapsto C)\in \Delta$ entails $
A\longmapsto B\wedge C\,$;
[**(Or)**]{}
$(A\longmapsto C) \,,\, (B\longmapsto C)\in \Delta$ entails $
A\vee B\longmapsto C\,$.
[*Proof*]{} - [*Reflexivity*]{} amounts to $P(A|A)=1$ for every possible event.
[*Left Logical Equivalence*]{} and [*Right weakening*]{} trivially follow from elementary properties of conditional probability.
[*Cut*]{}: from $P(C|A\wedge B)=P(B|A)=1$ it follows that $$P(C|A)=P(C|A\wedge
B)P(B|A)+P(C|A\wedge B^c)P(B^c|A)=$$ $$=P(C|A\wedge B)P(B|A) =1\,.$$
[*Cautious Monotonicity*]{}: since $P(B|A)=P(C|A)=1$, we have that $$1= P(C|A\wedge
B)P(B|A)+
P(C|A\wedge B^c)P(B^c|A)=$$ $$=P(C|A\wedge B)P(B|A)\,,$$ hence $P(C|A\wedge B)=1$.
[*Equivalence*]{}: since at least one conditioning event must have positive probability, it follows that $A, B, C$ have positive probability; moreover, $$P(A\wedge C)=P(A)=P(A\wedge B)=P(B)\,,$$ which implies $P(A\wedge B\wedge
C)=P(A)=P(B)$, so $P(C|B)=1$.
[*And*]{}: since $$1\geq P(B\vee C|A)=P(B|A)+P(C|A)-P(B\wedge
C|A)=$$ $$=2-P(B\wedge C|A)\,,$$ it follows $P(B\wedge C|A)=1$.
[*Or*]{}: since $$P(C|A\vee B)=$$ $$=P(C|A)P(A|A\vee B)+P(C|B)P(B|A\vee B)-$$ $$-P(C|A\wedge B )P(A\wedge B|A\vee B)=$$ $$= P(A|A\vee B)+P(B|A\vee B)-$$ $$-P(C|A\wedge B )P(A\wedge B|A\vee B)\geq 1\,,$$ we get $P(C|A\vee B)=1$.
We consider now some “unpleasant” properties (cf., e.g., Lehmann and Magidor, 1992), that in fact do not necessarily hold also in our framework:
[**(Monotonicity)**]{}
$(A\subseteq B) \,,\, (B\longmapsto C) \in \Delta\,$ entails $
A\longmapsto C$
[**(Transitivity)**]{}
$(A\longmapsto B) \,,\, (B\longmapsto C) \in \Delta\,$ entails $
A\longmapsto C$
[**(Contraposition)**]{}
$(A\longmapsto B) \in \Delta\,$ entails $ B^c\longmapsto
A^c$
The previous example about Tweety shows that [*Transitivity*]{} can fail.
In the same example, if we add the evaluation $P(\varphi|\beta)=1$ (that is, a bird [*usually*]{} flies) to the initial ones, the assessment is still coherent (even if $P(\varphi|{\pi})=0$ and $\pi\subseteq \beta$), but [*Monotonicity*]{} can fail.
Now, consider the conditional probability $P$ defined as follows: $$P(B|A)=1 \;,\;
P(A^c|B^c)=\frac{1}{4}\;;$$ it is easy to check that it is coherent, and so [*Contraposition*]{} can fail.
Many authors (cf., e.g., again Lehmann and Magidor, 1992) claim (and we agree) that the previous unpleasant properties should be replaced by others, that we express below in our own notation and interpretation: we show that these properties hold in our framework. Since a widespread consensus among their “right” formulation is lacking, we will denote them as cs–(Negation Rationality), cs–(Disjunctive Rationality), cs–(Rational Monotonicity), where “cs” stands for “in a coherent setting”. Notice that, given a default rule $H\longmapsto E$, to say $(H\longmapsto E) \not\in \Delta$ means that the conditional event $E|H$ belongs to the set $\C
\setminus\Delta$.
[**cs–(Negation Rationality)**]{}
If $(A\wedge C \longmapsto B) \,,\, (A\wedge C^c \longmapsto B) \not\in \Delta$\
then $\Delta$ does not entail $ (A\longmapsto B)$
[*Proof*]{} - If $(A\wedge C \longmapsto B)$ and $(A\wedge C^c \longmapsto B)$ do not belong to $\Delta$, i.e. $P(B|A\wedge C)<1$ and $P(B|A\wedge C^c)<1$ imply $$P(B|A)=P(B|A\wedge C)P(C|A)+P(B|A\wedge C^c)P(C^c|A)<$$ $$<P(C|A)+P(C^c|A)=1\,.$$
[**cs–(Disjunctive Rationality)**]{}
If $(A \longmapsto C) \,,\, (B \longmapsto C) \not\in\Delta$\
then $\Delta$ does not entail $(A\vee B\longmapsto C)$
[*Proof*]{} - Starting from the equalities $$P(C|A\vee B)=$$ $$= P(C|A)P(A|A\vee B)
+P(C|A^c\wedge B)P(A^c\wedge B|A\vee B)$$ and $$P(C|A\vee B)=$$ $$=P(C|B)P(B|A\vee B) +P(C|A\wedge B^c)P(A\wedge B^c|A\vee B),$$ since we have $P(C|A)<1$ and $P(C|B)<1$, then $P(C|A\vee B)=1$ would imply (by the first equality) $P(A|A\vee B)=0$ and (by the second one) $P(B|A\vee B)=0$ (contradiction).
[**cs–(Rational Monotonicity)**]{}
If $(A \wedge B\longmapsto C) \,,\, (A \longmapsto B^c) \not\in\Delta$\
then $\Delta$ does not entail $ (A\longmapsto C)$
[*Proof*]{} - If it were $P(C|A)=1$, i.e. $$1=P(C|A\wedge B)P(B|A)+P(C|A\wedge
B^c)P(B^c|A)\,,$$ we would get either $$P(C|A\wedge B)=P(C|A\wedge B^c)=1$$ or one of the following $$P(C|A\wedge B)=P(B|A)= 1 \,,$$ $$P(C|A\wedge B^c)=P(B^c|A)= 1$$ (contradiction).
In conclusion, let us notice the simplicity of our approach (Occam’s razor...!), with respect to other well-known methodologies, such as, e.g. those given by Adams (1975), Benferhat, Dubois and Prade (1997), Goldszmidt and Pearl (1996), Lehmann and Magidor (1992), Schaub (1998).
DISCUSSION
==========
Thought-provoking comments of two anonymous reviewers suggested to us to add this further section.
Among coherence–based approaches to default reasoning (in the framework of imprecise probability propagation), that of Gilio (2000) deserves to be mentioned, even if we claim (besides the utmost simplicity of our definitions and results) many important semantic and syntactic differences.
First of all, our framework (see the very beginning of our Introduction) is clearly and rigorously settled: conditional events $E|H$ are [**not**]{} 3-valued entities whose third value is looked on as “undetermined” when $H$ is false, but they have been defined instead in a way which entails “automatically” (so-to-say) the axioms of conditional probability, which are those [*ruling coherence*]{} (the details, as already recalled in the Introduction, are in Coletti and Scozzafava, 1999).
In other words ([**french**]{} words, since we are in France), “tout se tient”, while in the aforementioned paper by Gilio a concept such as $E|H$ is interpreted sometimes as a 3-valued entity looked on as “undetermined” when $H$ is false, sometimes as an ordered pair of events, sometimes as a conditional assertion $H\,|\hspace{-2mm}\sim E$ (in the knowledge base).
Moreover, our notions of [*consistency*]{} and [*entailment*]{} are both different from his: in fact he gives a theorem (without proof) connecting the notion of consistency to that of Adams (1975).
The problem is that we do not understand Adams’ framework: in fact he requires probability to be [*proper*]{} (i.e., positive) on the given events, but (since the domain of a probability $P$ is an algebra) we need to extend $P$ from the given events to other events (by the way, coherence is nothing but complying with this need). In particular, these “new” events may have zero probability: it follows, according to Adams’ definition of [*conditional*]{} probability in the case of a conditioning event of zero probability, that we can easily get [*incoherent*]{} assessments (see the example below). By the way, in the section “[**Some preliminaries**]{}”, Gilio claims “We can frame our approach to the problem of propagating imprecise conditional probability assessments from the [*probabilistic logic*]{} point of view, see, e.g., Frisch and Haddawy ...”: unfortunately, Frisch and Haddawy definition of conditional probability coincides (for conditioning events which are null) with that of Adams, and so it violates coherence as well!
Not to mention that both Gilio and Adams (and many others: some of them are mentioned at the end of the previous section) base the concept of consistency on that of [*quasi conjunction*]{}, which is a particular conditional event (and our concept of conditional event is different from theirs); moreover we deem that the notion they give of [*verifiability*]{} of a conditional event $E|H$, that is $E \wedge H \neq
\emptyset$, is too weak – except in the case $H=\Omega$ – to express properly the relevant semantics.
Our discussion can be better illustrated by the following (very simple) example:
[**Example**]{} - Consider two (logically independent) events $H_1$ and $H_2$, and put $$E_1=H_1 \wedge H_2 \;,\; E_2=H_1^c \wedge H_2\,,$$ $$E_3=H_1^c \wedge H_2^c \;,\;
E=E_2 \;,\; H=H_3=\Omega\,.$$ Given $\alpha\,$, with $\,0<\alpha<1\,$, the assessment $$P(E_1|H_1)=P(E_2|H_2)=1\;,\;P(E_3|H_3)=\alpha$$ on $\C=\{E_1|H_1 , E_2|H_2 ,
E_3|H_3\}$ is coherent; the relevant probabilities of the atoms are $$P(H_1 \wedge
H_2) = P(H_1 \wedge H_2^c)=0\,,$$ $$P(H_1^c \wedge H_2^c)= \alpha \,,\, P(H_1^c \wedge
H_2)=1-\alpha\,,$$ so that the set $\Delta$ of default rules corresponding to $\{E_1|H_1 , E_2|H_2\}\,$ is consistent.
Does $\Delta$ entail $E|H\,$? A simple check shows that the only coherent assessment for this conditional event is $P(E|H)=1-\alpha$. Then the answer is NO, since we require (in the definition of entailment) that $1$ is (the only) coherent extension.
On the contrary, according to Gilio characterization of entailment – that is: $\Delta$ (our notation) entails $E|H\,$ iff $P(E^c|H)=1$ is not coherent – the answer to the previous question is YES, since the only coherent value of this conditional probability is $P(E^c|H)=\alpha\,$ (see the above computation).
For any $\epsilon>0$, consider now the assessment $$P(E_1|H_1)=1 \,,\,
P(E_2|H_2)=1-\epsilon\;,$$ so that $\{E_1|H_1 , E_2|H_2\}\,$ is consistent according to Adams, as can be easily checked giving the atoms the probabilities $$P(H_1 \wedge
H_2) = \epsilon \,,\, P(H_1 \wedge H_2^c)=0\,,$$ $$P(H_1^c \wedge H_2^c)=0 \,,\,
P(H_1^c \wedge H_2)=1-\epsilon\,,$$ (notice that the assessment is proper). But for any event $A \subset H_1 \wedge H_2^c$ we can extend $P$, according to his definition of conditional probability, as $$P(A|H_1 \wedge H_2^c)=P(A^c|H_1 \wedge H_2^c)=1\,,$$ which is [**not**]{} coherent!
Finally, there is no mention in Gilio’s paper of Negation Rationality, Disjunctive Rationality, and Rational Monotonicity (and, according to one of the reviewers, these properties do not hold “in default reasoning under coherent probabilities”, while in our setting they have been proved at the end of Section 5).
### Acknowledgements {#acknowledgements .unnumbered}
We thank an anonymous referee for signaling us a slight mistake in the proof of Disjunctive Rationality.
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abstract: |
We report the observation of intensity feedback random lasing at 645 nm in Disperse Orange 11 dye-doped PMMA (DO11/PMMA) with dispersed ZrO$_2$ nanoparticles (NPs). The lasing threshold is found to increase with concentration, with the lasing threshold for 0.1 wt% being $75.8 \pm 9.4$ MW/cm$^2$ and the lasing threshold for 0.5 wt% being $121.1 \pm 2.1$ MW/cm$^2$, with the linewidth for both concentrations found to be $\approx 10$ nm. We also consider the material’s photostability and find that it displays fully reversible photodegradation with the photostability and recovery rate being greater than previously observed for DO11/PMMA without NPs. This enhancement in photostability and recovery rate is found to be explicable by the modified correlated chromophore domain model, with the NPs resulting in the domain free energy advantage increasing from 0.29 eV to 0.41 eV. Additionally, the molecular decay and recovery rates are found to be in agreement with previous measurements of DO11/PMMA \[Polymer Chemistry **4**, 4948 (2013)\]. These results present new avenues for the development of robust photodegradation-resistant organic dye-based optical devices.
PACS: 42.55.Mv,42.55.Zz,42.70.Hj, 42.70.Jk
author:
- 'Benjamin R. Anderson$^*$'
- Ray Gunawidjaja
- Hergen Eilers
bibliography:
- 'PrimaryDatabase.bib'
- 'ASLbib.bib'
title: 'Random Lasing and Reversible Photodegradation in Disperse Orange 11 Dye-Doped PMMA with Dispersed ZrO$_2$ Nanoparticles'
---
Introduction
============
Lasing in scattering media – known as random lasing (RL) – was first predicted by Letokhov and coworkers [@Letokhov67.01; @Letokhov67.02; @Letokhov66.01] in the late 1960’s and then experimentally observed by Lawandy *et al.* in 1994 [@Lawandy94.01]. RL differs from normal lasing in that random lasers operate without an external cavity, with scattering acting as the feedback mechanism [@Cao03.01; @Wiersma96.01; @Wiersma08.01]. Due to different scattering regimes in diffuse media, RL is found to have two distinct spectral classes: intensity feedback random lasing (IFRL) and resonant feedback random lasing (RFRL) [@Cao03.01; @Cao05.01; @Ignesti13.01]. IFRL is characterized by a single narrow emission peak (FWHM on the order of 10’s of nm) and is wholly determined by the diffusive nature of light [@Lawandy94.01; @Burin01.01; @Wiersma96.01; @Pinheiro06.01]. On the other hand, RFRL is characterized by multiple sub-nm width peaks [@Ling01.01; @Cao03.01; @Cao03.02; @Cao05.02; @Cao99.01; @Tureci08.01] with two proposed mechanisms: strong scattering resonances [@Molen07.01] and Anderson localization of light [@Cao00.01]. Based on these proposed mechanisms, models of RFRL have been developed using spin-glass modeling of light [@Angelani06.01], Levy-flight scattering [@Ignesti13.01], condensation of lasing modes [@Conti08.01; @Leonetti13.03], and strongly interacting lossy modes [@Tureci08.01].
Regardless of the microscopic mechanisms of the two regimes, their spectral characteristics can be described macroscopically in terms of active lasing modes, with RFRL representing a few distinct active lasing modes, and IFRL representing multiple overlapping active lasing modes [@Cao03.01; @Cao03.02; @Ling01.01; @Tureci08.01; @Ignesti13.01]. The different modal nature of the two regimes makes each attractive for different applications. Since RFRL has few modes –allowing for the creation of unique spectral signatures – it is attractive in the fields of authentication [@Zurich08.01], biological imaging, emergency beacons [@Hoang10.01; @Cao05.01], and random number generation [@Atsushi08.01; @Murphy08.01; @Mgrdichian08.01]. Also, the limited number of active modes in RFRL allows for a high degree of spectral control, as the pump beam can be modulated using a spatial light modulator (SLM) to activate only certain lasing modes, thus controlling the RFRL spectrum [@Leonetti12.01; @Cao05.01; @Leonetti13.02; @Leonetti12.02; @Leonetti12.03; @Andreasen14.01; @Bachelard12.01; @Bachelard14.01]. This spectral control is attractive for implementing optically based physically unclonable functions [@Anderson14.04; @Anderson14.05; @Eilers14.01] and the creation of bright tunable light sources [@Cao05.01].
In the case of IFRL, the many active modes leads to the emission having a low degree of spatial coherence [@Redding11.01], making it an attractive method for high-intensity low-coherence light sources [@Redding12.01]. Such light sources have applications in photodynamic therapy, tumor detection [@Hoang10.01; @Cao05.01], flexible displays, active elements in photonics devices [@Cao05.01], picoprojectors, cinema projectors [@Hecht12.01], and biological imaging [@Redding12.01; @Hecht12.01].
With the wide variety of possible applications for RL, work on RL has focused on discovering materials that have the following three characteristics: they provide desirable RL spectra, are relatively cheap and easy to work with, and are robust enough to use over a reasonable time frame. One such class of materials that fulfills the first two criteria are organic-dye-based materials. Unfortunately, most organic-dye-based systems are found to *irreversibly* photodegrade when exposed to intense radiation [@wood03.01; @taylo05.01; @Avnir84.01; @Knobbe90.01; @Kaminow72.01; @Rabek89.01], thus limiting their usefulness in optical devices. However, in the past two decades it has been discovered that some dye-doped polymers actually photodegrade *reversibly*, with the material self-healing once the illumination is turned off for a period of time. These materials include Rhodamine B and Pyrromethene dye-doped (poly)methyl-methacrylate (PMMA) optical fibers [@Peng98.01], disperse orange 11 (DO11) dye-doped PMMA [@howel04.01; @howel02.01] and styrene-MMA copolymer [@Hung12.01], anthraquinone-derivative-doped PMMA [@Anderson11.02], 8-hydroxyquinoline (Alq) dye-doped PMMA [@Kobrin04.01], and air force 455 (AF455) dye-doped PMMA [@Zhu07.01]. In all these studies the dye was doped into the polymer without any scattering particles, such that no RL was observed.
Given the large number of dye-doped polymers that display self-healing (but not RL) and the desirability of a self-healing organic dye-based random laser, we recently tested random lasers consisting of Rhodamine 6G dye-doped polyurethane with dispersed ZrO$_2$ (R6G+ZrO$_2$/PU) [@Anderson14.04] or Y$_2$O$_3$ (R6G+Y$_2$O$_3$/PU) nanoparticles (NP) for reversible photodegradation [@Anderson15.01; @Anderson15.03]. In those studies we found that R6G+ZrO$_2$/PU and R6G+Y$_2$O$_3$/PU display self-healing after photodegradation, with a recovery efficiency of 100% [@Anderson15.01; @Anderson15.03]. However, we also found that the photodegradation could not be called truly reversible as the RL wavelength and linewidth changed [@Anderson15.01; @Anderson15.03].
While the approach used in our recent study – of testing an already known RL system for self-healing – was successful at producing a self-healing organic dye based random laser, a different approach to the problem is to develop an already known self-healing system into a random laser. To this end we investigate RL in DO11/PMMA with dispersed ZrO$_2$ NPs. The choice to use DO11/PMMA is three-fold: (1) DO11 has previously been shown to be suitable as a laser dye [@howel02.01; @howel04.01], (2) the majority of organic-dye based RL studies have focused on Rhodamine dyes and therefore DO11 is a new and unique organic dye in RL studies, with its lasing wavelength being attractive for use with polymer optical fibers [@howel02.01] and (3) DO11/PMMA is the test bed system for self-healing research, with numerous studies performed to understand the phenomenon of self healing in DO11/PMMA.
These studies have been performed with different probe techniques including: absorption [@embaye08.01; @Anderson14.02], white light interferometry [@Anderson14.03], fluorescence [@Dhakal12.01; @raminithesis], photoconductivity [@Anderson13.02], transmittance microscopy [@Anderson11.01; @Anderson13.01], and amplified spontaneous emission (ASE) [@howel02.01; @howel04.01; @embaye08.01; @Ramini12.01; @Ramini13.01]. These techniques have been used to characterize the behavior of DO11/PMMA’s photodegradation and recovery under different wavelengths [@Anderson15.04], temperatures [@Ramini12.01; @Ramini13.01; @raminithesis; @andersonthesis], applied electric fields [@Anderson13.01; @andersonthesis; @Anderson14.01], co-polymer compositions [@Hung12.01], thicknesses [@Anderson14.01], concentrations [@Ramini12.01; @Ramini13.01; @raminithesis], and intensities [@Anderson11.01; @Anderson14.01; @Anderson14.02]. Based on all these studies a model has been developed to describe DO11/PMMA’s photodegradation and recovery called the correlated chromophore domain model (CCDM) [@Ramini12.01; @Ramini13.01; @raminithesis; @Anderson14.02; @andersonthesis].
The CCDM posits that dye molecules form linear isodesmic domains along polymer chains with molecular interactions – mediated by the polymer – resulting in increased photostability and self-healing. Within the domain model the decay rate $\alpha$, depends inversely on the domain size $N$, as
$$\alpha=\frac{\alpha_1}{N},\label{eqn:domdec}$$
and the recovery rate $\beta$, depends linearly on the domain size, $$\beta=\beta_1N,\label{eqn:domrec}$$ where $\alpha_1$ and $\beta_1$ are the unitary domain decay and recovery rates, respectively. While these rates describe the dynamics of a single domain, the macroscopically measured rates result from an ensemble average over the distribution of domains $\Omega(N)$, which depends on the density of dye molecules $\rho$, and the free energy advantage $\lambda$ [@Ramini12.01; @Ramini13.01; @raminithesis; @Anderson14.02; @andersonthesis].
Method
======
In order to produce a suitable DO11 based random lasing material we disperse ZrO$_2$ NPs into DO11 dye-doped PMMA. We begin by first fabricating the ZrO$_2$ NPs using forced hydrolysis followed by calcination at a temperature of 600 $^{\circ}$C for an hour [@Gunawidjaja13.01]. The ZrO$_2$ NPs are then functionalized by dispersing them in a 2.5 vol% solution of 3-(Trimethoxysilyl)propyl methacrylate in toluene, which is subsequently refluxed for 2 h [@Gunawidjaja11.01]. To prepare the dye-doped polymer, we first filter Methyl methacrylate (MMA) through a column of activated basic alumina to remove inhibitor. Next we dissolve 25 wt% PMMA into the inhibitor-free MMA and divide the MMA/PMMA solution into three batches for different dye concentrations. DO11 dye (TCI America, purity $>$98%) is added to the MMA/PMMA solution in concentrations of 0.1 wt%, 0.5 wt%, and 1.0 wt%. The functionalized ZrO$_2$ NPs are then added at a concentration of 10 wt% and the mixture is sonicated until it is homogeneous, at which point 0.25 wt% 2, 2’-azobis(2-methyl-propionitrile) is added and the mixture is further sonicated before being poured onto 1”$\times$1.5” glass slides. The samples are then covered and placed in an oven at 60-65 $^\circ$C for 2 h to cure. Once prepared the samples are characterized using SEM, absorption spectroscopy, transmission measurements, and mechanical measurements. The results of the relevant sample parameters are tabulated in Table \[tab:param\].
To measure the sample’s emission we use an intensity controlled random lasing system [@Anderson15.03] shown schematically in Figure \[fig:setup\]. The pump is a Spectra-Physics Quanta Ray Pro Q-switched frequency doubled Nd:YAG laser (532 nm, 10 Hz, 10 ns) with the emission stabilized using a motorized half-waveplate (HWP) and polarizing beamsplitter (PBS) combination with a Thorlabs Si photodiode (PD) providing feedback for the HWP. The stabilized pump beam is focused onto the sample using a spherical lens with a focal length of 50 mm. Once pumped, the sample emits light in the backward direction, which is collimated using the focusing lens and then reflected by a dichroic mirror (DCM) (cutoff wavelength of 550 nm) into an optical fiber connected to a Princeton Instruments Acton 2300i spectrometer with a Pixis 2K CCD detector. For reference the relevant experimental parameters are tabulated in Table \[tab:param\].
![Schematic of RL setup.[]{data-label="fig:setup"}](RLSetup)
[|ccl|]{}\
$l$$^1$ & $4.10 \pm 0.25$ & $\mu$m\
$l_a$$^2$ & $657\pm 20$ & $\mu$m\
$d_{NP}$$^3$ &$195 \pm 32$ &nm\
$\rho_{NP}$& $2.26 \times 10^{12}$ & cm$^{-3}$\
$L$ & $\approx 500$ & $\mu$m\
\
$\lambda_p$ &532 & nm\
$r_p$ & 10 & Hz\
$\Delta t$ & 10 & ns\
$A$ & $7.85\times10^{-3}$ & cm$^2$\
$\Delta\lambda$ & 0.27 & nm\
\
\
\
\
\
\
Results and discussion
======================
Random Lasing Properties
------------------------
We test DO11+ZrO$_2$ for RL using a NP concentration of 10 wt% and three different dye concentrations (0.1 wt%, 0.5 wt%, and 1.0 wt%) and find that the 0.1 wt% and 0.5 wt% samples display RL, while the 1.0 wt% samples are not found to produce RL. Figure \[fig:RLspec\] shows the emission spectra for the 0.1 wt% sample at several pump energies, with the emission narrowing into a single lasing peak as the pump energy passes the lasing threshold, while Figure \[fig:denscomp\] compares the normalized spectra of the 0.1 wt% sample (pump energy of 10 mJ) and the 1.0 wt% sample (pump energy of 60 mJ). Note that the 1.0 wt% sample is pumped at a level near it’s ablation threshold (i.e. any higher pump energies result in the material being ablated by a single pulse). From Figure \[fig:denscomp\] we find that at high pump energies the emission from the 0.1 wt% sample is characteristic of IFRL, while the emission from the 1.0 wt% sample is much broader. From the spectral shape of the 1.0 wt% emission, and it’s peak location of 647.5 nm, we conclude that the emission corresponds to a combination of amplified spontaneous emission (ASE) and fluorescence, as DO11/PMMA’s ASE wavelength is known to be $\approx 650$ nm [@howel02.01; @howel04.01; @embaye08.01]. Since we are primarily concerned with RL in DO11+ZrO$_2$/PMMA, the remainder of this study will focus only on the samples with dye concentrations of 0.1 wt% or 0.5 wt%.
![Random lasing spectra as a function of wavelength for different pump energies for a dye concentration of 0.1 wt%.[]{data-label="fig:RLspec"}](11514spec)
![Comparison of emission from 0.1 wt% sample and 1.0 wt% sample. Note that the 0.1 wt% sample is pumped with an energy of 10 mJ, while the 1.01 wt% sample is pumped with an energy of 60 mJ, which is near the ablation threshold.[]{data-label="fig:denscomp"}](103114BFcomp)
We characterize the RL properties of the 0.1 wt% and 0.5 wt% samples by considering three RL features: peak intensity, peak wavelength, and RL linewidth (e.g. FWHM). Figure \[fig:01\] shows the peak intensity and linewidth for the 0.1 wt% sample, while Figure \[fig:05\] shows the same quantities for the 0.5 wt% sample. From Figure \[fig:01\] we see that the transition to RL for the 0.1 wt% sample is quick, with the linewidth narrowing from 100 nm at 3 mJ to 10 nm at 9 mJ, and the intensity’s slope changes by a factor of $\approx 4.1\times$ once above the lasing threshold. While the transition for the 0.1 wt% sample is found to be abrupt, the transition to RL for the 0.5 wt% sample is more gradual. From Figure \[fig:05\] we find that the linewidth changes from 100 nm at 5 mJ to 10 nm at 25 mJ and the intensity’s slope only increases by a factor of $\approx 2.7\times$. This more gradual transition into lasing suggests that there is more competition between ASE and lasing [@Andreasen10.01; @Cao00.02] in the 0.5 wt% sample, than in the 0.1 wt%.
![Peak intensity and linewidth as a function of pump energy for a sample with a dye concentration of 0.1 wt% and NP concentration of 10 wt%. From both the peak intensity and linewidth we determine a lasing threshold of $75.8 \pm 9.4$ MW/cm$^2$.[]{data-label="fig:01"}](DO11d01Thresh)
![Peak intensity and linewidth as a function of pump energy for a sample with a dye concentration of 0.5 wt% and NP concentration of 10 wt%. From both the peak intensity and linewidth we determine a lasing threshold of $121.1 \pm 2.1$ MW/cm$^2$.[]{data-label="fig:05"}](DO11d05Thresh)
While Figures \[fig:01\] and \[fig:05\] can help us understand the underlying spectral properties of the sample’s emission, they also can be used to directly determine the sample’s lasing threshold. Using either the FWHM as a function of pump energy [@Cao03.01] or a bilinear fit to the peak intensity [@Vutha06.01; @Anderson14.04] the lasing threshold of each sample can be calculated with the 0.1 wt% sample having a lasing threshold of $75.8 \pm 9.4$ MW/cm$^2$ and the 0.5 wt% sample having a threshold of $121.1 \pm 2.1$ MW/cm$^2$. Note that these thresholds are much larger ($\approx 10\times$) than similar RL materials based on R6G [@Anderson14.04]. These large lasing thresholds are due to DO11 having a smaller gain coefficient than R6G [@howel02.01; @howel04.01; @Mysliwiec09.01] as well as our use of off resonance pumping ($\lambda_{pump}=532$ nm and $\lambda_{res}=470$ nm).
Based on the observed lasing thresholds – and the observation that the 1.0 wt% sample didn’t lase even with a pump energy of 60 mJ ($I=754$ MW/cm$^2$) – we find that the RL threshold of DO11+ZrO$_2$/PMMA increases with dye concentration, which is counter to measurements in other dyes [@Anderson14.04]. One possible explanation for this effect is the formation of dimers at higher concentrations. From studies in other organic dye materials it is known that dimer formation leads to a redshift in the absorption spectrum of the material [@Toudert13.01; @Gavrilenko06.01; @Arbeloa88.01]. This redshift can result in fluorescence quenching [@Penzkofer86.01; @Penzkofer87.01; @Bojarski96.01; @Setiawan10.01], which will decrease the material’s RL gain leading to the RL threshold increasing. Additionally, dimer formation can describe the difference in the shape of the linewidth as a function of pulse energy (i.e. the 0.1 wt% has a sudden decrease in linewidth and the 0.5 wt% has a slow decrease) as a similar effect has been observed in comparisons of RL in Rhodamine B monomers and dimers [@Marinho15.01].
The last RL property we consider is the peak wavelength as a function of pump energy, which is shown in Figure \[fig:wave\] for both the 0.1 wt% and the 0.5 wt% samples. Both sample’s begin with their peak emission near 625 nm, with the peak wavelength smoothly transitioning into a steady lasing wavelength after the lasing threshold. The 0.1 wt% sample is found to have its RL peak at 645 nm, while the 0.5 wt% sample is found to have its RL peak at 646 nm. These two results, along with the observation of the 1.0 wt% sample’s ASE wavelength of 647.5 nm suggests that as the dye concentration increases the peak emission is redshifted. This is a known effect caused by increased self-absorption due to the greater dye concentration [@Shuzhen09.01; @Ahmed94.01; @Shank75.01] and subsequent dimer formation [@Toudert13.01; @Gavrilenko06.01; @Arbeloa88.01].
![Peak wavelength as a function of pump energy for the 0.1 wt% and 0.5 wt% dye concentration samples. The low intensity emission is peaked near 625 nm and smoothly transitions with increasing pump energy to be centered at $\approx 645$ nm.[]{data-label="fig:wave"}](DO11wavelength)
Photodegradation and self-healing
---------------------------------
With DO11+ZrO$_2$/PMMA found to lase in the IFRL regime for low dye concentrations, we now turn to consider the effect of ZrO$_2$ NPs on DO11/PMMA’s ability to self heal. For these measurements we use a sample with a dye concentration of 0.1 wt% and a NP concentration of 10 wt%. We use a 7 mJ/pulse (time averaged intensity of $I_{avg}=8.9$ W/cm$^2$) beam for both degrading the sample and measuring the RL spectra. During decay, the beam is always incident on the sample, while during recovery the beam is blocked except when taking measurements of the sample’s RL spectrum. Spectral measurements during recovery involve exposing the sample to three pump pulses to determine the average RL spectrum. These measurements occur every ten minutes during recovery, which equates to a duty cycle of 0.05%. Figure \[fig:dec\] shows the measured RL spectra during decay at several times, with the peak blueshifting and becoming broader. The large background fluorescence in Figure \[fig:dec\] is due to pumping the sample only slightly above it’s lasing threshold (7 mJ pump, 5.9 mJ threshold).
![Random lasing spectra as a function of wavelength at different times during photodegradation for a pump energy of 7 mJ and a 0.1 wt% dye-concentration sample.[]{data-label="fig:dec"}](11414decspec)
From the spectra recorded during decay and recovery we determine the peak emission intensity as a function of time, shown in Figure \[fig:pint\]. The peak intensity is found to decay to 40% of its initial value during degradation and found to fully recover (within uncertainty) after the pump beam is turned off. This observation is consistent with ASE measurements of DO11/PMMA without dispersed NPs [@howel04.01; @howel02.01; @embaye08.01], where the ASE signal is found to fully recover after degradation.
![Peak intensity as a function of time during decay and recovery.[]{data-label="fig:pint"}](11414pint)
While we observe full reversibility for a degree of decay of up to 60%, we also perform decay measurements with extreme degrees of decay ($\approx 95$ %) and observe partial recovery. This suggests that as the degree of degradation increases past some threshold value full recovery is lost and the material incurs some irreversible damage. To quantify this threshold degree of degradation at which point reversibility is lost we are planning measurements to systematically vary the degree of degradation and measure the degree of recovery.
In addition to performing preliminary measurements of how reversible photodegradation changes with the degree of damage, we also consider how cycling through degradation and recovery effects the material’s self-healing. These measurements so far have consisted of two decay and recovery cycles with full reversibility observed in both cycles, which is consistent with DO11/PMMA without disperesed NPs [@howel02.01]. Further work is planned to consider how many cycles can be completed before full reversibility is lost.
### Random Lasing Intensity Decay and Recovery
To further characterize the influence of the dispersed NPs on DO11/PMMA’s photodegradation and self-healing we determine the decay and recovery parameters by fitting the peak intensity as a function of time to a simple model of the RL intensity. Assuming that only undamaged molecules (with fractional number density $n(t)$) participate in RL, the material’s laser gain will be proportional to $n(t)$ leading to a RL intensity of [@Menzel07.01],
$$I_{RL}(t)=I_0e^{\sigma n(t)},\label{eqn:int}$$
where $I_0$ is the initial peak intensity and $\sigma$ is the RL cross section. To model the population dynamics of the molecules we use Embaye *et al.*’s two-species non-interacting molecule model, in which undamaged molecules reversibly transition into a damaged state during degradation [@embaye08.01]. In this model the undamaged population’s fractional number density during decay ($t\leq t_D$) is
$$n(t)=\frac{\beta}{\beta+\alpha I}+\frac{\alpha I}{\beta+\alpha I}e^{-(\beta+\alpha I)t},$$
and during recovery ($t>t_D$) is,
$$n(t)=1-[1-n(t_D)]e^{-\beta(t-t_D)},\label{eqn:rec}$$
where $t_D$ is the time at which the pump is turned off, $\alpha$ is the decay parameter, $I$ is the pump intensity, and $\beta$ is the recovery rate. Using Equations \[eqn:int\]–\[eqn:rec\] we can model the RL peak intensity’s decay and recovery as a function of time and extract the relevant dynamical parameters, with $\alpha=3.16(\pm 0.10) \times 10^{-2}$ cm$^2$W$^{-1}$min$^{-1}$ and $\beta =3.75( \pm 0.18) \times 10^{-2}$ min$^{-1}$. The decay parameter, $\alpha$, is found to be smaller than the previously measured values for DO11/PMMA [@howel02.01; @howel04.01; @embaye08.01; @Ramini12.01; @Ramini13.01], which means that the addition of nanoparticles improves the materials photostability. Additionally, we find that the recovery rate of DO11+ZrO$_2$/PMMA is larger than any previous measurement [@howel02.01; @howel04.01; @embaye08.01; @Anderson11.01; @Anderson13.01; @Anderson14.01; @Anderson14.02; @Ramini12.01; @Ramini13.01; @Anderson15.04], implying that the addition of ZrO$_2$ NPs helps aid the recovery process. An explanation for these effects is that the introduction of NPs can change the free energy advantage $\lambda$, and density parameter, $\rho$, such that the average domain size is greater with NPs than without [@Ramini12.01; @Ramini13.01; @Anderson14.02].
While a precise determination of the domain parameters is beyond the scope of the current study, we can estimate the modified domain parameters by considering the effect of the average domain size on the recovery rates. Previously it was shown that the average domain size is given by [@Anderson14.02]:
$$\langle N \rangle=\frac{\beta_M\Omega_1(\rho,\lambda)(1+z\Omega_1(\rho,\lambda))}{\rho |z\Omega_1(\rho,\lambda)-1|^3}, \label{eqn:N}$$
where $\Omega_1(\rho,\lambda)$ is the density of unitary domains and $z=\exp\{\lambda/kT\}$ with $T$ being the temperature and $k$ being Boltzmann’s constant. Using the average domain size (Equation \[eqn:N\]) and Equations \[eqn:domdec\] and \[eqn:domrec\] the measured decay and recovery rates can be approximated as
$$\begin{aligned}
\alpha&\approx\frac{\alpha_1}{\langle N\rangle}, \label{eqn:alp}
\\\beta&\approx \beta_1\langle N \rangle,\label{eqn:bet}\end{aligned}$$
where once again $\alpha_1$ and $\beta_1$ are the unitary domain decay and recovery rates, respectively.
Therefore, assuming the unitary domain recovery rate is the same for DO11/PMMA both with and without NPs, we can determine the ratio of average domain sizes by taking the ratio of recovery rates between a sample with NPs and a sample without NPs:
$$\begin{aligned}
\frac{\beta}{\beta_0}=\frac{\langle N\rangle}{\langle N_{0}\rangle}
\\ =\frac{\rho_0\Omega_1(\rho,\lambda)}{\rho\Omega_1(\rho_0,\lambda_0)}\frac{1+z\Omega_{1}(\rho,\lambda)}{1+z_0\Omega_1(\rho_0,\lambda_0)}\frac{ |z_0\Omega_1(\rho_0,\lambda_0)-1|^3}{ |z\Omega_1(\rho,\lambda)-1|^3}\label{eqn:ratio}\end{aligned}$$
where the subscript 0 corresponds to the parameters for the system with no nanoparticles and no subscript corresponds to the system with nanoparticles. Comparing the recovery rate of DO11+ZrO$_2$/PMMA to a similarly dye-doped DO11/PMMA without NPs [@Ramini13.01; @raminithesis] we find a ratio of $\beta/\beta_0\approx 12.5$, which means that with the inclusion of NPs the domain size is an order of magnitude larger. Given this large difference, and the linear relationship between domain size and the density parameter, we conclude that primary influence of the NPs is on the free energy advantage. Assuming that the density parameter is unchanged by the introductions of NPs, we can numerically solve Equation \[eqn:ratio\] for the new free energy advantage and find $\lambda \approx 0.41$ eV, which is 0.12 eV larger than DO11/PMMA’s value of 0.29 eV [@Ramini12.01; @Ramini13.01; @Anderson14.02].
With the new free energy advantage determined, we can estimate the average domain size for our system and the unitary domain decay and recovery rates. Substituting the new free energy advantage into Equation \[eqn:N\] we find that the average domain size of our system is $\langle N \rangle = 375$. Using this domain size, along with Equations \[eqn:alp\] and \[eqn:bet\], we determine the unitary domain decay rate to be $\alpha_1\approx 11.86 \pm 0.38 $ cm$^2$W$^{-1}$min$^{-1}$ and the unitary domain recovery rate to be $\beta_1\approx 1.00(\pm0.10)\times 10^{-4}$ min$^{-1}$, which are both found to be within uncertainty of the measured values for DO11/PMMA [@Ramini13.01]. This agreement of the unitary domain decay and recovery rate between DO11/PMMA and DO11+ZrO$_2$/PMMA implies that we are correct in our original assumption that the NPs only influence the domain size (via the free energy advantage) and do not influence the underlying molecular interactions leading to reversible photodegradation. Additionally, the success of the CCDM to correctly account for the influence of NPs on DO11/PMMA’s decay and recovery is a strong indication that the CCDM is a robust description of reversible photodegradation for DO11 dye-doped polymers, with the addition of NPs resulting in the free energy advantage increasing.
One possible explanation for this increase in free energy advantage is that the introduction of NP’s affects the local electric field experienced by the dye molecules, thereby influencing the underlying interactions behind the free energy advantage. This effect on the local electric field arises due to the introduction ZrO$_2$ NP’s (with a dielectric constant of $\approx 4.88$) into the polymer (with dielectric constant $\approx 2.22$) resulting in an increased dielectric constant of the dye’s local environment. To estimate the magnitude of this effect on the local field factor we recall that for a spherical cavity in a uniform dielectric medium, with dielectric constant $\epsilon$, the local field factor is given by [@jacks96.01]:
$$L\propto \frac{3\epsilon}{2\epsilon+1}.\label{eqn:LFF}$$
Using the relevant concentrations of NPs, dye, and polymer we determine that the permitivity is approximately 10% larger, which using Equation \[eqn:LFF\] results in the local field factor becoming 3% larger. Since the dielectric energy of a system depends on the square of the local field factor, we conclude that the dielectric influence of the NPs increases the free energy advantage by 6%, or 0.018 eV, which is too small to account for the total change in energy calculated above. This suggests two different possibilities: (1) the change in the free energy advantage also includes a contribution from another unidentified effect, most likely related to the interactions of dye, polymer, and NPs or (2) our simplistic treatment of the dielectric constant and local field factor may underestimate the actual enhancement in the local electric field, especially since ZrO$_2$ is a transparent conductive oxide [@Brune98.01; @Naik13.01], which can lead to plasmonic effects that can drastically increase the electric field [@Naik13.01].
### Effect of photodegradation and recovery on linewidth and wavelength of random lasing
In addition to considering the decay and recovery dynamics of the RL intensity we also quantify the changes in the lasing peak location and linewidth during decay and recovery, as shown in Figure \[fig:wave\]. From Figure \[fig:wave\] we find that during degradation the lasing peak blueshifts and the FWHM increases. After the pump beam is blocked (except for measurements during recovery), both the lasing peak and FWHM are found to return to within uncertainty of their initial values, suggesting that the photodegradation process is truly *reversible*. This result is different than observed for R6G+NP/PU, where the peak intensity fully recovers, but the lasing wavelength and linewidth are irreversibly changed due to photodegradation [@Anderson15.01; @Anderson15.03].
![Peak wavelength and RL linewidth as a function of time during decay and recovery.[]{data-label="fig:wave"}](11414wavelength)
The other difference, observed in Figure \[fig:wave\], between the photodegradation and self-healing of DO11+ZrO$_2$/PMMA and R6G+NP/PU, is that the lasing wavelength is found to blueshift during decay for DO11+ZrO$_2$/PMMA, while it is found to redshift for R6G+NP/PU. In R6G+NP/PU, the redshifting of the lasing peak during photodegradation and recovery is attributed to photothermal heating and the formation of R6G dimers and trimers [@Anderson15.01; @Anderson15.03]. The observation of the opposite effect in DO11+ZrO$_2$/PMMA suggests a different mechanism responsible, with the most likely mechanism being related to the observation of a blueshift in DO11/PMMA’s absorbance spectrum during photodegradation [@embaye08.01; @Anderson13.01; @andersonthesis; @Anderson14.02]. The effect of a blueshift in the absorbance spectrum leads to the shorter wavelengths experiencing less loss which results in RL emission blueshifitng, such that the gain-to-loss ratio is maximized. This effect is also observed in the ASE spectrum of DO11 in different solvents, where a blueshift in the absorbance peak leads to a blueshift in the emission peak [@howel04.01]. Additionally, this blueshift will lead to a larger portion of the emission spectrum being amplified resulting in the linewidth increasing, which is observed in the RL spectrum and has been observed in the ASE spectrum [@howel04.01].
Conclusions
===========
Based on the observation of reversible photodegradation in DO11/PMMA and the desirability of robust organic-dye based random lasers for a variety of applications (such as speckle-free imaging [@Redding12.01], tunable light sources [@Cao03.01] and optical physically unclonable functions [@Anderson14.04; @Anderson14.05]), we investigate the emission properties of DO11+ZrO$_2$/PMMA under nanosecond optical pumping. We find that for dye concentrations of less than 1.0 wt%, DO11+ZrO$_2$/PMMA lases in the IFRL regime; while for a dye concentration of 1.0 wt%, no lasing is observed for pump intensities up to the ablation threshold ($I\approx 754$ MW/cm$^2$). The lasing threshold is found to increase with concentration, with the 0.1 wt% sample having a threshold intensity of $75.8 \pm 9.4$ MW/cm$^2$ and the 0.5 wt% sample having a threshold intensity of $121.1 \pm 2.1$ MW/cm$^2$. Both concentrations are found to have lasing wavelengths near 645 nm with a linewidth of approximately 10 nm. This lasing wavelength region is attractive for use with hydrocarbon based polymers as these polymers have an absorption minimum near 650 nm [@howel02.01].
Along with considering the random lasing properties of DO11+ZrO$_2$/PMMA, we also measure the materials photodegradation and recovery. We find that DO11+ZrO$_2$/PMMA photodegrades reversibly, with both the RL spectra before and after a photodegradation and recovery cycle being identical. During photodegradation the lasing peak is found to blueshift, widen, and decrease in intensity, while during recovery the opposite process is found to occur with the lasing peak returning to its initial intensity, location, and linewidth. This suggests that the observed degradation is truly reversible, which contrasts to measurements of R6G+NP/PU random lasers where the linewidth and peak wavelength are changed after decay and recovery [@Anderson15.01; @Anderson15.03].
While DO11+ZrO$_2$/PMMA is found to reversibly photodegrade like DO11/PMMA, the introduction of NPs into the dye-doped matrix is found to affect the decay and recovery rates, with DO11+ZrO$_2$/PMMA found to display increased photostability and to recover more quickly than similarly dye-doped DO11/PMMA. These changes are explicable within the CCDM [@raminithesis; @andersonthesis; @Ramini13.01; @Anderson14.02], with the NPs resulting in the free energy advantage increasing to an estimated value of 0.41 eV, but having little to no effect on the unitary domain decay and recovery rates, which are found to be in agreement with previous measurements of DO11/PMMA without NPs [@Ramini13.01]. While we have provided estimates of the CCDM model parameters for DO11+ZrO$_2$/PMMA, a more thorough study is required to determine the precise values. Therefore we are planning measurements of DO11+ZrO$_2$/PMMA’s decay and recovery for different temperatures, applied electric fields, and dye concentrations, which allow for the calculation of all CCDM parameters [@Ramini12.01; @Ramini13.01; @raminithesis; @Anderson14.02].
Finally, the observation of fully reversible photodegradation in DO11+ZrO$_2$/PMMA has promising prospects in the development of robust photodegradation resistant random lasers for real-world applications. We forsee using random lasers based on DO11+ZrO$_2$/PMMA in low duty-cycle applications such that the effects of photodegradation are mitigated by the self-healing mechanism of DO11+ZrO$_2$/PMMA, thus allowing prolonged use of the material. While we use a very low duty cycle during our recovery measurements – to minimize photodegration and maximize self-healing – we hypothesize that the material can function without significant photodegradation at higher duty cycles, dependent on the pump intensity. Further work is required to determine the actual break even duty cycle at which photodegradation and self-healing are balanced.
Acknowledgements
================
This work was supported by the Defense Threat Reduction Agency, Award \# HDTRA1-13-1-0050 to Washington State University.
|
---
abstract: 'As Machine Learning (ML) systems becomes more ubiquitous, ensuring the fair and equitable application of their underlying algorithms is of paramount importance. We argue that one way to achieve this is to proactively cultivate public pressure for ML developers to design and develop fairer algorithms — and that one way to cultivate public pressure while simultaneously serving the interests and objectives of algorithm developers is through gameplay. We propose a new class of games — “games for fairness and interpretability” — as one example of an incentive-aligned approach for producing fairer and more equitable algorithms. Games for fairness and interpretability are carefully-designed games with mass appeal. They are inherently engaging, provide insights into how machine learning models work, and ultimately produce data that helps researchers and developers improve their algorithms. We highlight several possible examples of games, their implications for fairness and interpretability, how their proliferation could creative positive public pressure by narrowing the gap between algorithm developers and the general public, and why the machine learning community could benefit from them.'
author:
- Eric Chu
- Nabeel Gillani
- Sneha Priscilla Makini
bibliography:
- 'main.bib'
title: Games for Fairness and Interpretability
---
<ccs2012> <concept> <concept\_id>10003120</concept\_id> <concept\_desc>Human-centered computing</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
Introduction
============
As ML increasingly permeates virtually all aspects of life — and unequally serves, or fails to serve, certain subsegments of the population [@caliskan2017semantics; @bolukbasi2016man; @genderShades] — there is a need for a deeper exploration of how ML algorithms can be made fairer and more interpretable. To achieve this, we believe effective public pressure will be one lever to better models. There are several examples from history of how public pressure has spurred changes to technology policies. The creation of dynamite; America’s use of the atomic bomb during the second world war; and the eugenics movement from the early 20th century are all examples of ethically dubious endeavors that were at least somewhat abated by a critical public response[^1].
However, recent stories about Facebook and Cambridge Analytica, driverless cars going rogue[^2], and even machine-powered labor displacement [@autorJobs] have hinted at the dangers of simply letting history unfold. In all of these instances, there were certainly changes to the underlying technological methods — but it is hard to deny the importance of collective public pressure in catalyzing dialogue to envision a new set of policies and practices surrounding these powertools. It is unlikely that methodological changes alone would have been sufficient. Public pressure is often reactive and arises in the wake of crises. To counter this, we ask: how can public pressure operate proactively in order to ensure ML can effectively ground itself in — and respond to — calls for fairness and interpretability?
To that end, some authors have recently sparked public conversation around the ethical pitfalls of machine learning [@weaponsMath; @automatingInequality; @algosOppression]. Furthermore, initiatives like Turingbox [@turingBox] and OpenML [@vanschoren2014openml] are actively seeking to create platforms and marketplaces where members of the scientific community and general public can audit ML algorithms to promote more fairness, transparency, and accountability. These efforts are important first steps towards generating proactive public pressure. However, they fail to directly align incentives between those who design and deploy algorithms and those who are affected by them. Why should an algorithm developer care about how a niche group of individuals rates the fairness or interpretability of his or her algorithms? Why should members of the general public spend their time trying to understand, let alone evaluate, these algorithms? It is unclear how sustainable current efforts to generate proactive public pressure will be without incentive alignment.
To align incentives between ML developers and the general public in a quest for more interpretable — and as a result, in due course, fairer — ML, we propose “games for fairness and interpretability”: networked games that as a byproduct of the game’s objectives, engage the general public in auditing algorithms while simultaneously generating valuable training sets for ML developers.
ML Powered Games
================
Inspired by Luis von Ahn’s Games with a Purpose (GWAP) framework [@von2008human; @von2008designing], we propose using ML-powered games to enhance model interpretability — which we view as an important step towards developing fairer ML.
Games with a Purpose
--------------------
Described as “human computation", the GWAP framework was designed for problems solvable by humans but beyond the capabilities of machines. Instead of relying on financial incentives or altruism, GWAPs simply rely on people’s desire for fun and entertainment. A successful GWAP can produce not only novel and creative solutions to difficult problems, but also provide large amounts of labeled data for training machine learning models. Since its inception, GWAPs have attracted hundreds of thousands of players in order to tackle problems ranging from protein folding [@khatib2011crystal] and RNA folding [@lee2014rna] to examining the human perception of correlation in scatter plots[^3]. The framework has also since been extended to machine learning, such as using active learning to select examples during gameplay [@barrington2012game].
The GWAP framework includes several different templates of games [@von2008designing]. *Output-agreement* games has two players attempt to produce the same output when shown the same input. In the ESP game, shown in Figure \[fig:esp\], the players are shown an image and asked to guess what words the other player would use to describe the image. A variation of the game includes taboo words for each image, thus requiring users to guess more uncommon words, in turn producing more interesting labeled data [@von2004labeling]. In *input-agreement* games, two players are each provided an input which may or may not be different; the players are asked to output descriptions of the inputs and then finally guess whether they were shown the same input. For instance, players in the Tagatune game are given song clips and asked to output tags, before finally guessing whether they had the same clip [@law2009input].
![An example of a Game With a Purpose (GWAP): the original ESP game.[]{data-label="fig:esp"}](imgs/esp_game.png){width="\linewidth"}
Designing Games for Fairness and Interpretability
-------------------------------------------------
While reputation-based incentives can create social pressure and motivate ML developers, we believe a well-designed game aligns incentives between ML developers and the consumers of ML (i.e. the general public). Due to the importance of labeled data for deep neural networks, we believe ML researchers will have strong incentives to upload their models if the games that leverage them can produce valuable training data or adversarial examples.
On the consumer side, GWAPs have shown that such games can reach large audiences. Furthermore, a larger audience is often a broader audience, thus allowing more diverse probing of the model. We believe that there is an appetite for ML games, due both to increasing media attention on ML and the growing capabilities of new models. Recent examples of games that engage a general audience in exploring ML include the text auto-complete “Talk to Transformer”[^4], a Pictionary-like game Quick, Draw![^5], word embedding-powered word association games[^6], and an endless text-adventure game built using a generative text model[^7].
We define “games for fairness and interpretability” as ML-powered games in which the output and / or interaction with human players is produced by a machine learning model. These games can also be networked to enable human-human interaction and competition. Games should be fun and engaging, provide insight into how the underlying machine learning models work, and produce data that helps models improve — in particular, so that the models are better-equipped to more equitably serve a diverse range of individuals and scenarios.
One might imagine a platform for such games, where once a game has been designed and open-sourced, its backend model could be swapped for any model with similar inputs and outputs. The platform could also serve as a public forum for widespread participation in, and discussion about, the evaluation of new ML models. This unique forum — one where both ML developers and members of the public are present — could serve as an important vehicle for a) enhancing broader familiarity with and awareness of ML and its applications, and perhaps eventually, b) creating proactive public pressure that motivates algorithm developers to build more interpretable and fairer ML.
[0.5]{} {width="85.00000%"}
[0.5]{} {width="80.00000%"}
Proposed Categories of Games
----------------------------
In the spirit of GWAPs, we describe possible categories of games in the following sections.
### Humans vs. AI
**Setup.** Player 1 provides an input, and Player 2 competes against an AI to guess the correct answer.
**Example game 1 — Guess Who?** Player 1 describes themselves, their interests, job, and other attributes through freeform short text. Player 2 and the AI attempt to guess the age, sex, and location of Player 1.
**Example game 2 – Codenames**. Inspired by the popular Codenames board game [@wiki:codenames] , the players are presented with a 5x5 grid of words. Player 1 is a “spymaster” who is also allowed to see the placement of bombs on the grid. The spymaster’s role is to give a one word clue, plus the number of words that matches the clue. Player 2’s goal is to guess the correct words; however, if he or she guesses a bomb, the game is over. The game is won if all the non-bomb words are guessed correctly. The goal is to finish the game in fewer rounds; saying a larger number allows the team to win more quickly, but it is also more difficult to come up with clues.
In our ML-powered variant, the AI also attempts to guess the words; if the AI’s guesses matches Player 2’s guesses, those guesses are invalid. Figure \[fig:codenames\] shows an example round.
**Data produced and insight into interpretability.** Player 1 will have to produce inputs that are recognizable by another human but undetectable or incorrectly classified by the AI. This requires a player to intuit the space of inputs that a model understands and in which cases it might fail. For instance, Player 1 may find that cultural references are harder for a ML model. Natural language processing models that can incorporate common sense reasoning and knowledge also remains an open area of research. The successful inputs and clues can be used as more robust training data. In addition, baseline models for the AI could be based on word embeddings, which have been shown to reflect implicit human biases around gender, race, occupation, etc. [@caliskan2017semantics]. These biases may be surfaced if the AI incorrectly relies on them to make predictions.
### Break the Bot
**Setup.** Each player is shown an input and the model’s output (e.g. a prediction). Each player is asked to make a small modification to the input. Whoever can cause the largest change in the model output, while using the smallest modification, receives more points.
**Example game 1 — Vandalize it!** The brittleness of deep neural networks has been illustrated in several computer vision systems. For example, graffiti on signs can significantly lower object recognition accuracy [@eykholt2018robust], while Rosenfeld et al. showed that adding an object to a scene could drastically change the ability to recognize all other objects [@rosenfeld2018elephant]. These deficiencies can have catastrophic effects on real-world systems.
In this self-driving car inspired game, players are shown street images overlaid with bounding boxes of detected objects. For example, a stop sign may be detected by the model with probability 0.85. The players’ goal is to change that prediction by making small edits to the sign and its surroundings. The game will give players tools to alter the angle, lighting, hue of the image, as well as add and subtract other objects and artificats. (The game will have to measure the ‘size’ of modifications in order to assign scores). Figure \[fig:viz\] shows an example of how the game might look.
**Example game 2 — Beat the Banker.** ML has begun to be used in higher-stakes situations, ranging from recidivism prediction to loan default rate prediction. Unfortunately, these systems have also been shown to be susceptible to demographic features and unfairness [@hardt2016equality]. In this game, the players are bankers. The input is a hypothetical set of demographic features of an individual, and the output is the predicted probability of that individual’s loan repayment. Faced with a loan rejection, the goal is to find seemingly innocuous changes that can make the loan approved. **Data produced and insight into interpretability.** These games provide adversarial examples and sensitivity analysis on model inputs. This is important as the field of adversarial examples is becoming increasingly important [@goodfellow2014explaining], especially as ML models become deployed in the real world [@kurakin2016adversarial], and obtaining those examples can often be difficult [@zhao2017generating]. ML researchers can also gain a greater understanding of how inputs may be modified in semantically meaningful ways, as well as if the observed model behavior is desirable (e.g. fair).
Games and Current Research Directions in Machine Learning
=========================================================
The previous section illustrates how thoughtfully-designed games might help align incentives between ML developers and the general public, cultivating public pressure and awareness — along with the new, more representative datasets — to promote fairer, more inclusive ML systems. We believe the time to develop games for fairness and interpretability is now, largely because they align with several current directions in ML research. We highlight some of these directions below and explore how members of these respective research communities may benefit from games for fairness and interpretability.
Fairness
--------
As ML models become more pervasive, there has been an increasing call for models that can prevent discrimination along sensitive attributes such as race and gender. Part of the problem is detecting that biases in models even exist in the first place. To that end, recent research has shown how word embeddings encode biases as measured by standard tests such as the Implicit Association Test [@caliskan2017semantics], with relationships between word embeddings reflecting negative stereotypes about gender [@bolukbasi2016man]. Other work highlights deficiencies in datasets used for facial recognition, resulting in models that fail more frequently for women and people with darker skin tones [@genderShades].
How can models handle these sensitive attributes? A naive approach of removing sensitive attributes may not prevent discrimination if the sensitive attributes are correlated with other attributes left in the dataset. Enforcing demographic parity, in which the outcome is uncorrelated with the sensitive attribute, is also problematic because it does not guarantee fairness, and the sensitive attribute may actually be important for prediction, making removal of all correlation unrealistic. Thus far, various approaches to formalize and operationalize fairness include using the 80% rule of “disparate impact” outlined by the US Equal Employment Opportunity Commission as a definition of discrimination [@feldman2015certifying], treating similar individuals similarly by enforcing a Lipschitz condition on similar individuals and the classifier predictions for those individuals [@dwork2012fairness], preprocessing the dataset through methods such as weighting and sampling [@kamiran2012data], and allowing use of the sensitive attribute but aiming for “equality of *opportunity*” through the notion of equalized odds [@hardt2016equality]. Certain frameworks also provide the ability for people to select the tradeoff between model performance and fairness. Other work has centered on learning transferable *fair* representations that can be reused across tasks [@zemel2013learning].
We believe ML fairness researchers would find value in the datasets produced by games for fairness and interpretability. For example, machine predictions from “Human vs. AI” games would provide clear insights into which kinds of biases certain algorithms harbor; “Break the Bot” games might shed light on how robust or brittle algorithms are to changes in the datasets they operate on.
Interpretability
----------------
While deep neural networks have found great success as powerful function approximators, they have also developed a reputation as black boxes. Interpretability may be a case of “you know it when you see it”, but recent work has attempted to make the problem more tractable by defining interpretability, explaining why it is important, and explaining when it is necessary [@doshi2017towards; @lipton2016mythos].
There has also been a wide range of methods focused on *introspection and visualization*, including (but not limited to) “inverting” intermediate representations to generate images [@mahendran2015understanding; @mordvintsev2015inceptionism], producing input feature attributions and saliency maps [@ribeiro2016should; @sundararajan2017axiomatic; @smilkov2017smoothgrad; @petsiuk2018rise; @bau2017network] vs. producing counterfactual explanations [@wachter2017counterfactual] vs. pointing to protoypical examples [@chen2019looks], local per-example explanations [@ribeiro2016should] vs. global explanations based on feature representations across the entire dataset [@bau2017network], clear-box approaches with access to model gradients [@sundararajan2017axiomatic; @smilkov2017smoothgrad] vs. black box approaches [@ribeiro2016should; @petsiuk2018rise]. These methods often highlight what parts of the input (e.g. a segment of the image, or a span of the text), were most important to the model’s decision.
While there is also work worth mentioning on (1) generating human readable explanations in natural langauge [@lei2016rationalizing; @hendricks2016generating], (2) distilling neural networks into more interpretable models such as decision trees [@frosst2017distilling], and (3) disentangling factors of variation for generative models [@higgins2016beta; @chen2016infogan; @kulkarni2015deep], a significant portion of the field has focused on the aforementioned introspection and visualization methods. At the core, many of the methods attempt to relate input or internal representations to the model outputs. However, there are questions around the reliability and intuitiveness of these explanations ([@jain2019attention; @kindermans2019reliability]). The games’ data can be analyzed through these methods, perhaps providing insight into how well current explanations match human intuitions. The “Break the Bot“ games would also produce valuable counterfactual data; analyzing changes in the outputs of their underlying models as a function of changes to inputs could provide a deeper understanding of how, exactly, these models are conducting their computations.
Conclusion
==========
As ML-powered technologies continue to proliferate, the threat of biased and opaque decision-making looms large. We believe public pressure is a powerful mechanism for inspiring changes in how algorithms are developed. Games for fairness and interpretability provide one means for engaging the public in probes of ML systems while simultaneously producing hard-to-source data that serves the interests of ML developers. We believe games are unique in their ability to engage different audiences and are thus a promising avenue in which to pursue complicated, multi-stakeholder challenges like building fairer ML systems.
Looking ahead, there are several open questions: who should be responsible for designing and developing games for fairness and interpretability? How will the games be deployed and marketed so as to recruit a diverse range of players? What new risks or threats might these games introduce? These are important questions that will require continuous exploration and reflection. We hope this paper serves as an initial stepping stone and inspires individuals both within and beyond the ML community to consider the potential power of games.
[^1]: https://www.bostonglobe.com/ideas/2018/03/22/computer-science-faces-ethics-crisis-the-cambridge-analytica-scandal-proves/IzaXxl2BsYBtwM4nxezgcP/story.html
[^2]: https://www.nytimes.com/2018/03/23/technology/uber-self-driving-cars-arizona.html
[^3]: http://guessthecorrelation.com/
[^4]: https://talktotransformer.com/
[^5]: https://quickdraw.withgoogle.com/
[^6]: http://robotmindmeld.com/
[^7]: https://www.aidungeon.io/
|
---
abstract: |
Data obtained over the last three solar cycles have been analysed to reveal the relationships between theintensity of the photospheric field measured along the line of sight by the WSO group at heliolatitudes from -75 to 75 degrees and the intensity of the interplanetary magnetic field and absolute values of the perturbations of the different characteristics of the solar wind at the Earth orbit, and geomagnetic parameters. provided by the OMNI team.
The heliospheric and geomagnetic data are found to be divided into two groups characterized by their response to variability of the solar magnetic field latitudinal structures on short and on long time scales.
---
Title: To the question about perturbations of solar-terrestrial characteristics.
Authors: E. A. Gavryuseva (Institute for Nuclear Research RAS)
Comments: 14 pages, 6 Postscript figures
*Keywords*: Sun; solar variability; magnetic field; interplanetary magnetic field; solar wind; geomagnetic perturbations; solar cycles
Introduction
============
Solar magnetic field plays the main role in the heliosphere. It has to be study carefully the relations between the solar wind, the perturbations of geosphere and global structure of the photosperic magnetic field. WSO and OMNI data were used for comparison of 30 years long data sets (see referenses in Gavryuseva 2018c) on a long and short time scale. This paper is concentrated on the study the relations of between absolute values of these characteristics.
This approach led to the conclusion that all solar wind data and geomagnetic perturbations that were examined divided into two groups characterized by sensitivity to the variability of the interplanetary magnetic field and photospheric field at different latitudes.
We use Wilcox Solar Observatory data for the photospheric magnetic fields http://wso.stanford.edu/synopticl.html, (Scherrer et al., 1977), OMNI data for solar wind parameters at the Earth’s orbit, and indices of geomagnetic activity for the period 1976-2004, to study the relations between the solar wind, geomagnetic disturbances and solar drivers at different solar latitudes.
In order to understand from which latitudinal zone the solar wind is originated and how it depends on the activity cycle it is necessary to know the latitudinal $SMF$ structure over at least 22 years.
The latitudinal structure of the $SMF$ has been deduced for the last 29 years since May 27, 1976 from the Wilcox Solar Observatory (WSO) data (Scherrer et al., 1977; Gavryuseva & Kroussanova, 2003, Gavryuseva & Gogoli, 2006, Gavryuseva, 2005, 2006, 2006a,b, 2008a,b, 2018 and references there). The structure in latitude and time of the 1-year running mean of the solar magnetic field with 1 Bartels Rotation (BR, 1 BR = 27 days) step is shown on the upper plot in Fig. 1 Gavryuseva, 2018c.
The solar wind and geomagnetic data were taken from the OMNI directory (http://nssdc.gsfc.nasa.gov/omniweb) which contains the Bartels mean values of the interplanetary magnetic field (IMF) and solar wind plasma parameters measured by various space-crafts near the Earth’s orbit, as well as geomagnetic and solar activity indices). First, daily averages are deduced from OMNI’s basic hourly values, and then the 27-day Bartels averages are deduced from the daily averages. The corresponding standard deviations are related to only these averages and do not include the variances in the higher resolution data.
The $IMF$ and solar wind parameters taken into account are the following:\
$B_x$, $B_y$, $B_z$ and $B = (B_x^2+B_y^2+B_z^2)^{1/2}$ are the components and magnitude of the interplanetary magnetic field, in nT;\
Proton density, $N_p$, in $N/cm^3$;\
Proton temperature, $T_p$, in degrees $K$;\
Plasma speed, $V_p$, in $km/s$;\
Electric field, in mV/m;\
Plasma beta, $N_{\beta}= [(T*4.16/10^5) + 5.34] * N_p/B^2$;\
Ratio $N_{\alpha}/N_{p}$;\
Flow Pressure, $P$ proportional to $N_p*V^2$, in nPa;\
Alfven Mach number, $M_a = (V*N_p^{0.5})/20*B$.\
The geomagnetic parameters taken into account are the following:\
$AE$-index;\
Planetary Geomagnetic Activity Index, $K_p-$ index;\
$DST$-index, in nT.\
Sunspot number ($SSN$) was used, as well, for a further comparison.
The $X$ axis directed along the intersection line of the ecliptic and solar equatorial planes to the Sun, the Z axis is directed perpendicular and north-ward from the solar equator, and the $Y$ axis completes the right-handed set.
The solar wind parameters analysed cover the same period as the WSO solar data with one Bartels rotation resolution. We call the set of these 16 parameters taken from the OMNI data base as “solar wind” ($SW$) data; they include the interplanetary magnetic field, solar wind and geomagnetic parameters and sun spot number ($SSN$).
Relationships between the Solar Magnetic Field Intensity/ and the Absolute Values of the OMNI Data/ on Long- and Short-Term Scales
==================================================================================================================================
Physical connections between the Sun and the interplanetary parameters could be attributed to the influence of the intensity of the solar magnetic field without taking into account its polarity ($|MF|$). The corresponding correlation coefficients between the 1-year mean values of the $SMF$ intensity and absolute values of the $SW$ data ($|SW|$) as functions of time delay in years and in latitude are shown in Fig. 1. The $K_{cor}(|MF|, |SW|)$ have an 11-year periodicity for all the $SW$ data except the $IMF$ intensity $B$, and this periodicity is slightly visible in the $K_{cor}$ for the absolute values of the $B_y$: $|B_y|$ and $AE$: $|AE|$.
There is a remarkable particularity of the latitudinal dependence of the\
$K_{cor}(|MF|, |SW|)$: the change of sign and a phase shift of the correlation coefficients at the heliographic latitude of about 50-55 degrees. This result can be interpreted as an evidence that the photospheric magnetic fields originated at the heliographic latitudes up to $\pm 55$ degrees propagate in the heliosphere and contribute to the perturbations of the solar wind and magnetosphere. The solar magnetic fields originated above $\pm 55$ degrees do not appear close to the Earth’s orbit (see also Gavryuseva, 2006c,f; 2008b; Gavryuseva & Gondoli, 2006).
The $K_{cor}(|MF|, |SW|)$ are symmetric respect to the equator. This is well illustrated in Fig. 2 for the latitudinal dependence of the $K_{cor}(|MF|, |SW|)$ for the fixed optimum delay between the $SMF$ intensity and the absolute values of the $SW$ data. Thise figures are analogous to Fig. 11, 12 for the original (not absolute) $SMF$ (Gavryuseva, 2018c) and $SW$ values where the $K_{cor}(MF, SW)$ are antisymmetric to the equator. There is a clear anti-correlation between the intensity of the photospheric field of the activity belts and $B$, $V_p$, $P$, $N_p$, $N_{\beta}$ and $M_a$. This confirms that during high activity periods the slow solar wind prevails. Positive correlations of the $|MF|$ with the absolute values of the $AE$, $K_p$ and $DST$ indices correspond to the statement that most of the geomagnetic perturbations are originated from the low and middle latitudes when the intensity of the magnetic field is high. Figure 3 shows the latitudinal dependence of the $K_{cor}$ for the fixed optimum delay between the absolute values of the $SMF$ and $SW$ data after the filtering of the variabilities longer than 4 years and shorter than 1 year. The $K_{cor}$ for -$B_y$, $T_p$, $V_p$, $P$, $N_p$, $AE$, $K_p$ and $-DST$ have very similar latitudinal dependence, and the strongest correlation with the $SMF$ intensity takes place at about 40 degrees in the southern hemisphere.
Owing to the symmetry of the solar magnetic field intensity $|MF|$ the latitudinal dependences of the $K_{cor}(|MF|, |SW|)$ and $K_{cor}(F|MF|, F|SW|)$ are also symmetric respect to the equator. These correlation coefficients are plotted in Fig. 4 together for all the $|SW|$ data for the long-term variability (on the left side) and for the short-term variability (on the right side). Figure 4 for the $K_{cor}$ of the absolute values of the $SMF$ and $SW$ data is analogous to the corresponding Figs. 14 and 15 for the original values of the $SMF$ and $SW$ data (Gavryuseva, 2018c). The difference between them permits to investigate the sensitivity of the $SW$ parameters to the $SMF$ polarity (and not only to the intensity of the SMF) and to the basic topology of the solar magnetic field.
Cross-Correlation between the OMNI Data
=======================================
The cross-correlation coefficients between the OMNI data provide an information limited to the relationships between them and could be useful for understanding why the $SW$ parameters are subdivided into two groups only.
The parameters of the interplanetary field, solar wind and geomagnetic activity have been numbered as follows:\
1: $B$, 2: $B_x$, 3: $B_y$, 4: $B_z$, 5: $T_p$, 6: $V_p$, 7: $E$, 8: $N_{\alpha}/N_p$, 9: $N_p$, 10: $P$, 11: $N_{\beta}$, 12: $M_a$, 13: $AE$, 14: $K_p$, 15: $DST$, 16: $SSN$.\
The cross-correlation was calculated and the corresponding correlation coefficients are plotted in Fig. 5. The corresponding numbers and short names of the parameters are shown along the $X$ and $Y$ axis. Asterisks correspond to the positive values of $K_{cor}$, and diamonds correspond to the negative values of $K_{cor}$. The size of the symbols is proportional to their values. On the top there are $K_{cor}$ for the original values of the $SW$ data (on the left side), and for their absolute values (on the right side). This makes difference for the $IMF$ components $B_x$, $B_y$ and $B_z$, for $E$ and for $DST$. On the bottom there are $K_{cor}$ for the residuals of the original values of the $FSW$ data (on the left side), and for the residuals of their absolute values (on the right side).
From the first plot on the left top strong correlations between the parameters No 9, 10, 11, 12, 13 and between No 5, 6, 14 are deduced. Not very strong, but significant correlations take place between No 5, 6, 9, 13. Strong anti-correlations take place between No 4 and 7; No 14 and 15; No 5, 6 and 15, etc. Then there are anti-correlations between No 8 and 9, 10, 11, 12, 13, etc.
Comparison between the other plots of Fig. 5 and the groups of the $SW$ data connected with the $B$ and $B_z$ (or $B_x$, $B_y$) permits to verify and to understand the existence of such groups. The study of such cross-relationships is very informative, but it does not provide a sufficient support to predict the $SW$ connection with the solar magnetic field while it helps to understand which of $SW$ parameters could respond in a similar way to the solar perturbations.
Direct comparison of the $SW$ data with the basic topology of the $SMF$ permits to study SMF – SW relationships.
An influence of the solar activity (or solar magnetic field intensity) on the $SW$ parameters was investigated by the analysis of their correlation on short subsequent intervals of time. Fig. 6 shows the correlation through solar cycles between the 3-year long sub-sets of the data corresponding to the absolute values of the photospheric field and to the absolute values of the solar wind and geomagnetic parameters. It is clearly visible that the main source of geomagnetic perturbations is concentrated in the helio-latitudinal zone from -55 to 55 degrees about
Some summary remarks
====================
Southward-directed interplanetary magnetic field is considered a primary cause of geomagnetic perturbations (Durney, 1961; Gonzales et al., 1994, 1999). As a consequence the orientation of the interplanetary magnetic field (Axford and McKenzie, 1997; Low, 1996; Parker, 1997; Smith, 1997) plays an important role.
The solar activity phenomena depend on the sunspot cycle, which can be characterized by the variability of the $SMF$ intensity in time and along the latitudes. The topology of the solar magnetic field influences the geomagnetic perturbations through the intensity and orientation of the interplanetary magnetic field and/or through other parameters of the solar wind. In this approach we could understand the presence of two groups of the OMNI data similarly sensitive to the basic topology of the magnetic field of the Sun (from the point of view of the dependence on latitude and phase-shift of the correlation of the coefficients with the mean latitudinal magnetic field).
The formal and complete study of the problem of solar-terrestrial relations has been performed and the connections between the processes on the way from the Sun to the Earth have been revealed. A useful information was deduced from the temporal behaviour and dependence of the correlation of the photospheric magnetic field and different parameters of interplanetary space and geomagnetosphere.
It was revealed directly from the experimental data that there are two groups of $SW$ parameter which respond in a similar way to the behaviour of solar characteristics. We found that the photospheric field influences the magnitude of the interplanetary field and, in the same way, the proton density, flow pressure, Alfven Mach number and plasma $\beta$ respond to the $SMF$. Moreover the $AE$-index behaves in a similar way as the above mentioned solar wind parameters.
On the contrary, regarding the planetary geomagnetic activity index $K_p$ we can deduce that solar activity events (CME, magnetic field intensity, sunspots, etc.) through perturbations of the $B_z$ component ($B_x$, $B_y$ components) of the $IMF$, the proton temperature $T_p$, plasma speed $V_p$, $N_{\alpha}/N_{p}$ ratio influence the $K_p$ index. The variations of the $-B_z$ ($B_y$) component produce the perturbations of the $DST$ index, and they are of opposite sign of the $K_p$ and $B_x$ time dependence.
It was also revealed from the experimental data that the solar magnetic fields and solar activity processes originated bellow $\pm 55$ degrees propagate up to the Earth orbit and produce the perturbations of the magnetosphere (Gavryuseva, 2006 c,f; 2008b, Gavryuseva & Godoli, 2006).
These results are useful for understanding the origin of solar wind and geomagnetic perturbations and for long-term predictions.
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank the WSO and OMNI teams for making available data of measurements of the solar magnetic field, solar wind and geomagnetic quantities. I am grateful to Prof. G. Godoli for his stimulating interest in these results and Profs. B. Draine, L. Paterno and E. Tikhomolov for help in polishing this paper and useful advises.
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---
abstract: 'We introduce a natural framework for dealing with Mourre theory in an abstract two-Hilbert spaces setting. In particular a Mourre estimate for a pair of self-adjoint operators $(H,A)$ is deduced from a similar estimate for a pair of self-adjoint operators $(H_0,A_0)$ acting in an auxiliary Hilbert space. A new criterion for the completeness of the wave operators in a two-Hilbert spaces setting is also presented.'
author:
- 'S. Richard$^1$[^1] and R. Tiedra de Aldecoa$^2$[^2]'
title: 'A few results on Mourre theory in a two-Hilbert spaces setting'
---
**2000 Mathematics Subject Classification:** 81Q10, 47A40, 46N50, 47B25, 47B47.
**Keywords:** Mourre theory, two-Hilbert spaces, conjugate operator, scattering theory
Introduction
============
It is commonly accepted that Mourre theory is a very powerful tool in spectral and scattering theory for self-adjoint operators. In particular, it naturally leads to limiting absorption principles which are essential when studying the absolutely continuous part of self-adjoint operators. Since the pioneering work of E. Mourre [@M80], a lot of improvements and extensions have been proposed, and the theory has led to numerous applications. However, in most of the corresponding works, Mourre theory is presented in a one-Hilbert space setting and perturbative arguments are used within this framework. In this paper, we propose to extend the theory to a two-Hilbert spaces setting and present some results in that direction. In particular, we show how a Mourre estimate can be deduced for a pair of self-adjoint operators $(H,A)$ in a Hilbert space $\H$ from a similar estimate for a pair of self-adjoint operators $(H_0,A_0)$ in a auxiliary Hilbert space $\H_0$.
The main idea of E. Mourre for obtaining results on the spectrum $\sigma(H)$ of a self-adjoint operator $H$ in a Hilbert space $\H$ is to find an auxiliary self-adjoint operator $A$ in $\H$ such that the commutator $[iH,A]$ is positive when localised in the spectrum of $H$. Namely, one looks for a subset $I\subset\sigma(H)$, a number $a\equiv a(I)>0$ and a compact operator $K\equiv K(I)$ in $\H$ such that $$\label{Mourre}
E^H(I)[iH,A]E^H(I)\ge aE^H(I)+K,$$ where $E^H(I)$ is the spectral projection of $H$ on $I$. Such an estimate is commonly called a Mourre estimate. In general, this positivity condition is obtained via perturbative technics. Typically, $H$ is a perturbation of a simpler operator $H_0$ in $\H$ for which the commutator $[iH_0,A]$ is easily computable and the positivity condition easily verifiable. In such a case, the commutator of the formal difference $H-H_0$ with $A$ can be considered as a small perturbation of $[iH_0,A]$, and one can still infer the necessary positivity of $[iH,A]$.
In many other situations one faces the problem that $H$ is not the perturbation of any simpler operator $H_0$ in $\H$. For example, if $H$ is the Laplace-Beltrami operator on a non-compact manifold, there is no candidate for a simpler operator $H_0$! Alternatively, for multichannel scattering systems, there might exist more than one single candidate for $H_0$, and one has to take this multiplicity into account. In these situations, it is therefore unclear from the very beginning wether one can find a suitable conjugate operator $A$ for $H$ and how some positivity of $[iH,A]$ can be deduced from a hypothetic similar condition involving a simpler operator $H_0$. Of course, these interrogations have found positive answers in various situations. Nevertheless, it does not seem to the authors that any general framework has yet been proposed.
The starting point for our investigations is the scattering theory in the two-Hilbert spaces setting. In this setup, one has a self-adjoint operator $H$ in a Hilbert space $\H$, and one looks for a simpler self-adjoint operator $H_0$ in an auxiliary Hilbert space $\H_0$ and a bounded operator $J:\H_0\to\H$ such that the strong limits $$\slim_{t\to\pm\infty}\e^{itH}J\e^{-itH_0}\varphi$$ exist for suitable vectors $\varphi\in\H_0$. If such limits exist for enough $\varphi\in\H_0$, then some information on the spectral nature of $H$ can be inferred from similar information on the spectrum of $H_0$. We refer to the books [@BW83] and [@Yaf92] for general presentations of scattering theory in the two-Hilbert spaces setting. Therefore, the following question naturally arises: If $A_0$ is a conjugate operator for $H_0$ such that holds with $(H_0,A_0)$ instead of $(H,A)$, can we define a conjugate operator $A$ for $H$ such that holds? Under suitable conditions, the answer is “yes”, and its justification is the content of this paper. In fact, we present a general framework in which a Mourre estimate for a pair $(H,A)$ can be deduced from a similar Mourre estimate for a pair $(H_0,A_0)$. In that framework, we suppose the operators $A_0$ and $A$ given [[*a priori*]{} ]{}, and then exhibit sufficient conditions on the formal commutators $[iH,A]$ and $[iH_0,A_0]$ guaranteeing the existence of a Mourre estimate for $(H,A)$ if a Mourre estimate for $(H_0,A_0)$ is verified (see the assumptions of Theorem \[fonctionrho\]). We also show how a conjugate operator $A$ for $H$ can be constructed from a conjugate operator $A_0$ for $H_0$.
Let us finally sketch the organisation of the paper. In Section \[Sec\_one\], we recall a few definitions (borrowed from [@ABG Chap. 7]) in relation with Mourre theory in the usual one-Hilbert space setting. In Section \[sec\_two\], we state our main result, Theorem \[fonctionrho\], on the obtention of a Mourre estimate for $(H,A)$ from a similar estimate for $(H_0,A_0)$. A complementary result on higher order regularity of $H$ with respect to $A$ is also presented. In the second part of Section \[sec\_two\], we show how the assumptions of Theorem \[fonctionrho\] can be checked for short-range type and long-range type perturbations (note that the distinction between short-range type and long-range type perturbations is more subtle here, since $H_0$ and $H$ do not live in the same Hilbert space). We also show how a natural candidate for $A$ can be constructed from $A_0$. In Section \[Example\], we illustrate our results with the simple example of one-dimensional Schrödinger operator with steplike potential. A more challenging application on manifolds will be presented in [@IRT11] (many other applications such as curved quantum waveguides, anisotropic Schrödinger operators, spin models, [[*etc.*]{}]{} are also conceivable). Finally, in Section \[Sec\_Comp\] we prove an auxiliary result on the completeness of the wave operators in the two-Hilbert spaces setting without assuming that the initial sets of the wave operators are equal to the subspace $\H_{\rm ac}(H_0)$ of absolute continuity of $H_0$ (in [@BW83] and [@Yaf92], only that case is presented and this situation is sometimes too restrictive as will be shown for example in [@IRT11]).
Mourre theory in the one-Hilbert space setting {#Sec_one}
==============================================
In this section we recall some definitions related to Mourre theory, such as the regularity condition of $H$ with respect to $A$, providing a precise meaning to the commutators mentioned in the Introduction. We refer to [@ABG Sec. 7.2] for more information and details.
Let us consider a Hilbert space $\H$ with scalar product $\langle\;\!\cdot\;\!,\;\!\cdot\;\!\rangle_\H$ and norm $\|\;\!\cdot\;\!\|_\H$. Let also $H$ and $A$ be two self-adjoint operators in $\H$, with domains $\dom(H)$ and $\dom(A)$. The spectrum of $H$ is denoted by $\sigma(H)$ and its spectral measure by $E^H(\;\!\cdot\;\!)$. For shortness, we also use the notation $E^H(\lambda;\varepsilon):=E^H\big((\lambda-\varepsilon,\lambda+\varepsilon)\big)$ for all $\lambda\in\R$ and $\varepsilon>0$.
The operator $H$ is said to be of class $C^1(A)$ if there exists $z\in\C\setminus\sigma(H)$ such that the map $$\label{C1}
\R\ni t\mapsto\e^{-itA}(H-z)^{-1}\e^{itA}\in\B(\H)$$ is strongly of class $C^1$ in $\H$. In such a case, the set $\dom(H)\cap\dom(A)$ is a core for $H$ and the quadratic form $
\dom(H)\cap\dom(A)\ni\varphi
\mapsto\langle H\varphi,A\varphi\rangle_\H-\langle A\varphi,H\varphi\rangle_\H
$ is continuous in the topology of $\dom(H)$. This form extends then uniquely to a continuous quadratic form $[H,A]$ on $\dom(H)$, which can be identified with a continuous operator from $\dom(H)$ to the adjoint space $\dom(H)^*$. Furthermore, the following equality holds: $$\big[A,(H-z)^{-1}\big]=(H-z)^{-1}[H,A](H-z)^{-1}.$$ This $C^1(A)$-regularity of $H$ with respect to $A$ is the basic ingredient for any investigation in Mourre theory. It is also at the root of the proof of the Virial Theorem (see for example [@ABG Prop. 7.2.10] or [@GG99]).
Note that if $H$ is of class $C^1(A)$ and if $\eta \in C^\infty_{\rm c}(\R)$ (the set of smooth functions on $\R$ with compact support), then the quadratic form $
\dom(A)\ni\varphi\mapsto\langle\bar\eta(H)\varphi,A\varphi\rangle_\H
-\langle A\varphi,\eta(H)\varphi\rangle_\H
$ also extends uniquely to a continuous quadratic form $[\eta(H)A,]$ on $\H$, identified with a bounded operator on $\H$.
We now recall the definition of two very useful functions in Mourre theory described in [@ABG Sec. 7.2]. For that purpose, we use the following notations: for two bounded operators $S$ and $T$ in a common Hilbert space we write $S\approx T$ if $S-T$ is compact, and we write $S\lesssim T$ if there exists a compact operator $K$ such that $S\le T+K$. If $H$ is of class $C^1(A)$ and $\lambda\in\R$ we set $$\varrho^A_H(\lambda)
:=\sup\big\{a\in\R \mid\exists\;\!\varepsilon>0~\,\hbox{s.t.}~\,a\;\!
E^H(\lambda;\varepsilon)\le E^H(\lambda;\varepsilon)[iH,A]
E^H(\lambda;\varepsilon)\big\}.$$ A second function, more convenient in applications, is $$\widetilde\varrho^A_H(\lambda)
:=\sup\big\{a\in\R \mid\exists\;\!\varepsilon>0~\,\hbox{s.t.}~\,a\;\!
E^H(\lambda;\varepsilon)\lesssim E^H(\lambda;\varepsilon)[iH,A]
E^H(\lambda;\varepsilon)\big\}.$$ Note that the following equivalent definition is often useful: $$\label{autre}
\widetilde\varrho^A_H(\lambda)
=\sup\big\{a\in\R\mid\exists\;\!\eta\in C^\infty_{\rm c}(\R)
\hbox{ real}~\,\hbox{s.t.}~\,\eta(\lambda)\neq0,~a\;\!\eta(H)^2\lesssim
\eta(H)[iH,A]\eta(H)\big\}.$$ It is commonly said that $A$ is conjugate to $H$ at the point $\lambda\in\R$ if $\widetilde\varrho^A_H(\lambda)>0$, and that $A$ is strictly conjugate to $H$ at $\lambda$ if $\varrho^A_H(\lambda)>0$. Furthermore, the function $\widetilde\varrho^A_H:\R\to(-\infty,\infty]$ is lower semicontinuous and satisfies $\widetilde\varrho^A_H(\lambda)<\infty$ if and only if $\lambda$ belongs to the essential spectrum $\sigma_{\rm ess}(H)$ of $H$. One also has $\widetilde\varrho^A_H(\lambda)\ge\varrho^A_H(\lambda)$ for all $\lambda\in\R$.
Another property of the function $\widetilde\varrho$, often used in the one-Hilbert space setting, is its stability under a large class of perturbations: Suppose that $H$ and $H'$ are self-adjoint operators in $\H$ and that both operators $H$ and $H'$ are of class $C^1_{\rm u}(A)$, [[*i.e.*]{} ]{}such that the map is $C^1$ in norm. Assume furthermore that the difference $(H-i)^{-1}-(H'-i)^{-1}$ belongs to $\K(\H)$, the algebra of compact operators on $\H$. Then, it is proved in [@ABG Thm. 7.2.9] that $\widetilde\varrho^A_{H'}=\widetilde\varrho^A_H$, or in other words that $A$ is conjugate to $H'$ at a point $\lambda\in\R$ if and only if $A$ is conjugate to $H$ at $\lambda$.
Our first contribution in this paper is to extend such a result to the two-Hilbert spaces setting. But before this, let us recall the importance of the set $\widetilde\mu^A(H)\subset\R$ on which $\widetilde\varrho^A_H(\;\!\cdot\;\!)>0$: if $H$ is slightly more regular than $C^1(A)$, then $H$ has locally at most a finite number of eigenvalues on $\widetilde\mu^A(H)$ (multiplicities counted), and $H$ has no singularly continuous spectrum on $\widetilde\mu^A(H)$ (see [@ABG Thm. 7.4.2] for details).
Mourre theory in the two-Hilbert spaces setting {#sec_two}
===============================================
From now on, apart from the triple $(\H,H,A)$ of Section \[Sec\_one\], we consider a second triple $(\H_0,H_0,A_0)$ and an identification operator $J:\H_0\to\H$. The existence of two such triples is quite standard in scattering theory, at least for the pairs $(\H,H)$ and $(\H_0,H_0)$ (see for instance the books [@BW83; @Yaf92]). Part of our goal in what follows is to show that the existence of the conjugate operators $A$ and $A_0$ is also natural, as was realised in the context of scattering on manifolds [@IRT11].
So, let us consider a second Hilbert space $\H_0$ with scalar product $\langle\;\!\cdot\;\!,\;\!\cdot\;\!\rangle_{\H_0}$ and norm $\|\;\!\cdot\;\!\|_{\H_0}$. Let also $H_0$ and $A_0$ be two self-adjoint operators in $\H_0$, with domains $\dom(H_0)$ and $\dom(A_0)$. Clearly, the $C^1(A_0)$-regularity of $H_0$ with respect to $A_0$ can be defined as before, and if $H_0$ is of class $C^1(A_0)$ then the definitions of the two functions $\varrho^{A_0}_{H_0}$ and $\widetilde \varrho^{A_0}_{H_0}$ hold as well.
In order to compare the two triples, it is natural to require the existence of a map $J\in\B(\H_0,\H)$ having some special properties (for example, the ones needed for the completeness of the wave operators, see Section \[Sec\_Comp\]). But for the time being, no additional information on $J$ is necessary. In the one-Hilbert space setting, the operator $H$ is typically a perturbation of the simpler operator $H_0$. And as mentioned above, the stability of the function $\widetilde\varrho_{H_0}^{A_0}$ is an efficient tool to infer information on $H$ from similar information on $H_0$. In the two-Hilbert spaces setting, we are not aware of any general result allowing the computation of the function $\widetilde\varrho_H^A$ in terms of the function $\widetilde\varrho_{H_0}^{A_0}$. The obvious reason for this being the impossibility to consider $H$ as a direct perturbation of $H_0$ since these operators do not live in the same Hilbert space. Nonetheless, the next theorem gives a result in that direction:
\[fonctionrho\] Let $(\H,H,A)$ and $(\H_0,H_0,A_0)$ be as above, and assume that
1. the operators $H_0$ and $H$ are of class $C^1(A_0)$ and $C^1(A)$, respectively,
2. for any $\,\eta\in C^\infty_{\rm c}(\R)$ the difference of bounded operators $J[iA_0,\eta(H_0)]J^*-[iA,\eta(H)]$ belongs to $\K(\H)$,
3. for any $\,\eta\in C^\infty_{\rm c}(\R)$ the difference $J\eta(H_0)-\eta(H)J$ belongs to $\K(\H_0,\H)$,
4. for any $\,\eta\in C^\infty_{\rm c}(\R)$ the operator $\eta(H)(JJ^*-1)\eta(H)$ belongs to $\K(\H)$.
Then, one has $\widetilde\varrho_H^A\ge\widetilde\varrho_{H_0}^{A_0}$. In particular, if $A_0$ is conjugate to $H_0$ at $\lambda\in\R$, then $A$ is conjugate to $H$ at $\lambda$.
Note that with the notations introduced in the previous section, Assumption (ii) reads $J[iA_0,\eta(H_0)]J^*\approx[iA,\eta(H)]$. Furthermore, since the vector space generated by the family of functions $\{(\;\cdot\;-z)^{-1}\}_{z\in\C\setminus\R}$ is dense in $C_0(\R)$ and the set $\K(\H_0,\H)$ is closed in $\B(\H_0,\H)$, the condition $J(H_0-z)^{-1}-(H-z)^{-1}J \in\K(\H_0,\H)$ for all $z\in\C\setminus\R$ implies Assumption (iii) (here, $C_0(\R)$ denotes the set of continuous functions on $\R$ vanishing at $\pm\infty$).
Let $\eta\in C^\infty_{\rm c}(\R;\R)$, and define $\eta_1,\eta_2\in C^\infty_{\rm c}(\R;\R)$ by $\eta_1(x):=x\;\!\eta(x)$ and $\eta_2(x):=x\;\!\eta(x)^2$. Under Assumption (i), it is shown in [@ABG Eq. 7.2.18] that $$\eta(H)[iA,H]\eta(H)=[iA,\eta_2(H)]-2\re\big\{[iA,\eta(H)]\eta_1(H)\big\}.$$ Therefore, one infers from Assumptions (ii) and (iii) that $$\begin{aligned}
&\eta(H)[iA,H]\eta(H)\\
&\approx J[iA_0,\eta_2(H_0)]J^*-2\re\big\{J[iA_0,\eta(H_0)]J^*\eta_1(H)\big\}\\
&=J[iA_0,\eta_2(H_0)]J^*-2\re\big\{J[iA_0,\eta(H_0)]\eta_1(H_0)J^*\big\}
-2\re\big\{J[iA_0,\eta(H_0)]\big(J^*\eta_1(H)-\eta_1(H_0)J^*\big)\big\}\\
&\approx J[iA_0,\eta_2(H_0)]J^*-2J\re\big\{[iA_0,\eta(H_0)]\eta_1(H_0)\big\}J^*\\
&=J\eta(H_0)[iA_0,H_0]\eta(H_0)J^*,\end{aligned}$$ which means that $$\label{Eone}
\eta(H)[iA,H]\eta(H)\approx J\eta(H_0)[iA_0,H_0]\eta(H_0)J^*.$$ Furthermore, if $a\in\R$ is such that $\eta(H_0)[iA_0,H_0]\eta(H_0)\gtrsim a \eta(H_0)^2$, then Assumptions (iii) and (iv) imply that $$\label{Etwo}
J\eta(H_0)[iA_0,H_0]\eta(H_0)J^*
\gtrsim aJ\eta(H_0)^2J^*
\approx a\eta(H)JJ^*\eta(H)
\approx a\eta(H)^2.$$ Thus, one obtains $\eta(H)[iA,H]\eta(H)\gtrsim a\eta(H)^2$ by combining and . This last estimate, together with the definition of the functions $\widetilde\varrho_{H_0}^{A_0}$ and $\widetilde\varrho_H^A$ in terms of the localisation function $\eta$, implies the claim.
As mentioned in the previous sections, the $C^1(A)$-regularity of $H$ and the Mourre estimate are crucial ingredients for the analysis of the operator $H$, but they are in general not sufficient. For instance, the nature of the spectrum of $H$ or the existence and the completeness of the wave operators is usually proved under a slightly stronger $C^{1,1}(A)$-regularity condition of $H$. It would certainly be valuable if this regularity condition could be deduced from a similar information on $H_0$. Since we have not been able to obtain such a result, we simply refer to [@ABG] for the definition of this class of regularity and present below a coarser result. Namely, we show that the regularity condition “$H$ is of class $C^n(A)$” can be checked by means of explicit computations involving only $H$ and not its resolvent. For simplicity, we present the simplest, non-perturbative version of the result; more refined statements involving perturbations as in Sections \[S1\] and \[S2\] could also be proved.
For that purpose, we first recall that $H$ is of class $C^n(A)$ if the map is strongly of class $C^n$. We also introduce the following slightly more general regularity class: Assume that $(\G,\H)$ is a Friedrichs couple, [[*i.e.*]{} ]{}a pair $(\G,\H)$ with $\G$ a Hilbert space densely and continuously embedded in $\H$. Assume furthermore that the unitary group $\{\e^{itA}\}_{t\in\R}$ leaves $\G$ invariant, so that the restriction of this group to $\G$ generates a $C_0$-group, with generator also denoted by $A$. In such a situation, an operator $T\in\B(\G,\H)$ is said to belong to $C^n(\A;\G,\H)$ if the map $$\R\ni t\mapsto\e^{-itA}T\e^{itA}\in\B(\G,\H)$$ is strongly of class $C^n$. Similar definitions hold with $T$ in $\B(\H,\G)$, in $\B(\G,\G)$ or in $\B(\H,\H)$ (in the latter case, one simply writes $T\in C^n(A)$ instead of $T \in C^n(A;\H,\H)$).
The next proposition (which improves slightly the result of [@MW11 Lemma 1.2]) is an extension of [@ABG Thm. 6.3.4.(c)] to higher orders of regularity of $H$ with respect to $A$. We use for it the notation $\G$ for the domain $\dom(H)$ of $H$ endowed with its natural Hilbert space structure. We also recall that if $H$ is of class $C^1(A)$, then $[iH,A]$ can be identified with a bounded operator from $\G$ to $\G^*$. It has been proved in [@GG99 Lemma 2] that if this operator maps $\G$ into $\H$, then $\{\e^{itA}\}_{t\in\R}$ leaves $\G$ invariant, and thus one has a $C_0$-group in $\G$.
\[reguln\] Let $H$ be of class $C^1(A)$, assume that $[iH,A]\in\B(\G,\H)$ and suppose that $[iH,A]\in C^n(A;\G,\H)$ for some integer $n\ge0$. Then $(H-z)^{-1}\in C^{n+1}(A;\H,\G)\subset C^{n+1}(A)$ for any $z\in\C\setminus\R$.
We prove the claim by induction on $n$. For $n=0$, one has $[iH,A]\in\B(\G,\H)\equiv C^0(A;\G,\H)$. It follows from the equality $$\label{step1}
\big[i(H-z)^{-1},A\big]=-(H-z)^{-1}[iH,A](H-z)^{-1}$$ and from the inclusion $(H-z)^{-1}\in\B(\H,\G)$ that $\big[i(H-z)^{-1},A\big]\in\B(\H,\G)$. Then, one infers that $(H-z)^{-1}\in C^1(A;\H,\G)$ by using [@ABG Prop. 5.1.2.(b)].
Now, assume that the statement is true for $n-1\geq 0$, namely, $[iH,A]\in C^n(A;\G,\H)$ and $(H-z)^{-1}\in C^n(A;\H,\G)$. Then, by taking into account account and the property of regularity for product of operators stated in [@ABG Prop. 5.1.5], one obtains that $\big[i(H-z)^{-1},A\big]\in C^n(A;\H,\G)$. This is equivalent to the inclusion $(H-z)^{-1}\in C^{n+1}(A;\H,\G)$, which proves the statement for $n$.
Usually, the regularity of $H_0$ with respect to $A_0$ is easy to check. On the other hand, the regularity of $H$ with respect to $A$ is in general rather difficult to establish, and various perturbative criteria have been developed for that purpose in the one-Hilbert space setting. Often, a distinction is made between so-called short-range and long-range perturbations. Roughly speaking, the difference between these types perturbations is that the two terms of the formal commutator $[A,H-H_0]=A(H-H_0)-(H-H_0)A$ are treated separately in the former situation while the commutator $[A,H-H_0]$ is really computed in the latter situation. In the first case, one usually requires more decay and less regularity, while in the second case more regularity but less decay are imposed. Obviously, this distinction cannot be as transparent in the general two-Hilbert spaces setting presented here. Still, a certain distinction remains, and thus we dedicate to it the following two complementary sections.
Short-range type perturbations {#S1}
------------------------------
We show below how the condition “$H$ is of class $C^1(A)$” and the assumptions (ii) and (iii) of Theorem \[fonctionrho\] can be verified for a class of short-range type perturbations. Our approach is to derive information on $H$ from some equivalent information on $H_0$, which is usually easier to obtain. Accordingly, our results exhibit some perturbative flavor. The price one has to pay is that a compatibility condition between $A_0$ and $A$ is necessary. For $z\in\C\setminus\R$, we use the shorter notations $R_0(z):=(H_0-z)^{-1}$, $R(z):=(H-z)^{-1}$ and $$\label{limpide}
B(z):=JR_0(z)-R(z)J\in\B(\H_0,\H).$$
\[C1short\] Let $H_0$ be of class $C^1(A_0)$ and assume that $\D\subset\H$ is a core for $A$ such that $J^*\D\subset\dom(A_0)$. Suppose furthermore that for any $z\in\C\setminus\R$ $$\label{hyp1}
\overline{B(z)A_0\upharpoonright\dom(A_0)}\in\B(\H_0,\H)
\qquad\hbox{and}\qquad\overline{R(z)(JA_0J^*-A)\upharpoonright\D}\in\B(\H).$$ Then, $H$ is of class $C^1(A)$.
Take $\psi\in\D$ and $z\in\C\setminus\R$. Then, one gets $$\begin{aligned}
&\big\langle R(\bar z)\psi,A\psi\big\rangle_\H
-\big\langle A\psi,R(z)\psi \big\rangle_\H\\
&=\big\langle R(\bar z)\psi,A\psi\big\rangle_\H
-\big\langle A\psi,R(z)\psi\big\rangle_\H
-\big\langle\psi,J[R_0(z),A_0]J^*\psi\big\rangle_\H
+\big\langle\psi,J[R_0(z),A_0]J^*\psi\big\rangle_\H\\
&=\big\langle B(\bar z)A_0J^*\psi,\psi,\big\rangle_\H
-\big\langle\psi,B(z)A_0J^*\psi\big\rangle_\H
+\big\langle\psi,J[R_0(z),A_0]J^*\psi\big\rangle_\H\\
&\qquad+\big\langle R(\bar z)(JA_0J^*-A)\psi,\psi\big\rangle_\H
-\big\langle\psi,R(z)(JA_0J^*-A)\psi\big\rangle_\H.\end{aligned}$$ Now, one has $$\big|\big\langle B(\bar z)A_0J^*\psi,\psi,\big\rangle_\H
-\big\langle\psi,B(z)A_0J^*\psi\big\rangle_\H\big|
\le{\rm Const.}\;\!\|\psi\|_\H^2$$ due to the first condition in , and one has $$\big|\big\langle R(\bar z)(JA_0J^*-A)\psi,\psi\big\rangle_\H
-\big\langle\psi,R(z)(JA_0J^*-A)\psi\big\rangle_\H\big|
\le{\rm Const.}\;\!\|\psi\|^2_\H$$ due to the second condition in . Furthermore, since $H_0$ is of class $C^1(A_0)$ one also has $$\big|\big\langle\psi,J[R_0(z),A_0]J^*\psi\big\rangle_\H\big|
\le{\rm Const.}\;\!\|\psi\|^2_\H.$$ Since $\D$ is a core for $A$, the conclusion then follows from [@ABG Lemma 6.2.9].
We now show how the assumption (ii) of Theorem \[fonctionrho\] is verified for a short-range type perturbation. Note that the hypotheses of the following proposition are slightly stronger than the ones of Proposition \[C1short\], and thus $H$ is automatically of class $C^1(A)$.
\[ass2\_short\] Let $H_0$ be of class $C^1(A_0)$ and assume that $\D\subset\H$ is a core for $A$ such that $J^*\D\subset\dom(A_0)$. Suppose furthermore that for any $z\in\C\setminus\R$ $$\label{c123}
\overline{B(z)A_0\upharpoonright\dom(A_0)}\in\K(\H_0,\H)
\qquad\hbox{and}\qquad\overline{R(z)(JA_0J^*-A)\upharpoonright\D}\in\K(\H).$$ Then, for each $\eta\in C^\infty_{\rm c}(\R)$ the difference of bounded operators $J[A_0,\eta(H_0)]J^*-[A,\eta(H)]$ belongs to $\K(\H)$.
Take $\psi,\psi'\in\D$ and $z\in\C\setminus\R$. Then, one gets from the proof of Proposition \[C1short\] that $$\begin{aligned}
&\big\langle\psi',J[A_0,R_0(z)]J^*\psi\big\rangle_\H
-\big\langle\psi',[A,R(z)]\psi\rangle_\H\\
&=\big\langle B(\bar z) A_0J^*\psi',\psi,\big\rangle_\H
-\big\langle\psi',B(z)A_0J^*\psi\big\rangle_\H\\
&\qquad+\big\langle R(\bar z)(JA_0J^*-A)\psi',\psi\big\rangle_\H
-\big\langle\psi',R(z)(JA_0J^*-A)\psi\big\rangle_\H.\end{aligned}$$ By the density of $\D$ in $\H$, one then infers from the hypotheses that $J[A_0,R_0(z)]J^*-[A,R(z)]$ belongs to $\K(\H)$.
To show the same result for functions $\eta\in C^\infty_{\rm c}(\R)$ instead of $(\;\!\cdot\;\!-z)^{-1}$, one needs more refined estimates. Taking the first resolvent identity into account one obtains $$B(z)=\big\{1+(z-i)R(z)\big\}B(i)\big\{1+(z-i)R_0(z)\big\}.$$ Thus, one gets on $\D$ the equalities $$\label{llabol}
B(z)A_0 J^*=\big\{1+(z-i)R(z)\big\}B(i)A_0\big\{1+(z-i)R_0(z)\big\}J^*
+\big\{1+(z-i)R(z)\big\}B(i)(z-i)[R_0(z),A_0]J^*,$$ where $$[R_0(z),A_0]=\big\{1+(z-i)R_0(z)\big\}R_0(i)[A_0,H_0]R_0(i)\big\{1+(z-i)R_0(z)\big\}.$$ Obviously, these equalities extend to all of $\H$ since they involve only bounded operators. Letting $z=\lambda+i\mu$ with $|\mu|\le 1$, one even gets the bound $$\big\|B(z)A_0J^*\big\|_{\B(\H)}
\le{\rm Const.}\;\!\bigg(1+\frac{|\lambda+i(\mu-1)|}{|\mu|}\bigg)^4.$$ Furthermore, since the first and second terms of extend to elements of $\K(\H)$, the third term of also extends to an element of $\K(\H)$. Similarly, the operator on $\D$ $$R(z)(JA_0J^*-A)\equiv\big\{1+(z-i)R(z)\big\}R(i)(JA_0J^*-A)$$ extends to a compact operator in $\H$, and one has the bound $$\big\|R(z)(JA_0J^*-A) \big\|_{\B(\H)}
\le{\rm Const.}\;\!\bigg(1+\frac{|\lambda+i(\mu-1)|}{|\mu|}\bigg).$$
Now, observe that for any $\eta\in C^\infty_{\rm c}(\R)$ and any $\psi,\psi'\in\D$ one has $$\begin{aligned}
&\big\langle\psi',J[A_0,\eta(H_0)]J^*\psi\big\rangle_\H
-\big\langle\psi',[A,\eta(H)]\psi\big\rangle_\H\nonumber\\
&=\big\langle\big\{J\overline\eta(H_0)
-\overline\eta(H)J\big\}A_0 J^*\psi',\psi\big\rangle_\H
-\big\langle\psi',\big\{J\eta(H_0)-\eta(H)J\big\}
A_0J^*\psi\big\rangle_\H.\nonumber\\
&\qquad+\big\langle\overline\eta(H)(JA_0J^*-A)\psi',\psi\big\rangle_\H
-\big\langle\psi',\eta(H)(JA_0J^*-A)\psi\big\rangle_\H.\label{lopp}\end{aligned}$$ Then, by expressing the operators $\eta(H_0)$ and $\eta(H)$ in terms of their respective resolvents (using for example [@ABG Eq. 6.1.18]) and by taking the above estimates into account, one obtains that $\big\{J\eta(H_0)-\eta(H)J\big\}A_0J^*$ and $\eta(H)(JA_0J^*-A)$ are equal on $\D$ to a finite sum of norm convergent integrals of compact operators. Since $\D$ is dense in $\H$, these equalities between bounded operators extend continuously to equalities in $\B(\H)$, and thus the statement follows by using .
As mentioned just after Theorem \[fonctionrho\], the requirement $B(z)\in\K(\H_0,\H)$ for all $z\in\C\setminus\R$ implies the assumption (iii) of Theorem \[fonctionrho\]. Since an a priori stronger requirement is imposed in the first condition of , it is likely that in applications the compactness assumption (iii) will follow from the necessary conditions ensuring the first condition in .
Before turning to the long-range case, let us reconsider the above statements in the special situation where $A= JA_0J^*$. This case deserves a particular attention since it represents the most natural choice of conjugate operator for $H$ when $A_0$ is a conjugate operator for $H_0$. However, in order to deal with a well-defined self-adjoint operator $A$, one needs the following assumption:
\[bobo\] There exists a set $\D\subset\dom(A_0J^*)\subset \H$ such that $JA_0J^*$ is essentially self-adjoint on $\D$, with corresponding self-adjoint extension denoted by $A$.
Assumption \[bobo\] might be difficult to check in general, but in concrete situations the choice of the set $\D$ can be quite natural. We now show how the assumptions of the above propositions can easily be checked under Assumption \[bobo\]. Recall that the operator $B(z)$ was defined in .
\[bobone\] Let $H_0$ be of class $C^1(A_0)$, suppose that Assumption \[bobo\] holds for some set $\D\subset\H$, and for any $z\in\C\setminus\R$ assume that $$\overline{B(z)A_0\upharpoonright\dom(A_0)}\in\B(\H_0,\H).$$ Then, $H$ is of class $C^1(A)$.
All the assumptions of Proposition \[C1short\] are verified.
\[compact\] Let $H_0$ be of class $C^1(A_0)$, suppose that Assumption \[bobo\] holds for some set $\D\subset\H$, and for any $z\in\C\setminus\R$ assume that $$\label{cond_cor_comp}
\overline{B(z)A_0\upharpoonright\dom(A_0)}\in\K(\H_0,\H).$$ Then, for each $\eta\in C^\infty_{\rm c}(\R)$ the difference of bounded operators $J[A_0,\eta(H_0)]J^*-[A,\eta(H)]$ belongs to $\K(\H)$.
All the assumptions of Proposition \[ass2\_short\] are verified.
As mentioned above the choice $A = JA_0J^*$ is natural when $A_0$ is a conjugate operator for $H_0$. With that respect the second conditions in and quantify how much one can deviate from this natural choice.
The most important consequence of Mourre theory is the obtention of a limiting absorption principle for $H_0$ and $H$. Rather often, the space defined in terms of $A_0$ (resp. $A$) in which holds the limiting absorption principle for $H_0$ (resp. $H$) is not adequate for applications. In [@ABG Prop. 7.4.4] a method is given for expressing the limiting absorption principle for $H_0$ in terms of an auxiliary operator $\Phi_0$ in $\H_0$ more suitable than $A_0$. Obviously, this abstract result also applies for three operators $H$, $A$ and $\Phi$ in $\H$, but one crucial condition is that $(H-z)^{-1}\dom(\Phi)\subset\dom(A)$ for suitable $z\in\C$. In the next lemma, we provide a sufficient condition allowing to infer this information from similar information on the operators $H_0$, $A_0$ and $\Phi_0$ in $\H_0$. Note that $\Phi$ does not need to be of the form $J\Phi_0J^*$ but that such a situation often appears in applications.
\[passage\] Let $z\in\C\setminus\{\sigma(H_0)\cup\sigma(H)\}$. Suppose that Assumption \[bobo\] holds for some set $\D\subset\H$. Assume that $$\overline{B(\bar z) A_0\upharpoonright\dom(A_0)}\in\B(\H_0,\H).$$ Furthermore, let $\Phi_0$ and $\Phi$ be self-adjoint operators in $\H_0$ and $\H$ satisfying $(H_0-z)^{-1}\dom(\Phi_0)\subset\dom(A_0)$ and $J^*(\Phi-i)^{-1}-(\Phi_0-i)^{-1}J^*=(\Phi_0-i)^{-1}B$ for some $B\in\B(\H,\H_0)$. Then, one has the inclusion $(H-z)^{-1}\dom(\Phi)\subset\dom(A)$.
Let $\psi\in\D$ and $\psi'\in\H$. Then, one has $$\begin{aligned}
&\big\langle A\psi,(H-z)^{-1}(\Phi-i)^{-1}\psi'\big\rangle_\H\\
&=\big\langle\big\{(H-\bar z)^{-1}J-J(H_0-\bar z)^{-1}\big\}A_0J^*\psi,
(\Phi-i)^{-1}\psi'\big\rangle_\H
+\big\langle J(H_0-\bar z)^{-1}A_0J^*\psi,(\Phi-i)^{-1}\psi'\big\rangle_\H\\
&=-\big\langle B(\bar z)A_0J^*\psi,
(\Phi-i)^{-1}\psi'\big\rangle_\H
+\big\langle(H_0-\bar z)^{-1}A_0J^*\psi,(\Phi_0-i)^{-1}J^*\psi'\big\rangle_{\H_0}\\
&\qquad+\big\langle(H_0-\bar z)^{-1}A_0J^*\psi,
(\Phi_0-i)^{-1}B\psi'\big\rangle_{\H_0}.\end{aligned}$$ So, $
\big|\big\langle A\psi,(H-z)^{-1}(\Phi-i)^{-1}\psi'\big\rangle_\H\big|
\le{\rm Const.}\;\!\|\psi\|_\H
$, and thus $(H-z)^{-1}(\Phi-i)^{-1}\psi'\in\dom(A)$, since $A$ is essentially self-adjoint on $\D$.
Long-range type perturbations {#S2}
-----------------------------
In the case of a long-range type perturbation, the situation is slightly less satisfactory than in the short-range case. One reason comes from the fact that one really has to compute the commutator $[A,H-H_0]$ instead of treating the terms $A(H-H_0)$ and $(H-H_0)A$ separately. However, a rather efficient method for checking that “$H$ is of class $C^1(A)$” has been put into evidence in [@GM08 Lemma. A.2]. We start by recalling this result and then we propose a perturbative type argument for checking the assumption (ii) of Theorem \[fonctionrho\]. Note that there is a missprint in the hypothesis $1$ of [@GM08 Lemma A.2]; the meaningless condition $\sup_n\|\chi_n\|_{\dom(H)}<\infty$ has to be replaced by $\sup_n\|\chi_n\|_{\B(\dom(H))}<\infty$.
Let $\D\subset\H$ be a core for $A$ such that $\D\subset\dom(H)$ and $H\D\subset\D$. Let $\{\chi_n\}_{n\in\N}$ be a family of bounded operators on $\H$ such that
1. $\chi_n\D\subset\D$ for each $n\in\N$, $\slim_{n\to\infty}\chi_n=1$ and $\,\sup_n\|\chi_n\|_{\B(\dom(H))}<\infty$,
2. for all $\psi\in\D$, one has $\slim_{n\to\infty}A\chi_n\psi=A\psi$,
3. there exists $z\in\C\setminus\sigma(H)$ such that $\chi_n R(z)\D\subset\D$ and $\chi_nR(\bar z)\D\subset\D$ for each $n\in\N$,
4. for all $\psi\in\D$, one has $\slim_{n\to\infty}A[H,\chi_n]R(z)\psi=0$ and $\slim_{n\to\infty}A[H,\chi_n]R(\bar z)\psi=0$.
Finally, assume that for all $\psi\in\D$ $$\big|\langle A\psi,H\psi\rangle_\H-\langle H\psi,A\psi\rangle_\H\big|
\le{\rm Const.}\;\!\big(\|H\psi\|^2+\|\psi\|^2\big).$$ Then, $H$ is of class $C^1(A)$.
In the next statement we provide conditions under which the assumption (ii) of Theorem \[fonctionrho\] is verified for a long-range type perturbation. One condition is that for each $z\in\C\setminus\R$ the operator $B(z)$ belongs to $\K(\H_0,\H)$, which means that the hypothesis (iii) of Theorem \[fonctionrho\] is also automatically satisfied. We stress that no direct relation between $A_0$ and $A$ is imposed; the single relation linking $A_0$ and $A$ only involves the commutators $[H_0,A_0]$ and $[H,A]$. On the other hand, the condition on $H_0$ is slightly stronger than just the $C^1(A_0)$-regularity.
\[ass2\_long\] Let $H_0$ be of class $C^1(A_0)$ with $[H_0,A_0]\in\B\big(\dom(H_0),\H_0\big)$ and let $H$ be of class $C^1(A)$. Assume that the operator $J\in\B(\H_0,\H)$ extends to an element of $\B\big(\dom(H_0)^*,\dom(H)^*\big)$, and suppose that for each $z\in\C\setminus\R$ the operator $B(z)$ belongs to $\K(\H_0,\H)$ and that the difference $J[H_0,A_0]J^*-[H,A]$ belongs to $\K\big(\dom(H),\dom(H)^*\big)$. Then, for each $\eta\in C^\infty_{\rm c}(\R)$ the difference of bounded operators $$J[A_0,\eta(H_0)]J^*-[A,\eta(H)]$$ belongs to $\K(\H)$.
By taking the various hypotheses into account one gets for any $z\in\C\setminus\R$ that $$\begin{aligned}
&J[A_0,R_0(z)]J^*-[A,R(z)]\\
&=JR_0(z)[H_0,A_0]R_0(z)J^*-R(z)[H,A]R(z)\\
&=\big\{JR_0(z)-R(z)J\big\}[H_0,A_0]R_0(z)J^*
+R(z)J[H_0,A_0]\big\{R_0(z)J^*-J^*R(z)\big\}\\
&\qquad+R(z)\big\{J[H_0,A_0]J^*-[H,A]\big\}R(z)\\
&=B(z)[H_0,A_0]R_0(z)J^*+R(z)J[H_0,A_0]B(\bar z)^*
+R(z)\big\{J[H_0,A_0]J^*-[H,A]\big\}R(z),\end{aligned}$$ with each term on the last line in $\K(\H)$. Now, by taking the first resolvent identity into account, one obtains $$B(z)[H_0,A_0]R_0(z)J^*
=\big\{1+(z-i)R(z)\big\}B(i)\big\{1+(z-i)R_0(z)\big\}[H_0,A_0]R_0(i)
\big\{1+(z-i)R_0(z)\big\}J^*$$ and $$R(z)J[H_0,A_0]B(\bar z)^*
=\big\{1+(z-i)R(z)\big\}R(i)J[H_0,A_0]\big\{1+(z-i)R_0(z)\big\}B(-i)^*
\big\{1+(z-i)R(z) \big\}$$ as well as $$\begin{aligned}
&R(z)\big\{J[H_0,A_0]J^*-[H,A]\big\}R(z)\\
&=\big\{1+(z-i)R(z)\big\}R(i)\big\{J[H_0,A_0]J^*-[H,A]\big\}R(i)
\big\{1+(z-i)R(z)\big\}.\end{aligned}$$ Thus, by letting $z=\lambda+i\mu$ with $|\mu|\le 1$, one gets the bound $$\big\|J[A_0,R_0(z)]J^*-[A,R(z)]\big\|_{\B(\H)}
\le{\rm Const.}\;\!\bigg(1+\frac{|\lambda+i(\mu-1)|}{|\mu|}\bigg)^3.$$ One concludes as in the proof of Proposition \[ass2\_short\] by expressing $J[A_0,\eta(H_0)]J^*-[A,\eta(H)]$ in terms of $J[A_0,R_0(z)]J^*-[A,R(z)]$ (using for example [@ABG Eq. 6.2.16]), and then by dealing with a finite number of norm convergent integrals of compact operators.
As mentioned before the statement, no direct relation between $A_0$ and $A$ has been imposed, and thus considering the special case $A=JA_0J^*$ is not really relevant. However, it is not difficult to check how the quantity $J[H_0,A_0]J^*-[H,A]$ looks like in that special case, and in applications such an approach could be of interest. However, since the resulting formulas are rather involved in general, we do not further investigate in that direction.
One illustrative example {#Example}
========================
To illustrate our approach, we present below a simple example for which all the computations can be made by hand (more involved examples will be presented elsewhere, like in [@IRT11], where part of the results of the present paper is used). In this model, usually called one-dimensional Schrödinger operator with steplike potential, the choice of a conjugate operator is rather natural, whereas the computation of the $\varrho$-functions is not completely trivial due to the anisotropy of the potential. We refer to [@Akt99; @AJ07; @Chr06; @CK85; @Ges86] for earlier works on that model and to [@Ric05] for a $n$-dimensional generalisation.
So, we consider in the Hilbert space $\H:=\ltwo(\R)$ the Schrödinger operator $H:=-\Delta+V$, where $V$ is the operator of multiplication by a function $v\in C(\R;\R)$ with finite limits $v_\pm$ at infinity, [[*i.e.*]{} ]{}$v_\pm:=\lim_{x\to\pm\infty}v(x)\in\R$. The operator $H$ is self-adjoint on $\H^2(\R)$, since $V$ is bounded. As a second operator, we consider in the auxiliary Hilbert space $\H_0:=\ltwo(\R)\oplus\ltwo(\R)$ the operator $$H_0:=(-\Delta+\vg)\oplus(-\Delta+\vd),$$ which is also self-adjoint on its natural domain $\H^2(\R)\oplus\H^2(\R)$. Then, we take a function $j_+\in C^\infty(\R;[0,1])$ with $j_+(x)=0$ if $x\le1$ and $j_+(x)=1$ if $x\ge2$, we set $j_-(x):=j_+(-x)$ for each $x\in\R$, and we define the identification operator $J\in\B(\H_0,\H)$ by the formula $$J(\varphi_-,\varphi_+):=j_-\varphi_-+j_+\varphi_+,\quad(\varphi_-,\varphi_+)\in\H_0.$$ Clearly, the adjoint operator $J^*\in\B(\H,\H_0)$ is given by $J^*\psi=(j_-\psi,j_+\psi)$ for any $\psi\in\H$, and the operator $JJ^*\in\B(\H)$ is equal to the operator of multiplication by $j_-^2+j_+^2$.
Let us now come to the choice of the conjugate operators. For $H_0$, the most natural choice consists in two copies of the generator of dilations on $\R$, that is, $A_0:=(D,D)$ with $D$ the generator of the group $$\big(\e^{itD}\psi\big)(x):=\e^{t/2}\psi(\e^t x),\quad\psi\in\S(\R),~t,x\in\R,$$ where $\S(\R)$ denotes the Schwartz space on $\R$. In such a case, the map with $(H,A)$ replaced by $(H_0,A_0)$ is strongly of class $C^\infty$ in $\H_0$. Moreover, the $\varrho$-functions can be computed explicitly (see [@ABG Sec. 8.3.5] for a similar calculation in an abstract setting):
$$\widetilde\varrho^{A_0}_{H_0}(\lambda)
=\varrho^{A_0}_{H_0}(\lambda)
=\begin{cases}
+\infty & \hbox{if }\,\lambda<\min\{\vg,\vd\}\\
2\big(\lambda-\min\{\vg,\vd\}\big)
& \hbox{if }\,\min\{\vg,\vd\}\le\lambda<\max\{\vg,\vd\}\\
2\big(\lambda-\max\{\vg,\vd\}\big) & \hbox{if }\,\lambda\ge\max\{\vg,\vd\}.
\end{cases}$$
For the conjugate operator for $H$, two natural choices exist: either one can use again the generator $D$ of dilations in $\H$, or one can use the (formal) operator $JA_0J^*$ which appears naturally in our framework. Since the latter choice illustrates better the general case, we opt here for this choice and just note that the former choice would also be suitable and would lead to similar results. So, we set $\D:=\S(\R)$ and $j:=j_-+j_+$, and then observe that $JA_0J^*$ is well-defined and equal to $$\label{explicit}
JA_0J^*=jDj$$ on $\D$. This equality, the fact that $j$ is of class $C^1(D)$, and [@ABG Lemma 7.2.15], imply that $JA_0J^*$ is essentially self-adjoint on $\D$. We denote by $A$ the corresponding self-adjoint extension.
We are now in a position for applying results of the previous sections such as Theorem \[fonctionrho\]. First, recall that $H_0$ is of class $C^1(A_0)$ and observe that the assumption (iv) of Theorem \[fonctionrho\] is satisfied with the operator $J$ introduced above. Similarly, one easily shows that the assumption (iii) of Theorem \[fonctionrho\] also holds. Indeed, as mentioned after the statement of Theorem \[fonctionrho\], the assumption (iii) holds if one shows that $B(z)\in\K(\H_0,\H)$ for each $z\in\C\setminus\R$. But, for any $(\varphi_-,\varphi_+)\in\H_0$, a direct calculation shows that $B(z)(\varphi_-,\varphi_+)=B_-(z)\varphi_-+B_+(z)\varphi_+$, with $$B_\pm(z):=(H-z)^{-1}\big\{[-\Delta,j_\pm]+j_\pm(V-v_\pm)\big\}(-\Delta+v_\pm-z)^{-1}
\in\K(\H).$$ So, one readily concludes that $B(z)\in\K(\H_0,\H)$.
Thus, one is only left with showing the assumption (ii) of Theorem \[fonctionrho\] and the $C^1(A)$-regularity of $H$. We first consider a short-range type perturbation. In such a case, with $A$ defined as above, we know it is enough to check the condition of Corollary \[compact\]. For that purpose, we assume the following stronger condition on $v:$ $$\label{set1}
\lim_{|x|\to\infty}|x|\big(v(x)-v_\pm\big)=0,$$ and observe that for each $(\varphi_-,\varphi_+)\in\S(\R)\oplus\S(\R)$ and $z\in\C\setminus\R$ we have the equality $$B(z)A_0(\varphi_-,\varphi_+)=B_-(z)D\varphi_-+B_+(z)D\varphi_+.$$ Then, taking into account the expressions for $B_-(z)$ and $B_+(z)$ as well as the above assumption on $v$, one proves easily that $\overline{B_\pm(z)D\upharpoonright\dom(D)}\in\K(\H)$, which implies . Collecting our results, we end up with:
Assume that $v\in C(\R;\R)$ satisfies , then the operator $H$ is of class $C^1(A)$ and one has $\widetilde\varrho_H^A\ge\widetilde\varrho_{H_0}^{A_0}$. In particular, $A$ is conjugate to $H$ on $\,\R\setminus\{\vg,\vd\}$.
We now consider a long-range type perturbation and thus show that the assumptions of Proposition \[ass2\_long\] hold with $A$ defined as above. For that purpose, we assume that $v\in C^1(\R;\R)$ and that $$\label{set2}
\lim_{|x|\to\infty}|x|\;\!v'(x)=0.$$ Then, a standard computation taking the inclusion $(H-z)^{-1}\D\subset\dom(A)$ into account shows that $H$ is of class $C^1(A)$ with $$\label{form_com}
[A,H]=\big[j(-i\nabla)\;\!\id\;\!j,-\Delta\big]-ij^2\;\!\id\;\!v'
+\frac i2\big[j^2,-\Delta\big],$$ where $\id$ is the function $\R\ni x\mapsto x\in\R$. Then, using and , one infers that $J[H_0,A_0]J^*-[H,A]$ belongs to $\K\big(\dom(H),\dom(H)^*\big)$. Furthermore, simple considerations show that $J$ extends to an element of $\B\big(\dom(H_0)^*,\dom(H)^*\big)$. These results, together with the ones already obtained, permit to apply Proposition \[ass2\_long\], and thus to get:
Assume that $v\in C^1(\R;\R)$ satisfies , then the operator $H$ is of class $C^1(A)$ and one has $\widetilde\varrho_H^A\ge\widetilde\varrho_{H_0}^{A_0}$. In particular, $A$ is conjugate to $H$ on $\,\R\setminus\{\vg,\vd\}$.
Completeness of the wave operators {#Sec_Comp}
==================================
One of the main goal in scattering theory is the proof of the completeness of the wave operators. In our setting, this amounts to show that the strong limits $$\label{principal}
W_\pm(H,H_0,J):=\slim_{t\to\pm\infty}\e^{itH}J\e^{-itH_0}P_{\rm ac}(H_0)$$ exist and have ranges equal to $\H_{\rm ac}(H)$. If the wave operators $W_\pm(H,H_0,J)$ are partial isometries with initial sets $\H_0^\pm$, this implies in particular that the scattering operator $$S:=W_+(H,H_0,J)^*\;\!W_-(H,H_0,J)$$ is well-defined and unitary from $\H_0^-$ to $\H_0^+$.
When defining the completeness of the wave operators, one usually requires that $\H_0^\pm=\H_{\rm ac}(H_0)$ (see for example [@BW83 Def. III.9.24] or [@Yaf92 Def. 2.3.1]). However, in applications it may happen that the ranges of $W_\pm(H,H_0,J)$ are equal to $\H_{\rm ac}(H)$ but that $\H_0^\pm\neq\H_{\rm ac}(H_0)$. Typically, this happens for multichannel type scattering processes. In such situations, the usual criteria for completeness, as [@BW83 Prop. III.9.40] or [@Yaf92 Thm. 2.3.6], cannot be applied. So, we present below a result about the completeness of the wave operators without assuming that $\H_0^\pm=\H_{\rm ac}(H_0)$. Its proof is inspired by [@Yaf92 Thm. 2.3.6].
\[comp\_serge\] Suppose that the wave operators defined in exist and are partial isometries with initial set projections $P_0^\pm$. If there exists $\widetilde J\in\B(\H,\H_0)$ such that $$\label{tralalere}
W_\pm\big(H_0,H,\widetilde J\big)
:=\slim_{t\to\pm\infty}\e^{itH_0}\widetilde J\e^{-itH}P_{\rm ac}(H)$$ exist and such that $$\label{froufrou}
\slim_{t\to\pm\infty}\big(J\widetilde J-1\big)\e^{-itH}P_{\rm ac}(H)=0,$$ then the equalities $\Ran\big(W_\pm(H,H_0,J)\big)=\H_{\rm ac}(H)$ hold. Conversely, if $\Ran\big(W_\pm(H,H_0,J)\big)=\H_{\rm ac}(H)$ and if there exists $\widetilde J\in\B(\H,\H_0)$ such that $$\label{trainversTokyo}
\slim_{t\to\pm\infty}\big(\widetilde JJ-1\big)\e^{-itH_0}P_0^\pm=0,$$ then $W_\pm\big(H_0,H,\widetilde J\big)$ exist and holds.
\(i) By using the chain rule for wave operators [@Yaf92 Thm. 2.1.7], we deduce from the definitions - that the limits $$W_\pm \big(H,H,J\widetilde J\big)
:=\slim_{t\to\pm\infty}\e^{itH}J\widetilde J\e^{-itH}P_{\rm ac}(H)$$ exist and satisfy $$\label{prod_wave}
W_\pm\big(H,H,J\widetilde J\big)= W_\pm(H,H_0,J)W_\pm\big(H_0,H,\widetilde J\big).$$ In consequence, the equality $$\slim_{t\to\pm\infty}\big(\e^{itH}J\widetilde J\e^{-itH}P_{\rm ac}(H)
-P_{\rm ac}(H)\big)=0,$$ which follow from , implies that $W_\pm\big(H,H,J\widetilde J\big)P_{\rm ac}(H)=P_{\rm ac}(H)$. This, together with and the equality $W_\pm\big(H_0,H,\widetilde J\big)=W_\pm\big(H_0,H,\widetilde J\big)P_{\rm ac}(H)$, gives $$W_\pm(H,H_0,J)W_\pm\big(H_0,H,\widetilde J\big)
=W_\pm\big(H,H,J\widetilde J\big)P_{\rm ac}(H)
=P_{\rm ac}(H),$$ which is equivalent to $$W_\pm\big(H_0,H,\widetilde J\big)^*W_\pm(H,H_0,J)^*=P_{\rm ac}(H).$$ This gives the inclusion $
\Ker\big(W_\pm(H,H_0,J)^*\big)\subset\H_{\rm ac}(H)^\perp
$, which together with the fact that the range of a partial isometry is closed imply that $$\H=\Ran\big(W_\pm(H,H_0,J)\big)\oplus\Ker\big(W_\pm(H,H_0,J)^*\big)
\subset\H_{\rm ac}(H)\oplus\H_{\rm ac}(H)^\perp
=\H.$$ So, one must have $\Ran\big(W_\pm(H,H_0,J)\big)=\H_{\rm ac}(H)$, and the first claim is proved.
\(ii) Conversely, consider $\psi\in\H_{\rm ac}(H)$. Then we know from the hypothesis $\Ran\big(W_\pm(H,H_0,J)\big)=\H_{\rm ac}(H)$ that there exist $\psi_\pm\in P_0^\pm\H_0$ such that $$\label{reserved}
\lim_{t\to\pm\infty}\big\|\e^{-itH}\psi-J\e^{-itH_0}P_0^\pm\psi_\pm\big\|_\H=0.$$ Together with , this implies that the norm $$\begin{aligned}
&\big\|\e^{itH_0}\widetilde J\e^{-itH}\psi-P_0^\pm\psi_\pm \big\|_{\H_0}\\
&\le\big\|\e^{itH_0}\widetilde J
\big(\e^{-itH}\psi-J\e^{-itH_0}P_0^\pm\psi_\pm\big)\big\|_{\H_0}
+\big\|\e^{itH_0}\widetilde JJ\e^{-itH_0}P_0^\pm\psi_\pm
-P_0^\pm\psi_\pm\big\|_{\H_0}\\
&\le{\rm Const.}\;\!\big\|\e^{-itH}\psi-J\e^{-itH_0}P_0^\pm\psi_\pm\big\|_\H
+\big\|\big(\widetilde J J-1\big)\e^{-itH_0}P_0^\pm \psi_\pm \big\|_{\H_0}\end{aligned}$$ converges to $0$ as $t\to\pm\infty$, showing that the wave operators exist.
For the relation , observe first that gives $$\slim_{t\to\pm\infty}\big(J\widetilde J-1\big)J\e^{-itH_0}P_0^\pm
=\slim_{t\to\pm\infty}J\big(\widetilde JJ-1\big)\e^{-itH_0}P_0^\pm
=0.$$ Together with , this implies that the norm $$\begin{aligned}
&\big\|\big(J\widetilde J-1\big)\e^{-itH}\psi\big\|_\H\\
&\le\big\|\big(J\widetilde J-1\big)\big(J\e^{-itH_0}P_0^\pm\psi_\pm
-\e^{-itH}\psi\big)\big\|_\H
+\big\|\big(J\widetilde J-1\big)J\e^{-itH_0}P_0^\pm\psi_\pm\big\|_\H\\
&\le{\rm Const.}\;\!\big\|\e^{-itH}\psi-J\e^{-itH_0}P_0^\pm\psi_\pm\big\|_\H
+\big\|\big(J\widetilde J-1\big)J\e^{-itH_0}P_0^\pm\psi_\pm\big\|_\H\end{aligned}$$ converges to $0$ as $t \to \pm \infty$, showing that also holds.
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[^1]: On leave from Université de Lyon; Université Lyon 1; CNRS, UMR5208, Institut Camille Jordan, 43 blvd du 11 novembre 1918, F-69622 Villeurbanne-Cedex, France. Supported by the Japan Society for the Promotion of Science (JSPS) and by “Grants-in-Aid for scientific Research”.
[^2]: Supported by the Fondecyt Grant 1090008 and by the Iniciativa Cientifica Milenio ICM P07-027-F “Mathematical Theory of Quantum and Classical Magnetic Systems”.
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abstract: 'The response of semiconductor materials to external magnetic fields is a reliable approach to probe intrinsic electronic and spin-dependent properties. In this study, we investigate the common Zeeman splitting features of novel wurtzite materials, namely InP, InAs, and GaAs. We present values for the effective g-factors of different energy bands and show that spin-orbit coupling effects, responsible for the spin splittings, also have noticeable contributions to the g-factors. Within the Landau level picture, we show that the nonlinear Zeeman splitting recently explained in magneto photoluminescence experiments for InP nanowires by Tedeschi et al. \[arXiv:1811.04922 (2018)\] are also present in InAs, GaAs and even in the conventional GaN. Such nonlinear features stem from the peculiar coupling of the A and B valence bands, as a consequence of the interplay between the wurtzite crystal symmetry and the breaking of time-reversal symmetry by the external magnetic field. Moreover, we develop an analytical model to describe the experimental nonlinear Zeeman splitting and apply it to InP and GaAs data. Extrapolating our fitted results, we found that the Zeeman splitting of InP reaches a maximum value, which is a prediction that could be probed at higher magnetic fields.'
author:
- 'Paulo E. Faria Junior'
- Davide Tedeschi
- Marta De Luca
- Benedikt Scharf
- Antonio Polimeni
- Jaroslav Fabian
title: 'Common nonlinear features and spin-orbit coupling effects in the Zeeman splitting of novel wurtzite materials'
---
Introduction {#sec:intro}
============
Novel III-V semiconductor compounds with wurtzite (WZ) crystal structure, such as InP[@DeLuca2014NL; @Tedeschi2016NL; @DeLuca2017APR], InAs[@Moller2012Nanotech; @Rota2016NL] and GaAs[@Furthmeier2016NatComm; @DeLuca2017NL], can nowadays be synthesized as nanowhiskers or nanowires (NWs)[@Dubrovskii2009Semiconductors; @Carofff2009NatNano] with large diameters. In contrast to the widely studied zinc-blende (ZB) phase[@Vurgaftman2001JAP] – the most stable crystal structure of non-nitride III-V compounds – there are still many unknown, or at least not completely understood, properties of these WZ materials, especially regarding spin-dependent phenomena[@Zutic2004RMP]. For instance, spin-orbit coupling (SOC) parameters and effective g-factors in WZ NWs control the physics behind the exotic Majorana bound states in semiconductor/superconductor setups[@Lutchyn2010PRL; @Oreg2010PRL; @Das2012NatPhys; @Albrecht2016Nature], spin-laser operation[@Chen2014NatNano; @FariaJunior2017PRB], spin-relaxation mechanisms[@Kammermeier2018PRB; @Dirnberger2018] and can be drastically modified under lateral quantum confinement[@Winkler2017PRL; @Campos2018PRB].
One of the possibilities to probe the intrinsic spin properties of a semiconductor system is to investigate their response under external magnetic fields, for instance, coupled to optical excitation in magneto photoluminescence (PL) experiments. Recent studies investigated the Zeeman splitting (ZS) from magneto PL[@DeLuca2013ACSNano; @DeLuca2014NL; @DeLuca2017NL; @Tedeschi2018] and extracted effective g-factors using the conventional linear dispersion of the WZ ZS[@Cho1976PRB; @Venghaus1977PRB]. Despite the successful description of the ZS for the magnetic field oriented [*perpendicular*]{} to the NW axis (with \[0001\] growth direction), this theoretical modeling has two main disadvantages for the magnetic field oriented [*parallel*]{} the NW axis: (i) the effective g-factors of electrons and holes cannot be probed separately because of the optical transitions, and (ii) this theory does not account for the unconventional nonlinear ZS observed. Such nonlinear features have recently been observed in quantum dots[@Jovanov2012PRB; @Oberli2012PRB] and quantum wells[@Kotlyar2001PRB; @Durnev2012PhysicaE], i. e., semiconductor systems with strong quantum confinement.
On the other hand, the case of WZ NWs is quite different since the NWs used in these recent experiments have large diameter and effectively behave as a bulk material[@MishraAPL2007; @FariaJunior2012JAP; @Dacal2016SciRep; @DeLuca2017APR] with negligible lateral quantum confinement. Particularly for InP WZ, it was unambiguously shown in the study of Tedeschi et al.[@Tedeschi2018] that these nonlinear features originate from the peculiar coupling of Landau levels (LLs) from different energy states in the valence band, specifically between A and B bands. Although this nonlinear ZS has also been observed in InGaAs[@DeLuca2013ACSNano] and GaAs[@DeLuca2017NL], it remains to be shown that indeed these nonlinear features have the same origin and could be described in a compact analytical way. Furthermore, InAs WZ NWs have been investigated by recent transport experiments[@Das2012NatPhys; @Albrecht2016Nature; @Vaitiekenas2018PRL; @Iorio2018arXiv]. However, to the best of our knowledge, no theoretical attempt has ever been made to compute the effective g-factors in such material.
In this paper, we analyze the ZS of novel III-V WZ materials, namely InP, InAs and GaAs. We provide the values for effective g-factors of different energy bands and highlight important contributions due to SOC effects originating from the interband SOC interaction. Turning to the LL physics, we apply the theoretical approach presented in the study of Tedeschi et al.[@Tedeschi2018] to show that the nonlinear ZS arises solely from the mixing within the valence band and it is indeed a common feature present in the studied materials. Based on this common mechanism responsible for the nonlinear features, we developed an analytical model that reliably fits the available experimental data, especially for InP WZ. We then extrapolate our fitted results and show that the nonlinear feature acts as a limiting effect to the maximum value of the ZS for InP. Under higher magnetic fields, such features could be observed experimentally in order to test the limits of our suggested model.
We organize this paper as follows: In Sec. \[sec:gfactor\] we discuss the effective g-factors calculations and the role of SOC effects. In Sec. \[sec:LLs\] we show the common nonlinear features arising in the valence band from the LL coupling. The effective analytical model for the nonlinear ZS is presented in Sec. \[sec:modelZS\] and we draw our conclusions in Sec. \[sec:conclusions\]. In the Appendix, we discuss the LL spectra for ZB materials.
Effective g-factors and spin-orbit coupling effects {#sec:gfactor}
===================================================
In order to evaluate the effective g-factors within the $k \cdot p$ framework, we use the standard perturbative approach[@Roth1959PR; @Hermann1977PRB; @LewYanVoon2009] that accounts for the coupling between different energy bands. Here we focus on the energy bands around the band gap at the $\Gamma$-point of WZ crystals, namely the conduction band (CB) and the top three valence bands, labeled A, B and C from highest to lowest energy. In Fig. \[fig:WZ\](a) we depict the bulk WZ band structure and identify the labels for the different energy bands. Within this $k \cdot p$ perturbative approach, each band (two-fold degenerate) is described by an effective Zeeman term of the form $$H_{\text{ZS}}(B_{\alpha})=\frac{\mu_{B}}{2}B_{\alpha}g_{\alpha}\tau_{\alpha},\;\alpha=x,y,z \, ,$$ in which the matrices $\tau_\alpha$ are the Pauli matrices for the two-fold degenerate $\Gamma$-point states and the effective g-factor $g_\alpha$ is obtained after evaluating $$g_{\alpha}\tau^{nm}_\alpha=g_{0}\sigma^{nm}_{\alpha} -i\frac{2m_{0}}{\hbar^{2}}\underset{l\neq n,m}{\sum}\frac{\Pi_{\beta}^{nl}\Pi_{\gamma}^{lm}-\Pi_{\gamma}^{nl}\Pi_{\beta}^{lm}}{E_{n}-E_{l}}\,,
\label{eq:g}$$ with $g_0$ ($m_0$) being the bare electron g-factor (mass), $n,m$ being the states of a specific energy band at $\Gamma$-point, $E_{n(l)}$ are the energy values at $\Gamma$-point and $\left\{ \alpha,\beta,\gamma\right\} =\left\{ x,y,z \right\}$ (or cyclic permutations). The matrix elements of the Pauli matrices acting on the spin 1/2 are given by $\sigma^{ab}_{\alpha} = \left\langle a \left|\sigma_{\alpha}\right| b \right\rangle$ and the matrix elements for the $\vec{\Pi}$ operator are given by $\Pi^{ab}_{\alpha}=\left\langle a\left|\Pi_\alpha\right|b\right\rangle$, with the $\vec{\Pi}$ operator written as $$\vec{\Pi}=\frac{\hbar}{m_{0}}\vec{p}+\frac{\hbar^{2}}{4m_{0}^{2}c^{2}}\left[\vec{\sigma}\times\vec{\nabla}V(\vec{r})\right] \, ,
\label{eq:Pi}$$ in which the second term describes the SOC contribution. In Fig. \[fig:WZ\](b) we show the direction of magnetic fields with respect to the WZ crystal structure.
To compute the g-factors in Eq. (\[eq:g\]) we must specify a particular $k \cdot p$ Hamiltonian that contains the coupling among the different energy bands. For the CB, A, B and C bands including spin, the most general 8$\times$8 $k \cdot p$ Hamiltonian that includes both orbital \[1st term in Eq. (\[eq:Pi\])\] and SOC \[2nd term in Eq. (\[eq:Pi\])\] terms is given in Ref. \[\]. It is convenient to notice that the matrix elements $\Pi^{ab}_{\alpha}$ can be easily obtained by looking at the Hamiltonian terms $H_{kp}^{(1)}$ and $H_{kSO}^{(1)}$ (shown in the Appendix B of Ref. \[\]). Furthermore, an additional parameter present in the Hamiltonian is the k-independent SOC ($\sim\left[\vec{\nabla}V(\vec{r})\times\vec{p}\right]\cdot\vec{\sigma}$) between conduction and valence bands denoted by $\Delta_4$ (sometimes also called $\Delta_{sz}$[@Fu2008JAP]). The inclusion of $\Delta_4$ prevents the analytical diagonalization of the Hamiltonian at the $\Gamma$-point.
![(Color online) (a) Schematic band structure for bulk WZ crystals around the $\Gamma$-point indicating the different energy bands: CB, A, B and C (from top to bottom). The energies at $\Gamma$-point are indicated by the arrows. (b) Scheme of the magnetic field configurations [*parallel*]{} (B$_\text{z}$) and [*perpendicular*]{} (B$_\text{x}$) to the NW axis in typical magneto PL experiments. The laser excitation and the collected PL signal are parallel to the NW axis. The inset shows the WZ crystal structure with the orientation of the c-axis in \[0001\] direction.[]{data-label="fig:WZ"}](fig1.eps)
The most straightforward way to calculate the g-factors is to evaluate Eq. (\[eq:g\]) numerically, especially if the Hamiltonian does not allow analytical solutions. However, to unambiguously identify the contribution of SOC, we present here the analytical expressions for CB and A band g-factors assuming $\Delta_4 = 0$, that allows the analytical diagonalization of the WZ Hamiltonian at the $\Gamma$-point (see Sec. II.B of Ref. \[\] for instance). Rewriting the total g-factor of Eq. (\[eq:g\]) as $g_{\alpha}=g_{0}+\frac{2m_{0}}{\hbar^{2}}\left(L_{\alpha}+\lambda_{\alpha}\right)$ we can identify the orbital contributions in $L_{\alpha}$ and the SOC effects in $\lambda_{\alpha}$. These terms for CB and A bands read: $$\begin{aligned}
\label{eq:gCBx}
L_{x}^{\text{CB}} & =\sqrt{2}P_{1}P_{2}ab\left(\frac{1}{\Delta_{\text{C}}}-\frac{1}{\Delta_{\text{B}}}\right) \, ,\\
\lambda_{x}^{\text{CB}} & =\frac{1}{\Delta_{\text{B}}}\bigg[2a\beta_{1}\left(a\beta_{1}-\sqrt{2}b\beta_{2}\right)\nonumber \\
& \quad\quad\quad+bP_{1}\left(2b\beta_{2}-\sqrt{2}a\beta_{1}\right)+2a^{2}\beta_{1}P_{2}\bigg]\nonumber \\
& +\frac{1}{\Delta_{\text{C}}}\bigg[2b\beta_{1}\left(b\beta_{1}+\sqrt{2}a\beta_{2}\right)\nonumber \\
& \quad\quad\quad+aP_{1}\left(2a\beta_{2}+\sqrt{2}b\beta_{1}\right)+2b^{2}\beta_{1}P_{2}\bigg] \, , \nonumber\end{aligned}$$
$$\begin{aligned}
\label{eq:gCBz}
L_{z}^{\text{CB}} & =P_{2}^{2}\left(\frac{b^{2}}{\Delta_{\text{C}}}+\frac{a^{2}}{\Delta_{\text{B}}}-\frac{1}{\Delta_{\text{A}}}\right) \, , \\
\lambda_{z}^{\text{CB}} & =\frac{1}{\Delta_{\text{A}}}\beta_{1}\left(2P_{2}-\beta_{1}\right)\nonumber \\
& +\frac{1}{\Delta_{\text{B}}}\left[\left(a\beta_{1}-\sqrt{2}b\beta_{2}\right)^{2}+2aP_{2}\left(a\beta_{1}-\sqrt{2}b\beta_{2}\right)\right] \nonumber \\
& +\frac{1}{\Delta_{\text{C}}}\left[\left(b\beta_{1}+\sqrt{2}a\beta_{2}\right)^{2}+2bP_{2}\left(b\beta_{1}+\sqrt{2}a\beta_{2}\right)\right] \, , \nonumber\end{aligned}$$
$$\begin{aligned}
\label{eq:gAz}
L_{z}^{A} & = -\frac{P_{2}^{2}}{\Delta_{\text{A}}}+2A_{7}^{2}\left(\frac{b^{2}}{\Delta_{\text{AB}}}+\frac{a^{2}}{\Delta_{\text{AC}}}\right) \, , \\
\lambda_{z}^{A} & =-\frac{1}{\Delta_{\text{A}}}\left(\beta_{1}^{2}-2\beta_{1}P_{2}\right) \nonumber \\
& +\frac{1}{\Delta_{\text{AB}}}\left[\left(b\alpha_{1}+\sqrt{2}a\alpha_{2}\right)^{2}-2bA_{7}\left(2a\alpha_{2}+\sqrt{2}b\alpha_{1}\right)\right] \nonumber \\
& +\frac{1}{\Delta_{\text{AC}}}\left[\left(a\alpha_{1}-\sqrt{2}b\alpha_{2}\right)^{2}+2aA_{7}\left(2b\alpha_{2}-\sqrt{2}a\alpha_{1}\right)\right] \, , \nonumber\end{aligned}$$
with the energy differences given by $\Delta_{\text{A}}=E_{\text{CB}}-E_{\text{A}}$, $\Delta_{\text{B}}=E_{\text{CB}}-E_{\text{B}}$, $\Delta_{\text{C}}=E_{\text{CB}}-E_{\text{C}}$, $\Delta_{\text{AB}}=E_{\text{A}}-E_{\text{B}}$ and $\Delta_{\text{AC}}=E_{\text{A}}-E_{\text{C}}$. The values $E_{\text{CB}}$, $E_{\text{A}}$, $E_{\text{B}}$ and $E_{\text{C}}$ are the energies at $\Gamma$-point for the bands considered, indicated in Fig. \[fig:WZ\](a). The $a$ and $b$ coefficients read $a = \delta/\sqrt{\delta^{2}+2(\Delta_{3})^{2}}$ and $b = \sqrt{2}\Delta_{3}/\sqrt{\delta^{2}+2(\Delta_{3})^{2}}$ with $\delta=(\Delta_{1}-\Delta_{2})/2+\sqrt{(\Delta_{1}-\Delta_{2})^{2}/4+2(\Delta_{3})^{2}}$. The energy parameters $\Delta_1$ and $\Delta_{2,3}$ represent the crystal field splitting energy and the valence band SOC energies in WZ, respectively. The parameters $P_1$ and $P_2$ couple conduction and valence bands via the linear momentum operator, $A_7$ is the intra valence band coupling also mediated by $\vec{p}$, $\beta_1$ and $\beta_2$ are SOC terms between conduction and valence bands while $\alpha_1$ and $\alpha_2$ are SOC terms within the valence band only. For the precise definition of these couplings, please refer to the Appendix B of Ref. \[\].
The SOC corrections to the g-factors, shown in Eqs. (\[eq:gCBx\])-(\[eq:gAz\]), take into account the same parameters that control the spin splitting of the energy bands (see for instance Eq. (10) in Ref.\[\] for the CB spin splitting parameters). Therefore, since the spin splittings of the energy bands are different (either in magnitude or k-dependence), so are the SOC corrections to the g-factors. We point out that $g_x^{\text{A}}$ is not shown simply because it is zero due to the symmetry of the A bands (they do not couple via the $\Pi_z$ operator to any other band and their different spin projections also do not couple by $\sigma_x$). Furthermore, we emphasize that by removing the SOC contribution (setting $\lambda_{\alpha}=0$) we recover the known result for the conduction band presented by Hermann and Weisbuch[@Hermann1977PRB].
[1.0]{}[@ llrrrr]{} & & & & band & $g_{x}$ & $g_{z}$ & $g_{x}$ & $g_{z}$ InP$^a$ & CB & 1.72 & 1.81 & 1.29 (25) & 1.61 (11) & A & 0.0 & -3.30 & 0.0 & -3.05 (8) & B & -3.74 & 5.35 & -3.94 (5) & 5.12 (4) & C & 5.46 & 0.24 & 5.10 (7) & 0.47 (97) InAs$^a$ & CB & -5.49 & -5.33 & -6.82 (24) & -6.23 (17) & A & 0.0 & -23.71 & 0.0 & -22.90 (3) & B & -19.69 & -10.81 & -19.06 (3.1) & -8.97 (17) & C & 14.20 & 7.57 & 14.07 (1) & 7.70 (2) GaAs$^b$ & CB & 0.33 & 0.46 & & & A & 0.0 & -10.19 & & & B & -9.17 & 7.22 & & & C & 9.50 & -3.43 & &
\[tab:gfactors\]
Now we turn to the calculated values of the effective g-factors for InP, InAs, and GaAs WZ. For InP and InAs we used the parameters (which contain SOC effects) available in Ref. \[\] and for GaAs we used the parameters (without any SOC effects) from Ref. \[\]. Please see footnote \[\] for a brief information on the $k \cdot p$ parameters used. We emphasize here that the SOC effects we refer to in the $k \cdot p$ Hamiltonians are related to the terms that contribute to the g-factor as shown in Eqs. (\[eq:gCBx\])-(\[eq:gAz\]) and the additional interband k-independent SOC term $\Delta_4$. Furthermore, in both $k \cdot p$ models[@FariaJunior2016PRB; @Cheiw2011PRB] the usual SOC in the valence band is included via the parameters $\Delta_2$ and $\Delta_3$. In Table \[tab:gfactors\] we show the calculated g-factors in the absence of SOC and with SOC (if the $k \cdot p$ model allows). As a general trend, $g_x = g_y \neq g_z$ (highlighting the anisotropy of the WZ structure), and the valence bands have larger g-factors (in absolute value) than the CB. Taking into account the SOC effects, only available for InP and InAs, we notice that their correction to the g-factor values are, in general, not negligible and with values within the typical experimental precision. For instance, the influence of SOC can reach contributions of $\sim 25 \%$ in CB g-factors and it is larger for $g_x^{\text{CB}}$ than $g_z^{\text{CB}}$. The reason for this larger SOC effect in $g_x^{\text{CB}}$ is due to the contribution of the parameter $P_1$ in $\lambda_x^{\text{CB}}$ that is absent in $\lambda_z^{\text{CB}}$ (and additionally, $P_1 > P_2$). It is also worth mentioning the g-factors for ZB phase. Restricting ourselves to the CB, the most commonly investigated case, the ZB g-factor is $g^*=1.26$ for InP, $g^*=-14.9$ for InAs and $g^*=-0.44$ for GaAs (with values taken from Ref. \[\]). We notice that indeed the effective g-factor values are quite different for ZB and WZ crystal phases.
Let us now compare our theoretical g-factors values to experiments. In Table \[tab:gcomp\] we compare our calculated WZ g-factors with the available magneto PL experimental data for InP and GaAs NWs (with diameters large enough to be considered a bulk system). Let us first discuss the InP case. For both $g_x^{\text{CB}}$ and $g_z^{\text{CB}} - g_z^{\text{A}}$, our calculated values including the SOC contributions provide an excellent agreement to the reported experimental values. We also point out the apparent inconsistency between the two experimental g-factors by showing the range of magnetic field used in the fitting. Since for magnetic fields along the z direction the ZS is nonlinear, the larger the range used in the fitting the smaller the g-factors will be in order to account for the sublinear features. Therefore, we emphasize that experimentally determined g-factors should be fitted only at the limit of magnetic field values where the linear regime holds. For GaAs, although the comparison for $g_x^{\text{CB}}$ looks reasonable, the theoretical value obtained for $g_z^{\text{CB}} - g_z^{\text{A}}$ is nearly twice as large as the experimental value. This clearly indicates that the $k \cdot p$ parameters for GaAs are not completely consistent and further theoretical efforts are required to build a more realistic model.
[1]{}[@ llrr]{} & & $g_{x}^{\text{CB}}$ & $g_{z}^{\text{CB}}-g_{z}^{\text{A}}$ InP & expt. Ref. \[\] & $1.3^{a}$ & $4.4^{a}$ & expt. Ref. \[\] & $1.4^{b}$ & $3.4^{b}$ & this work, with SOC$^{c}$ & 1.29 & 4.66 & this work, no SOC$^{c}$ & 1.72 & 5.26 GaAs & expt. & $0.28^{d}$ & $5.4^{e}$ & this work, no SOC$^{f}$ & 0.33 & 10.83
\[tab:gcomp\]
Although there is, to the best of our knowledge, no magneto PL reported in pure InAs WZ NWs, it is important to mention that this material has recently been investigated in several transport experiments[@Das2012NatPhys; @Albrecht2016Nature; @Vaitiekenas2018PRL; @Iorio2018arXiv]. Specifically, the conductance experiments by Vaitiekėnas et al.[@Vaitiekenas2018PRL] using 100 nm InAs WZ NWs showed that for negative gate voltages the effective g-factor of CB electrons saturates to $|g^*| \sim 5$. This is significantly different from the bulk InAs ZB value of $g^*=-14.9$ but much closer to our predicted value for InAs WZ of $g_z^\text{CB}=-6.23(-5.33)$ with (without) SOC effects. Furthermore, the recent theoretical analysis of ZB NWs by Winkler et al.[@Winkler2017PRL] showed that orbital effects play a large role in the effective g-factors of different subbands. However the lowest subband still retains much of the bulk information, specially at 100 nm (see for instance Figs.(b),(d) in Ref. \[\]). Although our theoretical value for the InAs WZ g-factor seems consistent with recent experiments and, more importantly, it is substantially different than the ZB value, further investigations that account for the electrostatic environment and quantum confinement in WZ NWs are still required, as pointed out in the conclusions of Ref. \[\].
Nonlinear features in Landau levels {#sec:LLs}
===================================
Following the theoretical approach discussed in Ref. \[\] to model the nonlinear ZS of InP WZ, we show in this section that indeed such nonlinear features are a common trend and also appear in the LLs of InAs and GaAs WZ. To calculate the LLs, we use the general description of an external magnetic field within the $k \cdot p$ framework which considers the envelope function approximation (EFA), combined with the minimal coupling and the Zeeman term[@LewYanVoon2009; @Pryor2006PRL; @vanBree2012PRB]. The mathematical procedure of the EFA leads to the general Hamiltonian $$H = H_{\textrm{bulk}}\!\left[\vec{k}\rightarrow-i\vec{\nabla}+\frac{e}{\hbar}\vec{A}(\vec{r})\right]+g_{0}\frac{\mu_{B}}{2}\vec{\Sigma}\cdot\vec{B} \, ,
\label{eq:HkpBmag}$$ in which $H_{\textrm{bulk}}(\vec{k})$ is the $k \cdot p$ bulk Hamiltonian, the replacement $\vec{k}\rightarrow-i\vec{\nabla}+\frac{e}{\hbar}\vec{A}(\vec{r})$ takes into account the EFA procedure, $g_0$ is the bare electron g-factor and $\vec{\Sigma}$ is a vector of the Pauli matrices describing the Zeeman term for the spins of the bulk Bloch functions[@Pryor2006PRL; @vanBree2012PRB]. By choosing the vector potential with a single spatial dependence, we can solve the Hamiltonian (\[eq:HkpBmag\]) numerically by introducing the finite differences approach[@Chuang1997SST], similar to a quantum well treatment. The solution provides the LL spectra of the system, with energies denoted by $E_{\lambda}(k_{B},k_{A})$ and wave functions $$\psi_{\lambda,k_{B},k_{A}}(\vec{r})=\frac{e^{i\left(k_{B}r_{B}+k_{A}r_{A}\right)}}{\sqrt{\Omega}}\sum_{l}f_{\lambda,k_{B},k_{A},l}(\rho)\,u_{l}(\vec{r}) \, ,
\label{eq:psi_sp}$$ in which $\lambda$ is the LL label, $f_{\lambda,k_{B},k_{A},l}(\rho)$ is the envelope function, the summation in $l$ runs over the bulk basis states denoted by $u_{l}(\vec{r})$, $\Omega$ is the area of the system perpendicular to the confinement direction, $k_B$ is the wave vector parallel to the magnetic field, $k_A$ is parallel to the vector potential and the spatial dependence of the vector potential is denoted by the coordinate $\rho$. For the two directions of magnetic field investigated here \[indicated in Fig. \[fig:WZ\](b)\] we have $\vec{B} = B \hat{x} \Rightarrow \vec{A} = B y \hat{z}, k_B = k_x, k_A = k_z, \rho = y$ and $\vec{B} = B \hat{z} \Rightarrow \vec{A} = B x \hat{y}, k_B = k_z, k_A = k_y, \rho = x$. To simplify the notation we use $\vec{B} = B \hat{x}$ as B$_\text{x}$ and $\vec{B} = B \hat{z}$ as B$_\text{z}$ in the remainder of the paper. We note that in experimental papers[@DeLuca2013ACSNano; @DeLuca2014NL; @DeLuca2017NL; @Tedeschi2018], B$_\text{x}$ and B$_\text{z}$ are typically called Voigt and Faraday configurations, respectively. For the numerical implementation of the Hamiltonian (\[eq:HkpBmag\]) we considered the system to have a size of $L=200 \; \text{nm}$ with 401 discretization points (with approximately 1 point every 0.5 nm). For InP and InAs we used the bulk 8$\times$8 $k \cdot p$ model from Ref. \[\] and for GaAs we used the 6$\times$6 $k \cdot p$ model from Ref. \[\].
![(Color online) Calculated Landau level spectra for the A band at $k_B = k_A = 0$ as function of the magnetic field for (a) InP, (b) InAs and (c) GaAs. The color code indicates the contribution of the B valence band to the total state. The upper (lower) branch of the topmost Landau level is indicated by the 1+ (1-). (d) Zeeman splitting for the topmost Landau level. We denote with positive (negative) values in the x-axis the magnetic field configuration B$_\text{z}$ along z (B$_\text{x}$ along x) direction.[]{data-label="fig:LLs"}](fig2.eps)
![(Color online) Probability density of the envelope functions at 30 T for the first LL (indicated by the labels 1+ and 1- in Fig. \[fig:LLs\]) along B$_\text{x}$ and B$_\text{z}$ for (a-d) InP, (e-h) InAs and (i-l) GaAs. The B band envelope functions are multiplied by a factor of 8, as indicated in the legend. The spin notation for the energy bands is identified with respect to the leading contribution of the bulk basis states[@note:basis].[]{data-label="fig:probdens"}](fig3.eps)
Due to the coupling of the A and B bulk energy bands induced by the external magnetic field, the valence band LLs show a markedly nonlinear behavior[@Tedeschi2018]. This common feature can be seen in Figs. \[fig:LLs\](a)-(c) for InP, InAs and GaAs, respectively. We notice the clear nonlinear features in the different LLs whenever the B band mixing is present (indicated by the color code). Although for the B$_\text{x}$ configuration this mixing is drastically reduced, partially because of the zero g-factor of the A band, the A-B mixing is indeed responsible for the slight nonzero ZS observed. As revealed in Ref. \[\], the topmost LL (with branches indicated by 1+ and 1- in Fig. \[fig:LLs\]) provides the main contribution to the excitonic effects and therefore the nonlinear features in the magneto PL for the B$_\text{z}$ configuration originate from the mixing of A and B valence bands. To highlight the nonlinear features, we show in Fig. \[fig:LLs\](d) the ZS for the topmost LL in InP (solid lines), InAs (dashed lines) and GaAs (short dashed lines). We point out here that although the 6$\times$6 Hamiltonian for GaAs WZ[@Cheiw2011PRB] is not sufficient to describe the linear Zeeman splitting correctly, since the $P_1$ and $P_2$ parameters are not included, the nonlinear features in the LLs are clearly visible. In fact, this is an additional support to the fact that these nonlinear features are beyond the linear g-factor approach, which is mainly ruled by the $P_1$ and $P_2$ parameters as shown in Eqs. (\[eq:gCBx\])-(\[eq:gAz\]).
In order to complete our analysis of the common nonlinear features of the valence band, we now discuss how the LL coupling manifests in the envelope functions. We show in Fig. \[fig:probdens\] the probability density of the envelope functions for the topmost LL (the 1+ and 1- branches shown in Fig. \[fig:LLs\]) in both magnetic field configurations at 30 T. Due to the interplay of the WZ symmetry and the external magnetic field, the mixing of A and B bands has a peculiar form that couples envelope functions with different numbers of nodes, i. e., 0 node for A bands and 2 nodes for B bands (further details on this coupling can be found in Ref. \[\] and its Supplemental Material). In fact, due to strong SOC in InAs we also notice the contribution of B states with 1 node, see Figs. \[fig:probdens\](f) and \[fig:probdens\](h). Moreover, for the B$_\text{z}$ configuration the coupling between envelope functions is spin dependent, which means that the nonlinear feature is associated to one specific type of circular polarization (due to the conservation of angular momentum, spins in conduction and valence band define the allowed transitions of circularly polarized light[@FariaJunior2015PRB; @FariaJunior2017PRB]). Indeed, this is exactly the case observed in recent magneto PL experiments in InP WZ by Tedeschi et al.[@Tedeschi2018] that identified a strong nonlinear feature arising for a specific circular polarization of the PL spectra. On the other hand for the B$_\text{x}$ configuration, both spin components contribute equally and, as a consequence, the output light cannot be resolved in different circular polarizations.
GaN wurtzite
------------
As a well established WZ compound in the family of the nitrides and recognized in the 2014 Nobel Prize in Physics for the efficient blue light emitting diodes[@Akasaki2015RMP; @Amano2015RMP; @Nakamura2015RMP], we discuss here the case of GaN. We focus on the LL spectra of the valence bands, since a detailed discussion the effective g-factors in WZ GaN has been performed by Rodina and Meyer[@Rodina2001PRBb]. In Fig. \[fig:GaN\](a) we show the LL spectra for the A band of GaN. Due to the small SOC and crystal field energies of few meV in GaN, the mixing of A and B bands increases in comparison to the III-V WZ materials discussed above (notice the color code scale in Figs. \[fig:LLs\] and \[fig:GaN\]). For the ZS of the top most LL, shown in Fig. \[fig:GaN\](b), we notice that the nonlinear features are present for small values of magnetic field ($<5$ T) but the resulting ZS is $\sim$0.1 meV, which seems to be within the experimental error to be properly distinguished. Therefore, the overall behavior of the ZS can be modeled using a linear dispersion as indicated in the experimental study of Rodina et al.[@Rodina2001PRBa]. Finally, we show in Figs. \[fig:GaN\](c)-(f) the probability densities for the envelope functions of the LL branches 1+ and 1- at 30 T for the magnetic field along B$_\text{x}$ and B$_\text{z}$, and found that the same coupling mechanisms take place as discussed for Fig. \[fig:probdens\].
![(Color online) (a) Calculated Landau level spectra for the A band at $k_B = k_A = 0$ as function of the magnetic field for GaN. (b) Zeeman splitting for the topmost Landau level. The color code and axis notation follow Fig. \[fig:LLs\]. (c)-(f) Probability density of the envelope functions at 30 T for the first LL, indicated by the labels 1+ and 1- in Fig. \[fig:GaN\](a), along B$_\text{x}$ and B$_\text{z}$. The B band envelope functions are multiplied by a factor of 4, as indicated in the legend.[]{data-label="fig:GaN"}](fig4.eps)
Effective model for the nonlinear Zeeman splitting {#sec:modelZS}
==================================================
Based on the LL calculations, we showed that the physical mechanism behind the nonlinear ZS in the valence band is a common feature in WZ materials due to the mixing of A and B bands induced by magnetic field. It would be valuable to incorporate these features in an analytical expression that could be used to fit the experimental data beyond the linear ZS regime[@Cho1976PRB; @Venghaus1977PRB]. In order to capture the nonlinear effects present in the LL branch with spin up, we can restrict ourselves to the important contributions of the A-B mixing by using the basis set $\left\{ f_{0,\text{A}\Uparrow}\left|\textrm{A}\Uparrow\right\rangle, f_{2,\text{B}\Uparrow}\left|\textrm{B}\Uparrow\right\rangle \right\}$, in which the subindices 0 and 2 refer to the number of nodes in the envelope functions (see Fig. \[fig:probdens\]). We neglect here the minor contribution of the envelope function with 1 node, since it appears only in InAs due to strong SOC. Therefore, for the coupling between A$\Uparrow$ and B$\Uparrow$ LL branches, we can write the following $2 \times 2$ Hamiltonian: $$H=\left[\begin{array}{cc}
0 & 0\\
0 & E_{\text{B}}
\end{array}\right]+\frac{\mu_{B}}{2}B\left[\begin{array}{cc}
g_{\text{A}} & 0\\
0 & g_{\text{B}}
\end{array}\right]+B\left[\begin{array}{cc}
d_{\text{A}} & d_{\text{AB}}\\
d_{\text{AB}} & d_{\text{B}}
\end{array}\right] \, ,
\label{eq:Heff}$$ in which the first term indicates the energy separation between A and B valence bands in the bulk case (we set the energy of the A band to zero), the second term is the ZS due to the g-factor contribution and the third term is the coupling Hamiltonian that mixes A and B bands, which arises from the second-order $k \cdot p$ term[@Chuang1996PRB; @FariaJunior2016PRB]. Here, we assume these couplings to be parametrized by the variables $d_\text{A}$, $d_\text{B}$ and $d_\text{AB}$. Diagonalizing the Hamiltonian (\[eq:Heff\]) we find the energy for the A$\Uparrow$ branch as
$$E_{\text{A}\Uparrow}(B)=\frac{1}{2}\left[E_{\text{B}}+\frac{\mu_{B}}{2}Bg_{+}+d_{+}B+\sqrt{\left(E_{\text{B}}-\frac{\mu_{B}}{2}Bg_{-}-d_{-}B\right)^{2}+4d_{\text{AB}}^{2}B^{2}}\right] \, ,
\label{eq:Aup}$$
with $g_{\pm}=g_{\text{A}}\pm g_{\text{B}}$ and $d_{\pm}=d_{\text{A}}\pm d_{\text{B}}$. For the A$\Downarrow$ branch that does not couple to any other states, unlike A$\Uparrow$, we have simply the linear dependence in $B$, i. e., $$E_{\text{A}\Downarrow}(B)=-\frac{\mu_{B}}{2}Bg_{\text{A}}+Bd_{\text{A}} \, ,
\label{eq:Adw}$$ in which the first term is related to the ZS with opposite sign in the g-factor as compared to the A$\Uparrow$ branch and the second term is the energy shift of the LL branch with the same form as given in Eq. (\[eq:Heff\]).
In order to model the ZS obtained from the experimental PL peaks, we must take into account not only the ZS of the valence but also of the conduction band, since they are coupled via the optical transition. The total ZS can be written as the difference of the ZS of the conduction and valence bands, i. e., $\text{ZS}_\text{CB} - \text{ZS}_\text{A}$[@Cho1976PRB; @Venghaus1977PRB; @vanBree2012PRB]. For the conduction band we can use the linear g-factor ZS, $\text{ZS}_\text{CB} = \mu_BBg^{\text{CB}}_z$, and for the A band we use $\text{ZS}_\text{A} = E_{\text{A}\Uparrow}(B)-E_{\text{A}\Downarrow}(B)$ \[shown in Eqs. (\[eq:Aup\]) and (\[eq:Adw\])\]. Finally, the total ZS is given by $$\text{ZS}(B)=\mu_{B}B\left(g_{z}^{\text{CB}}-g_{z}^{\text{A}}\right)-\frac{1}{2}\left[\left(E_{\text{B}}-\frac{\mu_{B}}{2}Bg_{-}-d_{-}B\right)+\sqrt{\left(E_{\text{B}}-\frac{\mu_{B}}{2}Bg_{-}-d_{-}B\right)^{2}+4d_{\text{AB}}^{2}B^{2}}\right] \, ,
\label{eq:ZSBz}$$
in which the unknown parameters are only $d_{-}$ and $\left| d_{\text{AB}} \right|$ if we assume the values for the effective g-factors and the energy separation of A and B bands given by theory or found experimentally by other means. We emphasize that if we set the coupling parameter $d_{\text{AB}}$ to zero in Eq. (\[eq:ZSBz\]), we recover the linear ZS already established in the literature by Refs. \[\]. Furthermore, if we set the energy separation of A and B bands to zero ($E_{\text{B}} \rightarrow 0$, then all the terms in Eq. (\[eq:ZSBz\]) become linear in the magnetic field and the nonlinearities vanish. This condition would be equivalent to the case of ZB crystals that have degenerate heavy and light hole bands at the $\Gamma$-point and therefore would present a linear ZS (see for instance the experimental ZS of InP ZB in the Supplemental Material of Ref. \[\]).
![(Color online) Comparison between the fitting of Eq. (\[eq:ZSBz\]) to the experimental Zeeman splitting for (a) InP and (b) GaAs. We use the calculated g-factors of Table \[tab:gfactors\] and the theoretical values of $E_{\text{B}} = -35.4 \; \text{meV}$ for InP[@FariaJunior2016PRB] and $E_{\text{B}} = -99.4 \; \text{meV}$ for GaAs[@Cheiw2011PRB]. The fitting procedure provides $d_{-} = 0.16 \; \text{meV/T}$, $|d_{\text{AB}}| = 0.43 \; \text{meV/T}$ for InP and $d_{-} = 17.63 \; \text{meV/T}$, $|d_{\text{AB}}| = 2.82 \; \text{meV/T}$ for GaAs. The experimental data for InP is taken from Ref. \[\] and for GaAs from Ref. \[\][]{data-label="fig:effZS"}](fig5.eps)
Applying the effective analytical ZS of Eq. (\[eq:ZSBz\]) to the magneto PL data of InP[@Tedeschi2018] and GaAs[@DeLuca2017NL], we show in Fig. \[fig:effZS\] that this model successfully captures the experimental trends, particularly for InP WZ. In the fitting, we assumed the g-factors and energy separations to be known from theory and obtained the values for $d_{-}$ and $|d_{\text{AB}}|$ (given in the caption of Fig. \[fig:effZS\]). For GaAs we notice that the fitted parameters $d_{-}$ and $|d_{\text{AB}}|$ are nearly one order of magnitude larger than the values obtained for InP. We assign this feature to the overestimation of the GaAs g-factors in comparison with the experimental data in the linear regime, shown in Table \[tab:gcomp\]. We emphasize that it is beyond the scope of this study to provide reliable interband couplings of GaAs WZ since additional theoretical efforts are required, such as [*ab initio*]{} calculations with the correct conduction band ordering and a proper fitting of the $k \cdot p$ parameters, possibly including the SOC effects. Finally, we show that extrapolating our fitted curve up to 40 T, we observe that the ZS of InP reaches a maximum value and then starts to decrease. This indicates that the nonlinear features act as a limiting factor to the maximum ZS that can be observed. For GaAs this feature is not visible due to the overestimated g-factors. Therefore, additional experimental data at magnetic fields higher than 30 T[@note:bmag] could provide useful insight and also test the limits of the effective model presented in this study.
Conclusions {#sec:conclusions}
===========
In summary, we theoretically investigated the common features of the Zeeman splitting in novel III-V wurtzite materials, namely InP, InAs and GaAs, using the $k \cdot p$ method. Specifically, we calculated the effective g-factors for the important energy bands around the band gap at the $\Gamma$-point (CB, A, B and C) and showed that spin-orbit coupling effects have appreciable contribution to the total g-factor values (contributing up to $\sim$20% of CB, for instance). Our calculated values for InP and InAs g-factors are in very good agreement with the available experimental values. Within the Landau level picture, following the prescription of Tedeschi et al.[@Tedeschi2018], we showed that the nonlinear Zeeman splitting for the B$_\text{z}$ direction is a common feature of wurtzite materials due to the mixing of A and B valence bands induced by the external magnetic field. Relying on the main mechanism behind the origin of this nonlinear feature allowed us to develop an effective analytical description of the Zeeman splitting that describes the experimental data with very good agreement, particularly for InP WZ. By extrapolating our fitted model, we found that the nonlinear Zeeman splitting of InP WZ reaches a maximum value that could be investigated experimentally under magnetic fields higher than 30 T. We also investigated the conventional wurtzite material GaN and showed that the nonlinear features are very weak to be visible experimentally. For zinc-blende materials, discussed in the Appendix, we showed that the valence band Zeeman splittings follow a strong linear behavior, specially for InP and GaAs.
Furthermore, our study shows that the $k \cdot p$ approach is very versatile but it requires reliable parameter sets for quantitative comparison with the experimental data. For instance, the calculated g-factors we presented for GaAs do not provide a good description of the experimental data, indicating that further theoretical efforts in extracting reliable $k \cdot p$ parameters with the correct inclusion of SOC terms are needed. With the ongoing interest in these novel III-V WZ materials, with recent reports on high-quality samples of GaP[@Halder2018APL] and GaSb[@Namazi2018AFM], we believe our findings could guide future experiments and motivate further theoretical efforts to characterize these materials.
![(Color online) Calculated Landau level spectra for valence band at $k_B = k_A = 0$ as function of the magnetic field for (a) InP, (b) InAs and (c) GaAs with zinc blende structure. The upper and lower branches of the first (thick solid lines) and second (thick dashed lines) topmost Landau levels are indicated by the 1$\pm$ and 2$\pm$, respectively. Zeeman splitting for the first (solid lines) and second (dashed lines) topmost Landau levels for (d) InP, (e) InAs and (f) GaAs. Probability densities at 30 T for LL=1 and LL=2 for (g)-(j) InP, (l)-(n) InAs and (o)-(r) GaAs.[]{data-label="fig:ZB"}](fig6.eps)
Acknowledgements {#acknowledgements .unnumbered}
================
The authors acknowledge financial support to the Alexander von Humboldt Foundation, Capes (grant No. 99999.000420/2016-06), SFB 1277 (B05), SFB 1170 “ToCoTronics”, the ENB Graduate School on Topological Insulators, Awards2014 and Avvio alla Ricerca (Sapienza Università di Roma). P.E.F.J. is grateful to D. R. Candido, T. Frank, J. Lee and T. Campos for helpful dicussions.
Appendix: Zinc-blende materials {#appendix-zinc-blende-materials .unnumbered}
===============================
The LL formalism discussed in Sec. \[sec:LLs\] can also be applied to ZB materials. Using the conventional Luttinger-Kohn $k \cdot p$ Hamiltonian[@Luttinger1955PR] for the valence band combined with the Zeeman term[@Novik2005PRB], we calculate the LL spectra for InP, InAs and GaAs with ZB crystal structure assuming a magnetic field along the \[001\] axis. The effective mass parameters are taken from Ref. \[\] and the $\kappa$ parameters from Ref. \[\]. In Fig. \[fig:ZB\] we present our calculations for the LL spectra, the ZS and the probability densities focusing on the first and second topmost LL branches (denoted by 1$\pm$ and 2$\pm$), that have probability densities with majority contribution of zero nodes. Specifically, in Figs. \[fig:ZB\](a)-(c) we show the LL spectra highlighting the LL=1 branches (thick solid lines) and LL=2 branches (thick dashed lines). In Figs. \[fig:ZB\](d)-(f) we show the ZS for LL=1 and LL=2 branches. Finally, in Figs. \[fig:ZB\](g)-(r) we show the probability densities at B = 30 T. Although in these topmost LLs in ZB there is also mixing of the basis states for heavy and light hole bands (degenerate at $\Gamma$-point), the ZS for the topmost LL is linear for InP and GaAs and slightly nonlinear for InAs, but not as pronounced as in WZ for the top most LLs. A small nonlinear ZS can also be seen for the LL=2. Finally, we point out that in typical magneto-PL experiments the topmost LL would be accessed via the optical transition and therefore the ZS for InP and GaAs would have just a linear dependence with magnetic field \[please refer to Eq.(\[eq:ZSBz\]) and the discussion below it for the case of degenerate bands with $E_{\text{B}} = 0$\].
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---
abstract: 'A general class of (multiple) hypergeometric type integrals associated with the Jacobi theta functions is defined. These integrals are related to theta hypergeometric series via the residue calculus. In the one variable case, theta function extensions of the Meijer function are obtained. A number of multiple generalizations of the elliptic beta integral [@spi:elliptic] associated with the root systems $A_n$ and $C_n$ is described. Some of the $C_n$-examples were proposed earlier by van Diejen and the author, but other integrals are new. An example of the biorthogonality relations associated with the elliptic beta integrals is considered in detail.'
address: 'Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow Region 141980, Russia.'
author:
- 'V.P. Spiridonov'
title: Theta hypergeometric integrals
---
[^1]
*Dedicated to Mizan Rahman*
Introduction
============
Exact integration formulas and integral representations of functions are important from various points of view. Such representations serve sometimes as definitions of functions, but more often they are needed for the better understanding of properties of functions defined beforehand. Due to numerous applications (see [@aar:special]), the Euler beta integral $$\label{euler}
\int_0^1 x^{\alpha-1}(1-x)^{\beta-1}dx=\frac{\Gamma(\alpha)\Gamma(\beta)}
{\Gamma(\alpha+\beta)}, \qquad \mbox{Re }\alpha,\, \mbox{Re }\beta>0,$$ where $\Gamma(z)$ is the standard gamma function, plays a fundamental role in classical analysis. Various $q$-generalizations of (\[euler\]) involving $q$-gamma functions have been proposed within the theory of basic hypergeometric series [@gas-rah:basic]. Recently, a “third floor" of the hierarchy of beta-type integrals, which is related to the elliptic gamma function, has been discovered in [@spi:elliptic] (in the one variable case) and in [@die-spi:elliptic; @die-spi:selberg] (multiple extensions). For a brief review of results in this direction, see [@die-spi:review]. In this paper, we discuss a general class of hypergeometric type integrals associated with the Jacobi theta functions and propose several new multiple elliptic beta integrals admitting exact evaluations.
An infinite hierarchy of multiple gamma functions was proposed by Barnes long time ago [@bar:multiple]: $$\Gamma_r^{-1}(u;\mathbf{\omega}) =
e^{\sum_{i=0}^r\gamma_{ri}\frac{u^i}{i!}}\;u \prod_{n_1,\ldots,
n_r=0}^\infty{}' \left(1+\frac{u}{\Omega}\right)
e^{\sum_{i=1}^r(-1)^i\frac{u^i}{i\Omega^i}},
\label{m-gamma}$$ where $\gamma_{ri}$ are some constants analogous to the Euler constant and $\Omega=n_1\omega_1+\ldots+n_r\omega_r$ (if some of the ratios $\omega_i/\omega_k$ are real, then they must be positive). The prime in the product sign means that the point $n_1=\ldots=n_r=0$ is skipped. The function (\[m-gamma\]) satisfies a collection of $r$ first order difference equations $$\frac{\Gamma_r(u+\omega_j;\mathbf{\omega})}
{\Gamma_r(u;\mathbf{\omega})}=
\frac{1}{\Gamma_{r-1}(u;\mathbf{\omega}(j))},
\qquad j=1,\ldots,r,$$ where $\mathbf{\omega}(j)=(\omega_1,\ldots,\omega_{j-1},\omega_{j+1},\ldots,
\omega_r)$ and $\Gamma_1(u;\omega)=
\rho(\omega) \omega^{u/\omega}\Gamma(u/\omega)$ for some constant $\rho(\omega)$ (for a brief account of this function, see also appendix A in [@jim-miw:quantum]).
Following Barnes’ analysis, in [@jac:basic] Jackson has considered the generalized gamma functions in a slightly different way and proposed the $q$-gamma function and the elliptic gamma function. We recall the definition of the latter. Taking two complex variables $q$ and $p$ such that $|q|, |p|<1$, we compose the following (convergent) Jackson double infinite product: $$(z;q,p)_\infty=\prod_{j,k=0}^\infty(1-zq^jp^k).
\label{d-product}$$ Two first order $q$- and $p$-difference equations for this product $$\frac{(z;q,p)_\infty}{(qz;q,p)_\infty}=(z;p)_\infty, \qquad
\frac{(z;q,p)_\infty}{(pz;q,p)_\infty}=(z;q)_\infty,
\label{eq-1}$$ where $(z;p)_\infty=\prod_{k=0}^\infty(1-zp^k)$, are of major importance. Replacing $z$ by $pz^{-1}$ in the first equation and by $qz^{-1}$ in the second, we get $$\frac{(qp(qz)^{-1};q,p)_\infty}{(qpz^{-1};q,p)_\infty}
=(pz^{-1};p)_\infty,\qquad
\frac{(qp(pz)^{-1};q,p)_\infty}{(qpz^{-1};q,p)_\infty}
=(qz^{-1};q)_\infty.
\label{eq-2}$$
We define a theta function as follows: $$\theta(z;p)=(z;p)_\infty(pz^{-1};p)_\infty.
\label{theta}$$ It is related to the standard Jacobi $\theta_1$-function [@whi-wat:course] in a simple way $$\begin{aligned}
\nonumber
&& \theta_1(u;\sigma,\tau)=
-i\sum_{n=-\infty}^\infty (-1)^np^{(2n+1)^2/8}q^{(n+1/2)u}
\\ \nonumber && \makebox[2em]{}
=p^{1/8} iq^{-u/2}\: (p;p)_\infty\: \theta(q^{u};p), \quad
u\in\mathbb{C},
\label{theta1}\end{aligned}$$ where we assume that $q=e^{2\pi i\sigma}, p=e^{2\pi i\tau}$. Sometimes, for brevity, it is convenient to drop $q$ and $p$ or the modular parameters $\sigma$ and $\tau$ in the notations for theta functions and elliptic gamma functions, as well as for the elliptic analogs of shifted factorials to be defined below.
The function $\theta_1(u)$ is entire, odd, $\theta_1(-u)=-\theta_1(u)$, and doubly quasiperiodic $$\theta_1(u+\sigma^{-1}) = -\theta_1(u), \qquad
\theta_1(u+\tau\sigma^{-1}) = -e^{-\pi i\tau-2\pi i\sigma u}
\theta_1(u).
\label{quasi}$$ These transformation properties of the $\theta_1$-function are extensively used in our formalism. For the $\theta(z;p)$ function, they take the form $$\label{fun-rel}
\theta(pz;p)=\theta(z^{-1};p)=-z^{-1}\theta(z;p).$$
Now we multiply the left-hand sides and right-hand sides of the first identities in (\[eq-1\]) and (\[eq-2\]), respectively, and do the same with the second identities. This yields the difference equations $$\Gamma(qz;q,p)=\theta(z;p)\Gamma(z;q,p), \qquad
\Gamma(pz;q,p)=\theta(z;q)\Gamma(z;q,p),
\label{eq-3}$$ for the elliptic gamma function $\Gamma(z;q,p)$; in the explicit form, we have $$\Gamma(z;q,p) = \prod_{j,k=0}^\infty\frac{1-z^{-1}q^{j+1}p^{k+1}}{1-zq^jp^k}.
\label{ell-gamma}$$ Despite of the fact that the general idea of associating a generalized gamma function with the elliptic theta function was formulated in the well-known paper [@jac:basic], it did not get much attention. However, Jackson’s double infinite product was used explicitly in the mathematical physics literature on integrable models of statistical mechanics starting with the Baxter’s work on the eight vertex model; see [@bax:partition]. The name “elliptic gamma function" for the product (\[ell-gamma\]) was proposed by Ruijsenaars in the recent paper [@rui:first], where he reintroduced the function $\Gamma(z;q,p)$ anew and started a systematic investigation of its properties. A further detailed analysis of this function was performed by Felder and Varchenko in [@fel-var:elliptic].
In order to compare the elliptic gamma function with the Barnes multiple gamma function, in (\[ell-gamma\]) we put $$z=e^{2\pi i\frac{ u}{\omega_2}},\quad
q=e^{2\pi i\frac{ \omega_1}{\omega_2}},\quad
p=e^{2\pi i\frac{ \omega_3}{\omega_2}},$$ where $\omega_i$ are some constants satisfying the same constraints for the quasiperiods as in (\[m-gamma\]). Then it is not difficult to see that the set of zeros and poles of $\Gamma(z;q,p)$, viewed as a meromorphic function of the variable $u$, coincides with the set of zeros and poles of the following combination of Barnes $\Gamma_3$-functions: $$\frac{\Gamma_3(u;\omega_1,\omega_2,\omega_3)\Gamma_3(u-\omega_2;\omega_1,
-\omega_2,\omega_3)}{\Gamma_3(\omega_1+\omega_3-u;\omega_1,\omega_2,\omega_3)
\Gamma_3(\omega_1+\omega_3-\omega_2-u;\omega_1,-\omega_2,\omega_3)}.
\label{ell-g-bar1}$$ This means that the ratio of $\Gamma(z;q,p)$ and (\[ell-g-bar1\]) is an entire function of $u$, and this function is seen to be given by an exponential of some polynomial of $u$ of the third degree.
For arbitrary complex $s$, the elliptic shifted factorials are defined as ratios of elliptic gamma functions $$\theta(z;p;q)_s = \frac{\Gamma(zq^s;q,p)}{\Gamma(z;q,p)}.$$ We use also the following shorthand notation: $$\begin{aligned}
&& \Gamma(t_1,\ldots,t_k;q,p)\equiv \prod_{j=1}^k \Gamma(t_j;q,p),
\\ &&
\theta(t_1,\ldots,t_k;p;q)_n\equiv\prod_{j=1}^k\prod_{\ell=0}^{n-1}
\theta(t_jq^{\ell};p), \quad n\in\mathbb{N}.\end{aligned}$$
The elliptic beta integral [@spi:elliptic] is the first exact integration formula involving the elliptic gamma function. We conclude this section by an explicit description of it.
Let five complex parameters $t_m, m=0, \dots,4,$ satisfy the inequalities $|t_m|<1,\; |pq|<|A|,$ where $A\equiv\prod_{r=0}^4t_r.$ Define the elliptic beta integral as the following contour integral: $$\label{ell-int}
\mathcal{N}_E(\mathbf{t})=
\int_\mathbb{T}\Delta_E(z,\mathbf{t})\frac{dz}{z},$$ where $\mathbb{T}$ is the positively oriented unit circle, and $$\label{weight}
\Delta_E(z,\mathbf{t}) = \frac{1}{2\pi i}
\frac{\prod_{m=0}^4\Gamma(zt_m, z^{-1}t_m; q,p)}
{\Gamma(z^2,z^{-2}, zA, z^{-1}A;q,p)}.$$ Then $$\label{result}
\mathcal{N}_E(\mathbf{t})=\frac{2\prod_{0\leq m<s\leq 4} \Gamma(t_mt_s;q,p)}
{(q;q)_\infty(p;p)_\infty\prod_{m=0}^4\Gamma(At_m^{-1};q,p)}.$$
Relation (\[ell-int\]) determines a new Askey-Wilson type integral representing an elliptic extension of the Nassrallah-Rahman integral. Indeed, if we set $p=0$, then the integral (\[ell-int\]) is reduced to the Nassrallah-Rahman $q$-beta integral [@nas-rah:projection; @rah:integral] which, in turn, is a one parameter extension of the celebrated Askey-Wilson $q$-beta integral [@ask-wil:some]. Theorem 1 was proved by the author with the help of an elliptic generalization of the method used by Askey in [@ask:beta] for proving the Nassrallah-Rahman integral. A large list of known plain and $q$-hypergeometric beta integrals was given in [@rah-sus:pearson].
As shown in [@die-spi:elliptic], a special finite-dimensional reduction of (\[ell-int\]) associated with the residue calculus results in the elliptic generalization of the Jackson sum for a terminating $_8\Phi_7$ basic hypergeometric series, which was discovered by Frenkel and Turaev in [@fre-tur:elliptic]. In [@spi:special], the integral (\[ell-int\]) was applied to the construction of a large family of continuous biorthogonal functions generalizing the Rahman’s $_{10}\Phi_9$ biorthogonal rational functions [@rah:integral; @rah:biorthogonality]. These functions are expressed through products of two $_{12}E_{11}$ elliptic hypergeometric series with different modular parameters (for the definition of an appropriate system of notations for such series, see [@spi:theta]). It is believed that in the theory of biorthogonal functions they play a role similar to that played by the Askey-Wilson polynomials [@ask-wil:some] in the theory of orthogonal polynomials. We describe these biorthogonal functions in the last section and in Appendix A of the present paper and give complete proofs of some results announced in [@spi:special]. An elliptic extension of Wilson’s discrete (finite-dimensional) set of biorthogonal rational functions [@wil:orthogonal] was constructed earlier by Zhedanov and the author in [@spi-zhe:spectral; @spi-zhe:classical]. The corresponding three terms recurrence relation generates the most general example in the pool of known terminating continued fractions expressed in terms of the series of hypergeometric type, namely, it is expressed via the $_{12}E_{11}$ series as well [@spi-zhe:gevp; @spi-zhe:theory]. The integral (\[ell-int\]) leads to integral representations for a terminating $_{12}E_{11}$ series and its particular bilinear form [@spi:special] (the proofs are given in Appendix B). All these results open new ways of exploration of the world of elliptic functions and modular forms, which complement recent progress reached in the classical setting by Milne [@mil:infinite].
Acknowledgments
This paper is dedicated to Mizan Rahman, a master of $q$-special functions from whom the author has learned much over ten years of regular contacts. The author is also grateful to J.F. van Diejen for many useful discussions during the work on [@die-spi:elliptic]-[@die-spi:review], and for remarks to this paper. Permanent encouragement from A.S. Zhedanov is highly appreciated as well. Special thanks go to C. Krattenthaler for explaining some of the Warnaar theorems [@war:summation], to V. Tarasov for emphasizing the importance of the Barnes work [@bar:multiple] and for drawing attention to the paper [@tar-var:geometry], and to S. Kharchev for a discussion of the properties of the generalized gamma functions. Some of the main results were obtained during the author’s stay at the Max-Planck-Institut für Mathematik (Bonn) in the summer of 2002. The author is indebted to this institute for warm hospitality and to Yu.I. Manin, A. Okounkov, and D. Zagier for several stimulating discussions at MPI of the subject of the present paper. The organizers of the workshops “Special Functions in the Digital Age" (Minneapolis, USA, July 22-August 2, 2002) and “Classical and Quantum Integrable Systems" (Protvino, Russia, January 9-11, 2003) are thanked for giving an opportunity to present there some results of this work. This research is supported in part by the Russian Foundation for Basic Research (RFBR) Grant No. 03-01-00781.
A general definition of theta hypergeometric integrals
======================================================
The right-hand side of (\[ell-int\]) belongs to a general class of integrals related to the series of hypergeometric type built from Jacobi theta functions. In accordance with the theory of general theta hypergeometric series developed in [@spi:theta], we give the following definition.
Let $C$ denote a smooth Jordan curve on the complex plane. Let $\Delta(y_1,\ldots,$ $y_n)$ be a meromorphic function of its arguments $y_1,\ldots,y_n$. Consider the (multiple) integrals $$\label{definition}
I_n=\int_{C}dy_1\dots\int_Cdy_n \; \Delta(y_1,\ldots,y_n)$$ and the ratios $$h_\ell(\mathbf{y})=\frac{\Delta(y_1,\ldots,y_\ell+1,\ldots,y_n)
}{\Delta(y_1,\ldots,y_\ell,\ldots,y_n)}.
\label{hl}$$
Then the integrals $I_n$ are called:
1\) the [*plain hypergeometric integrals*]{} if $$h_\ell(\mathbf{y})=R_\ell(\mathbf{y}),
\label{plain-int}$$ are rational functions of $y_1,\ldots, y_n$ for all $\ell=1,\ldots, n;$
2\) the [*$q$-hypergeometric integrals*]{} if $$h_\ell(\mathbf{y})=R_\ell(q^\mathbf{y}),
\label{q-int}$$ are rational functions of $q^{y_1},\ldots, q^{y_n}$, $q\in\mathbb{C},$ for all $\ell=1,\ldots, n;$
3\) the [*elliptic hypergeometric integrals*]{} if for all $\ell=1,\ldots,n$ the ratios $h_\ell(\mathbf{y})$ are elliptic functions of the variables $y_1,\ldots,y_n$ with periods $\sigma^{-1}$ and $\tau\sigma^{-1}, \, \text{Im}(\tau)>0$;
4\) the general [*theta hypergeometric integrals*]{} if $h_\ell(\mathbf{y})$ and $1/h_\ell(\mathbf{y})$ are meromorphic functions obeying the double quasiperiodicity conditions $$\begin{aligned}
\nonumber
&& h_\ell(y_1,\ldots,y_k+\sigma^{-1},\ldots,y_n)
=e^{\sum_{j=1}^na_{\ell k}(j)y_j+b_{\ell k}} h_\ell(\mathbf{y}),
\\ && h_\ell(y_1,\ldots,y_k+\tau\sigma^{-1},\ldots,y_n)
=e^{\sum_{j=1}^n c_{\ell k}(j)y_j+d_{\ell k}} h_\ell(\mathbf{y}),
\label{quasi-per}\end{aligned}$$ with the quasiperiodicity factors similar to those for the Weierstrass sigma function (which is related to $\theta_1(u)$ in a simple way, see [@whi-wat:course]).
If we assume that the variables $y_1,\ldots,y_n$ are discrete, $\mathbf{y}\in\mathbb{N}^n$, and replace integrals by sums $\sum_{\mathbf{y}\in\mathbb{N}^n}$, then we get the definitions of the plain and $q$-hypergeometric series, which go back to Horn [@ggr:general], and the definition of the elliptic hypergeometric series suggested in [@spi:theta], respectively. The theta hypergeometric series were defined in [@spi:theta] in a less general form because of the less general choice of quasiperiodicity factors. Evidently, if $a_{\ell k}(j)=b_{\ell k}=c_{\ell k}(j)=d_{\ell k}=0$, then the theta hypergeometric functions are reduced to the elliptic ones. The integrals (or series) defined in this way do not form an algebra because, in general, sums of hypergeometric integrals do not fit the taken definition.
The shifts $y_\ell\to y_\ell+1$ in (\[hl\]) may be replaced by translations by an arbitrary constant $y_\ell\to y_\ell +\omega_1,$ $\omega_1=const.$ However, we can replace $\omega_1$ by 1 after an appropriate rescaling of $y_\ell$, which results in a simple deformation of the contour $C$ in (\[definition\]).
Consider the case of $n=1$ in detail. The general rational function of $y$ can be represented in the form $$R(y)=\frac{\prod_{j=1}^n(1-a_j+y)\prod_{j=n+1}^r(a_j-1-y)}
{\prod_{j=1}^m(b_j-1-y)\prod_{j=m+1}^s(1-b_j+y)}\, x,$$ where $n,r,m,s$ are arbitrary integers, $x$ is an arbitrary complex constant, and $a_j, b_j$ describe the positions of the zeros and poles of $R(y)$. The equation $\Delta(y+1)=R(y)\Delta(y)$ has the following general solution: $$\Delta(y)=\frac{\prod_{j=1}^m\Gamma(b_j-y)
\prod_{j=1}^n\Gamma(1-a_j+y)}{\prod_{j=m+1}^s\Gamma(1-b_j+y)
\prod_{j=n+1}^r\Gamma(a_j-y)}\, x^{y}\varphi(y),
\label{mej}$$ where $\Gamma(y)$ is the standard gamma function and $\varphi(y)$ is an arbitrary periodic function, $\varphi(y+1)=\varphi(y)$. If we set $\varphi(y)=1$, then for an appropriate choice of the contour $C$ the integral $I_1$ (see (\[definition\])) is none other than the Meijer function [@erd:higher]. In this case we have no natural additional tools for fixing an infinite dimensional (functional) freedom contained in the solution $\Delta(y)$.
In the $q$-case, in a similar way we can write $$R(q^y)=\frac{\prod_{j=1}^n(1-t_jq^y)\prod_{j=n+1}^r(1-t_j^{-1}q^{-y})}
{\prod_{j=1}^m(1-w_j^{-1}q^{-y})\prod_{j=m+1}^s(1-w_jq^y)}\, x.$$ For $0<|q|<1$, the general meromorphic solution of the equation $\Delta(y+1)=R(q^y)\Delta(y)$ is $$\Delta(y)= \frac{\prod_{j=n+1}^r (t_j^{-1}q^{1-y};q)_\infty
\prod_{j=m+1}^s(w_jq^y;q)_\infty}
{\prod_{j=1}^n(t_jq^y;q)_\infty
\prod_{j=1}^m(w_j^{-1}q^{1-y};q)_\infty}\, x^{y}\varphi(y),
\label{q-mej}$$ where, again, $\varphi(y)$ is an arbitrary periodic function, $\varphi(y+1)=\varphi(y)$. In this case, for $\varphi(y)=1$ the integral $I_1$ describes a $q$-Meijer function, which was investigated by Slater in [@sla:generalized].
For $|q|>1$, the equation $\Delta(y+1)=R(q^y)\Delta(y)$ has the following general solution $$\Delta(y)= \frac{\prod_{j=1}^n(t_jq^{y-1};q^{-1})_\infty
\prod_{j=1}^m(w_j^{-1}q^{-y};q^{-1})_\infty}{\prod_{j=n+1}^r
(t_j^{-1}q^{-y};q^{-1})_\infty \prod_{j=m+1}^s(w_jq^{y-1};q^{-1})_\infty}
\, x^{y}\varphi(y),$$ that is we have an effective $q\to q^{-1}$ replacement and a reshuffling of parameters in (\[q-mej\]).
We remind the reader that $q=e^{2\pi i \sigma}$. The parameter $\sigma$ gives a second scale, which may be used for generating a natural additional restriction upon $\Delta(y)$. The function $q^y$ is periodic under the shift $y\to y+\sigma^{-1}$, and (\[q-mej\]) satisfies the equation $\Delta(y+\sigma^{-1})/\Delta(y)=x^{1/\sigma}\varphi(y+\sigma^{-1})/
\varphi(y)$. We can fix $\varphi(y)$ by demanding that $$\varphi(y+\sigma^{-1})=\tilde R(e^{2\pi i y})\varphi(y),$$ where $\tilde R$ is another rational function of its argument. In accordance with the periodicity condition $\varphi(y+1)=\varphi(y)$, we have $$\tilde R(e^{2\pi i y})=
\frac{\prod_{j=1}^{n'}(1-\tilde t_je^{-2\pi iy})
\prod_{j=n'+1}^{r'}(1-\tilde t_j^{-1}e^{2\pi iy})}
{\prod_{j=1}^{m'}(1-\tilde w_j^{-1}e^{2\pi iy})
\prod_{j=m'+1}^{s'}(1-\tilde w_je^{-2\pi iy})},$$ where $\tilde t_j$ and $\tilde w_j$ are arbitrary new parameters. Note that we cannot multiply the function $\tilde R$ by terms like $\rho e^{2\pi i k y}, k\in\mathbb{Z}, \rho\in\mathbb{C},$ if they are different from 1, because then the periodicity condition for $\varphi(y)$ will be broken. For $|q|<1$, the general meromorphic solution of the difference equation for $\varphi(y)$ is as follows: $$\varphi(y)= \frac{\prod_{j=m'+1}^{s'}(\tilde w_je^{-2\pi iy};
\tilde q)_\infty \prod_{j=n'+1}^{r'}(\tilde q\tilde t_j^{-1}e^{2\pi iy};
\tilde q)_\infty} {\prod_{j=1}^{m'} (\tilde q\tilde w_j^{-1}e^{2\pi iy};
\tilde q)_\infty \prod_{j=1}^{n'}(\tilde t_je^{-2\pi iy};
\tilde q)_\infty}\, \tilde \varphi(y),
\label{tq-mej}$$ where $\tilde q=e^{-2\pi i/\sigma}$ is the modular partner of $q$. Indeed, for $\text{Im}(\sigma)>0$ we have $\text{Im}(\sigma^{-1})<0$, and (\[tq-mej\]) is well defined. The function $\tilde\varphi(y)$ in (\[tq-mej\]) is an arbitrary elliptic function with periods $1$ and $\sigma^{-1}$. It is characterized uniquely by the position of its poles and zeros in the fundamental parallelogram of periods containing $2k-1$ free parameters, where $k$ is the order of $\tilde \varphi(y)$. Thus, the space of solutions is not too large: it becomes finite-dimensional (in the sense of the number of free parameters).
Consider the regime $|q|=1$. Denoting $\sigma=\omega_1/\omega_2$ and assuming that $\text{Re}(\sigma)>0$, we introduce the variable $u=y\omega_1$. Now it is possible to choose the parameters $t_j, \tilde t_j$, etc in a special way, so that the infinite products $(t_jq^y;q)_\infty$, $(\tilde t_je^{-2\pi i y};\tilde q)_\infty$, etc in (\[q-mej\]) and (\[tq-mej\]) combine into the double sine functions $S(u+g_j;\omega_1,\omega_2)$ for some $g_j$, where $$S(u;\omega_1,\omega_2) = \frac{(e^{2\pi i u/\omega_2}; q)_\infty}
{(e^{2\pi iu/\omega_1}\tilde q; \tilde q)_\infty},
\label{2d-sin}$$ is a well defined function for $|q|\to 1$. Indeed, it can be checked that the zeros and poles of (\[2d-sin\]) coincide with the zeros and poles of the function $\Gamma_2(\omega_1+\omega_2-u;\mathbf{\omega})/\Gamma_2(u;
\mathbf{\omega})$, which is a well-defined meromorphic function of $u$ for $\omega_1/\omega_2>0$.
In this case $\sigma$ is real, and if it is incommensurate with 1, then $\tilde\varphi(y)=1$ (i.e., the function $\Delta(y)$ is determined quite uniquely). For a description of the properties of the double sine function and some of its applications, see [@jim-miw:quantum; @kls:unitary; @nis-uen:integral; @rui:generalized]. In particular, the integrals introduced by Jimbo and Miwa in [@jim-miw:quantum] as solutions of some $q$-difference equations at $|q|=1$ provided the first examples of $q$-hypergeometric integrals for $q$ on the unit circle. Faddeev’s concept of the modular double for quantum groups (see [@fad:modular]) is also related to the function (\[2d-sin\]).
Thus, the world of $q$-Meijer functions appears to be reacher than in the plain hypergeometric case. The introduction of the additional equation involving shifts by $\sigma^{-1}$ brought some new non-trivial structures in the integrals and reduced the functional freedom in the definition of meromorphic function $\Delta(y)$ to an elliptic function $\tilde\varphi(y)$ containing a finite number of free parameters.
Now, we turn to the single variable elliptic hypergeometric integrals. The general elliptic function of order $r+1$ can be factorized as follows [@whi-wat:course]: $$\begin{aligned}
&& h(y)=e^\gamma\prod_{j=0}^r\frac{\theta_1(u_j+y;\sigma,\tau)}
{\theta_1(v_j+y;\sigma, \tau)}=e^\gamma
\frac{\theta(t_0q^y,\ldots,t_rq^y ;p)}
{\theta(w_0q^y,\ldots,w_rq^y;p)},
\label{n=1} \\
&& \makebox[4em]{}
\theta(t_0,\ldots, t_k;p)=\prod_{i=0}^k\theta(t_i;p),
\nonumber\end{aligned}$$ where $p=e^{2\pi i\tau }, \; \text{Im}(\tau)>0, \; q=e^{2\pi i\sigma}$. The parameter $\gamma$ is an arbitrary complex number, but $t_i\equiv q^{u_i}, \; w_i\equiv q^{v_i}$ satisfy the balancing constraint $$\sum_{i=0}^r(u_i-v_i)=0, \quad \text{or}\quad
\prod_{i=0}^rt_i=\prod_{i=0}^rw_i,
\label{balance}$$ which guarantees that the meromorphic function $h(y)$ is doubly periodic: $$h(y+\sigma^{-1})=h(y),\qquad h(y+\tau\sigma^{-1})=h(y).$$ For $\tau=\sigma$ (which requires that $\text{Im}(\sigma)> 0$), the function $h(y)$ gives an explicit form of $\tilde\varphi(y)$ in (\[tq-mej\]).
In order to find the integrand $\Delta(y)$, it is necessary to solve the first order difference equation $$\Delta(y+1)=h(y)\Delta(y)
\label{1-eq}$$ in the class of meromorphic functions. The theory of such equations was developed long ago (see, e.g., [@bar:linear]). Obviously, since $h(y)$ is factorized into the ratio of products of theta functions, it suffices to find a meromorphic solution of the equation $$f(y+1)=\theta(q^y;p)f(y),
\label{ell-g-eq}$$ which leads to various elliptic gamma functions [@jac:basic]. The simplest such function (\[ell-gamma\]) is defined from equation (\[ell-g-eq\]) only up to a periodic function $\varphi(y+1)=\varphi(y)$ and, moreover, it requires that $\text{Im}(\sigma)>0$ (or $|q|<1$), which was not assumed in (\[n=1\]).
We introduce the variable $z=q^y$, so that the shift $y\to y+1$ becomes equivalent to the multiplication $z\to qz$. Then the general solution of (\[1-eq\]) looks like this: $$\Delta(y)=\prod_{j=0}^r\frac{\Gamma(t_jz;q,p)}
{\Gamma(w_jz;q,p)} e^{\gamma y+\delta}\varphi(y),
\label{D}$$ where the balancing condition (\[balance\]) is assumed and $\varphi(y+1)=\varphi(y)$ is an arbitrary periodic function. Using the reflection formulas $$\begin{aligned}
\nonumber
&& \Gamma(pz, qz^{-1};q,p)=\Gamma(qz,pz^{-1};q,p)
\\ && \makebox[2em]{} =\Gamma(pqz,z^{-1};q,p)=\Gamma(z,pqz^{-1};q,p)=1,
\label{refl-eq}\end{aligned}$$ in (\[D\]) we can replace several elliptic gamma functions containing in the arguments $z$ by those with arguments containing $z^{-1}$. After that, $\Delta(y)$ would look closer to the integrands for the plain or $q$-Meijer functions, but in the elliptic case this does not increase generality because of the right-hand side of (\[refl-eq\]) is trivial.
In the region $\text{Im}(\sigma)<0$, that is, for $|q|>1$, the general solution of (\[1-eq\]) can be written in the form $$\Delta(y)=\prod_{j=0}^r\frac{\Gamma(w_jq^{y-1};q^{-1},p)}
{\Gamma(t_jq^{y-1};q^{-1},p)} e^{\gamma y+\delta}\varphi(y).
\label{D'}$$ Effectively, we have a permutation of parameters and a simple $q\to q^{-1}$ substitution in the elliptic gamma functions in (\[D\]) (cf. the definition of this function for $|q|>1$ given in [@fel-var:elliptic]).
Let us take $\varphi(y)=1$. Then the function (\[D\]) satisfies two simple difference equations of the first order: $$\begin{aligned}
\label{2-eq}
&&\Delta(y+\sigma^{-1})=e^{\gamma/\sigma}\Delta(y), \\
&&\Delta(y+\tau\sigma^{-1})=e^{\gamma\tau/\sigma}
\prod_{j=0}^r \frac{\theta(t_jq^y;q)}{\theta(w_jq^y;q)}\Delta(y).
\label{3-eq}\end{aligned}$$ Suppose that $1, \sigma^{-1}, \tau\sigma^{-1}$ are pairwise incommensurate. Then the system of three equations (\[1-eq\]), (\[2-eq\]), and (\[3-eq\]) determines $\Delta(y)$ uniquely up to a factor. As in the $q$-hypergeometric case, we can generalize equations (\[2-eq\]) and (\[3-eq\]), use them as natural tools for fixing the functional freedom in $\Delta(y)$, and get qualitatively different elliptic hypergeometric integrals in this way.
The ratio $\Delta(y+\tau\sigma^{-1})/\Delta(y)$ in (\[3-eq\]) is an elliptic function with periods 1 and $\sigma^{-1}$. Therefore, it is natural to demand that $\Delta(y+\sigma^{-1})/\Delta(y)$ be also an elliptic function with periods that, by symmetry, are equal to 1 and $\tau\sigma^{-1}$.
Suppose that $\Delta(y)$ satisfies equation (\[1-eq\]) and that 1, $\sigma^{-1}$, $\tau\sigma^{-1}$ are pairwise incommensurate. Denote $\tilde q=e^{-2\pi i/\sigma}$, $\tilde p=e^{2\pi i\tau/\sigma}.$ For simplicity, assume that $\text{Im}(\sigma)>0$ (i.e., $|q|<1$). If $\Delta(y+\tau\sigma^{-1})/\Delta(y)$ is an elliptic function with periods 1 and $\tau\sigma^{-1}$, then for $\text{Im}(\tau/\sigma)>0$ the most general form of the meromorphic function $\Delta(y)$ is as follows: $$\label{delta-ell}
\Delta(y)=\prod_{j=0}^r\frac{\Gamma(t_jq^y;q,p)}{\Gamma(w_jq^y;q,p)}
\prod_{j=0}^{n}\frac{\Gamma(\tilde t_je^{-2\pi iy};\tilde q,\tilde p)}
{\Gamma(\tilde w_je^{-2\pi iy};\tilde q,\tilde p)}\,e^{\gamma y+\delta},$$ where $\prod_{j=0}^rt_jw_j^{-1}=\prod_{j=0}^{n}\tilde t_j\tilde w_j^{-1}=1$. For $\text{Im}(\tau/\sigma)<0$, we have $$\label{delta-ell'}
\Delta(y)=\prod_{j=0}^r\frac{\Gamma(t_jq^y;q,p)}{\Gamma(w_jq^y;q,p)}
\prod_{j=0}^{n}\frac{\Gamma(\tilde w_je^{-2\pi iy}\tilde p^{-1};
\tilde q,\tilde p^{-1})} {\Gamma(\tilde t_je^{-2\pi iy}\tilde p^{-1};
\tilde q,\tilde p^{-1})}\,e^{\gamma y+\delta}.$$
First, observe that for $\text{Im}(\sigma)>0$ we have $\text{Im}(\sigma^{-1})<0,$ automatically, that is $|\tilde q|<1.$ Therefore, for $\text{Im}(\tau/\sigma)>0$ the function $\Gamma(z;\tilde q,\tilde p)$ is well defined.
The function $\Delta(y)$ in (\[D\]) gives the general solution of equation (\[1-eq\]). Suppose that $\Delta(y+\sigma^{-1})/\Delta(y)$ is an elliptic function of order $n+1$ with periods 1, $\tau\sigma^{-1}$. For $\text{Im}(\tau/\sigma)>0$, this demand is equivalent to the following equation for $\varphi(y)$: $$\frac{\varphi(y+\sigma^{-1})}{\varphi(y)}=
\prod_{j=0}^{n}\frac{\theta(\tilde t_je^{-2\pi iy}; \tilde p)}
{\theta(\tilde w_je^{-2\pi iy}; \tilde p)},
\label{phi-eq}$$ where $\prod_{j=0}^{n}\tilde t_j\tilde w_j^{-1}=1$. Note that we cannot multiply the right-hand side of this equation by any constant different from 1, since this would violate the condition $\varphi(y+1)=\varphi(y)$.
The meromorphic solution of (\[phi-eq\]) is $$\varphi(y)=\prod_{j=0}^{n}\frac{\Gamma(\tilde t_je^{-2\pi iy};\tilde q,
\tilde p)}{\Gamma(\tilde w_je^{-2\pi iy};\tilde q,\tilde p)}\,
\tilde \varphi(y),
\label{phi-form}$$ where $\tilde \varphi(y)$ is an elliptic function with periods 1 and $\sigma^{-1}$. We can write $$\tilde\varphi(y)=\prod_{j=1}^m\frac{\theta(a_je^{-2\pi iy};\tilde q)}
{\theta(b_je^{-2\pi iy};\tilde q)}=\prod_{j=1}^m\frac{\Gamma(a_je^{-2\pi iy}
\tilde p,b_je^{-2\pi iy};\tilde q,\tilde p)}
{\Gamma(a_je^{-2\pi iy},b_je^{-2\pi iy}\tilde p;\tilde q,\tilde p)},$$ where $\prod_{j=1}^ma_jb_j^{-1}=1$. Therefore, we can absorb the function $\tilde\varphi(y)$ into the ratio of elliptic gamma functions in (\[phi-form\]) by changing $n\to n+2m$ and identifying $\tilde t_k=\tilde p a_k, \tilde w_k=a_k$ for $k=n+1,\ldots, n+m$ and $\tilde t_k=\tilde b_k, \tilde w_k=\tilde p \tilde b_k$ for $k=n+m+1,\ldots,n+2m$. Since $n$, $\tilde t_j$, $\tilde w_j$ are arbitrary, without loss of generality we can set $\tilde\varphi(y)=1$, which yields the desired expression (\[delta-ell\]).
The function (\[delta-ell\]) satisfies the following equations: $$\begin{aligned}
\label{2-eq'}
&&\Delta(y+\sigma^{-1})=e^{\gamma/\sigma}
\prod_{j=0}^{n}\frac{\theta(\tilde t_je^{-2\pi iy};\tilde p)}
{\theta(\tilde w_je^{-2\pi i y};\tilde p)}\, \Delta(y), \\
&&\Delta(y+\tau\sigma^{-1})=e^{\gamma\tau/\sigma}
\prod_{j=0}^r \frac{\theta(t_jq^y;q)}{\theta(w_jq^y;q)}
\prod_{j=0}^{n}\frac{\theta(\tilde w_je^{-2\pi iy}\tilde p^{-1};\tilde q)}
{\theta(\tilde t_je^{-2\pi iy}\tilde p^{-1};\tilde q)}\, \Delta(y).
\label{3-eq'}\end{aligned}$$ The elliptic functions defined by the products $\prod_{j=0}^r$ and $\prod_{j=0}^{n}$ in (\[3-eq’\]) have different forms though both have periods 1 and $\sigma^{-1}$. They are related to each other by the modular transformation $\sigma\to -1/\sigma$ for the corresponding theta functions.
Now, we consider the region $\text{Im}(\tau/\sigma)<0$. Equations (\[1-eq\]) and (\[3-eq’\]) are well defined in this case. They can be used for the determination of $\Delta(y)$, and it can be checked that, indeed, the function (\[delta-ell’\]) provides their general solution. Equation (\[2-eq’\]) is replaced now by the following one: $$\frac{\Delta(y+\sigma^{-1})}{\Delta(y)}= e^{\gamma/\sigma} \prod_{j=0}^{n}
\frac{\theta(\tilde w_je^{-2\pi iy};\tilde p^{-1})}
{\theta(\tilde t_je^{-2\pi iy}; \tilde p^{-1})},
\label{phi-eq'}$$ that is, $\tilde p$ in (\[phi-eq\]) is changed to $\tilde p^{-1}$, and the parameters $\tilde t_j, \tilde w_j$ are replaced by $\tilde p^{-1}\tilde w_j$ and $\tilde p^{-1}\tilde t_j$, respectively. Using (\[D’\]), it is easy to construct $\Delta(y)$ satisfying (\[1-eq\]), (\[2-eq’\]), and (\[3-eq’\]) in the $|q|>1$ region as well.
In order to be able to work with $q$ on the unit circle $|q|=1$, we need another elliptic gamma function: a kind of the elliptic analog of the double sine function (\[2d-sin\]). Denote $$\begin{aligned}
\nonumber
&& q=e^{2\pi i\frac{\omega_1}{\omega_2}}, \qquad
\tilde q =e^{-2\pi i\frac{\omega_2}{\omega_1}},
\\
&& p=e^{2\pi i\frac{\omega_3}{\omega_2}}, \qquad
\tilde p=e^{2\pi i\frac{\omega_3}{\omega_1}},
\label{ell-bases}\end{aligned}$$ where $\omega_i$ are some complex numbers.
Suppose that $\omega_1/\omega_2>0$ and $\text{Im}(\omega_3/\omega_2)>0$ (i.e., $|p|<1$). Then we have $\text{Im}(\omega_3/\omega_1)=
(\omega_2/\omega_1)\text{Im}(\omega_3/\omega_2)>0$, that is, $|\tilde p|<1$ automatically. Therefore, in the analysis of equation (\[1-eq\]) for $|q|=1$ it is necessary to assume that $|p|, |\tilde p|<1$.
Let $|q|,|p|, |\tilde p|<1$. Then we define a new elliptic gamma function by the formula $$G(u;\mathbf{\omega})= \prod_{j,k=0}^\infty
\frac{(1-e^{-2\pi i\frac{u}{\omega_2}}q^{j+1}p^{k+1})
(1-e^{2\pi i \frac{u}{\omega_1}}{\tilde q}^{j+1}{\tilde p}^k)}
{(1-e^{2\pi i \frac{u}{\omega_2}}q^jp^k)
(1-e^{-2\pi i \frac{u}{\omega_1}}{\tilde q}^j{\tilde p}^{k+1})}.
\label{ell-d}$$
In the limit $p\to 0$ taken in such a way that, simultaneously, $\tilde p\to 0$, we get $G(u;\omega_1,\omega_2,\omega_3)\to
S^{-1}(u;\omega_1,\omega_2)$, where the double sine function $S(u;\mathbf{\omega})$ is fixed in (\[2d-sin\]).
The function $G(u;\mathbf{\omega})$ satisfies the following three difference equations: $$\begin{aligned}
&& G(u+\omega_1;\mathbf{\omega})=\theta(e^{2\pi i\frac{u}{\omega_2}};p)
G(u;\mathbf{\omega}),
\label{ell-1eq} \\
&& G(u+\omega_2;\mathbf{\omega})=\theta(e^{2\pi i\frac{u}{\omega_1}};\tilde p)
G(u;\mathbf{\omega}),
\label{ell-2eq} \\
&& G(u+\omega_3;\mathbf{\omega})=
\frac{\theta(e^{2\pi i\frac{u}{\omega_2}};q)}
{\theta(e^{2\pi i\frac{u}{\omega_1}}\tilde q;\tilde q)}
G(u;\mathbf{\omega})
\nonumber \\ && \makebox[5.5em]{}
= S(u;\omega_1,\omega_2)
S(\omega_1+\omega_2-u;\omega_1,\omega_2)G(u;\mathbf{\omega}).
\label{ell-3eq}\end{aligned}$$ For pairwise incommensurate $\omega_1$, $\omega_2$, $\omega_3$, these equations determine the meromorphic function $G(u;\mathbf{\omega})$ uniquely up to multiplication by a constant, which follows from the nonexistence of triply periodic functions.
The first equation requires that $|p|<1$, the second one requires $|\tilde p|<1$, and both of them do not impose any constraint upon $q$. The third equation (\[ell-3eq\]) involves only the function $S(u;\omega_1,\omega_2)$, which is well defined for $\omega_1/\omega_2>0$, that is, $|q|=|\tilde q|=1$. This means that the function $G(u;\mathbf{\omega})$ may be well defined in this unit circle region as well.
In essence, the original elliptic gamma function (\[ell-gamma\]) has the same properties as the function (\[ell-g-bar1\]). In a similar way, the function (\[ell-d\]) can be expressed as the following combination of the Barnes $\Gamma_3$-functions up to an exponential of some polynomial in $u$ of the third degree: $$\begin{aligned}
\nonumber
&& \frac{\Gamma_3(u;\omega_1,\omega_2,\omega_3)
\Gamma_3(\omega_3-\omega_1-u;-\omega_1,
-\omega_2,\omega_3)}{\Gamma_3(\omega_1+\omega_3-u;\omega_1,\omega_2,\omega_3)
\Gamma_3(u-\omega_1-\omega_2;-\omega_1,-\omega_2,\omega_3)}
\\ && \makebox[2em]{} \times
\Gamma_2(\omega_3-\omega_2-u;-\omega_2,\omega_3)
\Gamma_2(\omega_3-u;\omega_1,\omega_3).
\label{ell-g-bar2}\end{aligned}$$ From this representation it follows that, indeed, (\[ell-d\]) is well defined for real $\omega_1, \omega_2$ with $\omega_1/\omega_2>0$ (and any complex $\omega_3$), like in the double sine function case. A more detailed analysis of this correspondence and an investigation of other properties of the function $G(u;\mathbf{\omega})$ will be given elsewhere. In particular, it is expected that $G(u;\mathbf{\omega})$ is the key function for an elliptic extension of the modular doubling principle for $q$-deformed algebras [@fad:modular; @kls:unitary].
As a result, for $|q|=1$ we get a solution $\Delta(y)$ of equation (\[1-eq\]) by the mere replacement of $\Gamma(q^y;q,p)$ in (\[D\]) by $G(y\omega_1;\mathbf{\omega})$. In the rest of this paper we limit ourselves to the case where $|q|<1$. Note that the region $|p|=1$ is not well defined in the elliptic functions setting. In a sense, the region of real $\omega_3/\omega_2$ is reachable only at the level of the original Barnes multiple gamma functions.
A theta analog of the Meijer function
=====================================
The integral corresponding to (\[D\]) may be considered as a kind of an elliptic extension of a particular Meijer function. The general Jacobi theta functions analog of the Meijer function appears in the case where $h(y)$ is a quasiperiodic function corresponding to the fourth case of the definition given at the beginning of the previous section, see (\[quasi-per\]).
Let $P_3(y)=\sum_{i=1}^3\alpha_iy^i$ be an arbitrary polynomial of the third degree obeying the property $P_3(0)=0$. The function defined by the integral $$G_r^s\left({\mathbf{t} \atop \mathbf{w}};\mathbf{\alpha};q,p\right)
=\int_C\frac{\prod_{j=0}^{s}\Gamma(t_jq^y;q,p)}
{\prod_{k=0}^r\Gamma(w_kq^y;q,p)} e^{P_3(y)}dy,
\label{e-meijer}$$ where $C$ is some contour on the complex plane, may be called a theta analog of the Meijer function whenever the integral is well defined. Note that no constraints are imposed in (\[e-meijer\]) upon the integers $r, s$ and the complex parameters $t_j, w_k$
We have the following equation for the integrand $\Delta(y)$ of (\[e-meijer\]): $$\frac{\Delta(y+1)}{\Delta(y)}=h(y)=e^{P_2(y)}
\frac{\theta(t_0q^y,\ldots,t_sq^y ;p)}
{\theta(w_0q^y,\ldots,w_rq^y;p)},
\label{theta-int}$$ where $P_2(y)=P_3(y+1)-P_3(y)$ is a polynomial in $y$ of the second degree. From the considerations of [@spi:theta] it follows that this $h(y)$ is the most general function such that $h$ is meromorphic in $y$ (together with its inverse $1/h(y)$) and satisfies the quasiperiodicity conditions $$h(y+\sigma^{-1})=e^{ay+b}h(y),\qquad
h(y+\tau\sigma^{-1})=e^{cy+d} h(y)
\label{qua}$$ for some constants $a,b,c,d$. The function $h(y)$ may also be interpreted as a general meromorphic modular Jacobi form in the sense of Eichler and Zagier [@eic-zag:theory].
However, the integral (\[e-meijer\]) is not the most general integral leading to (\[theta-int\]). Using appropriate modifications of the integrands (\[delta-ell\]) and (\[delta-ell’\]) and replacing $y$ by $y/\omega_1$, we arrive at the general theta analog of the Meijer function.
In the definitions (\[ell-bases\]) of the bases, assume that $|q|, |p|<1$. Then, for $|\tilde p|<1$, the integral $$\begin{aligned}
\nonumber
\lefteqn{G_{rm}^{sn}\left({\mathbf{t},\mathbf{\tilde t}\atop \mathbf{w},
\mathbf{\tilde w}};\mathbf{\alpha};\mathbf{\omega}\right) } &&
\\ &&
=\int_C\frac{\prod_{k=0}^{s}\Gamma(t_ke^{\frac{2\pi iy}{\omega_2}};q,p)
\prod_{j=0}^{n}\Gamma(\tilde t_je^{-\frac{2\pi iy}{\omega_1}};
\tilde q,\tilde p)}{\prod_{k=0}^r\Gamma(w_ke^{\frac{2\pi iy}{\omega_2}};q,p)
\prod_{j=0}^m\Gamma(\tilde w_je^{-\frac{2\pi iy}{\omega_1}};
\tilde q,\tilde p)}\, e^{P_3(y)}dy
\label{gen-int}\end{aligned}$$ is called the general theta hypergeometric integral of one variable whenever it is well defined. For $|\tilde p|>1$, we set $$\begin{aligned}
\nonumber
\lefteqn{G_{rm}^{sn}\left({\mathbf{t},\mathbf{\tilde t}\atop \mathbf{w},
\mathbf{\tilde w}};\mathbf{\alpha};\mathbf{\omega}\right) }&&
\\ &&
=\int_C\frac{\prod_{k=0}^{s}\Gamma(t_ke^{\frac{2\pi iy}{\omega_2}};q,p)
\prod_{j=0}^{m}\Gamma(\tilde w_je^{-\frac{2\pi iy}{\omega_1}}\tilde p^{-1};
\tilde q,\tilde p^{-1})}{\prod_{k=0}^r
\Gamma(w_ke^{\frac{2\pi iy}{\omega_2}};q,p)
\prod_{j=0}^n\Gamma(\tilde t_je^{-\frac{2\pi iy}{\omega_1}}\tilde p^{-1};
\tilde q,\tilde p^{-1})}\, e^{P_3(y)}dy.
\label{gen-int'}\end{aligned}$$ There are no constraints upon the integers $r, s, n, m \in\mathbb{N}$ and the complex parameters $t_j,\tilde t_j, w_k, \tilde w_k$.
Both integrands of (\[gen-int\]) and (\[gen-int’\]) satisfy the equations $\Delta(y+\omega_i)/\Delta(y)=h_i(y)$, $i=1,2,3,$ where $h_i$ are some quasiperiodic functions: $h_i(y+\omega_k)=e^{a_{ik} y+b_{ik}}h(y)$, $i\neq k$, with $a_{ik}, b_{ik}$ being some constants related to the parameters $\mathbf{t}, \mathbf{\tilde t},\mathbf{w},\mathbf{\tilde w},
\mathbf{\alpha}$ and $\mathbf{\omega}$. The integral (\[gen-int’\]) was determined by the condition that it has the same functions $h_1(y),
h_3(y)$ as (\[gen-int\]). For a special choice of parameters $\mathbf{t}, \mathbf{\tilde t},\mathbf{w},
\mathbf{\tilde w},\mathbf{\alpha}$, in the limits $|p|, |\tilde p|\to 0$ or $|p|, |\tilde p|^{-1}\to 0$ the function $G_{rm}^{sn}\left({\mathbf{t},\mathbf{\tilde t}\atop \mathbf{w},
\mathbf{\tilde w}};\mathbf{\alpha};\mathbf{\omega}\right)$ is reduced to the general $q$-hypergeometric integral considered in the previous section, see (\[q-mej\]) and (\[tq-mej\]).
The general single variable theta hypergeometric series is defined by the following formula [@spi:theta]: $$\begin{aligned}
\nonumber
\lefteqn{ {_{s+1}E_r}\left({t_0,\ldots, t_{s} \atop w_1,\ldots,w_r};
\mathbf{\alpha}; q,p\right) } &&
\\ && \makebox[4em]{}
= \sum_{n=0}^\infty \frac{\theta(t_0,t_1,\ldots,t_{s};p;q)_n}
{\theta(q,w_1,\ldots,w_r;p;q)_n}\, e^{P_3(n)}.
\label{_rE_s}\end{aligned}$$ Actually, these series are slightly more general than those introduced in [@spi:theta], because in that paper we considered only the case where $\alpha_3=0$, but the generalization to (\[\_rE\_s\]) is straightforward. We note that the presence of cubics of the independent variable $y$ in (\[e-meijer\]) or $n$ in (\[\_rE\_s\]) is natural since we are working at the level of the Barnes multiple gamma function (\[m-gamma\]) of the third order.
Writing (\[\_rE\_s\]) in the form of the sum $\sum_{n=0}^\infty c_n$ with $c_0=1$, we easily see that $c_{n+1}/c_n=h(n)$, where $h(n)$ is given by (\[theta-int\]) with $w_0=q$ and $y=n$. This coincidence is not artificial. Consider the sequence of poles of the integrand in (\[e-meijer\]) located at $y=y_0+n, n\in\mathbb{N}$, for some $y_0$. We denote by $\kappa c_n, c_0=1,$ the residues of these poles. As $y\to y_0+n$, we have $\Delta(y)\to \kappa c_n/(y-y_0-n) +O(1)$. Now it is not difficult to see that $$\lim_{y\to y_0+n}\frac{\Delta(y+1)}{\Delta(y)}=\frac{c_{n+1}}{c_n}=
\lim_{y\to y_0+n} h(y)=h(y_0+n).$$ In particular, this means that the sums of the residues in the integral (\[e-meijer\]) that appear from appropriate deformations of the contour $C$, form the theta hypergeometric series (\[\_rE\_s\]) for some choices of the parameters.
In accordance with the classification introduced in [@spi:theta], the elliptic hypergeometric series correspond, by definition, to $h(n)$ equal to an elliptic function of $n$. Such series are called also the balanced theta hypergeometric series. They are defined by the following constraints imposed upon (\[\_rE\_s\]): $$s=r, \qquad \alpha_3=\alpha_2=0, \qquad \prod_{j=0}^rt_j=\prod_{j=0}^r w_j.
\label{theta-balance}$$ Similarly, the integral (\[e-meijer\]) will be called the [*elliptic*]{} (or [*balanced theta*]{}) hypergeometric integral if the conditions (\[theta-balance\]) are satisfied. Evidently, in this case $h(y)$ in (\[theta-int\]) becomes an elliptic function of $y$.
When $h(y)$ is an elliptic function of $y$ and of all parameters $u_j, v_j$ (we remind the reader that $t_j=q^{u_j}, w_j=q^{v_j}$), we call (\[\_rE\_s\]) and (\[e-meijer\]) the [*totally elliptic hypergeometric series*]{} and integrals, respectively. As was shown in [@spi:theta], in addition to the balancing requirement, such a property imposes the following constraints on the parameters $$\label{well-poised-2}
t_jw_j=\rho=const, \qquad j=0,\ldots,r,$$ which are known as the [*well-poisedness*]{} conditions in the theory of basic hypergeometric series [@gas-rah:basic]. The explicit form of the integrand $\Delta(y)$ for well-poised balanced theta hypergeometric integrals is $$\Delta(y)=\prod_{j=0}^{m-1}\frac{\Gamma(t_jz;q,p)}
{\Gamma(\rho t_j^{-1}z;q,p)}\; \frac{\Gamma(\rho^{\frac{m+1}{2}}
\prod_{j=0}^{m-1}t_j^{-1}\, z;q,p)}{\Gamma(\rho^{\frac{1-m}{2}}
\prod_{j=0}^{m-1}t_j\, z;q,p)}\,e^{\gamma y},
\label{wp-int}$$ where we have denoted $z=q^y$ and $\gamma=\alpha_1$. The parameter $\rho$ is redundant, it can be eliminated by the rescalings $t_i\to \rho^{1/2}t_i, z\to \rho^{-1/2}z$, but we keep it for further needs. Observe that without loss of generality one of the parameters in (\[e-meijer\]) can be set equal to one by a shift of $y$.
Without the balancing condition, a theta hypergeometric series $_{r+1}E_r$ is said to be [*well-poised*]{} if the constraints (\[well-poised-2\]) are valid with $w_0=q$, and [*very-well-poised*]{} if, in addition to (\[well-poised-2\]), we have $$\begin{aligned}
\nonumber
&& t_{r-3}=t_0^{1/2}q,\quad t_{r-2}=-t_0^{1/2}q, \\
&& t_{r-1}=t_0^{1/2}qp^{-1/2}, \quad t_{r} =- t_0^{1/2}qp^{1/2}.
\label{very-well-poised-2}\end{aligned}$$ Such series take a simpler form $$\begin{aligned}
\nonumber
\lefteqn{
_{r+1}E_r\left({t_0,t_1,\ldots, t_{r-4},qt_0^{1/2},-qt_0^{1/2},
qp^{-1/2}t_0^{1/2},-qp^{1/2}t_0^{1/2} \atop
qt_0/t_1,\ldots,qt_0/t_{r-4},t_0^{1/2},-t_0^{1/2},
p^{1/2}t_0^{1/2},-p^{-1/2}t_0^{1/2} };\mathbf{\alpha};q,p\right) } &&
\\ && \makebox[5em]{}
= \sum_{n=0}^\infty \frac{\theta(t_0q^{2n};p)}{\theta(t_0;p)}
\prod_{m=0}^{r-4}\frac{\theta(t_m;p;q)_n}{\theta(qt_0/t_m;p;q)_n}\, (-q)^n
e^{P_3(n)}.
\label{vwp-1}\end{aligned}$$ The essence of (\[very-well-poised-2\]) consists in the replacement of the product of four $\theta(t_iz;p)$ by one theta function $\propto\theta(t_0q^2z^2;p)$ (this corresponds to doubling the argument of the $\theta_1$-function). Very-well-poised series play a distinguished role in applications, in particular, they admit an appropriate generalization of the Bailey chains technique of generating infinite sequences of summation or transformation formulae [@spi:bailey].
In the case of integrals, we call (\[e-meijer\]) the [*very-well-poised*]{} theta hypergeometric integral if, in addition to conditions (\[well-poised-2\]), eight parameters $t_i$ are fixed in the following way: $$\begin{aligned}
(t_{m-8},\ldots,t_{m-1})=(\pm(pq)^{1/2}, \pm q^{1/2}p, \pm p^{1/2}q,\pm pq).
\label{vwp}\end{aligned}$$ These constraints lead to squaring the argument of the elliptic gamma function $$\prod_{j=m-8}^{m-1}\Gamma(t_jz;q,p)=\Gamma(pqz^{2};q,p)=\frac{1}{\Gamma(z^{-2};q,p)}
\label{doubling}$$ (such a relation was used already in [@spi:elliptic] in the derivation of the elliptic beta integral (\[ell-int\])). As a result, the integrand of the very-well-poised balanced theta hypergeometric integral takes the form $$\Delta(y)=\prod_{j=0}^{m-9}\frac{\Gamma(t_jz;q,p)}
{\Gamma(\rho t_j^{-1}z;q,p)}\, \frac{\Gamma(\rho^{\frac{m+1}{2}}
p^{-6}q^{-6}\prod_{j=0}^{m-9}t_j^{-1}z;q,p)\,e^{\gamma y} }
{\Gamma(z^{-2},(\rho/pq)^2 z^2, \rho^{\frac{1-m}{2}}p^6q^6
\prod_{j=0}^{m-9}t_j\, z;q,p)}.
\label{vwp-int}$$ After imposing conditions (\[vwp\]), the parameter $\rho$ is no longer redundant, and its choice plays an important role. If we fix it as $\rho=pq,$ then $\Delta(y)$ takes a particularly symmetric form $$\Delta(y)=\frac{\prod_{j=0}^{m-9}\Gamma(t_jz,t_jz^{-1};q,p)}
{\Gamma(z^2,z^{-2}, Az, Az^{-1};q,p)}\,e^{\gamma y},
\label{vwp-int-fin}$$ where $A=(pq)^{\frac{13-m}{2}} \prod_{j=0}^{m-9}t_j$. Clearly, the cases where $m$ is odd or even differ from each other in a qualitative way. The choice $m=13$ gives the simplest expression for $\Delta(y)$ and plays a distinguished role. Other simple choices, $m=9$ or $11$, correspond to particular subcases of the situation with $m=13$. For $m=13$ and $\gamma =0$ we get the integrand $\Delta_E$ of the elliptic beta integral (\[weight\]), that is the simplest very-well-poised elliptic hypergeometric integral turns out to be exactly calculable when $C$ is taken to be a special cycle corresponding to the unit circle on the $z$-plane.
The sums of the residues of the function (\[vwp-int-fin\]) for $m=13$ are expected to form a $_{14}E_{13}$ theta hypergeometric series. However, the very-well-poisedness condition (\[vwp\]) results in the cancellation of theta functions in the corresponding ratios of the series coefficients $h(n)$. As a result, we get only a $_{10}E_9$ very-well-poised elliptic hypergeometric series which, for $\gamma=0$, corresponds to the left-hand side of (\[ft-sum\]) or (\[e-milne\]) at $n=1$ (for more details, see [@die-spi:elliptic; @spi:theta]). This shift of indices $m\to m-4$ brings in one more intriguing point related to the origins of the very-well-poisedness condition. It is necessary to find some deeper algebraic geometry explanations of the fact that in the single variable case “the nice things" (summation or integration formulae) are related to the number 14, the order of the initial elliptic function $h(y)$. As was shown in [@spi:theta; @spi:modularity], in the multiple case this order raises to higher numbers, but in quite an intriguing way as well.
Since $\Gamma((pq)^{1/2}z,(pq)^{1/2}z^{-1};q,p)=1$, we may drop two parameters $\pm(pq)^{1/2}$ in the condition (\[vwp\]). For $\rho=pq, \gamma=0,$ this yields $$\Delta(y)=\frac{\prod_{j=0}^{m-7}\Gamma(t_jz,t_jz^{-1};q,p)}
{\Gamma(z^2,z^{-2},-Az,-Az^{-1};q,p)},
\label{-A}$$ where $A=(pq)^\frac{11-m}{2} \prod_{j=0}^{m-7}t_j$. For $m=11$ this expression looks similar to (\[weight\]), but the different sign in front of $A$ changes the things drastically, and it is not known whether the corresponding integral gets any closed form expression.
In a more general setting, we can impose balancing and very-well-poisedness conditions upon general theta hypergeometric integrals (\[gen-int\]) and (\[gen-int’\]). However, at the moment it is not known whether the simplest integrals appearing in this way admit exact evaluation.
As far as the multivariable integrals of hypergeometric type are concerned, the general form of $\Delta(\mathbf{y})$ in the plain and $q$-hypergeometric cases can be deduced from the Ore-Sato theorem for Horn’s series (see, e.g., [@ggr:general] for a detailed discussion). The general form of the multiple elliptic hypergeometric series or integrals is not established yet. We formulate it as an open problem—to find an elliptic or general theta functions analog of the Ore-Sato characterization theorem. In the following sections we give a series of examples of multivariable extensions of the very-well-poised balanced theta hypergeometric integrals associated with the root systems $A_n$ and $C_n$.
As to the further possible generalizations, it is natural to consider integrals of hypergeometric type for arbitrary algebraic curves or general Abelian varieties. Both would involve Riemann theta functions of many variables, appropriate generalizations of gamma functions and theta hypergeometric series. Some preliminary discussion of ideas in this direction can be found in [@spi:modularity], in particular, a special subcase of the $_8\Phi_7$ Jackson summation formula was generalized there to Riemann surfaces of arbitrary genus.
Known $C_n$ elliptic beta integrals
===================================
The following multivariable generalization of the Euler beta integral (\[euler\]) has been introduced by Selberg [@aar:special]: $$\begin{aligned}
\lefteqn{\int_0^1\cdots\int_0^1
\prod_{1\leq j\leq n} x_j^{\alpha-1} (1-x_j)^{\beta-1}
\prod_{1\leq j<k\leq n}|x_j-x_k|^{2\gamma}\; dx_1\cdots dx_n } &&
\nonumber \\
&& =\prod_{1\leq j\leq n} \frac{\Gamma(\alpha+(j-1)\gamma)
\Gamma(\beta+(j-1)\gamma) \Gamma (1+j\gamma)}
{\Gamma (\alpha+\beta+(n+j-2)\gamma) \Gamma (1+\gamma)},
\label{Sint}\end{aligned}$$ where $\text{Re}(\alpha ),\, \text{Re} (\beta) >0$, and $\text{Re}(\gamma ) > - \min (1/n,
\text{Re} (\alpha )/(n-1) ,\text{Re} (\beta)/(n-1) )$. This integral has found many important applications in mathematical physics.
At the first glance, the integrand of (\[Sint\]) does not fit the definition of the hypergeometric integrals introduced in the preceding section. However, we can rescale $y_i\to y_i\epsilon $ in (\[definition\]), and choose rational functions $R_i(\mathbf{y})$ appropriately so that the limit $\epsilon\to 0$ becomes well defined and yields a system of linear differential equations of the first order. As a result, we get $\partial\Delta(\mathbf{y})/
\partial y_i=R_i(\mathbf{y})\Delta(\mathbf{y})$ and, apparently, (\[Sint\]) satisfies these conditions.
Two types of multidimensional generalizations of the elliptic beta integral (\[ell-int\]) to the root system $C_n$ were proposed by van Diejen and the author in [@die-spi:elliptic; @die-spi:selberg]. One of them reduces in a special limit to the Selberg integral (\[Sint\]). Here we describe these elliptic Selberg integrals explicitly. For brevity, we drop the bases $p,q$ in the notation for elliptic gamma functions from now on. We introduce the Type I integrand as $$\begin{aligned}
\Delta^I ({\bf z};C_n)&=& \frac{1}{(2\pi i)^n}
\prod_{1\leq j<k\leq n}
\Gamma^{-1}(z_jz_k,z_jz_k^{-1},z_j^{-1}z_k,z_j^{-1}z_k^{-1}) \nonumber \\
&&\times\prod_{j=1}^n\frac{\prod_{r=0}^{2n+2}\Gamma(t_rz_j,t_rz_j^{-1})}
{\Gamma(z_j^2,z_j^{-2}, Az_j,Az_j^{-1})},
\label{esintA}\end{aligned}$$ where $t_r\in\mathbb{C},\, r=0,\ldots,2n+2,$ are free parameters and $A\equiv\prod_{r=0}^{2n+2}t_r$. The Type II integrand has the form $$\begin{aligned}
\Delta^{II} (\mathbf{z};C_n)&=&\frac{1}{(2\pi i)^n}\prod_{1\leq j<k\leq n}
\frac{\Gamma(tz_jz_k,tz_jz_k^{-1},tz_j^{-1}z_k,tz_j^{-1}z_k^{-1})}
{\Gamma(z_jz_k,z_jz_k^{-1},z_j^{-1}z_k,z_j^{-1}z_k^{-1})} \nonumber \\
&& \times \prod_{j=1}^n\frac{\prod_{r=0}^4\Gamma(t_rz_j,t_rz_j^{-1})}
{\Gamma(z_j^2,z_j^{-2}, B z_j, B z_j^{-1})},
\label{esintB}\end{aligned}$$ where $t, t_r\in \mathbb{C},\, r=0,\ldots,4,$ are free parameters and $B\equiv t^{2n-2}\prod_{s=0}^4t_s$. By $\mathbb{T}$ we denote the unit circle with positive orientation.
Consider the first type of the $C_n$ multivariable elliptic beta integral. We take $|p|,|q|<1$ and $|t_r|<1,\, r=0,\ldots, 2n+2,$ and assume that $|pq|<|A|$. Then $$\begin{aligned}
\lefteqn{\int_{\mathbb{T}^n}\Delta^I ({\bf z};C_n)\frac{dz_1}{z_1}
\cdots \frac{dz_n}{z_n} } && \nonumber \\ &&
=\frac{2^n n!}{(p;p)_\infty^n(q;q)_\infty^n}
\frac{\prod_{0\leq r<s\leq 2n+2}\Gamma(t_rt_s)}
{\prod_{r=0}^{2n+2}\Gamma(t_r^{-1}A)}.
\label{SintA}\end{aligned}$$
The second type of the multiple elliptic beta integral has the following form. We take $|p|,|q|, |t|<1$ and $|t_r|<1,\, r=0,\ldots, 4,$ and assume that $|pq|<|B|$. Then $$\begin{aligned}
\lefteqn{\int_{\mathbb{T}^n}\Delta^{II} ({\bf z};C_n)\frac{dz_1}{z_1}
\cdots \frac{dz_n}{z_n} } && \nonumber \\
&& = \frac{2^n n!}{(p;p)_\infty^n(q;q)_\infty^n}
\prod_{j=1}^n\frac{\Gamma(t^j)}{\Gamma(t)}
\frac{\prod_{0\leq r<s\leq 4}\Gamma(t^{j-1}t_rt_s)}
{\prod_{r=0}^4\Gamma(t^{1-j}t_r^{-1}B)}.
\label{SintB}\end{aligned}$$
The elliptic Selberg integral of Type II (\[SintB\]) can be deduced from the Type I integral with the help of some interesting trick [@die-spi:selberg]. The type I integral (\[esintA\]) was proved in [@die-spi:selberg] under a vanishing hypothesis, namely, that its left-hand side vanishes on the hypersurface of parameters $A=t_{2n+2}$ after an appropriate deformation of $\mathbb{T}$ to an integration contour $C$ that separates the sequences of poles in $\mathbf{z}$ converging to zero from those diverging to infinity. As $p\to 0$, both Type I and II integrals are reduced to Gustafson’s well-known $q$-Selberg integrals [@gus:some1; @gus:some2], which are related (for particular choices of parameters) to the Macdonald-Morris constant term identities and the Macdonald polynomials for various root systems [@mac:constant], including the Koornwinder polynomials [@kor:pol].
As was shown in [@die-spi:elliptic; @die-spi:modular], the sums of residues of the functions (\[esintA\]) and (\[esintB\]) form some multiple elliptic hypergeometric series. In particular, the multivariable $_{10}E_9$ sum conjectured by Warnaar in [@war:summation] can be deduced from (\[SintB\]). The residue calculus for (\[SintA\]) yields an elliptic extension of the basic hypergeometric series summation formula proved in [@den-gus:beta] and [@mil-lil:consequences]. Recursive proofs of these multivariable Frenkel-Turaev sums were given by Rosengren in [@ros:proof; @ros:elliptic]. The property of well-poisedness (or total ellipticity, in the case of elliptic hypergeometric series) plays an important role in such summation formulas. First examples of (plain) multiple hypergeometric series well-poised on classical groups were introduced by Biedenharn, Holman, and Louck in [@hbl:hypergeometric].
A new $C_n$ integration formula
===============================
We define $$\begin{aligned}
\nonumber
&& \Delta^{III}(\mathbf{z};C_n) = \frac{1}{(2\pi i)^n}
\prod_{1\leq i<j\leq n}z_j\theta(z_iz_j^{-1},z_i^{-1}z_j^{-1};p)
\\ && \makebox[2em]{} \times
\prod_{i=1}^n\prod_{\nu=\pm 1}\frac{\Gamma(z_i^\nu x_i,z_i^\nu t_1,
z_i^\nu t_2,z_i^\nu t_3,z_i^\nu t/x_i)}
{\Gamma(z_i^{2\nu},z_i^\nu A)},
\label{delta-c3}\end{aligned}$$ where $A=tt_1t_2t_3q^{n-1}$.
Impose the following restrictions upon parameters: $|x_i|,$ $|t_k|<1,$ $|t|<|x_i|$, where $i=1,\ldots, n$, $k=1,2,3$, and $|pq|<|A|$. Then $$\begin{aligned}
\nonumber
&& \int_{\mathbb{T}^n}\Delta^{III}(\mathbf{z};C_n)
\frac{dz_1}{z_1}\cdots\frac{dz_n}{z_n}
=\frac{2^n}{(p;p)_\infty^n(q;q)_\infty^n}
\prod_{1\leq i<j\leq n}x_j\theta\left(x_i/x_j,t/x_ix_j;p\right)
\\ && \makebox[4em]{} \times
\Gamma^n(t)\prod_{i=1}^n\left(\frac{\prod_{1\leq r<s\leq 3}
\Gamma(t_rt_sq^{i-1})}{\Gamma(A/x_i,Ax_i/t)} \prod_{k=1}^3\frac{
\Gamma(x_it_k,tt_k/x_i)}{\Gamma(Aq^{1-i}/t_k)}\right).
\label{c3}\end{aligned}$$
Consider the determinant $$\begin{aligned}
\nonumber
&& \det_{1\leq i,j\leq n}\left(\int_\mathbb{T}
\Delta_E(z,x_i,t_1q^{n-j},t_2q^{j-1},t_3,tx_i^{-1})\frac{dz}{z}\right)
\\ && \makebox[2em]{}
=\frac{1}{(2\pi i)^n}\int_{\mathbb{T}^n}\frac{dz_1}{z_1}\cdots
\frac{dz_n}{z_n}\prod_{i=1}^n G_i(z_i)G_i(z_i^{-1})\; D(\mathbf{z}),
\label{det1}\end{aligned}$$ where $\Delta_E$ is the integrand of the elliptic beta integral (\[weight\]) with an appropriate choice of the parameters. The expression standing on the right-hand side of (\[det1\]) appears after taking the integral signs outside of the determinant symbol, so that we get a multiple integral with $$\begin{aligned}
&& G_i(z_i)=\frac{\Gamma(z_ix_i,z_it_1,z_it_2,z_it_3,z_it/x_i)}
{\Gamma(z_i^2,z_iA)},
\\
&& D(\mathbf{z})=\det_{1\leq i,j\leq n}\left(\theta(z_it_1,
z_i^{-1}t_1;p;q)_{n-j}\theta(z_it_2,z_i^{-1}t_2;p;q)_{j-1}\right).\end{aligned}$$ The determinant $D(\mathbf{z})$ can be rewritten as follows $$\begin{aligned}
&& D(\mathbf{z})=\frac{\prod_{i=1}^n \theta(z_it_2,z_i^{-1}t_2;p;q)_{n-1}}
{t_2^{2\binom{n}{2}} q^{4\binom{n}{3}} }
\\ && \makebox[4em]{} \times
\det_{1\leq i,j \leq n}
\left(\frac{\theta(z_it_1,z_i^{-1}t_1;p;q)_{n-j}}
{\theta(q^{2-n}/z_it_2,q^{2-n}z_i/t_2;p;q)_{n-j}}\right).\end{aligned}$$ In [@war:summation], Warnaar computed the following elliptic generalization of a Krattenthaler determinant [@kra:major]: $$\begin{aligned}
&& \det_{1 \leq i,j \leq n} \left(
\frac{ \theta(aX_i,ac/X_i;p;q)_{n-j} }
{ \theta(bX_i,bc/X_i;p;q)_{n-j} } \right)
\nonumber \\ && \makebox[4em]{}
= a^{\binom{n}{2}}q^{\binom{n}{3}}
\prod_{1\leq i<j\leq n}X_j\theta(X_iX_j^{-1},cX_i^{-1}X_j^{-1};p)
\nonumber \\ && \makebox[6em]{}
\times \prod_{i=1}^n \frac{\theta(b/a,abcq^{2n-2i};p;q)_{i-1}}
{\theta(bX_i,bc/X_i;p;q)_{n-1}}.
\label{e-kratt} \end{aligned}$$ Using this identity for $X_i=z_i$, $a=t_1, b=q^{2-n}/t_2$, and $c=1$, we find $$\begin{aligned}
\nonumber
&& D(\mathbf{z})=(t_1t_2^2)^{\binom{n}{2}}q^{3\binom{n}{3}}
\prod_{1\leq i<j\leq n}z_j\theta(z_iz_j^{-1},z_i^{-1}z_j^{-1};p)
\\ && \makebox[4em]{} \times
\prod_{i=1}^n \theta(q^{2-n}/t_1t_2,t_1q^{n+2-2i}/t_2;p;q)_{i-1}.
\label{det-aux}\end{aligned}$$ As a result, the determinant (\[det1\]) yields an expression proportional to the left-hand side of the $C_n$ multiple integral in question (\[c3\]).
Now we substitute the result of computation of the elliptic beta integral (\[ell-int\]) into the determinant (\[det1\]). This yields $$\begin{aligned}
\nonumber
&& \text{l.h.s. of (\ref{det1})} = \frac{2^n}{(q;q)_\infty^n(p;p)_\infty^n}
\\ && \makebox[1em]{} \times
\prod_{i=1}^n\frac{\Gamma(x_it_1,t_1t/x_i,x_it_2,t_2t/x_i,
x_it_3,t,t_1t_2q^{n-1},t_1t_3q^{n-i},t_2t_3q^{i-1}, t_3t/x_i)}
{\Gamma(A/x_i,A/t_3,x_iA/t,Aq^{1-i}/t_2,Aq^{i-n}/t_1)}
\nonumber \\ && \makebox[1em]{}
\times \det_{1\leq i,j\leq n}\left(\theta(x_it_1,t_1t/x_i;p;q)_{n-j}
\theta(x_it_2,t_2t/x_i;p;q)_{j-1}\right).
\label{rhs}\end{aligned}$$
By (\[e-kratt\]), the determinant in the last line takes the form $$\begin{aligned}
\nonumber
&& (t_1t_2^2t)^{\binom{n}{2}}q^{3\binom{n}{3}}
\prod_{1\leq i<j\leq n}x_j\theta\left(x_i/x_j,t/x_ix_j;p
\right) \\ && \makebox[4em]{} \times
\prod_{i=1}^n \theta(q^{2-n}/t_1t_2t,t_1q^{n+2-2i}/t_2;p;q)_{i-1}.\end{aligned}$$ Equating the resulting expression in (\[rhs\]) with the right-hand side of (\[det1\]), we get the desired integral evaluation (\[c3\]).
This is the first nontrivial multiple elliptic beta integral with a complete proof. It is not symmetric in $p$ and $q$, unlike all other cases considered in the present paper. This fact suggests that there should exist yet another integral of similar nature that would be symmetric in $p,q$.
The above method of computation of the taken $C_n$ integral represents a next step in the logical development of applications of determinant formulas to multiple basic hypergeometric series; see, e.g., the paper [@gus-kra:determinant] of Gustafson and Krattenthaler, which was followed by Schlosser [@sch:summation; @sch:nonterminating] and Warnaar [@war:summation]. Similar considerations for computing some multiple $q$-hypergeometric integrals were given by Tarasov and Varchenko in [@tar-var:geometry].
An elliptic beta integral for the $A_n$ root system
===================================================
In this section we conjecture a multiple elliptic beta integral for the $A_n$ root system, which will be used in the next section for derivation of other nontrivial $A_n$ integrals.
Let $z_i,\: i=1,\ldots,n,$ $t_k,\: k=1,\ldots,$ $n+1,$ and $f_j,\: j=1,\ldots,$ $n+2,$ be independent complex variables ($n$ is an arbitrary positive integer). We denote $A\equiv \prod_{k=1}^{n+1}t_k$, $B\equiv\prod_{j=1}^{n+2}f_j$, and $$\Delta^I(\mathbf{z};A_n)=\frac{1}{(2\pi i)^n}
\frac{\prod_{k=1}^{n+1}\left(\prod_{i=1}^{n+1}\Gamma(t_iz_k^{-1})
\prod_{j=1}^{n+2}\Gamma(f_jz_k)\right)}
{\prod_{i,j=1;\: i\neq j}^{n+1}\Gamma(z_iz_j^{-1})
\prod_{k=1}^{n+1}\Gamma(ABz_k)},
\label{delta-an}$$ where $z_1z_2\cdots z_{n+1}=1$.
Suppose that the parameters $t_k, f_j$ satisfy the constraints $|t_k|, |f_j|<1, |pq|<|AB|$. Then the following integration formula is conjectured to hold true: $$\begin{aligned}
\nonumber
&& \int_{\mathbb{T}^n}\Delta^I(\mathbf{z};A_n)
\frac{dz_1}{z_1}\cdots\frac{dz_n}{z_n}
=\frac{(n+1)!}{(q;q)_\infty^n(p;p)_\infty^n}
\\ && \makebox[2em]{} \times
\frac{\Gamma(A)\prod_{j=1}^{n+2}\Gamma(f_j^{-1}B)\:
\prod_{k=1}^{n+1}\prod_{j=1}^{n+2}\Gamma(t_kf_j)}
{\prod_{k=1}^{n+1}\Gamma(t_kB)\:\prod_{j=1}^{n+2}\Gamma(f_j^{-1}AB)}.
\label{int-an} \end{aligned}$$
For $n=1$ this conjecture is reduced to the elliptic beta integral (\[ell-int\]). For arbitrary $n$ and $p=0$, we get a Gustafson integral proved in [@gus:some1]. Let us show that the two sides of (\[int-an\]) satisfy one and the same difference equation.
Let $I_n(\mathbf{t},\mathbf{f})$ denote either side of (\[int-an\]). Then this function satisfies the $q$-difference equation $$\sum_{r=1}^{n+1}
\frac{\theta(Bt_r;p)}{\theta(A;p)}
\prod_{\stackrel{j=1}{j \neq r}}^{n+1}
\frac{\theta(ABt_j;p)}{\theta(t_rt_j^{-1};p)}
I_n(t_1,\ldots,qt_r,\ldots,t_{n+1},\mathbf{f})=I_n(\mathbf{t},\mathbf{f})
\label{an-eq}$$ and its partner obtained by the permutation of $q$ and $p$.
Denote the function (\[delta-an\]) by $\Delta^I(\mathbf{z};t_1,\ldots,t_{n+1};A_n)$. It is not difficult to see that $$\frac{\Delta^I(\mathbf{z};t_1,\ldots,qt_r,\ldots,t_{n+1};A_n)}
{\Delta^I(\mathbf{z};t_1,\ldots,t_{n+1};A_n)}
=\prod_{k=1}^{n+1}\frac{\theta(t_rz_k^{-1};p)}{\theta(ABz_k;p)},$$ so that equation (\[an-eq\]) for the left-hand side of (\[int-an\]) is satisfied if the following theta functions identity is fulfilled: $$\sum_{r=1}^{n+1} \frac{\theta(Bt_r;p)}{\theta(A;p)}
\prod_{\stackrel{j=1}{j \neq r}}^{n+1}
\frac{\theta(ABt_j;p)}{\theta(t_rt_j^{-1};p)}
\prod_{k=1}^{n+1}\frac{\theta(t_rz_k^{-1};p)}{\theta(ABz_k;p)}=1.
\label{id1}$$ For $n=1$ this identity is equivalent to the well-known relation for products of four theta functions $$\begin{aligned}
\nonumber
\lefteqn{\theta(xw,x/w,yz,y/z;p) -\theta(xz,x/z,yw,y/w;p)}
\makebox[8em]{} && \\
&& =yw^{-1}\theta(xy,x/y,wz,w/z;p)
\label{ident}\end{aligned}$$ and equation (\[an-eq\]) coincides with that used in [@spi:elliptic] for proving the integral (\[ell-int\]).
For $n>1$, identity (\[id1\]) can be established with the help of the Liouville theorem, much as in the arguments presented in [@die-spi:selberg]. A simpler proof follows from the general theta functions identity given in [@whi-wat:course]. As was shown by Rosengren in [@ros:elliptic], that identity can be rewritten as the following generalized partial fractions expansion of a ratio of theta functions: $$\prod_{k=1}^n\frac{\theta(t/b_k;p)}{\theta(t/a_k;p)}
=\sum_{r=1}^n\frac{\theta(t a_1\cdots a_n /a_rb_1\cdots b_n;p)}
{\theta(t/a_r,a_1\cdots a_n/b_1\cdots b_n;p)}
\frac{\prod_{j=1}^n\theta(a_r/b_j;p)}
{\prod_{\stackrel{j=1}{j \neq r}}^n\theta(a_r/a_j;p)},
\label{id2}$$ where $a_1\cdots a_n\neq b_1\cdots b_n$. Here we replace $n$ by $n+1$ and substitute $a_k=t_k^{-1}$, $b_k=z_k^{-1}$, and $t=AB$. As a result, we get an identity which is seen to coincide with (\[id1\]) due to the relation $z_1\cdots z_{n+1}=1$.
In a similar way, for the right-hand side of (\[int-an\]) we get $$\frac{I_n(t_1,\ldots,qt_r,\ldots,t_{n+1}, \mathbf{f} ) }
{I_n(\mathbf{t},\mathbf{f})}=
\frac{\theta(A;p)}{\theta(t_rB;p)}
\prod_{j=1}^{n+2}\frac{\theta(t_rf_j;p)}{\theta(ABf_j^{-1};p)},$$ and in this case equation (\[an-eq\]) becomes equivalent to the identity $$\frac{\prod_{j=1}^{n+2}\theta(ABf_j^{-1};p)}
{\prod_{j=1}^{n+1}\theta(ABt_j;p)}= \sum_{r=1}^{n+1}
\frac{ \prod_{j=1}^{n+2}\theta(t_rf_j;p) }
{\prod_{\stackrel{j=1}{j \neq r}}^{n+1} \theta(t_rt_j^{-1};p) }
\frac{1}{\theta(ABt_r;p)}.
\label{id3}$$ If we substitute here $f_{n+2}=B/f_1\cdots f_{n+1}$ and divide both sides by $\theta(ABf_{n+2}^{-1};p)$, then we get (\[id2\]) with $n$ replaced by $n+1$ and with $a_j=t_j^{-1}, b_j=f_j, t=AB.$
The equation derived above works in the space of parameters $t_k$, whereas in [@gus:some1] Gustafson used an equation in the variables $f_j$ for proving the $p=0$ case of the integral (\[int-an\]). It is of interest that the latter equation does not admit a straightforward elliptic generalization, namely, the corresponding partial fraction expansion cannot be lifted to the theta functions level.
Another argument in favor of the validity of formula (\[int-an\]) consists in the fact that, via the residue calculus, it generates a multivariable $_{10}E_9$ elliptic hypergeometric series sum for the $A_n$ root system, which was proved by Rosengren in [@ros:elliptic] and which was considered independently by the author in [@spi:modularity].
The residue calculus for the integral (\[int-an\]) yields the following summation formula: $$\begin{aligned}
\nonumber
\lefteqn{ \sum_{\stackrel{0\leq \lambda_j \leq N_j}{j=1,\ldots, n}}
q^{\sum_{j=1}^nj\lambda_j}
\prod_{j=1}^n\frac{\theta(t_jq^{\lambda_j+|\lambda|};p)}{\theta(t_j;p)}
\prod_{1\leq i<j \leq n} \frac{\theta(t_it_j^{-1}q^{\lambda_i-\lambda_j};p)}
{\theta(t_it_j^{-1};p)} } &&
\\ \nonumber
&& \times \prod_{i,j=1}^n\frac{\theta(t_it_j^{-1}q^{-N_j};p;q)_{\lambda_i}}
{\theta(qt_it_j^{-1};p;q)_{\lambda_i}}
\prod_{j=1}^n\frac{\theta(t_j;p;q)_{|\lambda|}}
{\theta(t_jq^{1+N_j};p;q)_{|\lambda|}}
\\ \nonumber
&& \times \frac{\theta(b,c;p;q)_{|\lambda|}}
{\theta(q/d, q/e;p;q)_{|\lambda|}}
\prod_{j=1}^n \frac{\theta(dt_j, et_j;p;q)_{\lambda_j}}
{\theta(t_jq/b, t_jq/c;p;q)_{\lambda_j}}
\\
&&
=\frac{\theta(q/bd,q/cd;p;q)_{|N|}}{\theta(q/d,q/bcd;p;q)_{|N|}}
\prod_{j=1}^n\frac{\theta(t_jq,t_jq/bc;p;q)_{N_j}}
{\theta(t_jq/b,t_jq/c;p;q)_{N_j}},
\label{e-milne}\end{aligned}$$ where $|\lambda|=\lambda_1+\cdots+\lambda_n$, $|N|=N_1+\cdots+N_n$, and $bcde=q^{1+|N|}$. For $n=1$ this is the Frenkel-Turaev sum [@fre-tur:elliptic], and for $p=0$ it is reduced to the Milne’s multiple $_8\Phi_7$ sum for the $A_n$ root system [@mil:multidimensional].
We scale $t_i$ for $i=1,\ldots, n$ from the region $|t_i|<1$ to $|t_i|>1$, and keep $|t_{n+1}|, |f_j|<1$ together with the condition $|pq|<|AB|$. During this procedure, some poles of the integrand $\Delta^I(\mathbf{z};A_n)$ in (\[int-an\]) go out of the unit disk and, on the contrary, some of them cross over $\mathbb{T}$ entering inside. The outgoing poles are located at the following points: $z_k=\{t_iq^{\lambda_i}, i=1,\ldots,n\}$ for each $k=1,\ldots, n$, and the number of such poles is determined by the conditions $|t_iq^{\lambda_i}|>1$. The ingoing poles correspond to the points $z_1\cdots z_n=\{t_i^{-1}q^{-\lambda_i},
i=1,\ldots,n\}$.
We denote by $C$ a deformed contour of integration such that none of the poles mentioned above crosses over $C$ during the change of parameters. By analyticity, the value of the integral (\[int-an\]) is not changing when $C$ replaces $\mathbb{T}$, that is the right-hand side of (\[int-an\]) remains the same. If we start to deform the contour $C$ back to $\mathbb{T}$, we start to pick up residues from the poles by the Cauchy theorem. As a result, the following formula arises: $$\int_{C^n}\Delta^I(\mathbf{z};A_n)\prod_{k=1}^n\frac{dz_k}{z_k}
=\sum_{j=0}^{n}\int_{\mathbb{T}^j} R_j(z_1,\ldots,z_j)\prod_{k=1}^j
\frac{dz_k}{z_k},
\label{res}$$ where $R_n=\Delta^I(\mathbf{z};A_n)$, and $R_j(z_1,\ldots,z_j)$ for $j<n$ are sums of the residues of $\Delta^I(\mathbf{z};A_n)$ corresponding to the poles crossing $C$.
We shall not derive explicit expressions for all coefficients $R_j$ as it was done for the $C_n$ integrals in [@die-spi:elliptic; @die-spi:modular]. For our purposes, it suffices to pick up only the residues that diverge in the limits $f_j\to q^{-N_j}t_j^{-1}$, $N_j\in \mathbb{N}$, for all $j=1,\ldots, n$ simultaneously. First, consider the residues appearing from the poles $z_j= t_jq^{\lambda_j}$, where $\lambda_j$ are some integers such that $|t_jq^{\lambda_j}|>1$. Straightforward computations yield $$\begin{aligned}
\nonumber
&& R_0^{div}(\mathbf{\lambda})\equiv\prod_{j=1}^n
\lim_{z_j\to t_jq^{\lambda_j}}
(1-t_jq^{\lambda_j}z_j^{-1})\; \Delta^I(\mathbf{z};A_n)
\\ \nonumber && \makebox[2em]{}
= \prod_{\stackrel{i,j=1}{i\neq j}}^n\Gamma^{-1}(t_it_j^{-1}
q^{\lambda_i-\lambda_j})
\frac{\prod_{i=1}^{n+1}\Gamma(t_iDq^{|\lambda|})
\prod_{j=1}^{n+2}\Gamma(f_jD^{-1}q^{-|\lambda|})}
{\prod_{k=1}^n\Gamma(t_kq^{\lambda_k}Dq^{|\lambda|},
D^{-1}q^{-|\lambda|}t_k^{-1}q^{-\lambda_k})}
\\ \nonumber && \makebox[4em]{} \times
\frac{\prod_{k=1}^n\left(\prod_{\stackrel{i=1}{i\neq k}}^{n+1}
\Gamma(t_it_k^{-1}q^{-\lambda_k}) \prod_{j=1}^{n+2}
\Gamma(f_jt_kq^{\lambda_k})\right)}
{\prod_{k=1}^n\Gamma(AB t_kq^{\lambda_k})\, \Gamma(ABD^{-1}q^{-|\lambda|})}
\\ && \makebox[4em]{} \times
\prod_{k=1}^n\frac{(-1)^{\lambda_k}q^{\lambda_k(\lambda_k+1)/2}}
{(q;q)_\infty(p;p)_\infty\theta(q;p;q)_{\lambda_k}},
\label{residue}\end{aligned}$$ where $D=A/t_{n+1}$. The factors $\Gamma(f_jt_jq^{\lambda_j})$ provide the required divergence in the limits $f_j\to q^{-N_j}t_j^{-1}$. We write $R_0^{div}(\mathbf{\lambda})= \kappa_n
\Delta(\mathbf{\lambda};A_n)$, where $$\begin{aligned}
&& \kappa_n = \prod_{1\leq i<j\leq n}\Gamma^{-1}(t_it_j^{-1},
t_i^{-1}t_j)\frac{\prod_{i=1}^{n+1}\Gamma(t_iD)
\prod_{j=1}^{n+2}\Gamma(f_jD^{-1})}
{\prod_{k=1}^n\Gamma(t_kD,t_k^{-1}D^{-1})}
\\ && \makebox[2em]{} \times
\frac{\prod_{k=1}^n\left(\prod_{\stackrel{i=1}{i\neq k}}^{n+1}
\Gamma(t_it_k^{-1}) \prod_{j=1}^{n+2}\Gamma(f_jt_k)\right)}
{(q;q)_\infty^n(p;p)_\infty^n\prod_{k=1}^n\Gamma(AB t_k)\,\Gamma(ABD^{-1})},\end{aligned}$$ and after a chain of simplifying calculations, $\Delta(\mathbf{\lambda};A_n)$ takes the form $$\begin{aligned}
&& \Delta(\mathbf{\lambda};A_n)=
q^{\sum_{j=1}^nj\lambda_j}\prod_{1\leq i<j\leq n}
\frac{\theta(t_it_j^{-1}q^{\lambda_i-\lambda_j};p)}
{\theta(t_it_j^{-1};p)}
\\ && \makebox[4em]{} \times
\prod_{j=1}^n\left(\frac{\theta(t_jD;p;q)_{|\lambda|}}
{\theta(qf_j^{-1}D;p;q)_{|\lambda|}}
\frac{\theta(t_jDq^{|\lambda|+\lambda_j};p)}{\theta(t_jD;p)}
\prod_{k=1}^n\frac{\theta(f_jt_k;p;q)_{\lambda_k}}
{\theta(qt_j^{-1}t_k;p;q)_{\lambda_k}}\right)
\\ && \makebox[4em]{} \times
\frac{\theta(cD,qD/AB;p;q)_{|\lambda|}}
{\theta(qD/d,qD/e;p;q)_{|\lambda|}}
\prod_{k=1}^n\frac{\theta(dt_k,et_k;p;q)_{\lambda_k}}
{\theta(qt_k/c,ABt_k;p;q)_{\lambda_k}},\end{aligned}$$ where we have denoted $c=t_{n+1},d=f_{n+1},e=f_{n+2}$. Now we substitute $f_j=q^{-N_j}t_j^{-1}$ (which assumes that $AB=q^{-|N|}cde$) in this expression and introduce the parameter $b\equiv q^{1+|N|}/cde$. As a result, the function $\Delta(\mathbf{\lambda};A_n)$ becomes equal to the summand on the left-hand side of (\[e-milne\]) after the transformations $t_j\to t_j/D$, $b\to b/D,$ $c\to c/D$, $d\to Dd$, $e\to De$.
Now, we find the total number of residues of such type. There is permutational symmetry between the variables $z_1,\ldots, z_n$. Therefore, there are $n!$ ways to satisfy the equalities $z_k=t_jq^{\lambda_j}$ using each $t_j$ only once. The residues of the other outgoing poles located at $z_k=\{t_iq^{\lambda_i}, i=1,\ldots,n\}$, where at least one $t_i$ enters twice, do not diverge at $f_j\to q^{-N_j}t_j^{-1}$ for some $j$.
We pass to the ingoing poles. It is not difficult to verify that the residues of $\Delta^I(\mathbf{z};A_n)$ for the poles located at $z_n=t_j^{-1}q^{-\lambda_j}/z_1\cdots z_{n-1}$ for some fixed $j$ (or, equivalently, for $z_{n+1}=t_jq^{\lambda_j}$) are equal to the residues for the poles at $z_n=t_jq^{\lambda_j}$. Among the remaining poles in the variables $z_1,\ldots,z_{n-1}$, we must consider only the outgoing ones since only they may diverge as $f_k\to q^{-N_k}t_k^{-1}$ with $k=1,\ldots, n, k\neq j$. There are $n$ ways to fix the variable $z_k$ for which we shall consider ingoing poles, there are $n$ ways to fix the parameter $t_j$ in the equation $z_{n+1}=t_jq^{\lambda_j}$, and there are $(n-1)!$ appropriate outgoing poles with the required residue divergence. As a result, the contribution of these combined ingoing and outgoing poles is equal to $n^2 (n-1)!$, and the total number of diverging residues (\[residue\]) is equal to $(n+1)!$. Roughly speaking, the incoming poles imitate the $(n+1)$st independent contour of integration over $z_{n+1}$, which enters symmetrically with $z_1,\ldots,z_n$, and there are $(n+1)!$ ways to order these variables in the residue calculus.
As has already been mentioned, (\[res\]) is equal to the right-hand side of (\[int-an\]). Now we divide both these expressions by $(n+1)!\kappa_n$ and take the limits as $f_j\to q^{-N_j}t_j^{-1},
j=1,\ldots,n.$ Since $\kappa_n\to \infty$ in this limit, only the residues considered above survive in (\[res\]), and their sum is given by the elliptic Milne series (\[e-milne\]). As to the right-hand side, we get $$\begin{aligned}
&& \lim_{f_j\to q^{-N_j}t_j^{-1}}\frac{\text{r.h.s. of (\ref{int-an})}}
{(n+1)!\kappa_n} = \frac{\theta(q/bd,q/cd;p;q)_{|N|}}
{\theta(qD/d,q/Dbcd;p;q)_{|N|}}
\\ && \makebox[4em]{} \times
\prod_{j=1}^n\frac{\theta(qt_jD,qt_j/Dbc;p;q)_{N_j}}
{\theta(qt_j/c,qt_j/b;p;q)_{N_j}},\end{aligned}$$ which coincides with the right-hand side of (\[e-milne\]) after the appropriate changes of parameters indicated above. The theorem is proved.
Formula (\[int-an\]) generates the following symmetry transformation for integrals: $$\begin{aligned}
\nonumber
&& \prod_{j=1}^{n+2}\frac{\Gamma(Bf_j^{-1})}{\Gamma(t^{n+1}Bf_j^{-1})}
\int_{\mathbb{T}^n}\frac{\prod_{k=1}^{n+1}\prod_{j=1}^{n+2}
\Gamma(tf_jz_k^{-1},s_jz_k)\prod_{j=1}^n dz_j/z_j}
{\prod_{\stackrel{i,j=1}{i\neq j}}^{n+1}
\Gamma(z_iz_j^{-1})\prod_{k=1}^{n+1}\Gamma(t^{n+1}Sz_k,tBz_k^{-1})}
\\ &&
=\prod_{j=1}^{n+2}\frac{\Gamma(Ss_j^{-1})}{\Gamma(t^{n+1}Ss_j^{-1})}
\int_{\mathbb{T}^n}\frac{\prod_{k=1}^{n+1}\prod_{j=1}^{n+2}
\Gamma(ts_jz_k^{-1},f_jz_k)\prod_{j=1}^n dz_j/z_j}
{\prod_{\stackrel{i,j=1}{i\neq j}}^{n+1}
\Gamma(z_iz_j^{-1})\prod_{k=1}^{n+1}\Gamma(t^{n+1}Bz_k,tSz_k^{-1})}.
\label{an-trans}\end{aligned}$$ Here $t, f_j, s_j, j=1, \ldots, n+2,$ are free independent variables, $B=\prod_{j=1}^{n+2}f_j, S=\prod_{j=1}^{n+2}s_j$, and it is assumed that $|t|,|f_j|, |s_j|<1,$ $|pq|<|t^{n+1}B|, |t^{n+1}S|.$ In order to derive this identity, it is necessary to consider the $2n$-tuple integral $$\begin{aligned}
&& \frac{1}{(2\pi i)^n} \int_{\mathbb{T}^{2n}}
\frac{\prod_{k=1}^{n+1}\prod_{j=1}^{n+2}\Gamma(f_jz_k,s_jw_k^{-1})
\prod_{i,k=1}^{n+1}\Gamma(tz_k^{-1}w_i)}
{\prod_{\stackrel{i,j=1}{i\neq j}}^{n+1}
\Gamma(z_iz_j^{-1},w_iw_j^{-1})\prod_{k=1}^{n+1}\Gamma(t^{n+1}Bz_k,
t^{n+1}Sw_k^{-1})}
\\ && \makebox[4em]{} \times
\frac{dz_1}{z_1}\cdots\frac{dz_n}{z_n}
\frac{dw_1}{w_1}\cdots\frac{dw_n}{w_n},\end{aligned}$$ where $z_1\cdots z_{n+1}=w_1\cdots w_{n+1}=1$. Integration with respect to the variables $z_k$ with the help of (\[int-an\]) makes this expression proportional to the left-hand side of (\[an-trans\]) (after the replacements $w_k\to z_k^{-1}$). Changing the order of integration, which is allowed because the integrand is bounded on $\mathbb{T}$, we arrive at the integral standing on the right-hand side of (\[an-trans\]), up to some coefficient. After cancelling common factors, we get the required identity. For $p=0$ this reduces to the Denis-Gustafson transformation formula [@den-gus:beta], which describes a Bailey transformation for a terminating multivariable $_{10}\Phi_9$ series. It is natural to expect that in our case formula (\[an-trans\]) yields a Bailey transformation for some terminating $A_n$ multiple $_{12}E_{11}$ series appearing from sums of residues of the corresponding integrals.
Some other $A_n$ integrals
==========================
Now from conjecture (\[int-an\]) we derive a number of different multiple $A_n$-integrals. Denote $$\begin{aligned}
\nonumber
&& \Delta^{II}({\bf z};A_n)=
\frac{1}{(2\pi i)^n}\prod_{1\leq i<j\leq n+1}
\frac{\Gamma(tz_iz_j,sz_i^{-1}z_j^{-1})}
{\Gamma(z_iz_j^{-1},z_i^{-1}z_j)}
\\ && \makebox[4em]{} \times
\prod_{j=1}^{n+1}\frac{\Gamma(t_1z_j,t_2z_j,t_3z_j,t_4z_j^{-1},t_5z_j^{-1})}
{\Gamma(z_j(ts)^{n-1}\prod_{j=1}^5 t_j)}.
\label{a2}\end{aligned}$$
Suppose the validity of the conjectured $A_n$ and $C_n$ multiple elliptic beta integrals (\[int-an\]) and (\[SintA\]), respectively. Then, the following two integration formulae are true. For odd $n=2m-1$, we have $$\begin{aligned}
\nonumber
&& \int_{\mathbb{T}^n}\Delta^{II}({\bf z};A_n)
\frac{dz_1}{z_1}\ldots \frac{dz_n}{z_n}
= \frac{ (n+1)! }{ (q;q)_\infty^n (p;p)_\infty^n }
\\ && \makebox[4em]{} \times
\frac{ \Gamma(t^m,s^m,s^{m-1}t_4t_5)\prod_{1\leq i<j\leq 3}
\Gamma(t^{m-1}t_it_j) }
{ \prod_{k=4}^5\Gamma(t^{2m-2}s^{m-1}t_1t_2t_3t_k) }
\nonumber \\ && \makebox[4em]{} \times
\prod_{j=1}^m \frac{ \prod_{i=1}^3\prod_{k=4}^5
\Gamma((ts)^{j-1}t_it_k) }
{ \prod_{1\leq i<\ell\leq 3} \Gamma((ts)^{m+j-2}t_it_\ell t_4t_5) }
\nonumber \\ && \makebox[4em]{} \times
\prod_{j=1}^{m-1} \frac{ \Gamma((ts)^j,t^js^{j-1}t_4t_5)
\prod_{1\leq i<\ell\leq 3}\Gamma(t^{j-1}s^jt_it_\ell) }
{ \prod_{k=4}^5\Gamma(t^{m+j-2}s^{m+j-1}t_1t_2t_3 t_k) }.
\label{int-an-odd}\end{aligned}$$
For even $n=2m$, we have $$\begin{aligned}
\nonumber
&&\int_{\mathbb{T}^n}\Delta^{II}({\bf z};A_n)
\frac{dz_1}{z_1}\ldots \frac{dz_n}{z_n}
= \frac{ (n+1)! }{ (q;q)_\infty^n (p;p)_\infty^n }
\\ && \makebox[4em]{} \times
\frac{ \prod_{i=1}^3\Gamma(t^mt_i)\prod_{k=4}^5\Gamma(s^mt_k)\,
\Gamma(t^{m-1}t_1t_2t_3) }
{ \Gamma(t^{2m-1}s^{m-1}\prod_{i=1}^5t_i,t^{2m-1}s^m t_1t_2t_3) }
\nonumber \\ && \makebox[4em]{} \times
\prod_{j=1}^m \frac{ \Gamma((ts)^j,t^js^{j-1}t_4t_5)
\prod_{i=1}^3\prod_{k=4}^5 \Gamma((ts)^{j-1}t_it_k) }
{ \prod_{k=4}^5\Gamma(t^{m+j-2}s^{m+j-1}t_k^{-1}\prod_{i=1}^5t_i)}
\nonumber \\ && \makebox[4em]{} \times
\prod_{j=1}^m \prod_{1\leq i<\ell\leq3 }
\frac{\Gamma(t^{j-1}s^jt_it_\ell)}
{\Gamma((ts)^{m+j-1}t_it_\ell t_4t_5)}.
\label{int-an-even}\end{aligned}$$
The proofs follow the procedure used by Gustafson in [@gus:some2] for proving the $p=0$ cases of the integrals (\[int-an-odd\]) and (\[int-an-even\]). We start with the case of odd $n=2m-1$. Consider the following $(4m-1)$-tuple integral: $$\begin{aligned}
\nonumber
\lefteqn{ \int_{\mathbb{T}^{4m-1}} \frac{\prod_{i=1}^{2m}\prod_{j=1}^m
\Gamma(t^{1/2}z_iw_j,t^{1/2}z_iw_j^{-1},
s^{1/2}z_i^{-1}x_j,s^{1/2}z_i^{-1}x_j^{-1})}
{\prod_{i,j=1;\: i\neq j}^{2m}\Gamma(z_iz_j^{-1}) \prod_{\nu=\pm1}
\prod_{1\leq i<j\leq m}\Gamma(w_i^\nu w_j^\nu,w_i^\nu w_j^{-\nu},
x_i^\nu x_j^\nu,x_i^\nu x_j^{-\nu})} }&&
\\ && \times
\prod_{i=1}^{2m}\frac{\Gamma(z_i(ts)^{m-2}\prod_{k=1}^5 t_k)}
{\Gamma(z_i(ts)^{2m-2}\prod_{k=1}^5 t_k)}
\prod_{k=1}^{2m-1}\frac{dz_k}{z_k}\prod_{j=1}^m\Biggl(\frac{dw_j}{w_j}
\frac{dx_j}{x_j} \label{n=2m-1} \\ && \times
\prod_{\nu=\pm1} \Biggl( \frac{\prod_{k=1}^3\Gamma(t^{-1/2}t_kw_j^\nu)
\Gamma(x_j^\nu t^{m-2}s^{-1/2}t_1t_2t_3)
\prod_{k=4}^5\Gamma(s^{-1/2}t_kx_j^\nu)}
{\Gamma(w_j^\nu t^{m-3/2}t_1t_2t_3,x_j^\nu t^{m-2}s^{m-3/2}\prod_{k=1}^5t_k,
w_j^{2\nu},x_j^{2\nu})} \Biggr)\Biggr),
\nonumber\end{aligned}$$ where $\prod_{i=1}^{2m}z_j=1$. Using the exact $C_n$ integration formula of type $I$ (see (\[SintA\])), first we take integrals in (\[n=2m-1\]) with respect to the variables $w_j, j=1,\ldots,m,$ and after that with respect to $x_j, j=1,\ldots,m$. The resulting integral is equal to the left-hand side of (\[int-an-odd\]) up to the factor $$\begin{aligned}
&& (2\pi i)^{4m-1}\frac{2^{2m}(m!)^2}{(p;p)_\infty^{2m}(q;q)_\infty^{2m}}
\frac{\Gamma(s^{-1}t_4t_5)}{\Gamma(s^{m-1}t_4t_5)}
\\ && \times
\prod_{1\leq i<k\leq 3}\frac{\Gamma(t^{-1}t_it_k)}{\Gamma(t^{m-1}t_it_k)}
\prod_{k=4,5}\frac{\Gamma(t^{m-2}s^{-1}t_k^{-1}\prod_{i=1}^5t_k)}
{\Gamma(t^{m-2}s^{m-1}t_k^{-1}\prod_{i=1}^5t_k)}.\end{aligned}$$
In this two step procedure, we need the following restrictions upon the parameters: $$|t|<1, \quad |t_{1,2,3}|<|t|^{1/2}, \quad |pq|<|t^{m-3/2}t_1t_2t_3|$$ and $$|s|<1,\quad |t_{4,5}|<|s|^{1/2},\quad
|pq|<|t^{m-2}s^{m-3/2}\prod_{k=1}^5t_k|,$$ respectively. However, the resulting expression can be extended analytically to the region $|t_k|<1, \, k=1,\ldots,5$, $|pq|<|(ts)^{2m-2}
\prod_{k=1}^5t_k|$ without changing the integral value.
Since the integrand function in (\[n=2m-1\]) is bounded on the unit circle, we can change the order of integrations. First, we take the integrals over $z_i,\: i=1,\ldots,2m-1,$ using the $A_n$-formula (\[int-an\]). Then we apply formula (\[SintA\]) in order to take the integrals over $x_j, j=1,\ldots,m$. Finally, we apply the intrinsic elliptic Selberg integral (\[SintB\]) for taking the integrals over $w_j,\: j=1,\ldots,m;$ this leads to the following expression: $$\begin{aligned}
&& \frac{(2\pi i)^{4m-1}(2m)! (2^mm!)^2}{((p;p)_\infty(q;q)_\infty)^{4m-1}}
\frac{\Gamma(t^m,s^m,t_4t_5s^{-1})} {\Gamma((ts)^m,t^ms^{m-1}t_4t_5)}
\prod_{k=4}^5\frac{\Gamma(t^{m-2}s^{-1}t_k^{-1}\prod_{i=1}^5t_i)}
{\Gamma(t^{2m-2}s^{m-1}t_k^{-1}\prod_{i=1}^5t_i)}
\\ && \times
\prod_{j=1}^m\frac{\Gamma((ts)^j,t^js^{j-1}t_4t_5)
\prod_{1\leq i<k\leq 3}\Gamma(t^{j-2}s^{j-1}t_it_k)
\prod_{k=4}^5 \prod_{i=1}^3\Gamma((ts)^{j-1}t_kt_i)}
{\prod_{k=1}^3\Gamma((ts)^{m+j-2}t_k^{-1}\prod_{i=1}^5t_i)
\prod_{k=4}^5 \Gamma(t^{m+j-3}s^{m+j-2}t_k^{-1}\prod_{i=1}^5t_i)}.\end{aligned}$$ As a result, we get the needed integration formula for odd $n=2m-1$.
In order to prove (\[int-an-even\]), we consider the $4m$-tuple integral $$\begin{aligned}
\nonumber
&& \int_{\mathbb{T}^{4m}} \frac{\prod_{i=1}^{2m+1}\prod_{j=1}^m
\Gamma(t^{1/2}z_iw_j,t^{1/2}z_iw_j^{-1},
s^{1/2}z_i^{-1}x_j,s^{1/2}z_i^{-1}x_j^{-1})}
{\prod_{i,j=1;\: i\neq j}^{2m+1}\Gamma(z_iz_j^{-1}) \prod_{\nu=\pm1}
\prod_{1\leq i<j\leq m}\Gamma(w_i^\nu w_j^\nu,w_i^\nu w_j^{-\nu},
x_i^\nu x_j^\nu,x_i^\nu x_j^{-\nu})}
\\ && \makebox[2em]{}\times
\prod_{i=1}^{2m+1}\frac{\Gamma(t_3z_i,s^{m-1}t_4t_5z_i,
t^{m-1}t_1t_2z_i^{-1})}{\Gamma(z_i(ts)^{2m-1}\prod_{k=1}^5 t_k)}
\prod_{j=1}^m\Biggl( \frac{dw_j}{w_j}\frac{dx_j}{x_j}
\label{n=2m} \\ && \makebox[2em]{}\times
\prod_{\nu=\pm1} \Biggl(\frac{\prod_{k=1}^2\Gamma(t^{-1/2}t_kw_j^\nu,
s^{-1/2}t_{k+3}x_j^\nu)}{\Gamma(t^{m-1/2}t_1t_2w_j^\nu,
s^{m-1/2}t_4t_5x_j^\nu, w_j^{2\nu},x_j^{2\nu})} \Biggr)\Biggr)
\prod_{k=1}^{2m}\frac{dz_k}{z_k},
\nonumber\end{aligned}$$ where $\prod_{i=1}^{2m+1}z_j=1$. Repeating the same trick as in the case of odd $n$ (that is, integrating successively with respect to the variables $w_j$ and $x_j$ and then changing the order of integrations in this expression), we get (\[int-an-even\]).
In a similar way, we can establish elliptic analogs of the $A_n$ basic hypergeometric integrals of Gustafson and Rakha [@gus-rak:beta].
Suppose the validity of the $A_n$ and $C_n$ multiple elliptic beta integrals (\[int-an\]) and (\[SintA\]), respectively. Denote $$\begin{aligned}
\nonumber
&& \Delta^{III}({\bf z};A_n)=
\frac{1}{(2\pi i)^n}\prod_{1\leq i<j\leq n+1}
\frac{\Gamma(tz_iz_j)}{\Gamma(z_iz_j^{-1},z_i^{-1}z_j) }
\\ && \makebox[4em]{} \times
\frac{\prod_{i=1}^{n+1}\left(\prod_{k=1}^{n+1}\Gamma(t_kz_i^{-1})
\prod_{k=n+2}^{n+4}\Gamma(tt_kz_i)\right)}
{\prod_{j=1}^{n+1}\Gamma(Az_j^{-1}) },
\label{a3}\end{aligned}$$ where $A=t^{n+2}\prod_{i=1}^{n+4}t_i$ and $\prod_{j=1}^{n+1}z_j=1$. Then the following two integration formulae are true. For odd $n=2l-1$, we have $$\begin{aligned}
&& \int_{\mathbb{T}^n}\Delta^{III}({\bf z};A_n)
\frac{dz_1}{z_1}\ldots\frac{dz_n}{z_n}
= \frac{ (n+1)! }{ (q;q)_\infty^n (p;p)_\infty^n }
\frac{\Gamma(t^l,\prod_{k=1}^{2l}t_k)}
{\Gamma(t^l\prod_{k=1}^{2l}t_k)}
\label{int-an-o} \\ && \makebox[1em]{} \times
\frac{ \prod_{i=1}^{2l}\prod_{j=2l+1}^{2l+3}\Gamma(tt_it_j)
\prod_{1\leq i<j\leq 2l} \Gamma(tt_it_j)
\prod_{2l+1\leq i<j\leq 2l+3}\Gamma(t^{l+1}t_it_j)}
{\prod_{i=1}^{2l}\Gamma(t^{2l+1}t_i^{-1}\prod_{k=1}^{2l+3}t_k)
\prod_{i=2l+1}^{2l+3}\Gamma(t^{l+1}t_i^{-1}\prod_{k=1}^{2l+3}t_k) }.
\nonumber\end{aligned}$$
For even $n=2l$, we have $$\begin{aligned}
\nonumber
&&\int_{\mathbb{T}^n}\Delta^{III}({\bf z};A_n)
\frac{dz_1}{z_1}\ldots\frac{dz_n}{z_n}
= \frac{ (n+1)! }{ (q;q)_\infty^n (p;p)_\infty^n }
\frac{\Gamma(\prod_{k=1}^{2l+1}t_k,t^{l+2}\prod_{k=2l+2}^{2l+4}t_k)}
{\Gamma(t^{l+2}\prod_{k=1}^{2l+4}t_k)}
\\ && \makebox[1em]{} \times
\frac{ \prod_{i=1}^{2l+1}\prod_{j=2l+2}^{2l+4}\Gamma(tt_it_j)
\prod_{1\leq i<j\leq 2l+1} \Gamma(tt_it_j)
\prod_{i=2l+2}^{2l+4}\Gamma(t^{l+1}t_i)}
{\prod_{i=1}^{2l+1}\Gamma(t^{2l+2}t_i^{-1}\prod_{k=1}^{2l+4}t_k)
\prod_{i=2l+2}^{2l+4}\Gamma(t^{l+1}t_i\prod_{k=1}^{2l+1}t_k) }.
\label{int-an-e} \end{aligned}$$
In accordance with the procedure used in [@gus-rak:beta], we consider the $(3l-1)$-tuple integral $$\begin{aligned}
&& \int_{\mathbb{T}^{3l-1}}
\frac{\prod_{i=1}^{2l}\prod_{j=1}^l\Gamma(t^{1/2}z_iw_j,t^{1/2}z_iw_j^{-1})
\prod_{i=0}^{2l}\prod_{j=1}^{2l}\Gamma(t_iz_j^{-1})}
{\prod_{i,j=1; i\neq j}^{2l}\Gamma(z_iz_j^{-1})
\prod_{j=1}^{2l} \Gamma(t^l\prod_{i=0}^{2l}t_iz_j^{-1})}
\\ && \makebox[1em]{} \times
\prod_{\nu=\pm1}\prod_{1\leq i<j\leq l}\Gamma^{-1}
(w_i^\nu w_j^{\nu},w_i^\nu w_j^{-\nu})
\prod_{j=1}^l
\frac{\prod_{k=2l+1}^{2l+3}\Gamma(t^{1/2}t_kw_j^\nu)}
{\Gamma(w_j^{2\nu},t^{l+3/2}\prod_{k=2l+1}^{2l+3}t_kw_j^\nu)}
\\ && \makebox[1em]{} \times
\frac{dw_1}{w_1}\ldots\frac{dw_l}{w_l}\frac{dz_1}{z_1}
\ldots\frac{dz_{2l-1}}{z_{2l-1}},\end{aligned}$$ where $\prod_{i=1}^{2l}z_i=1$ and $t_0=t^{l+1}\prod_{k=2l+1}^{2l+3}t_k$. Integrating with respect to the variables $w_j$ with the help of formula (\[SintA\]), we get the left-hand side of (\[int-an-o\]) up to some factor. Changing the order of integration, we can integrate over $z_i$ using (\[int-an\]) (where it is necessary to change $z_k$ to $z_k^{-1}$) and then over $w_j$ using (\[SintA\]). Equating two expressions, we arrive at formula (\[int-an-o\]).
In a similar way, in the case of even $n=2l$ we consider the $(3l+1)$-tuple integral $$\begin{aligned}
&& \int_{\mathbb{T}^{3l+1}}
\frac{\prod_{i=1}^{2l+1}\left(\prod_{j=1}^{l+1}
\Gamma(t^{1/2}z_iw_j,t^{1/2}z_iw_j^{-1})
\prod_{j=1}^{2l+1}\Gamma(t_iz_j^{-1})\right)}
{\prod_{i,j=1; i\neq j}^{2l+1}\Gamma(z_iz_j^{-1})
\prod_{j=1}^{2l+1} \Gamma(t^{l+1}\prod_{i=1}^{2l+1}t_iz_j)}
\\ && \makebox[1em]{} \times
\prod_{\nu=\pm1}\prod_{1\leq i<j\leq l+1}\Gamma^{-1}
(w_i^\nu w_j^{\nu},w_i^\nu w_j^{-\nu})
\prod_{j=1}^{l+1}
\frac{\prod_{k=2l+1}^{2l+5}\Gamma(t^{1/2}t_kw_j^\nu)}
{\Gamma(w_j^{2\nu},t^{l+5/2}\prod_{k=2l+1}^{2l+5}t_kw_j^\nu)}
\\ && \makebox[1em]{} \times
\frac{dw_1}{w_1}\ldots\frac{dw_{l+1}}{w_{l+1}}\frac{dz_1}{z_1}
\ldots\frac{dz_{2l}}{z_{2l}},\end{aligned}$$ where $\prod_{i=1}^{2l+1}z_i=1$ and $t_{2l+5}=t^l\prod_{k=1}^{2l+1}t_k$. Repeating the same trick as in the preceding case, we get the desired integration formula (\[int-an-e\]).
Sums of residues for the derived integrals (\[int-an-odd\])-(\[int-an-e\]) form elliptic hypergeometric series on the $A_n$ root system that differ from the series (\[e-milne\]) introduced in [@ros:elliptic; @spi:modularity]. We skip their consideration and formulate only a conjecture concerning the elliptic extension of Theorem 1.2 in [@gus-rak:beta].
Suppose that $N$ is a positive integer and $\prod_{k=1}^n t_k=q^{-N}$. Then $$\begin{aligned}
\nonumber
&&
\sum_{\stackrel{\lambda_k=0,\ldots,N}{\lambda_1+\ldots+\lambda_n=N}}
\frac{
\prod_{1\leq i<j\leq n}\theta(tt_it_j)_{\lambda_i+\lambda_j}
\prod_{i=1}^n\prod_{j=n+1}^{n+3}\theta(tt_it_j)_{\lambda_i}
\prod_{i,j=1}^n\theta(t_it_j^{-1})_{-\lambda_j}
}{
\prod_{i,j=1;i\neq j}^n\theta(t_it_j^{-1})_{\lambda_i-\lambda_j}
\prod_{j=1}^n\theta(t^{n+1}t_j^{-1}\prod_{k=1}^{n+3}t_k)_{-\lambda_j}
} \\ && \makebox[2em]{}
= \left\{ \begin{aligned} \frac{\theta(1)_{-N}}{\theta(t^{n/2})_{-N}
\prod_{n+1\leq i<j\leq n+3}\theta(t^{(n+2)/2}t_it_j)_{-N}},
& \quad n \quad \text{is even,} \\
\frac{\theta(1)_{-N}}{\prod_{i=n+1}^{n+3}\theta(t^{(n+1)/2}t_i)_{-N}
\theta(t^{(n+3)/2}\prod_{i=n+1}^{n+3}t_i)_{-N} },
& \quad n \quad \text{is odd,} \\
\end{aligned} \right.
\label{new-sums} \end{aligned}$$ where $\theta(a)_\lambda\equiv\theta(a;p;q)_\lambda$.
These summation formulas are expected to follow from the residue calculus for the integrals (\[int-an-o\]) and (\[int-an-e\]). Some evidence in favor of conjecture (\[new-sums\]) is provided by the following theorem.
Denote $t=q^g, t_i=q^{g_i},i=1,\ldots,n+3$ (so that $\sum_{j=1}^{n}g_j+N=0$). The series $\sum_{\mathbf{\lambda}}c(\mathbf{\lambda})$ standing on the left-hand side of (\[new-sums\]) is a totally elliptic hypergeometric series, that is, the ratios of successive series coefficients $$h_k(\mathbf{\lambda})=\frac{c(\lambda_1,\ldots,\lambda_k+1,\ldots,
\lambda_n)}{c(\lambda_1,\ldots,\lambda_n)}$$ are elliptic functions of all unconstrained variables in the set $(\lambda_1,\ldots,\lambda_n, g,g_1,\ldots,$ $ g_{n+3})$. Moreover, the functions $h_k(\mathbf{\lambda})$ are $SL(2,\mathbb{Z})$ modular invariant. The ratios of the expressions standing on the two sides of (\[new-sums\]) are elliptic functions of $g$ and $n+2$ free parameters in the set $(g_1,\ldots,g_{n+3})$, and these ratios are modular invariant as well.
We skip the proof of this theorem, because it consists of quite long but straightforward computations whose structure was described in detail in [@spi:theta; @spi:modularity] in the process of similar considerations for different elliptic hypergeometric series summation formulas. Using the fact that there are no cusp forms of weights below 12, as in [@die-spi:elliptic; @spi:modularity] from this theorem we deduce that relations (\[new-sums\]) are valid in the small $\sigma$ expansion up to the terms of order of $\sigma^{12}$. This gives also yet another example in favor of the general conjecture of [@spi:theta] that all totally elliptic hypergeometric series are automatically modular invariant.
The integrals (\[int-an-odd\]) and (\[int-an-even\]) are expected to generate $A_n$ summation formulas similar to (\[new-sums\]). It is also natural to expect that all multiple elliptic beta integrals described above lead to integral representations for various multiple $_{12}E_{11}$ elliptic hypergeometric series generalizing the single variable formula announced in [@spi:special] (see Appendix B for the proof of it). We suppose that, sas in the $A_n$-case, there exist several types of elliptic beta integrals and elliptic hypergeometric series sums associated with the $D_n$ root system (see, e.g., [@ros:elliptic; @spi:modularity]), but their consideration lies beyond the scope of the present paper.
Relations to the generalized eigenvalue problems
================================================
Consider the very-well-poised theta hypergeometric series (\[vwp-1\]) with the additional constraint $(-q)^ne^{P_3(n)}=(qx)^n$ for some $x\in\mathbb{C}$. Special notation for such series was introduced in [@spi:bailey]: $$\begin{aligned}
\nonumber
\lefteqn{ _{r+1}V_r(t_0;t_1,\ldots,t_{r-4};q,p;x) }&&
\\ && \makebox[3em]{}
\equiv \sum_{n=0}^\infty \frac{\theta(t_0q^{2n};p)}{\theta(t_0;p)}
\prod_{m=0}^{r-4}\frac{\theta(t_m;p;q)_n}{\theta(qt_0t_m^{-1};p;q)_n}
\, (qx)^n.
\label{vwp-2}\end{aligned}$$ For (\[vwp-2\]), the balancing condition (\[theta-balance\]) is reduced to $$\prod_{m=1}^{r-4}t_m= t_0^{(r-5)/2}q^{(r-7)/2}.$$ As $p\to 0$, the $_{r+1}V_r$ series are reduced to $_{r-1}W_{r-2}$ very-well-poised $q$-hypergeometric series of the argument $qx$ (in the notation of [@gas-rah:basic]).
The balanced $_{12}V_{11}$ series with $x=1$ plays an important role in applications. For instance, the elliptic solutions of the Yang-Baxter equation derived by Date et al in [@djkmo:exactly1; @djkmo:exactly2] are expressed in terms of such series for a particular choice of parameters [@fre-tur:elliptic]. In all cases of the $_{12}V_{11}$ function to be considered below, we have $x=1$; therefore, we omit dependence on this unit argument from now on.
We denote $\mathcal{E}(\mathbf{t}) \equiv
{_{12}}V_{11}(t_0;t_1,\ldots,t_7;q,p),$ where $\prod_{m=1}^7t_m=t_0^3q^2$, and assume that this series terminates due to the condition $t_m=q^{-n},\, n\in\mathbb{N},$ for some $m$. In [@spi-zhe:spectral; @spi-zhe:classical], the following two contiguous relations for $\mathcal{E}(\mathbf{t})$ were derived: $$\begin{aligned}
\label{1_con}
\lefteqn{\mathcal{E}(\mathbf{t}) - \mathcal{E}(t_0;t_1,\ldots,t_5,q^{-1}t_6,
qt_7) } && \\ &&
=\frac{\theta(qt_0,q^2t_0,qt_7/t_6,t_6t_7/qt_0;p)}
{\theta(qt_0/t_6,q^2t_0/t_6,t_0/t_7,t_7/qt_0;p)}
\prod_{r=1}^5\frac{\theta(t_r;p)}{\theta(qt_0/t_r;p)}\,
\mathcal{E}(q^2t_0;qt_1,\ldots,qt_5,t_6,qt_7),
\nonumber\\ \nonumber
\lefteqn{\frac{\theta(t_7;p)}{\theta(t_6/qt_0,t_6/q^2t_0,t_6/t_7;p)}
\prod_{r=1}^5 \theta(t_rt_6/qt_0;p)\,
\mathcal{E}(q^2t_0;qt_1,\ldots,qt_5,t_6,qt_7) } &&
\\ && \makebox[4em]{}
+\frac{\theta(t_6;p)}{\theta(t_7/qt_0,t_7/q^2t_0,t_7/t_6;p)}
\prod_{r=1}^5 \theta(t_rt_7/qt_0;p)\,
\mathcal{E}(q^2t_0;qt_1,\ldots,qt_6,t_7)
\nonumber \\ && \makebox[8em]{}
=\frac{1}{\theta(qt_0,q^2t_0;p)}
\prod_{r=1}^5\theta(qt_0/t_r;p)\, \mathcal{E}(\mathbf{t}).
\label{2_con} \end{aligned}$$ Their combination yields $$\begin{aligned}
\nonumber
\lefteqn{
\frac{\theta(t_7,t_0/t_7,qt_0/t_7;p)}{\theta(qt_7/t_6,t_7/t_6;p)}
\prod_{r=1}^5\theta(qt_0/t_6t_r;p)\left(
\mathcal{E}(t_0;t_1,\ldots,t_5,q^{-1}t_6,qt_7) -
\mathcal{E}(\mathbf{t}) \right) } &&
\\ \nonumber
&& +\frac{\theta(t_6,t_0/t_6,qt_0/t_6;p)}
{\theta(qt_6/t_7,t_6/t_7;p)}\prod_{r=1}^5\theta(qt_0/t_7t_r;p)
\left(\mathcal{E}(t_0;t_1,\ldots,t_5,qt_6,q^{-1}t_7)-
\mathcal{E}(\mathbf{t})\right) \\
&& \makebox[4em]{}
+\theta(qt_0/t_6t_7;p)\prod_{r=1}^5\theta(t_r;p)\,
\mathcal{E}(\mathbf{t})=0.
\label{3_con}\end{aligned}$$ As $p\to 0$, these three equalities are reduced to the contiguous relations for the terminating very-well-poised balanced $_{10}\Phi_9$ series of Gupta and Masson [@gup-mas:contiguous]. Similar contiguous relations at the level of $_8\Phi_7$ functions were constructed earlier by Ismail and Rahman [@ism-rah:associated].
We change the parametrization of the $\mathcal{E}$-function and consider relation (\[3\_con\]) for the following function: $$R_{n}(z;q,p)\equiv {_{12}V_{11}}\left(\frac{t_3}{t_4};
\frac{q}{t_0t_4},\frac{q}{t_1t_4},\frac{q}{t_2t_4},t_3z,\frac{t_3}{z},q^{-n},
\frac{Aq^{n-1}}{t_4};q,p\right),
\label{R_n}$$ where $A=\prod_{r=0}^4 t_r$. We replace the parameter $t_0$ in (\[3\_con\]) by $t_3/t_4$; the variables $t_1,t_2,$ and $t_3$ are replaced by $q/t_0t_4,q/t_1t_4,$ and $q/t_2t_4$; the variables $t_4$ and $t_5$ by $q^{-n}$ and $Aq^{n-1}/t_4$; and the variables $t_6$ and $t_7$ by $t_3z$ and $t_3/z$, respectively. As a result, we see that $R_{n}(z;q,p)$ provides a particular solution of the following finite-difference equation: $$\mathcal{D}_\mu f(z)=0,\qquad
\mathcal{D}_\mu=V_\mu(z)(T-1)+V_\mu(z^{-1})(T^{-1}-1)+\kappa_\mu,
\label{eig}$$ where $T$ is the $q$-shift operator, $Tf(z)=f(qz)$, and $$\begin{aligned}
\label{V}
&& V_\mu(z)=\theta\left(\frac{t_4}{q\mu z},\frac{A\mu}{q^2z},\frac{t_4z}{q};p\right)
\frac{\prod_{r=0}^4\theta(t_rz;p)}{\theta(z^2,qz^2;p)},
\\
&& \kappa_\mu=\theta\left(\frac{A\mu}{qt_4},\mu^{-1};p\right)
\prod_{r=0}^3\theta\left(\frac{t_rt_4}{q};p\right).
\label{kappa}\end{aligned}$$ The functions $f(z)=R_n(z;q,p)$ solve (\[eig\]) for $\mu=q^{n}, n\in\mathbb{N}.$
Equation (\[eig\]) looks like a nonstandard eigenvalue problem with the “spectral parameter" $\mu$; indeed, it can be rewritten as the generalized eigenvalue problem $$\label{gevp}
\mathcal{D}_\eta f(z)=\lambda \mathcal{D}_\xi f(z),$$ where the spectral parameter $\lambda$ is $$\lambda=\frac{\theta\left(\frac{\mu A\eta}{qt_4},
\frac{\mu }{\eta};p\right)}{\theta\left(\frac{\mu A\xi}{qt_4},
\frac{\mu }{\xi};p\right)}
\label{lambda}$$ and the operators $\mathcal{D}_{\xi}$, $\mathcal{D}_{\eta}$ are obtained from $\mathcal{D}_{\mu}$ after the replacements of $\mu$ by arbitrary gauge parameters $\xi,\eta \in\mathbb{C},\, \xi\neq\eta p^k,
qt_4p^k/A\eta,$ $k\in\mathbb{Z}$. Application of the theta function identity (\[ident\]) to equation (\[gevp\]) yields $$\mathcal{D}_\eta-\lambda \mathcal{D}_\xi
=\frac{\theta(A\eta\xi/qt_4,\xi/\eta;p)}
{\theta(A\mu \xi/qt_4, \xi/\mu ;p)}\, \mathcal{D}_{\mu},$$ which shows that the gauge parameters $\xi,\eta$ drop out completely from the equation determining $f(z)$, $\mathcal{D}_\mu f(z)=0$.
A three term recurrence relation for the functions $R_n(z;q,p)$ was derived in [@spi-zhe:spectral; @spi-zhe:classical]. It appears from formula (\[3\_con\]) if there we replace $t_6$ by $q^{-n}$ and $t_7$ by $Aq^{n-1}/t_4$, and substitute $t_1\to q/t_0t_4,
t_2\to q/t_1t_4, t_3\to q/t_2t_4$, $t_4\to t_3z$, $t_5\to t_3/z$. After some work, this foirmula can be represented in the form $$\begin{aligned}
\nonumber
&& (\gamma(z)-\alpha_{n+1})B(Aq^{n-1}/t_4)\left(R_{n+1}(z;q,p)-R_n(z;q,p)
\right) \\ \nonumber && \makebox[2em]{}
+(\gamma(z)-\beta_{n-1})B(q^{-n})
\left(R_{n-1}(z;q,p)-R_n(z;q,p)\right)
\\ && \makebox[4em]{}
+\delta\left(\gamma(z)-\gamma(t_3)\right)R_n(z;q,p)=0,
\label{ttr}\end{aligned}$$ where $$\begin{aligned}
\label{B}
&& B(x)=\frac{\theta\left(x,\frac{t_3}{t_4x},
\frac{qt_3}{t_4x},\frac{qx}{t_0t_1},\frac{qx}{t_0t_2},
\frac{qx}{t_1t_2},\frac{q^2\eta x}{A},\frac{q^2x}{A\eta};p
\right)}{\theta\left(\frac{qt_4x^2}{A},\frac{q^2t_4x^2}{A};p\right)},
\\ \label{gamma}
&& \delta=\theta\left(\frac{q^2t_3}{A},\frac{q}{t_0t_4},
\frac{q}{t_1t_4},\frac{q}{t_2t_4},t_3\eta,\frac{t_3}{\eta};p\right),
\\ &&
\gamma(z)=\frac{\theta(z\xi,z/\xi;p)}{\theta(z\eta,z/\eta;p)},
\label{ab} \\ &&
\alpha_n=\gamma(q^n/t_4),\qquad \beta_n=\gamma(q^{n-1}A).\end{aligned}$$ Here $\xi$ and $\eta \neq \xi p^k,\xi^{-1}p^k,$ $k\in\mathbb{Z},$ are arbitrary gauge parameters (they are not related to $\xi,\eta$ in the difference equation, but we use the same notation). Substituting (\[B\])-(\[ab\]) in (\[ttr\]) and applying identity (\[ident\]), we see that the auxiliary gauge parameters $\xi, \eta$ drop out completely from the resulting recurrence relation.
Since $B(q^{-n})=0$ for $n=0$, the indeterminate $R_{-1}$ is not involved in (\[ttr\]) for $n=0$. We can say that $R_n(z;q,p)$ are generated by the three term recurrence relation (\[ttr\]) for the initial conditions $R_{-1}=0,R_0=1$. All recurrence coefficients in (\[ttr\]) depend linearly on the variable $\gamma(z)$, which absorbs $z$-dependence. Therefore, $R_n(z;q,p)$ are rational functions of $\gamma(z)$ with $n$ being the degree of polynomials in $\gamma(z)$ in the numerator and denominator of $R_n$. Moreover, the poles of these functions are located at $\gamma(z)=\alpha_1,\ldots,\alpha_n$.
For a particular choice of one of the parameters and a discretization of the values of $z$, the functions $R_n(z;q,p)$ yield elliptic generalizations of Wilson’s finite-dimensional $_9F_8$ and $_{10}\Phi_9$ rational functions [@wil:orthogonal]. They were derived in [@spi-zhe:spectral] from the theory of self-similar solutions of nonlinear integrable discrete time chains (for a brief review of the corresponding approach to special functions, see [@spi:solitons; @spi:factorization]). Discrete analogs of equations (\[eig\]), (\[gevp\]) valid for the latter finite-dimensional system of functions were derived in [@spi-zhe:gevp] with the help of self-duality. An equation satisfied by $_{10}\Phi_9$ functions, appearing from $R_n(z;q,p)$ in the $p\to 0$ limit, was investigated by Rahman and Suslov in [@rah-sus:classical]. The general three term recurrence relations (\[ttr\]) were considered in [@zhe:bio] and, in a different form related to $R_{II}$ continued fractions, in [@ism-mas:general].
The solutions of the generalized eigenvalue problems are known to be biorthogonal to each other; see, e.g., [@spi-zhe:spectral; @spi-zhe:theory; @zhe:bio] and the references therein. Here we would like to demonstrate that the elliptic beta integral (\[ell-int\]) serves as the biorthogonality measure for solutions of equation (\[gevp\]). Consider the scalar product $$\int_C\Delta_E(z;\mathbf{t}) \Psi(z)\left(
\mathcal{D}_\eta-\lambda \mathcal{D}_\xi\right)\Phi(z)\frac{dz}{z},
\label{scalar}$$ where $\Delta_E(z;\mathbf{t})$ is the integrand of (\[ell-int\]) and $\Phi(z), \Psi(z)$ are some complex functions. The expression (\[scalar\]) can be rewritten as $$\begin{aligned}
\nonumber
&& \int_{C}\Delta_E(z;\mathbf{t})(\kappa_\eta-V_\eta(z)-V_\eta(z^{-1}))
\Psi(z)\Phi(z)\frac{dz}{z}
\\ \nonumber && \makebox[2em]{}
+\int_{C_-}\Delta_E(q^{-1}z;\mathbf{t}) V_\eta(q^{-1}z)
\Psi(q^{-1}z)\Phi(z)\frac{dz}{z}
\\ && \makebox[2em]{}
+ \int_{C_+}\Delta_E(qz;\mathbf{t}) V_\eta(q^{-1}z^{-1})
\Psi(qz)\Phi(z)\frac{dz}{z} - \lambda \{\eta\to\xi\},
\label{scalar'}\end{aligned}$$ where $\{\eta\to\xi\}$ means the preceding expression with $\eta$ replaced by $\xi$. The integration contours $C_\pm$ are obtained from $C$ after the scaling transformations $z\to q^{\pm 1} z$. Suppose that the poles of $\Delta_E(z;\mathbf{t})$ and the singularities of the functions $\Phi(z), \Psi(q^{\pm 1}z)$ do not lie in the region swept by the contours $C_\pm$ during their deformations to $C$. Then (\[scalar’\]) takes the form $$\int_C\Delta_E(z;\mathbf{t}) \Phi(z)\left(
\mathcal{D}_\eta^T-\lambda \mathcal{D}_\xi^T\right)\Psi(z)\frac{dz}{z},
\label{scalar-conj}$$ where the adjoint (or transposed) operator $\mathcal{D}_\xi^T$ has the form $$\begin{aligned}
\nonumber
&& \mathcal{D}_\xi^T=\frac{\Delta_E(qz;\mathbf{t})}{\Delta_E(z;\mathbf{t})}
V_\xi(q^{-1}z^{-1})T+\frac{\Delta_E(q^{-1}z;\mathbf{t})}
{\Delta_E(z;\mathbf{t})}V_\xi(q^{-1}z)T^{-1}
\\ && \makebox[4em]{}
-V_\xi(z)-V_\xi(z^{-1})+\kappa_\xi.
\label{op-conj}\end{aligned}$$
Suppose that $\Phi_\lambda(z)$ is a solution of the equation $\left(\mathcal{D}_\eta-\lambda \mathcal{D}_\xi\right)\Phi(z)=0$ and $\Psi_{\lambda'}(z)$ solves the conjugate equation $\left(\mathcal{D}_\eta^T-\lambda'\mathcal{D}_\xi^T\right)\Psi(z)=0$ for some $\lambda'$. Both these functions can be multiplied by arbitrary functions $f(z)$ satisfying the condition of periodicity on the logarithmic scale, $f(qz)=f(z)$. After the replacement of $\Phi(z)$ and $\Psi(z)$ in (\[scalar\]) and (\[scalar-conj\]) by $\Phi_\lambda(z)$ and $\Psi_{\lambda'}(z)$, these expressions become equal to zero. In particular, (\[scalar-conj\]) yields the relation $$\begin{aligned}
\nonumber
&& \int_C\Delta_E(z;\mathbf{t}) \Phi_\lambda(z)\left(
\mathcal{D}_\eta^T-\lambda \mathcal{D}_\xi^T\right)\Psi_{\lambda'}(z)
\frac{dz}{z}
\\ && \makebox[4em]{}
=(\lambda'-\lambda)\int_C\Delta_E(z;\mathbf{t})
\Phi_\lambda(z)\mathcal{D}_\xi^T\Psi_{\lambda'}(z)\frac{dz}{z}=0,
\label{biort}\end{aligned}$$ which shows that for $\lambda'\neq\lambda$ the function $\Phi_\lambda(z)$ is orthogonal to $\mathcal{D}_\xi^T\Psi_{\lambda'}(z).$
We find a function $g(z)$ such that $$g^{-1}(z)\left(\mathcal{D}_\eta^T-\lambda \mathcal{D}_\xi^T\right)g(z)
=\mathcal{D}_\eta-\lambda \mathcal{D}_\xi.$$ After substitution of the known expressions for $\Delta_E(qz;\mathbf{t})/\Delta_E(z;\mathbf{t})$ and the spectral parameter (see $\lambda$ in (\[lambda\])), we get the following equation for $g(z)$: $$g(qz)=\frac{\theta\left(\frac{q}{t_4z},\frac{q\mu z}{t_4},
Az,\frac{A\mu}{q^2z};p\right)}{\theta\left(\frac{q^2z}{t_4},
\frac{\mu}{t_4 z},\frac{A}{qz},\frac{A\mu z}{q};p\right)}\, g(z),$$ which is solved easily: $$g(z)=\frac{\Gamma(\frac{q\mu z}{t_4},\frac{\mu q}{t_4 z},
Az,\frac{A}{z};q,p)}{\Gamma(\frac{q^2z}{t_4},\frac{q^2}{t_4 z},
\frac{A\mu z}{q}, \frac{A\mu}{qz};q,p)}.
\label{g-function}$$ Here we have neglected the arbitrary factor $f(qz)=f(z)$, which has already been mentioned. As a result, we get a direct relation between $\Phi_\lambda(z)$ and $\Psi_\lambda(z)$: $\Psi_\lambda(z)=g(z)\Phi_\lambda(z)$, where $g(z)$ depends on $\lambda$ as well.
We denote $\lambda_n\equiv \lambda|_{\mu=q^n}$ and $$g_n(z)\equiv g(z)|_{\mu=q^n}=
\frac{\theta\left(\frac{q^2z}{t_4},\frac{q^2}{t_4z};p;q\right)_{n-1}}
{\theta\left(Az,\frac{A}{z};p;q\right)_{n-1}}.$$ Substituting $\Phi_{\lambda_n}(z)=R_n(z;q,p)$ in the derived biorthogonality relations, we see that the functions $R_n(z;q,p)$ are formally orthogonal to $\mathcal{D}_\xi^T g_m(z)R_m(z;q,p)$ for $n\neq m$.
The general considerations of [@spi-zhe:theory; @zhe:bio] show that $R_n(z;q,p)$, which are rational functions of $\gamma(z)$ with the poles at $\gamma(z)=\alpha_1,\ldots,\alpha_n$, are orthogonal to other rational functions of $\gamma(z)$, which we denote as $T_m(z;q,p)$, with the poles at $\gamma(z)=\beta_1,\ldots,\beta_{n}$. The choice of $\alpha_n,\beta_n$ and the other recurrence coefficients in (\[ttr\]) determine $R_n$ and $T_n$ uniquely, so that permutation of all $\alpha_n$ with $\beta_n$ permutes $R_n$ and $T_n$. In our case, we see that parameters $\beta_n$ are obtained from $\alpha_n=\gamma(q^n/t_4)$ after the replacement of $t_4$ by $pq/A$. Equivalently, this replacement converts $\beta_n$ to $\alpha_n$. An important point is that the weight function $\Delta_E(z,\mathbf{t})$ is invariant under such a transformation. Therefore, we can get $T_n$ out of $R_n$ simply by the $t_4\to pq/A$ involution, which yields $$T_n(z;q,p)={_{12}V_{11}}\left(\frac{At_3}{q};\frac{A}{t_0},\frac{A}{t_1},
\frac{A}{t_2},t_3z,\frac{t_3}{z},q^{-n},\frac{Aq^{n-1}}{t_4};q,p\right),
\label{T_n}$$ where the dependence on $p$ in the parameters drops out due to the total ellipticity of this $_{12}V_{11}$ series. Comparing with the previous consideration, we see that $$\mathcal{D}_\xi^T g_n(z)R_n(z;q,p) = \rho_n T_n(z;q,p)$$ for some proportionality constants $\rho_n$, which are of no importance for us here.
Thus, the operator formalism developed above leads to the following formal biorthogonality relation: $$\int_C T_n(z;q,p)R_m(z;q,p)\Delta_E(z,\mathbf{t})\frac{dz}{z}=\tilde h_n
\delta_{nm}
\label{formal-ort}$$ for some constants $\tilde h_n$. Suppose that $C=\mathbb{T}$ and $|t_r|<1, |qp|<|A|$. Some poles of the functions $R_n,T_m$ cancel with zeros of $\Delta_E(z;\mathbf{t})$. The remaining ones approach $\mathbb{T}$ as the indices $n,m$ increase. Starting with sufficiently high values of $n$ and $m$, the contour $\mathbb{T}$ stops to satisfy the conditions used in the derivation of (\[scalar-conj\]) and must be deformed. As is shown in Appendix A, (\[formal-ort\]) is true if $C$ separates the points $z= t_{0,1,2,3}p^aq^b,$ $t_4p^{a}q^{b-m},$ and $A^{-1}p^{a+1}q^{b+1-n},$ $a,b\in\mathbb{N},$ from their partners with the inverse $z\to z^{-1}$ coordinates.
Relation (\[formal-ort\]) seems to remain true even if we multiply $R_n(z;q,p)$ or $T_n(z;q,p)$ by an arbitrary function $f(z)$ with the property $f(qz)=f(z)$. However, such nontrivial $f(z)$ must contain singularities which are crossed over when the contours $C_\pm$ are deformed to $C$ (otherwise $f(z)=const$). Therefore, the influence of such additional factors should be considered more carefully. Moreover, only for very special $f(z)$ the normalization constants $\tilde h_n$ may admit exact evaluation.
The weight function $\Delta_E(z,\mathbf{t})$ is symmetric in $q$ and $p$, whereas neither $R_n(z;q,p)$ nor $T_n(z;q,p)$ possess such a property. We can try to restore this symmetry using the freedom in the factor $f(z)=f(qz)$. Take $f(z)=R_k(z;p,q)$, $k\in\mathbb{N}$, that is, to the functions $R_n(z;q,p)$ themselves with the permuted bases $q$ and $p$. Then, the product $$R_{nk}(z)\equiv R_n(z;q,p)R_k(z;p,q)$$ satisfies two generalized eigenvalue problems: (\[eig\]) and the $p$-difference equation obtained from it by the permutation of $q$ and $p$. For (\[eig\]) we should have $\mu=q^n$, and for its partner $\mu=p^k$. The function (\[lambda\]) does not change under the substitution $\mu\to p\mu$. Therefore, the choice $\mu=q^np^k,\, n,k\in \mathbb{N},$ gives “the spectrum" for both generalized eigenvalue problems. The first factor of $R_{nk}(z)$ is a rational function of $\gamma(z;p)$ (we indicate the dependence on the base $p$ explicitly), but the second is rational in $\gamma(z;q)$. Therefore, for generic $q,p$ it is necessary to view the functions $R_{nk}(z)$ not as rational functions of some variable but as meromorphic functions of $z$.
In the same way, the series termination condition $t_6=q^{-n}$ in (\[R\_n\]) may be replaced by $t_6=q^{-n}p^{-k}$, which terminates simultaneously the $_{12}V_{11}$ series for $R_k(z;p,q)$. The property of the total ellipticity of the balanced $_{r+1}V_{r}(t_0;$ $t_1,\ldots,$ $t_{r-4};q,p)$ series plays a crucial role at this place: any parameter $t_1,\ldots,t_{r-5}$ may be multiplied by an arbitrary integer power of $p$ without changes (note that the parameter $t_0$ plays a distinguished role and the series are invariant under the transformation $t_0\to p^2t_0$).
As a result of the doubling of eigenvalue problems, the functions $R_{nk}(z)$ turn out to satisfy a quite unusual biorthogonality relation (\[ort2\]) characteristic of functions of [*two*]{} independent variables, which was announced in [@spi:special]. A rigorous consideration of all these biorthogonalities associated with the function $R_n(z;q,p)$ with complete proofs is given in Appendix A.
In fact, there is a deeper relationship between the structure of the elliptic beta integral (\[ell-int\]) and the biorthogonal functions $R_n(z;q,p), T_n(z;q,p)$ and the solutions of equation (\[eig\]) than it is indicated in this section. We hope to address this later on. As far as the multivariable generalizations are concerned, there are some multidimensional analogs of equation (\[eig\]) for the solutions of which the multiple elliptic beta integrals on root systems described in this paper may serve as biorthogonality measures. However, their consideration lies beyond the scope of the present work.
Proof of a biorthogonality relation
===================================
We define a pair of functions of $z\in\mathbb{C}$ as products of two terminating $_{12}V_{11}$ very-well-poised balanced theta hypergeometric series with the argument $x=1$ and [*different modular parameters*]{}: $$\begin{aligned}
\nonumber
&&R_{nm}(z)={_{12}V_{11}}\left(\frac{t_3}{t_4};
\frac{q}{t_0t_4},\frac{q}{t_1t_4},\frac{q}{t_2t_4},t_3z,\frac{t_3}{z},q^{-n},
\frac{Aq^{n-1}}{t_4};q,p\right) \\ \label{R_nm}
&&\makebox[3em]{}\times {_{12}V_{11}}\left(\frac{t_3}{t_4};
\frac{p}{t_0t_4},\frac{p}{t_1t_4},\frac{p}{t_2t_4},t_3z,\frac{t_3}{z},p^{-m},
\frac{Ap^{m-1}}{t_4};p,q\right), \\ \nonumber
&&T_{nm}(z)={_{12}V_{11}}\left(\frac{At_3}{q};\frac{A}{t_0},\frac{A}{t_1},
\frac{A}{t_2},t_3z,\frac{t_3}{z},q^{-n},\frac{Aq^{n-1}}{t_4};q,p\right)
\\ \label{T_nm}
&& \makebox[3em]{}\times {_{12}V_{11}}\left(\frac{At_3}{p};\frac{A}{t_0},\frac{A}{t_1},
\frac{A}{t_2},t_3z,\frac{t_3}{z},p^{-m},\frac{Ap^{m-1}}{t_4};p,q\right),\end{aligned}$$ where $n,m\in \mathbb{N}$. Obviously, these functions are symmetric with respect to the permutation of $p$ and $q$. The balancing conditions are used already in these series in order to express one of the parameters in terms of the others.
For $m\neq 0$ and the fixed parameters $q, t_r, r=0,\ldots,4,$ the limit as $p\to 0$ is not well defined for $R_{nm}(z)$ and $T_{nm}(z)$. The reason for this comes from the quasiperiodicity of $\theta(z;p)$ because the limits $z\to 0$ or $z\to\infty$ are not defined for it.
Now, we consider the following integral: $$\label{ort-int}
J_{mn,kl}\equiv\int_{C_{mn,kl}}T_{nl}(z)R_{mk}(z)
\Delta_E(z,\mathbf{t}) \frac{d z}{z},$$ where $\Delta_E(z,\mathbf{t})$ denotes the weight function (\[weight\]) and $C_{mn,kl}$ is some contour of integration. We want to show that a particular choice of $C_{mn,kl}$ (to be specified below) leads to the formula $$\label{ort}
J_{mn,kl}=h_{nl}\delta_{mn}\delta_{kl},$$ where $h_{nl}$ are some normalization constants.
An elliptic extension of the Bailey transformation for the $_{10}\Phi_9$ series, which was derived by Frenkel and Turaev in [@fre-tur:elliptic], has the following form (an alternative proof of it was given in [@spi:bailey]): $$\begin{aligned}
\nonumber
{_{12}V_{11}}(t_0;t_1,\dots,t_7;q,p) &=&
\frac{\theta(qt_0,qs_0/t_4,qs_0/t_5,qt_0/t_4t_5;p;q)_N}
{\theta(qs_0,qt_0/t_4,qt_0/t_5,qs_0/t_4t_5;p;q)_N} \\
&& \times {_{12}V_{11}}(s_0;s_1,\dots,s_7;q,p),
\label{ft-bailey}\end{aligned}$$ where $\prod_{m=1}^7t_m=t_0^3q^2$, $t_6=q^{-N}$, $N\in\mathbb{N}$, $$s_0=\frac{qt_0^2}{t_1t_2t_3},\quad s_1=\frac{s_0t_1}{t_0},\quad
s_2=\frac{s_0t_2}{t_0},\quad s_3=\frac{s_0t_3}{t_0},
\label{s-param}$$ and $s_4, s_5, s_6, s_7$ form an arbitrary permutation of $t_4, t_5, t_6, t_7$. Using this identity, we can rewrite the functions $R_{mk}(z)$ as follows $$\begin{aligned}
\nonumber
&& R_{mk}(z) = \kappa_m(p;q)\kappa_k(q;p) \\
&& \makebox[3em]{} \times
{_{12}V_{11}}\left(\frac{t_0}{t_4};\frac{q}{t_1t_4},\frac{q}{t_2t_4},
\frac{q}{t_3t_4},t_0z,\frac{t_0}{z},q^{-m},\frac{Aq^{m-1}}{t_4};q,p\right)
\nonumber \\
&& \makebox[3em]{} \times
{_{12}V_{11}}\left(\frac{t_0}{t_4};\frac{p}{t_1t_4},\frac{p}{t_2t_4},
\frac{p}{t_3t_4},t_0z,\frac{t_0}{z},p^{-k},\frac{Ap^{k-1}}{t_4};p,q\right),
\label{R-transf}\end{aligned}$$ where $$\kappa_m(p;q)=\frac{\theta(qt_3/t_4,t_0t_1,t_0t_2,A/qt_0;p;q)_m}
{\theta(qt_0/t_4,t_1t_3,t_2t_3,A/qt_3;p;q)_m}.$$
Substituting the explicit series expressions and in the definition of $J_{mn,kl}$ (\[ort-int\]) yields $$\begin{aligned}
\nonumber
J_{mn,kl}&=& \kappa_m(p;q)\kappa_{k}(q;p)
\sum_{r=0}^n\sum_{r'=0}^l\sum_{s=0}^{m}\sum_{s'=0}^k q^{r+s}p^{r'+s'}
\\ \nonumber
&& \times \frac{\theta(At_3q^{2r-1},t_0q^{2s}/t_4;p)}{\theta(At_3/q,t_0/t_4;p)}
\frac{\theta(At_3p^{2r'-1},t_0p^{2s'}/t_4;q)}{\theta(At_3/p,t_0/t_4;q)}
\\ \nonumber
&& \times \frac{\theta(At_3/q,A/t_0,A/t_1,A/t_2,q^{-n},Aq^{n-1}/t_4;p;q)_r}
{\theta(q,t_0t_3,t_1t_3,t_2t_3,At_3q^n,t_3t_4q^{1-n};p;q)_r}
\\ \nonumber
&& \times \frac{\theta(At_3/p,A/t_0,A/t_1,A/t_2,p^{-l},Ap^{l-1}/t_4;q;p)_{r'}}
{\theta(p,t_0t_3,t_1t_3,t_2t_3,At_3p^l,t_3t_4p^{1-l};q;p)_{r'}}
\\ \nonumber
&& \times\frac{\theta(t_0/t_4,q/t_1t_4,q/t_2t_4,q/t_3t_4,q^{-m},
Aq^{m-1}/t_4;p;q)_s}
{\theta(q,t_0t_1,t_0t_2,t_0t_3,t_0q^{m+1}/t_4,t_0q^{2-m}/A;p;q)_s}
\\ \nonumber
&& \times\frac{\theta(t_0/t_4,p/t_1t_4,p/t_2t_4,p/t_3t_4,p^{-k},
Ap^{k-1}/t_4;q;p)_{s'}}
{\theta(p,t_0t_1,t_0t_2,t_0t_3,t_0p^{k+1}/t_4,t_0p^{2-k}/A;q;p)_{s'}}\,
I_{rs,r's'},\end{aligned}$$ where $$\begin{aligned}
\nonumber
&I_{rs,r's'}& = \int_{C_{mn,kl}}\Delta_E(z,\mathbf{t})
\frac{\theta(zt_3,z^{-1}t_3;p;q)_r}{\theta(zA,z^{-1}A;p;q)_r}
\frac{\theta(zt_0,z^{-1}t_0;p;q)_s}
{\theta(zqt_4^{-1},z^{-1}qt_4^{-1};p;q)_s} \\
&& \times
\frac{\theta(zt_3,z^{-1}t_3;q;p)_{r'}}{\theta(zA,z^{-1}A;q;p)_{r'}}
\frac{\theta(zt_0,z^{-1}t_0;q;p)_{s'}}
{\theta(zpt_4^{-1},z^{-1}pt_4^{-1};q;p)_{s'}} \frac{d z}{z}.
\label{int3}\end{aligned}$$ We introduce the notation $$\tilde t_0=t_0q^sp^{s'},\quad \tilde t_1=t_1,\quad \tilde t_2=t_2,
\quad \tilde t_3=t_3q^rp^{r'}, \quad \tilde t_4=t_4q^{-s}p^{-s'},$$ so that $$\tilde A=\prod_{m=0}^4\tilde t_m=Aq^rp^{r'}.$$ Then, using the transformation property $$\theta(z;p;q)_l=(-z)^lq^{l(l-1)/2}\theta(z^{-1}q^{-l+1};p;q)_l
=\frac{(-z)^lq^{l(l-1)/2}}{\theta(qz^{-1};p;q)_{-l}},$$ we can rewrite the integral (\[int3\]) in the form $$\label{int4}
I_{rs,r's'}=\left(\frac{t_0t_4}{pq}\right)^{2ss'}
\left(\frac{t_3}{A}\right)^{2rr'}\frac{t_4^{2(s+s')}}{q^{s(s+1)}p^{s'(s'+1)}}
\int_{C_{mn,kl}}
\Delta_E(z,\mathbf{\tilde t}) \frac{d z}{z}.$$ But the integral on the right-hand side of (\[int4\]) coincides with the elliptic beta integral (\[ell-int\]), provided we identify $C_{mn,kl}=\mathbb{T}$ and impose the constraints $|\tilde t_m|<1$, $|pq|<|\tilde A|$. However, the values of the integers $r,s,r',s'\in\mathbb{N}$ are not limited; starting with their sufficiently large values, we shall have either $|\tilde A|=|q^rp^{r'}A|<|pq|$ or $|\tilde t_4|=|q^{-s}p^{-s'}t_4|>1$. Now is the moment to specify the contour $C_{mn,kl}$. We choose it in such a way that formula (\[ell-int\]) remains applicable. More precisely, let $C_{mn,kl}$ be a deformation of $\mathbb{T}$ such that it separates the poles at $z=t_{0,1,2,3}p^aq^{b},$ $t_4p^{a-k}q^{b-m},$ and $A^{-1}p^{a+1-l}q^{b+1-n},$ $a,b\in\mathbb{N}$, that lie inside $C_{mn,kl}$ and converge to zero, from the poles diverging to infinity, which are obtained from the poles inside $C_{mn,kl}$ by the inversion transformation $z\to z^{-1}$. The subscripts $m,n,k,l$ in the notation for such a contour indicate that, evidently, the shape of $C_{nm,kl}$ depends on the indices of the functions $T_{nl}(z), R_{mk}(z)$.
For such a contour $C_{mn,kl}$, we have $$\begin{aligned}
\nonumber
I_{rs,r's'} &=&
\left(\frac{t_0t_4}{pq}\right)^{2ss'}
\left(\frac{t_3}{A}\right)^{2rr'}\frac{t_4^{2(s+s')}}{q^{s(s+1)}p^{s'(s'+1)}}
\mathcal{N}_E(\mathbf{\tilde t}) \\ \nonumber
&=&\mathcal{N}_E(\mathbf{t})
\frac{\theta(t_1t_3,t_2t_3,t_3t_4;p;q)_r}
{\theta(A/t_0,A/t_1,A/t_2;p;q)_r} \\ \nonumber
&&\times \frac{\theta(t_0t_3;p;q)_{r+s}}{\theta(A/t_4;p;q)_{r+s}}
\frac{\theta(t_0t_1,t_0t_2,q^{1-r}t_0/A;p;q)_s}
{\theta(q/t_1t_4,q/t_2t_4,q^{1-r}/t_3t_4;p;q)_s}
\\ \nonumber
&& \times
\frac{\theta(t_1t_3,t_2t_3,t_3t_4;q;p)_{r'}}
{\theta(A/t_0,A/t_1,A/t_2;q;p)_{r'}} \\ \nonumber
&&\times \frac{\theta(t_0t_3;q;p)_{r'+s'}}{\theta(A/t_4;q;p)_{r'+s'}}
\frac{\theta(t_0t_1,t_0t_2,p^{1-r'}t_0/A;q;p)_{s'}}
{\theta(p/t_1t_4,p/t_2t_4,p^{1-r'}/t_3t_4;q;p)_{s'}},\end{aligned}$$ where the function $\mathcal{N}_E(\mathbf{t})$ is fixed in .
As a result of these manipulations, the quantity $J_{mn,kl}$ splits into a product of two double series each depending only on the indices $m, n$ and $k,l$ separately. After an application of the relation $(a;p;q)_{r+s}=(aq^r;p;q)_s(a;p;q)_r$ and various simplifications, we can write $$J_{mn,kl}=\mathcal{N}_E(\mathbf{t})J_{mn}(p;q)J_{kl}(q;p),$$ where $$\begin{aligned}
\nonumber
J_{mn}(p;q) &=& \kappa_m(p;q) \sum_{r=0}^nq^r
\frac{\theta(At_3q^{2r-1};p)}{\theta(At_3/q;p)}
\frac{\theta(At_3/q,q^{-n},Aq^{n-1}/t_4,t_3t_4;p;q)_r}
{\theta(q,At_3q^n,t_3t_4q^{1-n},A/t_4;p;q)_r}
\\ \label{8E7-aux}
&& \times {_{10}V_9}\left(\frac{t_0}{t_4};\frac{q}{t_3t_4},
t_0t_3q^r,\frac{q^{1-r}t_0}{A},\frac{Aq^{m-1}}{t_4},q^{-m};q,p\right).\end{aligned}$$
The constraint $t_2t_3=qt_0$ imposed upon relation (\[ft-bailey\]) converts the $_{12}V_{11}$-series on its left-hand side into a terminating $_{10}V_9$ series, whereas on the right-hand side only the first term of the corresponding $_{12}V_{11}$-series survives. As a result, we get the Frenkel-Turaev sum, or an elliptic generalization of the Jackson sum for terminating very-well-poised balanced $_8\Phi_7$-series: $$\begin{aligned}
\nonumber
\lefteqn{ {_{10}V_9}(t_0;t_1,t_4, t_5,t_6,t_7;q,p) } &&
\\ && \makebox[4em]{}
= \frac{\theta (qt_0,qt_0/t_1t_4,qt_0/t_1t_5,qt_0/t_4t_5;p;q)_N}
{\theta(qt_0/t_1t_4t_5,qt_0/t_1,qt_0/t_4,qt_0/t_5;p;q)_N},
\label{ft-sum}\end{aligned}$$ where $t_1t_4t_5t_6t_7 =qt_0^2$ and $t_6=q^{-N}$, $N\in \mathbb{N}$. Application of this sum to the $_{10}V_9$ series in yields $${_{10}V_9}(\ldots)=\frac{\theta(qt_0/t_4,t_1t_2,At_3q^{r-1},q^{-r};p;q)_m}
{\theta(t_0t_3,A/qt_0,Aq^r/t_4,q^{1-r}/t_3t_4;p;q)_m}.$$ Clearly, this expression vanishes for $m>r$. This means that $J_{mn}=0$ for $m>n$. For $m\leq n$, we get $$\begin{aligned}
\nonumber
J_{mn}(p;q)=\kappa_m(p;q)
\frac{\theta(At_3;p;q)_{2m}}{\theta(A/t_4;p;q)_{2m}}
\frac{\theta(Aq^{n-1}/t_4,qt_0/t_4,t_1t_2,q^{-n};p;q)_m}
{\theta(t_3t_4q^{1-n},t_0t_3,A/qt_0,At_3q^n;p;q)_m}
\\ \nonumber
\times (t_3t_4)^m\, {_8V_7}(At_3q^{2m-1};t_3t_4,Aq^{n+m-1}/t_4,q^{m-n};q,p).\end{aligned}$$ Applying the summation formula (\[ft-sum\]) to the latter $_8V_7$ series, we get $${_8V_7}(\ldots)=\frac{\theta(At_3q^{2m},q^{m-n+1},Aq^{m+n}/t_4,qt_3t_4;p;q)_{n-m}}
{\theta(q,Aq^{2m}/t_4,t_3t_4q^{m-n+1},At_3q^{m+n};p;q)_{n-m}},$$ which is equal to zero for $n>m$ due to the factor $\theta(q^{m-n+1};p;q)_{n-m}$. As a result, $J_{mn}(p;q)=h_n(p;q)\delta_{mn}$, where the normalization constants have the form $$\label{norm}
h_n(p;q)=\frac{\theta(A/qt_4;p)
\theta(q,qt_3/t_4,t_0t_1,t_0t_2,t_1t_2,At_3;p;q)_nq^{-n}}
{\theta(Aq^{2n-1}/t_4;p)
\theta(1/t_3t_4,t_0t_3,t_1t_3,t_2t_3,A/qt_3,A/qt_4;p;q)_n}.$$
The fact that $J_{mn}=0$ for $n\neq m$ provides the desired biorthogonality relation (\[ort\]). We summarize the result obtained in the form of the theorem that was announced in [@spi:special] (it is necessary to apply the series notation introduced in [@spi:theta; @spi:bailey] and permute the parameters $t_3$ and $t_4$ in [@spi:special] in order to match with the current presentation).
Let $t_m,$ $\Delta_E(z,{\bf t}),$ $\mathcal{N}_E({\bf t})$ be the same as in Theorem 1. Let $C_{mn,kl}$ denote a positively oriented contour separating the points $z=\{ t_{0,1,2,3}p^aq^b,$ $t_4p^{a-k}q^{b-m},$ $A^{-1}p^{a+1-l}q^{b+1-n}\}_{a,b\in\mathbb{N}}$ from the points with the inverse ($z\to z^{-1}$) coordinates. Then $R_{mk}(z)$ and $T_{nl}(z)$ satisfy the following biorthogonality relation $$\label{ort2}
\int_{C_{mn,kl}}T_{nl}(z)R_{mk}(z)
\Delta_E(z,{\bf t}) \frac{d z}{z}=
h_{nl}\mathcal{N}_E({\bf t})\delta_{mn}\delta_{kl},$$ where $h_{nl}$ are the normalization constants, $$\begin{aligned}
\nonumber
&& h_{nl}= \frac{\theta(A/qt_4;p)
\theta(q,qt_3/t_4,t_0t_1,t_0t_2,t_1t_2,At_3;p;q)_nq^{-n}}
{\theta(Aq^{2n}/qt_4;p)
\theta(1/t_3t_4,t_0t_3,t_1t_3,t_2t_3,A/qt_3,A/qt_4;p;q)_n} \\
&&\makebox[2em]{}\times \frac{\theta(A/pt_4;q)
\theta(p,pt_3/t_4,t_0t_1,t_0t_2,t_1t_2,At_3;q;p)_lp^{-l}}
{\theta(Ap^{2l}/pt_4;q)
\theta(1/t_3t_4,t_0t_3,t_1t_3,t_2t_3,A/pt_3,A/pt_4;q;p)_l}.
\label{norm2}\end{aligned}$$
As is clear from (\[R\_n\]) and (\[T\_n\]), we have $R_m(z;q,p)$ $= R_{m0}(z)$ and $T_n(z;q,p)= T_{n0}(z)$. These functions $R_m, T_n$ are equal to $_{12}V_{11}$ elliptic hypergeometric series with particular choices of the parameters.
The functions $R_m(z;q,p)$ and $T_n(z;q,p)$ satisfy the following biorthogonality condition $$\label{R_mT_n-ort}
\int_{C_{mn}}T_{n}(z;q,p)R_{m}(z;q,p)
\Delta_E(z,{\bf t}) \frac{d z}{z}= h_{n}\mathcal{N}_E({\bf t})\delta_{mn},$$ where the constants $h_n$ are fixed in (\[norm\]) and the contour $C_{mn}$ encircles the poles of the integrand located at $z=\{ t_{0,1,2,3}q^ap^b,
t_4p^aq^{b-m},A^{-1}p^{a+1}q^{b+1-n}\}_{a,b\in\mathbb{N}}$ and separates them from the poles with inverse $z\to z^{-1}$ coordinates.
The biorthogonal rational functions $R_m(z;q,p)$ and $T_n(z;q,p)$ describe elliptic generalizations of the Rahman set of continuous $_{10}\Phi_9$ functions [@rah:integral] to which they are reduced in the limit as $p\to 0$. Accordingly, in this limit, formula (\[R\_mT\_n-ort\]) is reduced to the Rahman biorthogonality condition.
Integral representations for $_{12}E_{11}$ series
=================================================
Here we derive an integral representation for the product of two terminating $_{12}E_{11}$ (more precisely, $_{12}V_{11}$) series with some particular choice of parameters. For this, we apply an elliptic generalization of the technique used in [@rah:integral] for the derivation of the contour integral representation for a terminating $_{10}\Phi_9$ series.
Suppose that five parameters $t_k, k=0,\ldots, 4,$ satisfy the conditions of Theorem 1. Denote by $m,n$ two positive integers and by $C_{mn}$ a positively oriented contour such that for all $a,b \in\mathbb{N}$ it separates the points $z=\{t_kp^aq^b,$ $A^{-1}q^{b+1-m}p^{a+1-n}\}$ from their partners with the inverse coordinates ($z\to z^{-1}$). Under these conditions, the following integral representation for the product of two $_{12}V_{11}$ terminating very-well-poised balanced theta hypergeometric series at $x=1$ holds true: $$\begin{aligned}
\nonumber
&&{_{12}V_{11}}\left(\frac{At_0}{q};\alpha,t_0t_1,t_0t_2,t_0t_3,t_0t_4,
q^{-m},\frac{A^2q^{m-1}}{\alpha};q,p\right) \\ \nonumber
&& \times {_{12}V_{11}}
\left(\frac{At_0}{p};\beta,t_0t_1,t_0t_2,t_0t_3,t_0t_4,
p^{-n},\frac{A^2p^{n-1}}{\beta};p,q\right) \\ \nonumber
&&=\frac{1}{\mathcal{N}_E({\bf t})}
\frac{\theta(At_0,\frac{A}{t_0};p;q)_m\theta(At_0,\frac{A}{t_0};q;p)_n}
{\theta(\frac{A}{\alpha t_0},\frac{At_0}{\alpha};p;q)_m
\theta(\frac{A}{\beta t_0},\frac{At_0}{\beta};q;p)_n} \\
&& \makebox[1em]{}
\times \int_{C_{mn}}
\Delta_E(z,{\bf t})\frac{\theta(\frac{Az}{\alpha},\frac{A}{\alpha z};p;q)_m
\theta(\frac{Az}{\beta},\frac{A}{\beta z};q;p)_n }
{\theta(Az,\frac{A}{z};p;q)_m\theta(Az,\frac{A}{z};q;p)_n}\frac{d z}{z},
\label{intrep}\end{aligned}$$ where $\alpha$ and $\beta$ are arbitrary complex parameters.
Under the conditions imposed upon the parameters in the formulation of this theorem, the following relations are true: $$\begin{aligned}
\nonumber
\lefteqn{
\int_{C_{ij}}\Delta_E(z,\mathbf{t})
\frac{\theta(zt_0,z^{-1}t_0;p;q)_i\theta(zt_0,z^{-1}t_0;q;p)_j}
{\theta(zA,z^{-1}A;p;q)_i\theta(zA,z^{-1}A;q;p)_j}\frac{dz}{z}
} && \\ &&
= \left(\frac{t_0}{A}\right)^{2ij}
\mathcal{N}_E(t_0q^ip^j,t_1,\ldots,t_4)
\nonumber \\ &&
=\frac{\theta(t_0t_1,\ldots,t_0t_4;p;q)_i\theta(t_0t_1,\ldots,t_0t_4;q;p)_j}
{\theta(A/t_1,\ldots,A/t_4;p;q)_i\theta(A/t_1,\ldots,A/t_4;q;p)_j}
\, \mathcal{N}_E(\mathbf{t}).
\label{identity}\end{aligned}$$
We multiply (\[identity\]) by the factor $$\begin{aligned}
&& q^i\frac{\theta(At_0q^{2i-1};p)}{\theta(At_0/q;p)}
\frac{\theta(At_0/q,\alpha,q^{-m},A^2q^{m-1}/\alpha;p;q)_i}
{\theta(q,At_0/\alpha,At_0q^m,\alpha q^{1-m}t_0/A;p;q)_i}
\\
&& \times p^j\frac{\theta(At_0p^{2j-1};q)}{\theta(At_0/p;q)}
\frac{\theta(At_0/p,\beta,p^{-n},A^2p^{n-1}/\beta;q;p)_j}
{\theta(p,At_0/\beta,At_0p^n,\beta p^{1-n}t_0/A;q;p)_j},\end{aligned}$$ where $\alpha$ and $\beta$ are arbitrary complex parameters, and sum over $i$ from 0 to $m$ and over $j$ from 0 to $n$. As a result, we get the relation $$\begin{aligned}
&&\int_{C_{mn}} \Delta_E(z,\mathbf{t})\,
{_{10}V_9}(At_0/q;t_0z,t_0z^{-1},\alpha,q^{-m},A^2q^{m-1}/\alpha;q,p)
\\ && \makebox[4em]{}
\times {_{10}V_9}(At_0/p;t_0z,t_0z^{-1},\beta,p^{-n},A^2p^{n-1}/\alpha;p,q)
\,\frac{dz}{z} \\
&&\makebox[2em]{} ={_{12}V_{11}}(At_0/q;\alpha,q^{-m},A^2q^{m-1}/\alpha,
t_0t_1,\ldots,t_0t_4;q,p) \\
&& \makebox[3em]{} \times
{_{12}V_{11}}(At_0/p;\beta,p^{-n},A^2p^{n-1}/\beta,t_0t_1,\ldots,t_0t_4;
p,q)\, \mathcal{N}_E(\mathbf{t}).\end{aligned}$$ Application of the Frenkel-Turaev sum to the $_{10}V_9$ series standing under the integral sign leads to (\[intrep\]).
For $n=0$ we get an integral representation of a single terminating $_{12}V_{11}$ series, which can be reduced further to the $_{10}\Phi_9$ $q$-series level [@rah:integral] by letting $p\to 0$. However, for $n\neq 0$ the limit as $p\to 0$ is not well defined, and formula (\[intrep\]) exists only at the elliptic level.
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|
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abstract: 'Reconstruction of the curvatures of radio wavefronts of air showers initiated by ultra high energy cosmic rays is discussed based on minimization algorithms commonly used. We emphasize the importance of the convergence process induced by the minimization of a non-linear least squares function that affects the results in terms of degeneration of the solutions and bias. We derive a simple method to obtain a satisfactory estimate of the location of the main point of emission source, which mitigates the problems previously encountered.'
address:
- 'SUBATECH IN2P3-CNRS/Université de Nantes/Ecole des Mines de Nantes, Nantes, France'
- 'Ecole des Mines de Nantes, Nantes, France '
author:
- Ahmed Rebai
- Tarek Salhi
- Pascal Lautridou
- Olivier Ravel
title: 'Ill-posed formulation of the emission source localization in the radio-detection experiments of extensive air showers'
---
UHECR ,radio-detection ,antennas ,non-convex analysis ,optimization,ill-posed problem.
Introduction
============
The determination of the nature of the ultra-high energy cosmic rays (UHECR) is an old fundamental problem in cosmic rays studies. Numerous are the difficulties. New promising approaches could emerge from the use of the radio-detection method which exploits, through antennas, the radio signal that accompanies the development of the extensive air shower (EAS). Several experimental prototypes like CODALEMA [@key-1] in France and LOPES [@key-2] in Germany shown the feasibility and the potential of the method to reconstruct EAS parameters, as the arrival direction, the impact location at ground, the lateral distribution function of the electric field, or the primary particle energy [@key-3; @key-4; @key-5; @key-6; @key-7]. However, the temporal radio wavefront characteristics remain still poorly determined [@key-8; @key-8-1], although its knowledge could be consider as one of the first steps in retrieving information about the EAS itself. The importance of this information resides in its potential sensitivity to the nature of the primary particle, especially because the existence of a curvate radio wavefront (a spherical wavefront) could provide the location of the main point of the emission source, and possibly an estimation of Xmax, event by event. Indeed, the arrival timing being defined by the maximum amplitude of the radio signal, it is more likely linked to a limited portion of the longitudinal development of the shower (and so especially at the point of maximum) [@key-8-2].
Moreover, the migration of present small scale radio-prototypes to large scale experiments spread over surfaces of several tens of $1000\: km^{2}$ using self-triggered antennas, is challenging. This technique is subjected to delicate limitations in regard to UHECR recognition, due to noises induced by human activities (high voltage power lines, electric transformers, cars, trains and planes) or by stormy weather conditions (lightning). Figure \[codanoise\] shows a typical reconstruction of sources obtained with the CODALEMA experiment [@key-9], by invoking a spherical wave minimization. Such patterns are also commonly observed in others radio experiments [@key-9-1; @key-9-2]. In most of the cases, one of the striking results is that these emission sources are reconstructed with great inaccuracy, although they are fixed and although the number of measured events is high. By extension, a cosmic event being a single realization of the detected observables (arrival time and peak amplitude on each antenna), interpretation of such methods of reconstruction for the identification of a point source can become even more delicate, even using statistical approaches.
![Typical result of reconstruction of two entropic emitters at ground, observed with the stand-alone stations of CODALEMA, through standard minimization algorithms. Despite the spreading of the reconstructed positions, these two transmitters are, in reality, two stationary point sources. []{data-label="codanoise"}](BKG_2011_06_21_source2){width="11cm" height="6cm"}
The commonly used technique relies on the minimization of an objective function which depends on the assumed shape of the wavefront, using the arrival times and locations of the antennas. The aim of this paper is to highlight that the minimization of such an objective function, incorporating a spherical wave front, can be an ill-posed problem. We will show that it originates from strong dependencies of the convergence of the minimization algorithms with initial parameters, from the existence of degenerations of the solutions (half lines) which can trap most of the common algorithms, and from the existence of offsets (bias) in the reconstructed positions. Finally, by avoiding more complex estimates based on advanced statistical theories, we got to deduce a simple method to obtain a significant estimate of the source location. We compared the exact results with our numerical reconstructions performed on a test array.
Reconstruction with common algorithms
=====================================
The performances of different algorithms has been tested using the simplest test array of antennas. Within the constraints imposed by the number of free parameters used for reconstruction, we choose an array of 5 antennas for which the antennas positions $\overrightarrow{r_{i}}=\left(x_{i},y_{i},z_{i}\right)$ are fixed (see Fig. \[how\_test\_estim\]) (this corresponds to a multiplicity of antennas similar to that sought at the detection threshold in current setups).
![Scheme of the antenna array used for the simulations. The antenna location is took from a uniform distribution of $1\: m$ width. []{data-label="how_test_estim"}](prototype_array){width="11cm" height="6cm"}
A source S with a spatial position $\overrightarrow{r_{s}}=\left(x_{s},y_{s},z_{s}\right)$ is set at the desired value. Assuming $t_{s}$ the unknown instant of the wave emission from S, c the wave velocity in the medium considered constant during the propagation, and assuming that the emitted wave is spherical, the reception time $t_{i}$ on each antenna $i\in\left\{ 1,\ldots,N\right\} $ can written: $$t_{i}=t_{s}+\frac{\sqrt{\left(x_{i}-x_{s}\right)^{2}+\left(y_{i}-y_{s}\right)^{2}+\left(z_{i}-z_{s}\right)^{2}}}{c}+G(0,\sigma_{t})$$ where $G(0,\sigma_{t})$ is the Gaussian probability density function centered to $t=0$ and of standard deviation $\sigma_{t}$. This latter parameter stand for the the global time resolution, which depends as well on technological specifications of the apparatus than on analysis methods.
The theoretical predictions are compared to the reconstructions given by the different algorithms. The latter are setup in two steps. First, a planar adjustment is made, in order to pres-tress the region of the zenith angle $\theta$ and azimuth angle $\phi$ of the source arrival direction. It specifies a target region in this subset of the phase space, reducing the computing time of the search of the minimum of the objective function of the spherical emission. Reconstruction of the source location is achieved, choosing an objective-function that measures the agreement between the data and the model of the form, by calculating the difference between data and a theoretical model (in frequentist statistics, the objective-function is conventionally arranged so that small values represent close agreement):
$$f(\vec{r_{s}},t_{s}^{*})=\frac{1}{2}\sum_{i=1}^{N}\left[\left\Vert \overrightarrow{r_{s}}-\overrightarrow{r_{i}}\right\Vert ^{2}-\left(t_{s}^{*}-t_{i}^{*}\right)^{2}\right]^{2}$$
The partial terms $\left\Vert \overrightarrow{r_{s}}-\overrightarrow{r_{i}}\right\Vert ^{2}-\left(t_{s}^{*}-t_{i}^{*}\right)^{2}$ represents the difference between the square of the radius calculated using coordinates and the square of the radius calculated using wave propagation time for each of the N antennas. The functional $f$ can be interpreted as the sum of squared errors. Intuitively the source positions $\overrightarrow{r_{s}}$ at the instant $t_{s}$ is one that minimizes this error.
In the context of this paper, we did not use genetic algorithms or multivariate analysis methods but we focused on three minimization algorithms, used extensively in statistical data analysis software of high energy physics[@key-10; @key-11]: Simplex, Line-Search and Levenberg-Marquardt (see table \[diff\_algo 1\]). They can be found in many scientific libraries as the Optimization Toolbox in Matlab, the MPFIT in IDL and the library Minuit in Root that uses 2 algorithms Migrad and Simplex which are based respectively on a variable-metric linear search method with calculation of the objective function first derivative and a simple search method. For the present study, we have used with their default parameters.
\[diff\_algo 1\]
[|>p[22mm]{}|>p[30mm]{}|>p[30mm]{}|>p[30mm]{}|]{} Minimization algorithms & Levenberg-Marquardt & Simplex & Line-Search[\
]{} Libraries & lsqnonlin - MPFIT & fminsearch - SIMPLEX & MIGRAD - lsqcurvefit[\
]{} Software & Optimization Toolbox Matlab - IDL & Optimization Toolbox Matlab - MINUIT-ROOT & Optimization Toolbox Matlab - MINUIT-ROOT[\
]{} Method Principles & Gauss-Newton method combined with trust region method & Direct search method & Compute the step-size by optimizing the merit function $f(x+t.d)$ [\
]{} Used information & Compute gradient $(\nabla f)_{k}$ and an approximate hessian $(\nabla^{2}f)_{k}$ & No use of numerical or analytical gradients & $f(x+t.d,\, d)$ where $d$ is a direction descent computed with gradient/hessian[\
]{} Advantages / Disadvantages & Stabilize ill-conditioned Hessian matrix / time consuming and local minimum trap & No reliable information about parameter errors and correlations & Need initialization with another method, give the optimal step size for the optimization algorithm then reduce the complexity[\
]{}
We tested three time resolutions with times values took within $3\,\sigma_{t}$ .
- $\sigma_{t}=0\, ns$ plays the role of the perfect theoretical detection and serves as reference;
- $\sigma_{t}=3\, ns$ reflects the optimum performances expected in the current state of the art;
- $\sigma_{t}=10\, ns$ stands for the timing resolution estimate of an experiment like CODALEMA [@key-1-1].
For every source distance and temporal resolution, one million events were generated. Antenna location was taken in a uniform distribution of $1\, m$ width. A blind search was simulated using uniform distribution of the initial $r_{s}$ values from $0.1\, km$ to $20\, km$. Typical results obtained with our simulations are presented in Figures \[LVM\_REC\_10km\] and \[Simpl\_1\_10km\]. The summary of the reconstructed parameters is given in table \[ResulSimul\].
![Results of the reconstruction of a source with a radius of curvature equal to 1 and 10 km with the LVM algorithm. For $R_{true}=1\, km$, the effect of the blind search leads to non-convergence of the LVM algorithm, when initialization values are greater than $R_{true}=1\, km$.[]{data-label="LVM_REC_10km"}](results_reconstriuction_R_1000.png "fig:"){width="6cm" height="6cm"} ![Results of the reconstruction of a source with a radius of curvature equal to 1 and 10 km with the LVM algorithm. For $R_{true}=1\, km$, the effect of the blind search leads to non-convergence of the LVM algorithm, when initialization values are greater than $R_{true}=1\, km$.[]{data-label="LVM_REC_10km"}](results_reconstriuction_R_10000.png "fig:"){width="6cm" height="6cm"}
![Results of the reconstruction of a source with a radius of curvature equal to 1 and 10 km with the Simplex algorithm.[]{data-label="Simpl_1_10km"}](fig_simplex_R1km_all.png "fig:"){width="6cm" height="6cm"} ![Results of the reconstruction of a source with a radius of curvature equal to 1 and 10 km with the Simplex algorithm.[]{data-label="Simpl_1_10km"}](fig_simplex_R10km_all.png "fig:"){width="6cm" height="6cm"}
\[ResulSimul\]
[|c|c|c|c|c|c|]{} [\
]{} $\sigma_{t}\,(ns)$ & $R_{true}(m)$ & $Algorithms$ & & & [\
]{} & & Levenberg-Marquardt & (10071) 1002 & (1081) 1081 & (5763) 102[\
]{} & & Simplex & & & [\
]{} & & Levenberg-Marquardt & (9960) 3082 & (2998) 2998 & (5781) 302[\
]{} & & Simplex & & & [\
]{} & & Levenberg-Marquardt & & & [\
]{} & & Simplex & & & [\
]{} & & Levenberg-Marquardt & (10071) 1003 & (934) 934 & (5763) 108[\
]{} & & Simplex & & & [\
]{} & & Levenberg-Marquardt & (9954) 3068 & (2874) 2874 & (5792) 495[\
]{} & & Simplex & & & [\
]{} & & Levenberg-Marquardt & & & [\
]{} & & Simplex & & & [\
]{} & & Levenberg-Marquardt & (10068) 985 & (964) 964 & (5767) 175[\
]{} & & Simplex & & & [\
]{} & & Levenberg-Marquardt & (9703) 2238 & (1620) 1620 & (6125) 877[\
]{} & & Simplex & & & [\
]{} & & Levenberg-Marquardt & & & [\
]{} & & Simplex & & & [\
]{}
Whatever the simulations samples (versus any source distances, arrival directions, time resolutions), (also with several detector configurations) and the three minimization algorithms, large spreads were generally observed for the source locations reconstructed. This suggests that the objective-function presents local minima. Moreover, the results depend strongly on initial conditions. All these phenomena may indicate that we are facing an ill-posed problem. Indeed, condition number calculations [@key-14] (see Fig. \[conditionement\]), which measures the sensitivity of the solution to errors in the data (as the distance of the source, the timing resolution, etc.), indicate large values ($>10^{4}$), when a well-posed problem should induce values close to 1.
To understand the observed source reconstruction patterns, we have undertaken to study the main features of this objective-function.
{width="11cm" height="6cm"} \[conditionement\]
Study of the objective-function for the spherical emission
==========================================================
To estimate the source position $X_{s}=(\vec{r_{s}},t_{s})$ using the sequence of arrival $t_{i}$, the natural method is to formulate an unconstrained optimization problem of type a non-linear least square [@key-12], starting from eq. 1. which can rewrite[^1] (see the notations listed in table \[table\_nomenclature\]):
$$f\left(X_{s}\right)=\frac{1}{2}\sum_{i=1}^{N}\left[\left\Vert \overrightarrow{r_{s}}-\overrightarrow{r_{i}}\right\Vert _{2}^{2}-\left(t_{s}^{*}-t_{i}^{*}\right)^{2}\right]^{2}=\frac{1}{2}\sum_{i=1}^{N}f_{i}^{2}(X_{s})$$
[|>p[15cm]{}|]{} $\vec{r_{s}}$, $\vec{r_{i}}$: position of the source, position of the $i^{th}$ antenna [\
]{}$t_{s}$, $t_{i}$: emission time of the signal, signal arrival time at the $i^{th}$ antenna [\
]{}$t_{s}^{*}$, $t_{i}^{*}$: reduced time variables (ie. $t^{*}=c.t$)[\
]{}$\sigma_{i}^{t}$: time resolution on the $i^{th}$ antenna [\
]{}$X_{s}$, $X_{i}$ spacio-temporal position of the source, of the $i^{th}$ antenna [\
]{}$\nabla f$, $\nabla^{2}f$: first and second derivative of the objective function $f$[\
]{}$M=I_{4}-2E_{44}=\begin{bmatrix}1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & -1
\end{bmatrix}$ : second order tensor related to the Minkowski metric[\
]{}$Q$, $L_{i}$: quadratic and linear form [\
]{}$<.|.>$: inner product [\
]{}$X^{T}$: transpose of a vector or a matrix X[\
]{}
\[table\_nomenclature\]
Several properties of the objective-function $f$ were studied: the coercive property to indicate the existence of at least one minima, the non-convexity to indicate the existence of several local minima, and the jacobian to locate the critical points. (Bias study, which corresponds to a systematic shift of the estimator, is postponed to another contribution). In mathematical terms, this analysis amounts to:
- Estimate the limits of $f$ to make evidence of critical points; obviously, the objective function $f$ is positive, regular and coercive. Indeed, $f$ tends to $+\infty$ when $\|X\|\to\pm\infty$, because it is a polynomial and contains positive square terms. So, $f$ admits at least a minimum.
- Verify the second optimality condition: the convexity property of a function on a domain for a sufficiently regular function is equivalent to positive-definiteness character of its Hessian matrix.
- Solve the first optimality condition: $\nabla f(X_{s})=0$ (jacobian) to find the critical points.
Convexity property
------------------
Using $f_{i}(X_{s})=(X_{s}-X_{i})^{T}.M.(X_{s}-X_{i})$ where $M$ designates the *Minkowski* matrix and given $\nabla f_{i}(X_{s})=2.M(X_{s}-X_{i})$, the $f$ gradient function can written (see appendix 1): $$\frac{1}{2}\nabla f(X_{s})=(\sum f_{i}(X_{s}))M.X_{s}-M.(\sum f_{i}(X_{s})X_{i})$$ The Hessian matrix, which is the $f$ second derivative can written: $$\nabla^{2}f(X_{s})=\sum\nabla f_{i}(X_{s}).\nabla f_{i}^{T}+\sum f_{i}.\nabla^{2}f_{i}$$ that becomes, replacing $\nabla f_{i}$ by its expression: $$\nabla^{2}f(X_{s})=(\sum f_{i}(X_{s})).M+2M.[N.X_{s}.X_{s}^{T}+\sum X_{i}X_{i}^{T}-X_{s}(\sum X_{i})^{T}-(\sum X_{i})X_{s}^{T}].M$$ Using a Taylor series expansion to order $2$ (see appendix $1$), an expanded form of the Hessian matrix, equivalent to the previous formula of the $f$ second derivative, is:
[ $$\frac{1}{2}Q\left(X_{s},X_{i}\right)=\left[\begin{array}{cccc}
\sum_{i}K_{i}+2\sum_{i}\left(x_{s}-x_{i}\right)^{2} & 2\sum_{i}\left(x_{s}-x_{i}\right)\left(y_{s}-y_{i}\right) & 2\sum_{i}\left(x_{s}-x_{i}\right)\left(z_{s}-z_{i}\right) & 2\sum_{i}\left(x_{s}-x_{i}\right)\left(t_{i}^{*}-t_{s}^{*}\right)\\
* & \sum_{i}K_{i}+2\sum_{i}\left(y_{s}-y_{i}\right)^{2} & 2\sum_{i}\left(y_{s}-y_{i}\right)\left(z_{s}-z_{i}\right) & 2\sum_{i}\left(y_{s}-y_{i}\right)\left(t_{i}^{*}-t_{s}^{*}\right)\\
* & * & \sum_{i}K_{i}+2\sum_{i}\left(z_{s}-z_{i}\right)^{2} & 2\sum_{i}\left(z_{s}-z_{i}\right)\left(t_{i}^{*}-t_{s}^{*}\right)\\
* & * & * & -\sum_{i}K_{i}+2\sum_{i}\left(t_{i}^{*}-t_{s}^{*}\right)^{2}
\end{array}\right]$$ ]{}
This latter allowed us to study the convexity of $f$ (see appendix 1). Indeed, because its mathematical form is not appropriate for a direct use of the convexity definition, we have preferred to use the property of semi-positive-definiteness of the Hessian matrix. Our calculus lead to the conclusion that:
- Using the criterion of Sylvester [@key-13] and the analysis of the principal minors of the Hessian matrix , we find that $f$ is not convex on small domains, and thus is likely to exhibit several local minima, according to $X_{s}$ and $X_{i}$. It is these minima, which induce convergence problems to the correct solution for the common minimization algorithms.
Critical points
---------------
The study of the first optimality condition (Jacobian = 0) gives the following system $\nabla f(X_{s})=0$ and allows finding the critical points and their phase-space distributions. Taking into account the following expression:
$\frac{1}{2}\nabla f(\bar{X_{s}})=(\sum f_{i}(\bar{X_{s}}))M.\bar{X_{s}}-M.(\sum f_{i}(\bar{X_{s}})X_{i})$
we get the relation:
$$\overline{X}_{s}=\sum_{i=1}^{N}\frac{f_{i}(\overline{X}_{s})}{\sum_{j}f_{j}(\overline{X}_{s})}\, X_{i}$$
This formula looks like the traditional relationship of a barycenter. Thus, we interpret it in terms of the antennas positions barycenter and its weights. The weight function $f_{i}$ expressing the space-time distance error between the position exact and calculated, the predominant direction will be the one presenting the greatest error between its exact and calculated position. The antennas of greatest weight will be those the closest to the source.
In practice (see appendix 2), because the analytical development of this optimality condition in a three-dimensional formulation is not practical, especially considering the nonlinear terms, we chose to study particular cases. We considered the case of a linear antennas array (1D) for which the optimality condition is easier to express with an emission source located in the same plane. This approach allows us to understand the origin of the observed degeneration which appears from the wave equation invariance by translation and by time reversal (known reversibility of the wave equation in theory of partial differential equations) and provides us a intuition of the overall solution. It also enlightens the importance of the position of the actual source relative to the antennas array (the latter point is linked to the convex hull of the antenna array and is the object of the next section). Our study led to the following interpretations:
- The iso-barycenter of the antenna array (of the lit antennas for a given event) plays an important role in explaining the observed numerical degeneration. The nature of the critical points set determines the convergence of algorithms and therefore the reconstruction result.
- There are strong indications, in agreement with the experimental results and our calculations (for 1D geometry), that the critical points are distributed on a line connecting the barycenter of the lit antennas and the actual source location. We used this observation to construct an alternative method of locating the source (section 4).
- According to the source position relative to the antenna array, the reconstruction can lead to an ill-posed or well-posed problem, in the sense of J. Hadamard.
Convex hull
-----------
In the previous section we pointed that to face a well-posed problem (no degeneration in solution set), it was necessary to add constraints reflecting the propagation law in the medium, the causality constraints, and a condition linking the source location and the antenna array, the latter inducing the concept of convex hull of the array of antennas. From appendix 2, we also saw that analytically the critical points evidence could become very complex from the mathematical point of view. Therefore, we chose again an intuitive approach to characterize the convex hull, by exploring mathematically the case of a linear array with an emission source located in the same plane. This is the subject of the appendix 3.
The results extend to a 2D antenna array, illuminated by a source located anywhere at ground, arguing that it is possible to separate the array into sub-arrays arranged linearly. The superposition of all the convex segments of the sub-arrays leads then to conceptualize a final convex surface, built by all the peripheral antennas illuminated (see Figure \[Det2D\]).
![Scheme of the reconstruction problem of spherical waves for our testing array of antennas (2D), with a source located at ground. For this configuration, the convex hull becomes the surface depicted in red. The result is the same for a source in the sky. []{data-label="Det2D"}](xfig_Array_2D.png){width="11cm" height="7cm"}
The generalization of these results to real practical experience (with a source located anywhere in the sky) was guided by our experimental observations (performed through minimization algorithms) that provide a first idea of what happens. For this, we chose to directly calculate numerically the objective function for both general topologies: a source inside the antenna array (ie. and at ground level) and an external source to the antenna array (in the sky ). As can be deduced from the results (see Figs. \[obj\_func\_inside\] and \[obj\_func\_outside\]), for a surface antenna array, the convex hull is the surface defined by the antennas illuminated. (An extrapolation of reasoning to a 3D network (such as Ice Cube, ANTARES,...) should lead, this time, to the convex volume of the setup).
![Plots of the objective-function versus $R$ and versus the phase space $(R,\, t)$, in the case of our testing array (2D), for a source on the ground and located inside the convex surface of the antenna array. This configuration leads to a single solution. In this case the problem is well-posed. []{data-label="obj_func_inside"}](source_inside_chi_sigma10ns "fig:"){width="6cm" height="6cm"} ![Plots of the objective-function versus $R$ and versus the phase space $(R,\, t)$, in the case of our testing array (2D), for a source on the ground and located inside the convex surface of the antenna array. This configuration leads to a single solution. In this case the problem is well-posed. []{data-label="obj_func_inside"}](source_inside_chi_sigma10ns_view "fig:"){width="6cm" height="6cm"}
![Plots of the objective-function versus $R$ and versus the phase space $(R,\, t)$, in the case of our testing array (2D) for a source outside the convex hull. This configuration leads to multiple local minima. All minima are located on the line joining the antenna barycenter to the true source. In this case the problem is ill-posed. []{data-label="obj_func_outside"}](source_outside_chi_sigma10ns_R1km_theta90.png "fig:"){width="6cm" height="6cm"} ![Plots of the objective-function versus $R$ and versus the phase space $(R,\, t)$, in the case of our testing array (2D) for a source outside the convex hull. This configuration leads to multiple local minima. All minima are located on the line joining the antenna barycenter to the true source. In this case the problem is ill-posed. []{data-label="obj_func_outside"}](source_outside_chi_sigma10ns_R1km_theta90_bis.png "fig:"){width="6cm" height="6cm"}
Our results suggest the following interpretations:
- If the source is in the convex hull of the detector, the solution is unique. In contrast, the location of the source outside the convex hull of the detector, causes degeneration of solutions (multiple local minimums) regarding to the constrained optimization problem. The source position, outside or inside the array, affects the convergence of reconstruction algorithms.
Proposed method of reconstruction
==================================
Taking into account of the previous results, in order to avoid the trap of the local minima with common algorithms, we chose to compute directly the values of the objective function on a grid, using a subset of phase space in the vicinity of the solution a priori, and assuming that the minimum of the objective function correspond to the best estimate of the position of the source of emission. The input parameters are set from the planar fit, which provides both windows in $\theta$ and $\phi$. By cons, this method, being based on the search of the minimum of the objective function using a grid calculation, the choice of the metric becomes crucial. On the zenith and azimuth angles, the metric can be adequately inferred from the value of the angular resolution obtained by the current detectors, or $0.1^{\text{\textdegree}}$ for $\theta$ and $\phi$. Looking at the space metrics, the latter can be inferred by considering the quantity $(c^{2}\sigma_{t}^{2})^{2}$, ie. around one meter. The direction-priori is given by the planar fit, while the quantity $r_{s}$ is left free in the range $0.1-20\: km$ (the upper bound being determined by the value of the curvature exploitable, given the temporal resolutions currently available).
A typical result obtained with our method is presented in Fig. \[Mindemin\] and a summary of the reconstructed parameters is given in table \[TabMindemin\] which have to be compared to those presented in the table \[ResulSimul\].
![Histogram of the reconstruction of a source located at 10 km from the detector array, and using the grid method associated to the search of the global minimum for each event. []{data-label="Mindemin"}](R10km_minofmin_10ns.png){width="11cm" height="6cm"}
[|l|l|l|l|l|]{} [\
]{} $\sigma_{t}(ns)$ & $R_{true}(m)$ & $R_{mean}(m)$ & $R_{mode}(m)$ & $\sigma^{R}(m)$ [\
]{} $0$ & 1000 & 1000 & 1000 & 0 [\
]{} & 3000 & 3000 & 3000 & 0 [\
]{} & 10000 & 10000 & 10000 & 0 [\
]{} 3 & 1000 & 1010 & 998 & 58 [\
]{} & 3000 & 2964 & 2700 & 214 [\
]{} & 10000 & 9806 & 9700 & 161 [\
]{} $10$ & 1000 & 987 & 902 & 150 [\
]{} & 3000 & 2780 & 2700 & 149 [\
]{} & 10000 & 9734 & 9700 & 50 [\
]{}
\[TabMindemin\]
Conclusion
==========
Experimental results indicated that the common methods of minimization of spherical wavefronts could induce a mis-localisation of the emission sources. In the current form of our objective function, a first elementary mathematical study indicates that the source localization method may lead to ill-posed problems, according to the actual source position. To overcome this difficulty, we developed a simple method, based on grid calculation of the objective function. This approach appears to provide, at worst, an estimate as good as for the common algorithms for locating the main point of the emission source, keeping in mind that this method is not optimal in the sense of optimization theories. However, further developments are without any doubt still necessary, maybe based on advanced statistical theories, like for instance by adding further information (as the signal amplitude or the functional of the radio lateral distribution). This could be achieved by trying a generalized objective function which includes these parameters. In addition, the interactions with other disciplines which face this problem could also provide tracks of work (especially regarding earth sciences which focus on technics of petroleum prospecting).
Appendix 1
==========
Symbolic calculus
-----------------
Keeping the same notation as in table \[table\_nomenclature\], the objective function can be written: $$f\left(X_{s}\right)=\frac{1}{2}\sum_{i=1}^{N}f_{i}^{2}\left(X_{s}\right)$$ with $f_{i}\left(X_{s}\right)=\left(X_{s}-X_{i}\right)^{T}\cdot M\cdot\left(X_{s}-X_{i}\right)=\left\Vert \overrightarrow{r_{s}}-\overrightarrow{r_{i}}\right\Vert ^{2}-\left(t_{s}^{*}-t_{i}^{*}\right)^{2}$.
The formula $\nabla f\left(X_{s}\right)=\sum f_{i}\left(X_{s}\right)\cdot\nabla f_{i}\left(X_{s}\right)$ is derived from the formula of a product derivation. Using the bi-linearity of the inner product, we show that $\nabla f_{i}\left(X_{s}\right)=2M\cdot\left(X_{s}-X_{i}\right)$. By injecting this formula into the formula of $\nabla f$ , we obtain the following formula: $$\begin{aligned}
\nabla f\left(X_{s}\right)= & \sum f_{i}\left(X_{s}\right)\cdot\nabla f_{i}\left(X_{s}\right)\\
= & \sum f_{i}\left(X_{s}\right)\cdot2M\cdot\left(X_{s}-X_{i}\right)\end{aligned}$$ It then leads to the following form: $$\frac{1}{2}\nabla f\left(X_{s}\right)=\left(\sum f_{i}\left(X_{s}\right)\right)M\cdot X_{s}-M\cdot\left(\sum f_{i}\left(X_{s}\right)X_{i}\right)$$ With the same method, the second derivative matrix (Hessian matrix) is given by the following formula : $$\nabla^{2}f\left(X_{s}\right)=\sum\nabla f_{i}\left(X_{s}\right)\cdot\nabla f_{i}\left(X_{s}\right)^{T}+\sum f_{i}\left(X_{s}\right)\cdot\nabla^{2}f_{i}\left(X_{s}\right)$$
By injecting in the previous formula the following formula of the second derivatives $\nabla^{2}f_{i}\left(X_{s}\right)=2M$ and by using the relation $\left(AB\right)^{T}=B^{T}A^{T}$, we get the following formula: $$\begin{aligned}
\nabla^{2}f\left(X_{s}\right)= & \sum\nabla f_{i}\left(X_{s}\right)\cdot\nabla f_{i}\left(X_{s}\right)^{T}+\sum f_{i}\left(X_{s}\right)\cdot\nabla^{2}f_{i}\left(X_{s}\right)\\
= & \sum2M\cdot\left(X_{s}-X_{i}\right)\cdot\left(2M\cdot\left(X_{s}-X_{i}\right)\right)^{T}+\sum f_{i}\left(X_{s}\right)\cdot2M\\
= & 4M\cdot\left[\sum\left(X_{s}X_{s}^{T}-X_{s}X_{i}^{T}-X_{i}X_{s}^{T}+X_{i}X_{i}^{T}\right)\right]\cdot M+2\left(\sum f_{i}\left(X\right)\right)\cdot M\\
= & 4M\cdot\left[NX_{s}X_{s}^{T}+\sum X_{i}X_{i}^{T}-X_{s}\left(\sum X_{i}\right)^{T}-\left(\sum X_{i}\right)X_{s}^{T}\right]\cdot M+2\left(\sum f_{i}\left(X_{s}\right)\right)\cdot M\end{aligned}$$ Both relationships correspond to the end-calculus forms given in the $3^{\textrm{rd}}$ section. These forms are easy to handle for symbolic calculus but not convenient for explicit calculation used for studying the convexity.
Taylor expansion and explicit calculus
--------------------------------------
An explicit form for the objective function first and second differential can be obtained using a Taylor expansion. Indeed, the function$f$ is an element of $C^{\infty}\left(\mathbb{R}^{4},\mathbb{R}\right)$ [^2] and is therefore differentiable in the sense of *Fréchet*.
Let $X_{s}=\left(\overrightarrow{r_{s}},t_{s}^{*}\right)^{T}$ be a fixed vector of $\mathbb{R}^{4}$ and $\overrightarrow{\varepsilon}=\left(\overrightarrow{h},t^{*}\right)^{T}$ another vector of $\mathbb{R}^{4}$. In order to simplify the calculus, we use the following notations :
$K_{i}=\left\Vert \overrightarrow{r_{s}}-\overrightarrow{r_{i}}\right\Vert _{2}^{2}-\left(t_{s}^{*}-t_{i}^{*}\right)^{2}$ a constant term when setting the vector $X_{s}$;
$L_{i}\left(\overrightarrow{\varepsilon}\right)=\left\langle \overrightarrow{r_{s}}-\overrightarrow{r_{i}}\mid\overrightarrow{h}\right\rangle -\left(t_{s}^{*}-t_{i}^{*}\right)\cdot t^{*}$ the linear form;
and $Q\left(\overrightarrow{h},t^{*}\right)=\left\Vert \overrightarrow{h}\right\Vert _{2}^{2}-t^{*2}$ the quadratic form.
The Taylor expansion leads to: $$\begin{aligned}
f\left(X_{s}+\overrightarrow{\varepsilon}\right)= & \frac{1}{2}\sum_{i}\left(\left\Vert \overrightarrow{r_{s}}+\overrightarrow{h}-\overrightarrow{r_{i}}\right\Vert _{2}^{2}-\left(t_{0}^{*}+t^{*}-t_{i}^{*}\right)^{2}\right)^{2}\\
= & \frac{1}{2}\sum_{i}\left(\left\langle \overrightarrow{r_{s}}+\overrightarrow{h}-\overrightarrow{r_{i}}\mid\overrightarrow{r_{s}}+\overrightarrow{h}-\overrightarrow{r_{i}}\right\rangle -\left(t_{s}^{*}+t^{*}-t_{i}^{*}\right)^{2}\right)^{2}\\
= & \frac{1}{2}\sum_{i}\left(\left\Vert \overrightarrow{r_{s}}-\overrightarrow{r_{i}}\right\Vert _{2}^{2}+\left\Vert \overrightarrow{h}\right\Vert _{2}^{2}+2\left\langle \overrightarrow{r_{s}}-\overrightarrow{r_{i}}\mid\overrightarrow{h}\right\rangle -\left(t_{s}^{*}-t_{i}^{*}\right)^{2}-t^{*2}-2t^{*}\left(t_{s}^{*}-t_{i}^{*}\right)\right)^{2}\end{aligned}$$
Using the multinomial expansion, the function $f$ can then be approximated by the second-order Taylor expansion following: $$f\left(\overrightarrow{r_{s}}+\overrightarrow{h},t_{s}^{*}+t^{*}\right)\approx\frac{1}{2}\sum_{i}K_{i}^{2}+2\sum_{i}K_{i}\cdot L_{i}\left(\overrightarrow{h},t^{*}\right)+2\sum_{i}L_{i}^{2}\left(\overrightarrow{h},t^{*}\right)+\left(\sum_{i}K_{i}\right)\cdot Q\left(\overrightarrow{h},t^{*}\right)$$
We identify from this formula:
the constant term $\frac{1}{2}\underset{i}{\sum}K_{i}^{2}$;
the linear term which is $\nabla f\left(X_{s}\right)^{T}\cdot\overrightarrow{\varepsilon}=\left[2\cdot\underset{i}{\sum}K_{i}\left(\begin{array}{c}
\overrightarrow{r_{s}}-\overrightarrow{r_{i}}\\
t_{i}^{*}-t_{s}^{*}
\end{array}\right)\right]^{T}\cdot\overrightarrow{\varepsilon}$ (the $f$ first differential in $\left(\overrightarrow{r_{s}},t_{s}^{*}\right)$ );
and the quadratic form at the point $X_{s}$: [ $$\frac{1}{2}Q\left(X_{s},X_{i}\right)=\left[\begin{array}{cccc}
\sum_{i}K_{i}+2\sum_{i}\left(x_{s}-x_{i}\right)^{2} & 2\sum_{i}\left(x_{s}-x_{i}\right)\left(y_{s}-y_{i}\right) & 2\sum_{i}\left(x_{s}-x_{i}\right)\left(z_{s}-z_{i}\right) & 2\sum_{i}\left(x_{s}-x_{i}\right)\left(t_{i}^{*}-t_{s}^{*}\right)\\
* & \sum_{i}K_{i}+2\sum_{i}\left(y_{s}-y_{i}\right)^{2} & 2\sum_{i}\left(y_{s}-y_{i}\right)\left(z_{s}-z_{i}\right) & 2\sum_{i}\left(y_{s}-y_{i}\right)\left(t_{i}^{*}-t_{s}^{*}\right)\\
* & * & \sum_{i}K_{i}+2\sum_{i}\left(z_{s}-z_{i}\right)^{2} & 2\sum_{i}\left(z_{s}-z_{i}\right)\left(t_{i}^{*}-t_{s}^{*}\right)\\
* & * & * & -\sum_{i}K_{i}+2\sum_{i}\left(t_{i}^{*}-t_{s}^{*}\right)^{2}
\end{array}\right]$$ ]{}
which is the $f$ Hessian matrix in $\left(\overrightarrow{r_{s}},t_{s}^{*}\right)$, or the second differential of $f$ also denoted $\nabla^{2}f\left(X_{s},X_{i}\right)$. The use of $*$ indicates that the coefficients above and below the diagonal are equal (*Schwarz* Lemma). The quadratic form represented by this matrix gives us the local second-order properties for the function $f$. To show that a critical point is a local minimum, it will suffice to verify that the Hessian matrix is definite positive in the vicinity of this point.
Convexity property
------------------
The convex analysis occupies a capital place in the problems of minimization. Indeed, an important theorem yet intuitive stated that if a convex function has a local minimum, it is automatically global. We will shows that the function $f$ is not convex in $\mathbb{R}^{4}$, i.e. that the Hessian matrix in non-positive define.
Let $\nabla^{2}f\left(X\right)$ the Hessian matrix, and let’s suppose $d$ a vector, since the function $f$ is twice differentiable, using the Sylvester’s criterion [@key-13] to characterize the convexity of $f$ , we can write the following equivalence: $$f\;\textrm{is convex}\Leftrightarrow\textrm{ Hessian is positive semi-definite}\Leftrightarrow\textrm{ All Hessian principal minors are just nonnegative}$$ $$f\;\textrm{is convex}\Leftrightarrow\forall d,\:\forall X,\; d^{T}\cdot\nabla^{2}f\left(X\right)\cdot d\geqslant0$$ So if we can find an element $X$ and $d$ such as $d^{T}\cdot\nabla^{2}f\left(X\right)\cdot d<0$, $f$ will be non-positive definite. For this, it is sufficient to find a single negative principal minor to demonstrate the Hessian matrix is non-positive definite. The objective function f will present then several local minimums and will be thus locally non-convex.
So let $Q$ the explicit expression of the Hessian and let us choose $d^{T}\:=\:(0\:0\:0\:1)$ then: $$d^{T}\cdot\nabla^{2}f(X)\cdot d=(0\:0\:0\:1)\cdot Q(X_{s},X_{i})\cdot\left(\begin{array}{c}
0\\
0\\
0\\
1
\end{array}\right)$$ $$=-\sum_{i}K_{i}+2\sum_{i}\left(t_{i}^{*}-t_{s}^{*}\right)^{2}$$ which is represent the principal minor of order $4$ of the Hessian.
For a family of fixed positions antennas and for a signal source with coordinates $X_{s}$ such as $y_{s}=z_{s}=t_{s}^{*}=0$, the negativity condition of the principal minor of order 4 can then written: $$\sum_{i}\left(x_{s}-x_{i}\right)^{2}>\sum_{i}\left(-y_{i}^{2}-z_{i}^{2}+3t_{i}^{*2}\right)$$ Now the left term tends to infinity when the source tends to infinity[^3]. It is written in terms of limits, $$\lim_{\left|x_{s}\right|\rightarrow+\infty}\sum_{i}\left(x_{s}-x_{i}\right)^{2}=+\infty\Leftrightarrow\forall A>0,\:\exists\eta>0\diagup\left|x_{s}\right|>\eta\Rightarrow\sum_{i}\left(x_{s}-x_{i}\right)^{2}>A$$ Taking a value $\underset{i}{\sum}\left(-y_{i}^{2}-z_{i}^{2}+3t_{i}^{2*}\right)$ of the constant $A$, it exist a real $\eta$ and therefore a $x_{s}$ such that $\underset{i}{\sum}\left(x_{s}-x_{i}\right)^{2}>\underset{i}{\sum}\left(-y_{i}^{2}-z_{i}^{2}+3t_{i}^{*2}\right)$. We deduce that the function is not convex in the vicinity of this point. It suffices to take $d^{T}=(0\:0\:0\:1)$ and $x_{s}=\eta+1$.
Appendix 2
==========
Degeneration line for a linear antenna array
--------------------------------------------
According to experimental data analysis and to our simulations (see Fig. \[codanoise\] and \[obj\_func\_outside\]), the results of the common minimization algorithms appear to fall on a half-line in the phase space $(x,y,z)$ which we shall call the degeneration line, which is linked to the existence of local minima. We present the mathematical development in the case of a linear array using an analysis-synthesis method. Then we try to generalize results to the higher dimension cases.
Let suppose $X_{s}=(x_{s},t_{s}^{*})$ a critical point of $f$, ie. $\nabla f(X_{s})=0$, for a linear array, the minimization problem with constraints can written:
$$\left\{ \begin{aligned}arg~min~f(x_{s},t_{s}^{*})=\frac{1}{2}\sum_{i=1}^{N}((x_{s}-x_{i})^{2}-(t_{s}^{*}-t_{i}^{*})^{2})^{2}~~1\leqslant i\leqslant N\\
Propagation~constraint:~|x_{s}-x_{i}|=|t_{s}^{*}-t_{i}^{*}|\\
Causality~constraint:~t_{s}^{*}<min_{i}(t_{i}^{*})
\end{aligned}
\right.$$
and Let suppose $L=\left(\begin{array}{c}
L\\
L
\end{array}\right)$ so that $X_{s}-L$ is also a a solution of the minimization problem, ie. $\nabla f(X_{s}-L)=0$)
The Jacobian of $f$ is written as: $$\vec{\nabla}f\left(x_{s},t^{*}\right)=2\left(\begin{array}{c}
\underset{i}{\sum}\left(x_{s}-x_{i}\right)\left(\left(x_{s}-x_{i}\right)^{2}-\left(t_{s}^{*}-t_{i}^{*}\right)^{2}\right)\\
\underset{i}{\sum}\left(t_{i}^{*}-t_{s}^{*}\right)\left(\left(x_{s}-x_{i}\right)^{2}-\left(t_{s}^{*}-t_{i}^{*}\right)^{2}\right)
\end{array}\right)$$ If $X_{s}$ being a critical point, this leads to two equations: $$\begin{cases}
\sum_{i}\left(x_{s}-x_{i}\right)\left(\left(x_{s}-x_{i}\right)^{2}-\left(t_{s}^{*}-t_{i}^{*}\right)^{2}\right)=0\quad\left(1\right)\\
\sum_{i}\left(t_{i}^{*}-t_{s}^{*}\right)\left(\left(x_{s}-x_{i}\right)^{2}-\left(t_{s}^{*}-t_{i}^{*}\right)^{2}\right)=0\quad\left(2\right)
\end{cases}$$ Assuming that $X_{s}-L$ being also a critical point, this leads to two equations: $$\begin{cases}
\sum_{i}\left(x_{s}-x_{i}-L\right)\left(\left(x_{s}-x_{i}-L\right)^{2}-\left(t_{s}^{*}-t_{i}^{*}-L\right)^{2}\right)=0\quad\left(3\right)\\
\sum_{i}\left(t_{i}^{*}-t_{s}^{*}+L\right)\left(\left(x_{s}-x_{i}-L\right)^{2}-\left(t_{s}^{*}-t_{i}^{*}-L\right)^{2}\right)=0\quad\left(4\right)
\end{cases}$$ By developing the equation $\left(3\right)$ and by using the equation $\left(1\right)$, then: $$\left(3\right)\Rightarrow~~\sum_{i}\left(x_{s}-x_{i}\right)\left[\left(x_{s}-x_{i}\right)^{2}-\left(t_{s}^{*}-t_{i}^{*}\right)^{2}-2L\left[\left(x_{s}-x_{i}\right)-\left(t_{s}^{*}-t_{i}^{*}\right)\right]\right]-L\sum_{i}\left(x_{s}-x_{i}\right)^{2}-\left(t_{s}^{*}-t_{i}^{*}\right)^{2}$$ $$\ldots~~+2L^{2}\sum_{i}\left[\left(x_{s}-x_{i}\right)-\left(t_{s}^{*}-t_{i}^{*}\right)\right]=0$$
$$\Rightarrow~~-L\sum_{i}\left(x_{s}-x_{i}\right)^{2}+L^{2}\sum_{i}\left[\left(x_{s}-x_{i}\right)-\left(t_{s}^{*}-t_{i}^{*}\right)\right]-L\sum_{i}\left(x_{s}-x_{i}\right)^{2}-\left(t_{s}^{*}-t_{i}^{*}\right)^{2}$$ $$\ldots~~+L\sum_{i}\left(x_{s}-x_{i}\right)\left(t_{s}^{*}-t_{i}^{*}\right)=0$$ The set of constraints requires that the term $\underset{i}{\sum}\left(x_{s}-x_{i}\right)^{2}-\left(t_{s}^{*}-t_{i}^{*}\right)^{2}$ is null. We get the simplified equation: $$L\sum_{i}\left(x_{s}-x_{i}\right)-\left(t_{s}^{*}-t_{i}^{*}\right)=\sum_{i}\left(x_{s}-x_{i}\right)\left(\left(x_{s}-x_{i}\right)-\left(t_{s}^{*}-t_{i}^{*}\right)\right)$$ In the cases where $x_{s}-x_{i}<0$ for all $i$, the set of constraints is equivalent to $\left(x_{s}-x_{i}\right)-\left(t_{s}^{*}-t_{i}^{*}\right)=0$. Thus, if one assumes that $x_{s}-x_{i}<0$ for all $i$, i.e that the source is outside the array convex hull (a segment), we find that previous implications are equivalences and thus that equation $\left(3\right)$ is verified. Operating in the same manner for the equation $\left(4\right)$, we obtain the following equations: $$\left(4\right)\Rightarrow\sum_{i}\left(t_{i}^{*}-t_{s}^{*}\right)\left[\left(x_{s}-x_{i}\right)^{2}-\left(t_{s}^{*}-t_{i}^{*}\right)^{2}-2L\left[\left(x_{s}-x_{i}\right)-\left(t_{s}^{*}-t_{i}^{*}\right)\right]\right]+L\sum_{i}\left(x_{s}-x_{i}\right)^{2}-\left(t_{s}^{*}-t_{i}^{*}\right)^{2}$$
$$\ldots-2L^{2}\sum_{i}\left[\left(x_{s}-x_{i}\right)-\left(t_{s}^{*}-t_{i}^{*}\right)\right]=0$$
$$\Rightarrow-2L\sum_{i}\left(t_{i}^{*}-t_{s}^{*}\right)\left(x_{s}-x_{i}\right)-2L\sum_{i}\left(t_{i}^{*}-t_{s}^{*}\right)^{2}+L\sum_{i}\left(x_{s}-x_{i}\right)^{2}-\left(t_{s}^{*}-t_{i}^{*}\right)^{2}$$ $$\ldots-2L^{2}\sum_{i}\left(x_{s}-x_{i}\right)-\left(t_{s}^{*}-t_{i}^{*}\right)=0$$ Using the set of constraints as above, we obtain the following equation: $$L\sum_{i}\left(x_{s}-x_{i}\right)-\left(t_{s}^{*}-t_{i}^{*}\right)=\sum_{i}\left(t_{i}^{*}-t_{s}^{*}\right)\left(\left(x_{s}-x_{i}\right)-\left(t_{s}^{*}-t_{i}^{*}\right)\right)$$ The same analysis as above gives us the condition that the source is out of the antennas convex hull. This degeneration is an important point because it determines the convergence of minimization algorithms. In this case the problem of the reconstruction is ill-posed.
The generalization of the previous calculation to higher dimensions is more delicate, insofar as there are infinitely many directions in which the source can move. The idea now is to translate the source, from its position $\overrightarrow{r_{s}}$, simultaneously in all directions $\overrightarrow{r_{s}}-\overrightarrow{r_{i}}$ and with the same distances. We define the unit vector on the direction source-antenna. It will be noted: $\overrightarrow{e_{i}}=\frac{\overrightarrow{r_{s}}-\overrightarrow{r_{i}}}{\left\Vert \overrightarrow{r_{s}}-\overrightarrow{r_{i}}\right\Vert _{2}}$. The translation spatial direction thus defined, is given by the vector $\overrightarrow{L}=\underset{i}{\sum}\overrightarrow{e_{i}}=\underset{i}{\sum}\frac{\overrightarrow{r_{s}}-\overrightarrow{r_{i}}}{\left\Vert \overrightarrow{r_{i}}-\overrightarrow{r_{s}}\right\Vert _{2}}=-\underset{i}{\sum}\frac{\overrightarrow{r_{i}}}{\left\Vert \overrightarrow{r_{s}}-\overrightarrow{r_{i}}\right\Vert _{2}}+\left(\underset{i}{\sum}\frac{1}{\left\Vert \overrightarrow{r_{s}}-\overrightarrow{r_{i}}\right\Vert _{2}}\right)\overrightarrow{r_{s}}$. Considering the reduced temporal variables, the wave required delay to traverse the distance induced by the translation $\left\Vert \overrightarrow{L}\right\Vert $. Let $V$ the vector of coordinates $V=\left(\overrightarrow{L},\left\Vert \overrightarrow{L}\right\Vert \right)^{T}$. We write the first order optimality condition for the vector of $\mathbb{\mathbb{R}}{}^{4}$: $X_{s}-V$: $$\begin{aligned}
\nabla f\left(X_{s}-V\right)= & \left(\sum f_{i}\left(X_{s}-V\right)\right)\cdot M\cdot\left(X_{s}-V\right)-M\cdot\left(\sum f_{i}\left(X_{s}-V\right)\cdot X_{i}\right)\end{aligned}$$ By introducing the condition $\nabla f\left(X_{s}\right)=0$ which implies that: $$\left(\sum f_{i}\left(X_{s}\right)\right)\cdot M\cdot X_{s}-M\cdot\left(\sum f_{i}\left(X_{s}\right)X_{i}\right)=0$$ we obtain: $$\begin{aligned}
\nabla f\left(X_{s}-V\right)= & N\left(V^{T}\cdot M\cdot V\right)\cdot M\cdot\left[X_{s}-V-\frac{1}{N}\sum_{i}X_{i}\right]\\
\, & 2M\left(\sum\left(X_{s}-X_{i}\right)^{T}M\cdot V\cdot\overrightarrow{X_{i}}\right)-2\left(\sum\left(X_{s}-X_{i}\right)^{T}M\cdot V\right)\cdot M\cdot\left(X_{s}-V\right)\\
\, & -\left(\sum f_{i}\left(X_{s}\right)\right)\cdot M\cdot V\end{aligned}$$ According to the imposed form of the vector $\overrightarrow{V}$, then: $$V^{T}\cdot M\cdot V=\left(\overrightarrow{L}\:\left\Vert \overrightarrow{L}\right\Vert _{2}\right)M\left(\overrightarrow{L}\:\left\Vert \overrightarrow{L}\right\Vert _{2}\right)^{T}=0$$ It remains then the following expression: $$\begin{aligned}
\nabla f\left(X_{s}-V\right)= & 2M\left(\sum\left(X_{s}-X_{i}\right)^{T}M\cdot V\cdot X_{i}\right)-2\left(\sum\left(X_{s}-X_{i}\right)^{T}M\cdot V\right)\cdot M\cdot\left(X_{s}-V\right)\\
\, & -\left(\sum f_{i}\left(X_{s}\right)\right)\cdot M\cdot V\end{aligned}$$ The resolution of this equation should lead to an analytical expression for the topology of critical points. We failed to develop it, but we can already see that the explicit development leads to cross terms that will make simplifications difficult. Therefore, we have tried again an intuitive approach based on the numerical simulations presented section 3.3.
Appendix 3
==========
Convex hull for a linear antenna array
--------------------------------------
Let us consider the sub-array of the 3 aligned upper antennas presented in Fig. \[how\_test\_estim\]). The figure \[Pb\_1D-1\] shows the physical principle of the reconstruction of the source.
![Scheme of the reconstruction problem of spherical waves for a 1D array of antennas. For this configuration, the convex hull is the segment shown in red. []{data-label="Pb_1D-1"}](setup_lineaire.png){width="11cm" height="6cm"}
Three situations must be considered:
- the source located inside the array;
- the source located outside the array but on the detector axis;
- the source located outside this main axis. The latter corresponds to the typical problems encountered with of the man-made emitters located on the ground.
The first situation leads to 2 half-lines cutting each other in a single point: the solution is unique (Figure \[Cstr\_1D\_inside\]) and the localization problem is well-posed. The source is unique and inside the line segment linking the nearest antennas to the source. This segment correspond to the convex hull within this geometry. We can also note that only the two antennas flanking the source then play a role in its localization. The problem writes:
$$\left\{ \begin{aligned}arg~min~f(x_{s},t_{s}^{*})=\frac{1}{2}\sum_{i=1}^{N}((x_{s}-x_{i})^{2}-(t_{s}^{*}-t_{i}^{*})^{2})^{2}\\
Propagation~constraint:~|x_{s}-x_{i}|=|t_{s}^{*}-t_{i}^{*}|\\
Causality~constraint:~t_{s}^{*}<min_{i}(t_{i}^{*})
\end{aligned}
\right.$$
![Phase space representation in the case of a linear array of three antennas (shown as green squares located at $x_{1}=-200\: m$, $x_{2}=0\: m$, $x_{3}=200\: m$). The source is located at $x_{s}=60\, m$ when the instant of the emission is taken as the time origin ($t_{s}=0\: s$). Because the source is outside this sub-array, the constraints on the positions of antenna 1 and 2 lead to the same equation $t_{s}=60-x_{s}$ (black line). Equation for the antenna 3 (blue line) leads to $t_{s}=-60+x_{s}$. The causality conditions restrict the initial lines to two half-lines (red lines). The source location (black star) is at the intercept of the both half-lines.[]{data-label="Cstr_1D_inside"}](Constraints_1D_inside_source.png){width="11cm" height="6cm"}
About the source on-axis, but outside the convex hull, the arrival times between the antennas, are no longer related to the source position, but to their locations. Whatever their positions, the time differences remain constant (for equally spaced antennas). It becomes impossible to distinguish between two different shifted sources by any length. The only relevant information lies in the direction of propagation of the wave (see figure \[Cstr\_1D\_outside\]). This result appears by a degeneration of solutions because all points located on the half-line starting from the first tagged antenna are solutions of the problem which is ill-posed. The source is outside the convex hull of the antenna array.
![Same as figure \[Cstr\_1D\_inside\] but for an on-axis source outside the linear array of three antennas. The whole constraints lead the same equation $t_{s}^{*}=60-x_{s}$. All points belonging to the lower half-line are solutions of the source localization problem (red dashed line), which becomes, in this case, ill-posed, and creates the degenerations. []{data-label="Cstr_1D_outside"}](Constraints_1D_outside_source.png){width="11cm" height="6cm"}
On the configuration where source located outside this antenna axis (problem in two dimensions), the solving starts with:
$$\left\{ \begin{aligned}arg~min~f(x_{s},y_{s},t_{s}^{*})=\frac{1}{2}\sum_{i=1}^{N}((x_{s}-x_{i})^{2}+y_{s}^{2}-(t_{s}^{*}-t_{i}^{*})^{2})^{2}\\
Propagation~constraint:~(x_{s}-x_{i})^{2}+y_{s}^{2}=(t_{s}^{*}-t_{i}^{*})^{2}\\
Causality~constraint:~t_{s}^{*}<min_{i}(t_{i}^{*})
\end{aligned}
\right.$$
![Reconstruction problem of spherical waves for a 1D array of antennas with the source outside the convex hull. The local minima are located at the intersection of the cones. []{data-label="Pb_2D"}](1D_constraints_outside_line.png){width="10cm" height="7cm"}
The constraint set reduces the problem of characterization of critical points to the search of the half-cones intersections induced by each antenna, in the 3 dimensional phase space $(x,\, y\,,\, t)$ and which presents a great similarity of constraints with the light cone used in special relativity (Fig. \[Pb\_2D\]). Intersection of the half-cones, two to two, induces multiple critical points which are local minima.
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[^1]: In practice of the minimization, it is usual to take into account errors on the measured parameters by putting them in the objective function denominator. In our theoretical study, it is assumed that the arrival times errors are the same for all the antennas ($\sigma_{t}=constant~\forall~i$). The present studied functional is generic and does not include errors, but as will see later, introduction of a multiplicative constant doesn’t change the results of our study.
[^2]: The function is also an element of the algebra $\mathbb{R}\left[X_{1},\ldots,X_{4}\right]$
[^3]: We say that the function is *coercive*
|
---
abstract: 'We demonstrate that a conjecture of Teh which relates the niveau filtration on Borel-Moore homology of real varieties and the images of generalized cycle maps from reduced Lawson homology is false. We show that the niveau filtration on reduced Lawson homology is trivial and construct an explicit class of examples for which Teh’s conjecture fails by generalizing a result of Schülting. We compare various cycle maps and in particular we show that the Borel-Haeflinger cycle map naturally factors through the reduced Lawson homology cycle map.'
author:
- Jeremiah Heller
- Mircea Voineagu
bibliography:
- 'filtrations.bib'
title: Remarks on filtrations of the homology of real varieties
---
Introduction
============
Let $X$ be a quasi-projective real variety. In [@Teh:real] the reduced Lawson homology groups $RL_{q}H_{n}(X)$ are introduced as homotopy groups of certain spaces of “reduced” algebraic cycles. For $q=0$ we have $RL_{0}H_{n}(X) = H_{n}(X({\mathbb{R}});{\mathbb{Z}}/2)$, where $H_{n}(X({\mathbb{R}});{\mathbb{Z}}/2)$ is the Borel-Moore homology. At the other extreme we have that $RL_{n}H_{n}(X)$ is a quotient of the Chow group $CH_{n}(X)$. There are generalized cycle maps $$cyc_{q,n}:RL_{q}H_{n}(X)\to H_{n}(X({\mathbb{R}});{\mathbb{Z}}/2)$$ and the images of these cycle maps form a filtration of the homology $$Im(cyc_{n,n}) \subseteq Im(cyc_{n-1,n}) \subseteq \cdots \subseteq Im(cyc_{0,n})=H_{n}(X({\mathbb{R}});{\mathbb{Z}}/2).$$ The first step of this filtration is the image of the Borel-Haeflinger cycle map $Im(cyc_{n,n}) =H_{n}(X({\mathbb{R}});{\mathbb{Z}}/2)_{alg}$ .
The construction of the reduced Lawson homology is based on Friedlander’s construction of Lawson homology groups for complex varieties. Friedlander-Mazur [@FM:filt] have conjectured a relationship between the filtration on singular homology of the space of complex points given by images of the generalized cycle map and the niveau filtration. Teh makes an analogous conjecture for the reduced Lawson groups.
\[tco\] Let $X$ be a smooth projective real variety. Then $Im(cyc_{q,n}) =N _{2n-q}H _n(X({\mathbb{R}});{\mathbb{Z}}/2)$ for any $0\leq q\leq n$.
Here $N_pH _n(X({\mathbb{R}});{\mathbb{Z}}/2)$ is the niveau filtration which is the sum over all images $$Im\left(H_n(V({\mathbb{R}});{\mathbb{Z}}/2)\rightarrow H_n(X({\mathbb{R}});{\mathbb{Z}}/2)\right)$$ such that $\dim V \leq p$.
In the complex case, Friedlander-Mazur’s conjecture is a very difficult and interesting question. It is known to be true with arbitrary finite coefficients (in particular with ${\mathbb{Z}}/2$-coefficients), as a corollary of Beilinson-Lichtenbaum conjecture. With integral coefficients, it is known that Suslin’s conjecture for a smooth quasi-projective variety $X$ implies Friedlander-Mazur’s conjecture for $X$.
Surprisingly, the real case is totally different. Our main result says that the analog of this conjecture over the field of real numbers is false.
Conjecture \[tco\] is false.
To see the failure of this conjecture we first observe that the niveau filtration on reduced Lawson homology is uninteresting (the case of Borel-Moore homology is $q=0$). Specifically we have that $$N_{j}RL_{q}H_{n}(X) =
\begin{cases}
RL_{q}H_{n}(X) & j\geq n \\
0 & \textrm{else} .
\end{cases}$$ This is a consequence of Corollary \[dge\] which asserts that the coniveau spectral sequence for the reduced morphic cohomology collapses. Conjecture \[tco\] is therefore equivalent to the surjectivity of the generalized cycle maps. It is known that in general $H_{n}(X({\mathbb{R}});{\mathbb{Z}}/2) \neq H_{n}(X({\mathbb{R}});{\mathbb{Z}}/2)_{alg}$ although it is difficult to find explicit examples. In Example \[example\] and Proposition \[genex\] we give an explicit class of such examples. Our examples are based on a decomposition given in Theorem \[decompthrm\] of the reduced cycle spaces of a blow-up with smooth center. This decomposition is a generalization to reduced cycle spaces of the main result of [@S:reelle].
As another application of thee decomposition we have the following theorem, which is also different from the complex case where the group of divisors modulo algebraic equivalence of an irreducible complex variety always injects into the corresponding homology group. Theorem \[intt\] gives examples of thin divisors that are non-trivial in the reduced Lawson group (see Remark \[thin\]).
\[intt\] There exists a smooth real variety $X\stackrel{birational}\sim \mathbb{P}^N _\mathbb{R}$ such that the cycle map on divisors $RL^1H^1(X)\rightarrow H^1(X({\mathbb{R}}),{\mathbb{Z}}/2)$ is not injective.
The collapse of the coniveau spectral sequence for reduced morphic cohomology is a consequence of local vanishing of motivic cohomology in degrees larger than the weight together with the vanishing theorem proved in [@HV:VT]. For the purposes of seeing that Conjecture \[tco\] is false one does not need the full strength of the collapsing, it suffices to use only the local vanishing of motivic cohomology. However, an interesting consequence of the collapse of this spectral sequence is that we can compute reduced morphic cohomology as the sheaf cohomology $$H^{n}_{Zar}(X;{\mathcal{RL}}^{q}{\mathcal{H}}^{0})= RL^{q}H^{n}(X).$$ As a consequence, we identify a family of birational invariants given by $$H^{0}_{Zar}(X;{\mathcal{RL}}^{q}{\mathcal{H}}^{0})= RL^{q}H^{0}(X),$$ for any $q\geq 0$. In case $q=\dim(X)$ we obtain that the number $s$ of connected components of $X(\mathbb{R})$ is a birational invariant of an algebraic nature (i.e. $RL^{\dim(X)}H^{0}(X)=({\mathbb{Z}}/2)^s$). The purely algebraic nature of $s$ forms part of the main result of [@CTP:real], where they use étale cohomology. The relation between reduced morphic cohomology and étale cohomology is discussed in the final section. As an application of these birational invariants we compute reduced Lawson homology of a real rational surface in Corollary \[comp\].
In Section \[cyclemaps\] we discuss various cycle maps. We show that there is basically one cycle map from the mod-$2$ motivic cohomology to mod-$2$ singular cohomology of the space of real points. As a consequence we see in Theorem \[bh\] that the Borel-Haeflinger cycle map factors through this cycle map.
Preliminaries
=============
Let $Y$ be a projective complex variety and ${\mathcal{C}}_q(Y)$ the Chow variety of effective $q$-cycles on $Y$. Write ${\mathcal{Z}}_q(Y) = ({\mathcal{C}}_q(Y)({\mathbb{C}}))^{+}$ for the group completion of this monoid. The group completion is done algebraically and ${\mathcal{Z}}_q(Y)$ is given the quotient topology. It turns out that this naive group completion is actually a homotopy group completion [@FG:cyc], [@LF:qproj]. When $U$ is quasi-projective with projectivization $U\subseteq \overline{U}$ then define ${\mathcal{Z}}_{q}(U) = {\mathcal{Z}}_{q}(\overline{U})/{\mathcal{Z}}_{q}(U_{\infty})$ where $U_{\infty} = \overline{U}\backslash U$ (and the quotient is a group quotient). This definition is independent of choice of projectivization [@LF:qproj], [@FG:cyc].
If $X$ is a real variety then $G={\mathbb{Z}}/2$ acts on ${\mathcal{Z}}_{q}(X_{{\mathbb{C}}})$ via complex conjugation. The space of real cycles on $X$ is defined to be the subgroup ${\mathcal{Z}}_{q}(X_{{\mathbb{C}}})^{G}$ of cycles invariant under conjugation. Write ${\mathcal{Z}}_{q}(X_{{\mathbb{C}}})^{av}$ for the subgroup generated by cycles of the form $\alpha+\overline{\alpha}$. The space of reduced cycles on $X$ is defined to be the quotient group $${\mathcal{R}}_{q}(X) = \frac{{\mathcal{Z}}_{q}(X_{{\mathbb{C}}})^{G}}{{\mathcal{Z}}_{q}(X_{{\mathbb{C}}})^{av}}.$$
Let $X$ be a quasi-projective real variety. The *reduced Lawson homology* of $X$ is defined by $$RL_{q}H_{q+i}(X) = \pi_{i}{\mathcal{R}}_{q}(X).$$
When $q=0$ we have that ${\mathcal{R}}_{0}(X) = {\mathcal{Z}}_{0}(X({\mathbb{R}}))/2$ so by the Dold-Thom theorem $RL_{0}H_{i}(X) = H_{i}(X({\mathbb{R}});{\mathbb{Z}}/2)$ is the Borel-Moore homology of $X({\mathbb{R}})$. In general $RL_{q}H_{q+i}(X)$ are all ${\mathbb{Z}}/2$-vector spaces. It is not known whether these are finitely-generated vector spaces or not however we do have the following vanishing theorem.
Let $X$ be a quasi-projective real variety. Then $$RL_{q}H_{q+i}(X) = 0$$ if $q+i>\dim(X)$.
There is also a space ${\mathcal{R}}^{q}(X) = {\mathcal{Z}}^{q}(X_{{\mathbb{C}}})^{G}/{\mathcal{Z}}^{q}(X_{C})^{av}$ of reduced algebraic cocycles on $X$ when $X$ is normal and projective. We refer to [@Teh:real] for the details of its construction. It is convenient to extend this definition to quasi-projective normal varieties, which is done in [@HV:VT] although not introduced formally as such an extension. We avoid difficulties with point-set topology by giving the extension as a simplicial abelian group. Define the simplicial abelian group of reduced cocyles on a quasi-projective normal real variety $$\widetilde{{\mathcal{R}}}^{q}(X) = \frac{\operatorname{\mathrm{Sing}}_{\bullet}{Z}/2^{q}(X_{{\mathbb{C}}})^{G}}{\operatorname{\mathrm{Sing}}{Z}_{\bullet}/2^{q}(X_{{\mathbb{C}}})^{av}}.$$ If $X$ is a projective, normal real variety then $\widetilde{R}^{q}(X) \xrightarrow{{\simeq}} \operatorname{\mathrm{Sing}}_{\bullet}{\mathcal{R}}^{q}(X)$ is a homotopy equivalence [@HV:VT Lemma 6.7].
Let $X$ be a normal quasi-projective variety. Define the reduced morphic cohomology of $X$ by $$RL^{q}H^{q-i}(X) = \pi_{i}\widetilde{{\mathcal{R}}}^{q}(X).$$
The reduced Lawson homology and reduced morphic cohomology are related by a Poincare duality. This is proved in [@Teh:real Theorem 6.2] for smooth projective varieties. We give a quick proof below which applies to both the projective and quasi-projective case. If $T$ is a topological abelian group we write $\widetilde{T} = \operatorname{\mathrm{Sing}}_{\bullet}T$ for the associated simplicial abelian group. If $M$ is a $G$-module and $\sigma$ is the nontrivial element of $G$ we write $N= 1+\sigma$ and define $M^{av}= Im(N)$. If in addition $M$ is $2$-torsion then we have the two fundamental short exact sequences of abelian groups $0\to M^{G} \to M \xrightarrow{N} M^{av} \to 0$ and $0\to M^{av} \to M^{G} \to M^{G}/M^{av} \to 0 $.
\[pd\] Let $X$ be a smooth quasi-projective real variety of dimension $d$. The inclusion $$\widetilde{{\mathcal{R}}}^{q}(X) \to \widetilde{{\mathcal{R}}}_{d}(X\times {\mathbb{A}}^{q})$$ is a homotopy equivalence of simplicial abelian groups. Consequently there is a natural isomorphism $RL^{q}H^{n}(X)= RL_{d-q}H_{d-q}(X)$.
This follows immediately from consideration of the following comparison diagrams of homotopy fiber sequences of simplicial abelian groups, where the displayed homotopy equivalences follow from [@F:algco Theorem 5.2] and [@HV:VT Corollary 4.20]
$$\xymatrix{
\widetilde{{\mathcal{Z}}}^{q}/2(X_{{\mathbb{C}}})^{G} \ar[r]\ar[d]^{{\simeq}} & \widetilde{{\mathcal{Z}}}^{q}/2(X_{{\mathbb{C}}}) \ar[d]^{{\simeq}}\ar[r]^{N} & \widetilde{{\mathcal{Z}}}^{q}/2(X_{{\mathbb{C}}})^{av}\ar[d] \\
\widetilde{{\mathcal{Z}}}_{d}/2(X_{{\mathbb{C}}}\times_{{\mathbb{C}}}{\mathbb{A}}_{{\mathbb{C}}}^{q}))^{G} \ar[r] & \widetilde{{\mathcal{Z}}}_{d}/2(X_{{\mathbb{C}}}\times_{{\mathbb{C}}}{\mathbb{A}}_{{\mathbb{C}}}^{q})) \ar[r]^{N} & \widetilde{{\mathcal{Z}}}_{d}/2((X_{{\mathbb{C}}}\times_{{\mathbb{C}}}{\mathbb{A}}_{{\mathbb{C}}}^{q})^{av}
}$$
and $$\xymatrix{
\widetilde{{\mathcal{Z}}}^{q}/2(X_{{\mathbb{C}}})^{av} \ar[r]\ar[d] & \widetilde{{\mathcal{Z}}}^{q}/2(X_{{\mathbb{C}}})^{G} \ar[r]\ar[d]^{{\simeq}} & \widetilde{{\mathcal{R}}}^{q}(X) \ar[d]\\
\widetilde{{\mathcal{Z}}}_{d}/2(X_{{\mathbb{C}}}\times_{{\mathbb{C}}}{\mathbb{A}}^{q}_{{\mathbb{C}}})^{av} \ar[r] & \widetilde{{\mathcal{Z}}}_{d}/2(X_{{\mathbb{C}}}\times_{{\mathbb{C}}}{\mathbb{A}}^{q}_{{\mathbb{C}}})^{G} \ar[r] & \widetilde{{\mathcal{R}}}_{d}(X\times{\mathbb{A}}^{q}_{{\mathbb{R}}}) .
}$$
The inclusion of algebraic cocycles into topological cocycles gives a generalized cycle map $$cyc_{q,n}:RL^{q}H^{n}(X) \to H^{n}(X({\mathbb{R}});{\mathbb{Z}}/2).$$ If $X$ is smooth and $q\geq \dim X$ then the cycle map $cyc_{q,n}$ is an isomorphism. For $X$ projective this follows from [@Teh:real Corollary 6.5, Theorem 8.1] (the results there are stated under the assumption that $X({\mathbb{R}})$ is nonempty and connected but this is unnecessary). The isomorphism for projective varieties implies the isomorphism for quasi-projective varieties (for example by using cohomology with supports and an argument as in [@HV:AHSS Corollary 4.2]).
There are operations called the $s$-map in both reduced Lawson homology and reduced morphic cohomology $s:RL_{q}H_{n}(X)\to RL_{q-1}H_{n}(X)$ and $s:RL^{q}H^{n}(X)\to RL^{q+1}H^{n}(X)$. Iterated compositions of $s$-maps give rise to generalized cycle maps in reduced Lawson homology $$cyc_{q,n}:RL_{q}H_{n}(X) \xrightarrow{s} RL_{q-1}H_{n}(X) \xrightarrow{s} \cdots \xrightarrow{s} RL_{0}H_{n}(X) = H_{n}(X({\mathbb{R}});{\mathbb{Z}}/2).$$ The $s$-map in morphic cohomology is compatible with the $s$-map in the sense that $cyc_{q,n}$ agrees with the composition $$cyc_{q,n}:RL^{q}H^{n}(X) \xrightarrow{s} RL^{q+1}H^{n}(X) \xrightarrow{cyc_{q,n}} H^{n}(X({\mathbb{R}});{\mathbb{Z}}/2).$$
Let $X$ be a normal quasi-projective real variety and $Z\subseteq X$ a closed subvariety. The *reduced morphic cohomology with supports* is defined in the usual way with $RL^qH^{q-i}(X)_Z = \pi_{i}\widetilde{{\mathcal{R}}}^{q}(X)_{Z}$ where $\widetilde{{\mathcal{R}}}^{q}(X)_{Z} = \operatorname{\mathrm{hofib}}(\widetilde{{\mathcal{R}}}^{q}(X) \to \widetilde{{\mathcal{R}}}^{q}(X-Z) )$.
\[reducedpurity\] Let $X$ be a smooth, quasi-projective real variety of dimension $d$ and $Z\subset X$ a closed smooth subvariety of codimension $p$. There are homotopy equivalences $\widetilde{\mathcal{R}}^q(X)_Z{\simeq}\widetilde{R}^{q-p}(Z)$. These induce natural isomorphisms $$RL^qH^n(X)_Z = RL^{q-p}H^{n-p}(Z).$$
This follows from the localization sequence for reduced Lawson homology [@Teh:real Corollary 3.14] together with Poincare duality between reduced Lawson homology and reduced morphic cohomology (see [@Teh:real Theorem 6.2] and Theorem \[pd\]).
Recall that a presheaf $F(-)$ of cochain complexes satisfies *Nisnevich descent* provided that for any smooth $X$, any étale map $f:Y\to X$, and open embedding $i:U\subseteq X$ such that $f: Y - f^{-1}(U)\to X - U$ is an isomorphism, we have a Mayer-Vietoris exact triangle (in the derived category of abelian groups): $$F(X) \to F(Y)\oplus F(U) \to F(f^{-1}(V)) \to F(X)[1].$$
\[boprops\] The presheaf $\widetilde{\mathcal{R}}^q(-)$ is homotopy invariant theory and satisfies Nisnevich descent.
It is homotopy invariant by [@Teh:real Theorem 5.13]. Nisnevich descent follows immediately from Theorem \[reducedpurity\].
Birational invariants and examples {#BIE}
==================================
We use the following decomposition theorem for spaces of reduced cycles on blow-ups with smooth center in order to obtain a decomposition of the cokernels of the cycle map from reduced Lawson homology. Later we use this decomposition to exhibit spaces whose cycle map has nontrivial cokernel i.e. it is not surjective. Recall that ${\mathcal{R}}_{-q}(X) = {\mathcal{R}}_{0}(X\times{\mathbb{A}}^{-q})$ for $q\geq 0$.
\[decompthrm\] Let $X$ be a smooth real projective variety and $Z\subset X$ a smooth closed irreducible subvariety of codimension $d>1$. Let $\pi:X^*\rightarrow X$ be the blow up of $X$ by the smooth center $Z$. Then, for any $0\leq q\leq dim(X)$ we have a homotopy equivalence of topological abelian groups $$\label{dec}
{\mathcal{R}}_q(X^*)\stackrel{h.e.}{\simeq} {\mathcal{R}}_q(X)\oplus {\mathcal{R}}_{q-d+1}(Z)\oplus {\mathcal{R}}_{q-d+2}(Z)\oplus \cdots \oplus {\mathcal{R}}_{q-1}(Z).$$ Moreover this decomposition is compatible with $s$-maps.
We follow the ideas used to prove [@Voin:2 Theorem 2.5] and the main theorem of [@S:reelle]. We work in $\mathcal{H}^{-1}AbTop$, the category of topological abelian groups with a CW-structure with inverted homotopy equivalences.
Recall that $\pi^{-1}(Z)\rightarrow Z$ is the projective bundle $p:\mathbb{P}(N _ZX)\rightarrow Z$ of dimension $d-1$. The decomposition in the statement of the theorem is given as follows. The first component of the map is $\pi_{*}$ (notice that $\pi _*\circ \pi^*=id$). The other maps are given via compositions $$\phi _l: {\mathcal{R}}_{q-d+1+l}(Z)\xrightarrow{p^*} {\mathcal{R}}_{q+l}(X^*)\xrightarrow{-\cap c _1(O(1))^l} {\mathcal{R}}_q(X^*),$$ where $p^*: {\mathcal{R}}_q(Z)\rightarrow {\mathcal{R}}_{q+d-1}(\mathbb{P}(N _ZX))\stackrel{i _*}{\rightarrow} {\mathcal{R}}_{q+d-1}(X^*)$.
Using the Mayer-Vietoris sequence (see Corollary \[boprops\]), we have that $${\mathcal{R}}_k(X^*)\stackrel{h.e.}{\rightarrow} {\mathcal{R}}_k(X)\oplus Ker(p _*),$$ with $p _*:{\mathcal{R}} _k(N _Z(X))\rightarrow {\mathcal{R}}_k(Z)$ and then identify in $\mathcal{H}^{-1}AbTop$ $$\label{khe}
Ker(p _*)\stackrel{h.e.}{\simeq}{\mathcal{R}}_{q-d+1}(Z)\oplus {\mathcal{R}}_{q-d+2}(Z)\oplus\cdots\oplus {\mathcal{R}}_{q-1}(Z).$$ Using the properties of Segre classes $s _l(N _ZX)\cap -:{\mathcal{R}}_k(Z)\rightarrow {\mathcal{R}}_{k+d-1-l}(Z)$ i.e. $s _l(N _Z(X))\cap -=0$ if $l<0$ and $s _0(N _Z(X))\cap -=id$, one can prove the projective bundle formula for the reduced cycle groups in the usual way, see [@Teh:real]. Observe that the projective bundle formula works also for negative indexes. We obtain $${\mathcal{R}}_k(N _Z(X))=\oplus _{0\leq l\leq d-1}{\mathcal{R}}_{k-d+1+l}(Z).$$ Now one can conclude the homotopy equivalence (\[khe\]).
The $s$-maps are compatible with all of the maps involved in the decomposition \[dec\] (see [@Teh:real]) and therefore the decomposition of the theorem is preserved by the $s$-maps.
The generalized cycle maps $cyc_{q,n}: RL_qH _n(X)\rightarrow H _n(X(\mathbb{R}),\mathbb{Z}/2)$, are defined as a composite of $s$-maps together with the Dold-Thom isomorphism. Write $$T _{q,n}:=\operatorname{\mathrm{Coker}}(cyc_{q,n}:RL_qH _n(X)\rightarrow H _n(X(\mathbb{R}),\mathbb{Z}/2))$$ and
$$K _{q,n}(X)=\ker(cyc _{q,n}: RL _qH _n(X)\rightarrow H _n(X(\mathbb{R}),\mathbb{Z}/2)).$$
Notice that $T _{q,n}=0 = K_{q,n}$, for $q\leq 0$.
\[bma\] Let $\pi:X^{*}\to X$ and $Z$ be as in the above theorem. Then $$T _{q,n}(X^*)=T _{q,n}(X)\oplus T _{q-1,n-1}(Z)\oplus...\oplus T _{q-d+1,n-d+1}(Z),$$ and $$K _{q,n}(X^*)=K _{q,n}(X)\oplus K _{q-1,n-1}(Z)\oplus...\oplus K _{q-d+1,n-d+1}(Z).$$
In the case $k=0$ the decomposition (\[dec\]) gives $${\mathcal{R}}_0(X^*(\mathbb{R}))\stackrel{h.e.}{\simeq} {\mathcal{R}}_0(X(\mathbb{R}))\oplus {\mathcal{R}}_{-d+1}(Z(\mathbb{R}))\oplus {\mathcal{R}}_{-d+2}(Z(\mathbb{R}))\oplus \cdots \oplus {\mathcal{R}}_{-1}(Z(\mathbb{R})),$$ therefore producing the decomposition of Borel-Moore homology $$H _k(X^*(\mathbb{R}),\mathbb{Z}/2)=H _k(X(\mathbb{R}),\mathbb{Z}/2)\oplus H _{k-d-1}(Z(\mathbb{R}), \mathbb{Z}/2)\oplus \cdots \oplus H _{k-1}(Z(\mathbb{R}),\mathbb{Z}/2).$$ The $s$-maps respect the decomposition (\[dec\]) and comparing the decomposition for $q=0$ and for $q>0$ yields the result.
If $Z(\mathbb{R})=\emptyset$, then $T _{q,n}(X^*)=T _{q,n}(X)$ for any $0\leq q\leq n\leq dim(X)$.
An analog of Corollary \[bma\] for $T_{q,q}$ was originally proven by Schülting in [@S:reelle]. There separate arguments are needed to give a decomposition algebraically and a decomposition topologically. An advantage that our uniform proof has is that is entirely algebraic, the homology of real points being expressed in terms of homotopy of the group of algebraic cycles ${\mathcal{R}}_0(X)$.
Using similar techniques one can prove a decomposition analogous to (\[dec\]) for the spaces of real cycles defining dos Santos equivariant Lawson homology groups.
\[bic\] The groups $T _{1,n}(X)$ and $K _{1,n}(X)$ are birational invariants for smooth projective real varieties.
By [@AKMW:birat Theorem 0.3.1] every birational map between smooth projective varieties factors as a composition of blow-ups and blow-downs with smooth centers. The result then follows from Corollary \[bma\].
We close the section with the following computation.
\[comp\] Let $X$ be a rational smooth projective surface i.e. $X\stackrel{birational}{\simeq}\mathbb{P}^2 _{\mathbb{R}}$. Then the cycle map $$cyc_{q,n}:RL_{q}H _n(X)\to H _n(X(\mathbb{R}),\mathbb{Z}/2)$$ is an isomorphism for $q \leq n$.
We have ${\mathcal{R}}_0(X)={\mathcal{R}}_0(X(\mathbb{R}))$. By Corollary \[bic\] we have that $T_{1,n}(X)$ and $K_{1,n}(X)$ are birational invariants and we know that $T_{1,n}({\mathbb{P}}^{2}_{{\mathbb{R}}}) = 0 = K_{1,n}({\mathbb{P}}^{2}_{{\mathbb{R}}})$. By Theorem \[dge\] the group $\pi_{0}{\mathcal{R}} _2(X)$ is a birational invariant.
For $X$ as in the previous corollary we have that $RL_{q}H_{n}(X) = 0$ for $n<q$ and $n>2$ and $RL_{0}H_{0}(X)= RL_{0}H_{2}(X) = RL_{1}H_{2}(X) = RL_{2}H_{2}(X) = {\mathbb{Z}}/2$.
Coniveau spectral sequences
===========================
In this section we show that the coniveau spectral sequence for reduced morphic cohomology collapses. We make use of [@CTHK:BO] for the Bloch-Ogus theorem identifying the $E_{2}$-term of this spectral sequence. Let $X$ be a smooth, quasi-projective real variety and write $X^{(p)}$ for the set of points $x\in X$ whose closure has codimension $p$. Let $h^*$ be a cohomology theory with supports. Define $h^{i}_{x}(X) = \operatorname*{\mathrm{colim}}_{ U\subseteq X}h^{i}_{\overline{x}\cap U}(U)$ and $h^{i}(k(x)) = \operatorname*{\mathrm{colim}}_{ U \subseteq \overline{x}}h^{i}(U)$ (where in both colimits, $U$ ranges over nonempty opens). One may form the Gersten complex $$0\rightarrow \bigoplus_{x\in X^{(0)}} h^{n}_{x}(X)\rightarrow \bigoplus_{x\in X^{(1)}}h^{n+1}_{x}(X) \rightarrow\cdots \to \bigoplus_{x\in X^{(p)}} h^{n+p}_{x}(X) \to \cdots .$$
This complex gives rise to the coniveau spectral sequence $$E_{1}^{p,q} = \bigoplus_{x\in X^{(p)}}h^{p+q}_{x}(X) \Longrightarrow h^{p+q}(X)$$ The associated filtration is $N^{p}h^{n}(X) = \cup_{Z}Im(h^{n}_{Z}(X) \to h(X))$, where the union is over closed subvarieties $Z\subseteq X$ of codimension $p$.
[@CTHK:BO Corollary 5.1.11, Theorem 8.5.1] Let $h^*$ be a cohomology theory with supports on $Sm/{\mathbb{R}}$ which satisfies Nisnevich excision and is homotopy invariant. Let ${\mathcal{H}}^{q}$ be the Zariski sheafification of the presheaf $U\mapsto h^{q}(U)$. Then the Gersten complex $E_{1}^{\bullet,q}$ is a flasque resolution of ${\mathcal{H}}^{q}$ and the coniveau spectral sequence has the form $$E^{p,q} _2=H^{p}_{Zar}(X;\mathcal{H}^q)\Longrightarrow H^{p+q}(X).$$ For every $q$, the group $H^0(X,\mathcal{H}^q)$ is a birational invariant for smooth proper varieties.
\[sm\] Let $X$ be a smooth quasi-projective real variety. For each $k$ we have spectral sequences $$E_{1}^{p,q}(k) = \bigoplus\limits_{x\in X^{(p)}}RL^{k-p}H^{q}(k(x)) \Longrightarrow RL^{k}H^{p+q}(X).$$ The $E_{2}$-terms are $E_{2}^{p,q}(k) = H_{Zar}^{p}(X;{\mathcal{RL}}^{k}{\mathcal{H}}^{q})$ and each $H_{Zar}^{0}(X;{\mathcal{RL}}^{k}{\mathcal{H}}^{q})$ is a birational invariant for smooth projective real varieties. Moreover, the $s$-maps induce maps of spectral sequences $\{E_{r}^{p,q}(k)\} \to \{E_{r}^{p,q}(k+1) \}$.
By Corollary \[boprops\] reduced morphic cohomology is homotopy invariant and satisfies Nisnevich excision. By Theorem \[reducedpurity\] we have for any $x\in X^{(p)}$ the isomorphism $RL^{k}H^{p+q}(X)_{x}=RL^{k-p}H^{q}(k(x))$. Thus the $E_{1}$-page of the coniveau spectral sequence can be rewritten in the displayed form. The $s$-maps are natural transformations and so induce maps of the exact couples defining the coniveau spectral sequence.
The following result gives us the collapsing of the coniveau spectral sequence. It is a consequence of the main vanishing theorem of [@HV:VT] and the local vanishing of equivariant morphic cohomology and morphic cohomology.
\[vanishing\] Let $R = {\mathcal{O}}_{T,t_{1},\cdots, t_{n}}$ be the semi-local ring of a smooth, real variety $T$ at the closed points $t_{1},\ldots, t_{n}\in T$. Then $$RL^{k}H^{q}(\operatorname{\mathrm{Spec}}R)= 0$$ if $q\neq 0$ and any $k>0$.
For convenience write $Y=\operatorname{\mathrm{Spec}}R$. Recall that by definition $$RL^{k}H^{q}(Y)=\operatorname*{\mathrm{colim}}RL^{k}H^{q}(U),$$ where the colimit is over all open $U\subseteq T$ such that all $t_{i}\in U$. Recall also that filtered colimits commute with homotopy groups and preserve exact sequences.
We need to see that $RL^{k}H^{k-s}(Y) = \pi_{s}{\mathcal{R}}^{k}(Y) = 0$ for $s\neq k$. The main vanishing result in [@HV:VT Theorem 6.10] implies that $\pi_{s}{\mathcal{R}}^{k}(Y) = 0$ for $s> k$.
Consider the homotopy fiber sequences of simplicial abelian groups $$\xymatrix{
\widetilde{{\mathcal{Z}}}^{k}/2(Y_{{\mathbb{C}}})^{G} \ar[r] & \widetilde{{\mathcal{Z}}}^{k}/2(Y_{{\mathbb{C}}}) \ar[r]^{N} & \widetilde{{\mathcal{Z}}}^{k}/2(Y_{{\mathbb{C}}})^{av}
}$$ and $$\xymatrix{
\widetilde{{\mathcal{Z}}}^{k}/2(Y_{{\mathbb{C}}})^{av} \ar[r] & \widetilde{{\mathcal{Z}}}^{k}/2(Y_{{\mathbb{C}}})^{G} \ar[r] & \widetilde{{\mathcal{R}}}^{k}(Y) .
}$$ Because $\pi_{s}\widetilde{{\mathcal{Z}}}^{k}/2(Y_{{\mathbb{C}}})^{G} = 0 = \pi_{s}\widetilde{{\mathcal{Z}}}^{k}/2(Y_{{\mathbb{C}}})$ if $s\leq k-1$ ([@FHW:sst Theorem 7.3] and [@HV:AHSS Lemma 3.22]) we see that $\pi_{s}\widetilde{{\mathcal{Z}}}^{k}/2(Y_{{\mathbb{C}}})^{av} = 0$ if $s\leq k-1$. Using the second homotopy fiber sequence we conclude that $\pi_{s}\widetilde{{\mathcal{R}}}^{k}(Y) = 0$ if $s\leq k-1$.
\[dge\] Let $X$ be a smooth quasi-projective real variety. For any $k$, the coniveau spectral sequence for reduced morphic cohomology satisfies $$E_{1}^{p,q}(k) = 0$$ for $q\neq 0$. Consequently $E_{2}^{p,0}(k) = E_{\infty}^{p,0}(k)$ and so we have natural isomorphisms $$H^{p}_{Zar}(X;{\mathcal{RL}}^{k}H^{0}) = RL^{k}H^{p}(X).$$ In particular $H^0(X,\mathcal{RL}^qH^0)=RL^qH^0(X)=\pi _q(\mathcal{R}^q(X))$ is a birational invariant for smooth projective real varieties.
\[shvan\] The case $k= \dim X$ tells us that ${\mathcal{H}}^{i}_{{\mathbb{R}}}= 0$ for any $i>0$ where ${\mathcal{H}}^{i}_{{\mathbb{R}}}$ is the Zariski sheafification of the presheaf $U\mapsto H^{i}(U({\mathbb{R}});{\mathbb{Z}}/2)$. This also follows from [@Sch:ret Theorem 19.2].
Corollary \[dge\] gives us birational invariants $RL^qH^0(X)$ for $0\leq q\leq d = \dim(X)$. If $q= d$ then we have $$RL^dH^0(X) = H^{0}(X({\mathbb{R}});{\mathbb{Z}}/2) = ({\mathbb{Z}}/2)^{\oplus s}$$ and therefore $s=s(X) = \#(\textrm{connected components of}\,\,X({\mathbb{R}}))$ is a birational invariant of an algebraic nature. This also follows from the main result of [@CTP:real] where they show that $H^0(X,\mathcal{H}^n _{et})=H^0(X(\mathbb{R}),\mathbb{Z}/2)$ for any $n\geq dim(X)+1$. Here $\mathcal{H} _{et}$ is the sheaf associated to the presheaf $U\rightarrow H^n _{et}(U, \mu^{\otimes n} _2)$.
At the other extreme if one takes $q=0$, $$RL^0H^0(X)= ({\mathbb{Z}}/2)^{\oplus r}$$ where $r = r(X)= \#(\textrm{geometrically irreducible components of}\,\,X)$ and so $r$ is also a birational invariant.
\[cmor\] By Corollary \[dge\] and Corollary \[sm\] the $s$-maps $$RL^qH^n(X)\stackrel{s}{\rightarrow}RL^{q+1}H^n(X)$$ are obtained as the map induced on Zariski sheaf cohomology by the sheafified $s$-maps $s:{\mathcal{RL}}^q{\mathcal{H}}^{0}\to{\mathcal{RL}}^{q+1}{\mathcal{H}}^{0}$. In particular we see that the generalized cycle map $$cyc_{q,n}:RL^{q}H^{n}(X) \to H^{n}(X({\mathbb{R}});{\mathbb{Z}}/2)$$ is obtained from the sheafified cycle map ${\mathcal{RL}}^{q}{\mathcal{H}}^{0} \to {\mathcal{H}}^{0}_{{\mathbb{R}}}$. In the last section we show that this cycle map is naturally related to the Borel-Haeflinger cycle map [@BH:cycle].
We finish by observing that Poincare duality gives the collapsing of the niveau spectral sequence for reduced Lawson homology (of possibly singular varieties).
\[niveau\] Let $X$ be a quasi-projective real variety. Write $X_{(p)}$ for the set of points $x\in X$ whose closure has dimension $p$. The niveau spectral sequence $$E^1 _{p,q}(k)=\oplus _{x\in X_{(p)}}RL_{k}H _{p+q}(k(x))\Longrightarrow RL_{k}H _{p+q}(X)$$ satisfies $E^{1}_{p,q}(k)=0$ for any $q\neq 0$ and therefore $E^{2}_{p,q}(k) = E^{\infty}_{p,q}(k)$.
The niveau spectral sequence is constructed as in [@BO:gersten]. Let $x\in X_{(p)}$. Using Corollary \[dge\] we see that for any $q\neq 0$ $$RL_{k}H _{p+q}(k(x)) = RL^{p-k}H^{-q}(k(x)) = 0 .$$
Filtrations in homology
=======================
Let $X$ be a quasi-projective real variety of dimension $d$. The generalized cycle map $\phi _{q,n}: RL_{q}H_{n}(X)\rightarrow H _n(X(\mathbb{R}),\mathbb{Z}/2)$ is the composition of $q$ iterations of the $s$-map together with the Dold-Thom isomorphism. Write $$RT_qH _n(X)=Im(\phi _{q,n}:RL_qH _n(X) \to H_{n}(X({\mathbb{R}});{\mathbb{Z}}/2).$$ This gives us a decreasing filtration of the homology of the space of real points and is called the topological filtration.
Associated to the niveau spectral sequence is the niveau filtration $$N_pRL_{k}H _n(X)=\sum_{\dim V \leq p} Im\left(RL_{k}H_n(V)\rightarrow RL_{k}H_n(X)\right).$$ Notice that in the complex case, if $Y$ is a complex variety of dimension $d$ then Weak Lefschetz theorem says that $$N_{n}H_n(Y({\mathbb{C}});{\mathbb{Z}})=N _{n+1}H_n(Y({\mathbb{C}});{\mathbb{Z}})= \cdots = N_{d}H _n(Y({\mathbb{C}});{\mathbb{Z}})).$$ In particular, in the complex case there are only $n+1$ steps in the filtration, and another $d-n$ are equal to the homology. In the real case, we don’t have this theorem and so apriori all one has is a filtration. $$N_{0}H _n(X({\mathbb{R}});{\mathbb{Z}}/2)\subseteq \cdots \subseteq N_{d}H _n(X({\mathbb{R}});{\mathbb{Z}}/2)=H _n(X({\mathbb{R}});{\mathbb{Z}}/2).$$
Teh has formulated the following conjecture which is made in analogy with a conjecture of Friedlander-Mazur [@FM:filt Conjecture p.71] for complex varieties.
\[Tq\] Let $X$ be a smooth projective real variety. Then $RT _qH _n(X)\subseteq N_{2n-q}H _n(X({\mathbb{R}});{\mathbb{Z}}/2)$ and moreover this containment is an equality $RT _qH _n(X) =N _{2n-q}H _n(X({\mathbb{R}});{\mathbb{Z}}/2)$ for any $0\leq q\leq n$.
From Proposition \[niveau\] we have the following equality: $$E^{\infty}_{n-q,q}(k)=N _{n-q}RL_{k}H_{n}(X)/N _{n-q-1}RL_{k}H_{n}(X)=0$$ for any $q\neq 0$. This means that for any $k$ we have
$$RL_{k}H_{n}(X) = N_dRL_{k}H_{n}(X) = \cdots = N_{n+1}RL_{k}H_{n}(X) = N_{n}RL_{k}H_{n}(X),$$ and $$0 = N_{-1}RL_{k}H_{n}(X) = N_{0}RL_{k}H_{n}(X) =\cdots = N_{n-1}RL_{k}H_{n}(X).$$
The first row of equalities contains the groups that appear in Conjecture \[Tq\]. Consequently the first part of the conjecture is obviously true because by the above we have that $N_{j}H_{n}(X({\mathbb{R}});{\mathbb{Z}}/2) = H_{n}(X({\mathbb{R}});{\mathbb{Z}}/2)$ for all $j\geq n$.
The second part of the conjecture is false because the $s$-maps are not always be surjective. Using the material from Section \[BIE\] we give an explicit example of this failure. Recall that we write $T_{q,n}(X) = \operatorname{\mathrm{Coker}}(cyc_{q,n}:RL_{q}H_{n}(X) \to H_{n}(X({\mathbb{R}});{\mathbb{Z}}/2))$.
\[example\] Let $Z\subseteq {\mathbb{P}}^{3} _{\mathbb{R}}$ be the smooth irreducible elliptic curve given by the equation $t^2x+ty^2-x^3=0$, $z=0$. Then $Z({\mathbb{R}})$ is well known to consist of 2 connected components (see for example [@BCR:rag Example 3.1.2]). Let $X^{*}$ be the blow-up of ${\mathbb{P}}^{3}$ along $Z$. Then $T_{2,2}(X^{*}) = T_{2,2}({\mathbb{P}}^{3}_{{\mathbb{R}}})\oplus T_{1,1}(Z)$ by Corollary \[bma\]. Because $RL_{1}H_{1}(Z) = \pi_{0}{\mathcal{R}}_{1}(Z) = {\mathbb{Z}}/2$ and $Z({\mathbb{R}})$ has two components we conclude that $T_{2,2}(X^{*}) = {\mathbb{Z}}/2$.
More generally we have the following.
\[genex\] For each $N\geq 3$, there is a smooth projective real variety $X$ of dimension $N$ (which is topologically connected) such that $T _{q,q}(X)\neq 0$ for all $2\leq q \leq N-1$.
Let $Z\subseteq {\mathbb{P}}^{N}$ be a smooth irreducible real curve such that $Z(\mathbb{R})$ has at least 2 connected components. Let $s$ denote the number of connected components of $Z({\mathbb{R}})$. Since $Z$ is irreducible we have that ${\mathcal{R}}_1(Z)=\mathbb{Z}/2$. Therefore $T_{1,1}(Z)=(\mathbb{Z}/2)^{s-1}$. Since $Z$ is a curve $T_{i,n}(Z) = 0$ for all other values of $i$ and $n$. We take $X\rightarrow {\mathbb{P}}^{N}$ to be the blow up of ${\mathbb{P}}^{N}$ along $Z$. Then $T_{q,q}(X) = T _{1,1}(Z)={\mathbb{Z}}/2^{s-1}$ by Corollary \[bma\] because $2\leq q \leq N-1 = \operatorname{\mathrm{codim}}(Z)$.
We also have a similar result for the kernel.
\[Kenex\] For each $N\geq 3$, there is a smooth projective real variety $X$, birational to ${\mathbb{P}}^{N}$, such that $K _{q,q}(X)\neq 0$ for all $2\leq q \leq N-1$.
Let $Z\subseteq {\mathbb{P}}^{N}$ be a smooth irreducible real curve such that $Z(\mathbb{R})=\emptyset$. Then $K_{1,1}(Z)=\mathbb{Z}/2$ and $K_{i,n}(Z) = 0$ for all other values of $i$ and $n$. Take $X\rightarrow {\mathbb{P}}^{N}$ to be the blow up of ${\mathbb{P}}^{N}$ along $Z$. We have $K_{q,q}(X) = K _{1,1}(Z)={\mathbb{Z}}/2$ by Corollary \[bma\].
As an interesting particular case we have the following which is different than the complex analog.
There exists a smooth real variety $X\stackrel{birational}\sim \mathbb{P}^N _\mathbb{R}$ such that the cycle map on divisors $RL^1H^1(X)\rightarrow H^1(X({\mathbb{R}}),{\mathbb{Z}}/2)$ is not injective.
\[thin\] A $k$-cycle is said to be thin if it is a sum of closed subvarieties $Z\subseteq X$ with $\dim Z({\mathbb{R}}) < k$. The kernel of the Borel-Haeflinger cycle map consists entirely of thin cycles by [@IS:real] and the composite $CH _q(X)\to RL _qH _q(X)\rightarrow H _q(X({\mathbb{R}});{\mathbb{Z}}/2)$ agrees with the Borel-Haeflinger cycle map by Theorem \[bh\]. This means that the proposition above gives examples of nonzero classes which are represented by thin cycles in $RL _qH _q(X)$.
Cycle Maps {#cyclemaps}
==========
Let $X$ be a smooth quasi-projective real variety. We discuss two natural cycle maps from motivic cohomology to the singular cohomology of $X({\mathbb{R}})$. Based on this, we show that Borel-Haeflinger map [@BH:cycle] factors through the reduced Lawson homology cycle map. We end the section with a discussion of the maps involved in the Suslin conjecture from the view of the methods in this section.
Recall that $G={\mathbb{Z}}/2$. If $M$ is a $G$-space we write $H^{i}_{G}(M;{\mathbb{Z}}/2)$ for the Borel cohomology with ${\mathbb{Z}}/2$-coefficients. The reduced morphic cohomology of $X$ comes equipped with a natural generalized cycle map to the singular cohomology of real points. Composing this with the canonical map from real morphic cohomology and its isomorphism with motivic cohomology (with ${\mathbb{Z}}/2$-coefficients) gives us the cycle map $$\label{themap}
H^{p}_{{\mathcal{M}}}(X;{\mathbb{Z}}/2(q)) \to RL^{q}H^{p-q}(X) \xrightarrow{cyc_{}}H^{p-q}(X({\mathbb{R}});{\mathbb{Z}}/2).$$ On the other hand the real morphic cohomology maps naturally to the Borel cohomology of the space of complex points. In turn there is a map $H^{n}_{G}(X({\mathbb{C}});{\mathbb{Z}}/2) \to \oplus H^{n-i}(X({\mathbb{R}});{\mathbb{Z}}/2)$ obtained by restricting to real points together with the decomposition $$H^n _G(X(\mathbb{R}),\mathbb{Z}/2)= H^{n}(X({\mathbb{R}})\times {\mathbb{R}}P^{\infty};{\mathbb{Z}}/2) = \bigoplus _{0\leq i\leq n}H^i(X(\mathbb{R}),\mathbb{Z}/2).$$ Composing with the appropriate projection gives us $$\label{amap}
H^{p}_{{\mathcal{M}}}(X;{\mathbb{Z}}/2(q)) \to H^{p}_{G}(X({\mathbb{C}});{\mathbb{Z}}/2) \xrightarrow{}H^{p-q}(X({\mathbb{R}});{\mathbb{Z}}/2).$$ We show that the cycle maps (\[themap\]) and (\[amap\]) agree with each other. Basically these agree because they can be seen as induced by maps of presheaves of cochain complexes and so Theorem \[MT\] applies.
Write $(Top)_{an}$ for the category of topological spaces homeomorphic to a finite dimensional CW-complex given the usual topology and $\phi:(Top)_{an}\to (Sm/{\mathbb{R}})_{Zar}$ for the map of sites induced by $X\mapsto X({\mathbb{R}})$.
\[thm:cycmaps\] Let $X$ be a smooth quasi-projective real variety. The cycle maps given by (\[themap\]) and (\[amap\]) agree. Moreover the intermediate maps in (\[amap\]) can be chosen so that the following diagram commutes $$\xymatrix{
H^p _{{\mathcal{M}}}(X,{\mathbb{Z}}/2(q))\ar[r]^{}\ar[d] & H^p _{et}(X,\mu^{\otimes q} _2) \ar[r]^{{\cong}}\ar[d] & H^p _G(X({\mathbb{C}}),{\mathbb{Z}}/2)\ar[d]\\
RL^pH^{p-q}(X)\ar[r]^{cyc} & H^{p-q}(X({\mathbb{R}}),{\mathbb{Z}}/2) & \ar[l] H^p _G(X({\mathbb{R}}),{\mathbb{Z}}/2).\\
}$$ for any $p,q\geq 0$.
Consider the following complexes of Zariski sheaves on $Sm/{\mathbb{R}}$: $$\begin{aligned}
\mathbb{Z}/2(q)(X) & = (z_{equi}({\mathbb{P}}_{{\mathbb{R}}}^{q/q-1},0)(X\times_{{\mathbb{R}}}\Delta^{\bullet}_{{\mathbb{R}}})\otimes{\mathbb{Z}}/2)[-2q] \\
\mathbb{Z}/2(q)^{sst}(X) & = \operatorname{\mathrm{Sing}}_{\bullet} ({\mathcal{Z}}^{q}/2(X_{{\mathbb{C}}})^{G})[-2q] \\
\mathbb{Z}/2(q)^{top}(X) & = {\mathrm{Hom}}_{cts}({X({\mathbb{C}})\times\Delta^{\bullet}_{top}},{{\mathcal{Z}}/2_{0}({\mathbb{A}}^{q}_{{\mathbb{C}}})})^{G}[-2q] \\
\mathbb{Z}/2(q)^{Bor}(X) & = {\mathrm{Hom}}_{cts}({X({\mathbb{C}})\times EG\times\Delta^{\bullet}_{top}},{{\mathcal{Z}}/2_{0}({\mathbb{A}}^{q}_{{\mathbb{C}}})})^{G}[-2q]\\
\mathbb{Z}/2(q)^{Bor} _{\mathbb{R}}(X) & = {\mathrm{Hom}}_{cts}({X({{\mathbb{R}}})\times EG\times\Delta^{\bullet}_{top}},{{\mathcal{Z}}/2_{0}({\mathbb{A}}^{q}_{{\mathbb{C}}})})^G[-2q], \\
{\mathcal{R}}(q)(X) &= (\operatorname{\mathrm{Sing}}_{\bullet} {\mathcal{Z}}^{q}/2(X_{{\mathbb{C}}})^{G}/\operatorname{\mathrm{Sing}}_{\bullet}{\mathcal{Z}}^{q}/2(X_{{\mathbb{C}}})^{av})[-2q]\end{aligned}$$
These complexes all satsify Nisnevich descent for standard reasons. See e.g. [@HV:VT Section 5] for the first three. Similarly the complex $\mathbb{Z}/2(q)^{Bor} _{\mathbb{R}}(X)$ because taking real points of a distinguished Nisnevich square of real varieties gives a homotopy pushout square of spaces. The last complex satisfies Nisnevich descent by Proposition \[boprops\]. As a consequence we have
$$\begin{aligned}
\mathbb{H}^{i}_{Zar}(Z; \mathbb{Z}/2(q)) & = H^{i}_{{\mathcal{M}}}(X;{\mathbb{Z}}/2(q)) \\
\mathbb{H}_{Zar}^{i}(X;\mathbb{Z}/2(q)^{sst}) & = L^{q}H{\mathbb{R}}^{i-q,q}(X;{\mathbb{Z}}/2) \\
\mathbb{H}_{Zar}^{i}(X;\mathbb{Z}/2(q)^{top}) & = H^{i-q,q}(X({\mathbb{C}});\underline{{\mathbb{Z}}/2}) \\
\mathbb{H}^{i}_{Zar}(X;\mathbb{Z}/2(q)^{Bor}) & = H^{i}_{G}(X({\mathbb{C}});{\mathbb{Z}}/2) \\
\mathbb{H}^{i}_{Zar}(X;\mathbb{Z}/2(q)^{Bor} _{\mathbb{R}}) & = H^{i}_{G}(X({\mathbb{R}});{\mathbb{Z}}/2) \\
\mathbb{H}_{Zar}^{i}(X;{\mathcal{R}}(q)) & = RL^{q}H^{i-q}(X),\end{aligned}$$
where $L^{q}H{\mathbb{R}}^{i-q,q}(X;{\mathbb{Z}}/2)$ denotes Friedlander-Walker’s real morphic cohomology [@FW:real] and $H^{i-q,q}(X({\mathbb{C}});\underline{{\mathbb{Z}}/2})$ is Bredon cohomology.
The map (\[themap\]) is induced by the map of complexes $$\label{2ndm}
\mathbb{Z}/2(q)\xrightarrow{1)} \mathcal{R}(q)\xrightarrow{2)} \phi _*\mathbb{Z}/2[-q].$$ The map 1) is given by the “usual” cycle map from motivic cohomology to reduced morphic cohomology. It is defined as the composite ${\mathbb{Z}}/2(q) \to {\mathbb{Z}}/2(q)^{sst}\to {\mathcal{R}}(q)$. The map 2) is obtained by adjunction from the composite $$\phi^*(\mathcal{R}(q)) \xrightarrow{{\simeq}} Map _{cts}(X(\mathbb{R})\times \Delta^* _{top}, {\mathcal{R}}_0(\mathbb{A}^q _{\mathbb{R}}))[-2q]\xrightarrow{\simeq}\mathbb{Z}/2[-q]$$ which arises because ${\mathcal{R}} _0(\mathbb{A}^q _{\mathbb{R}})\simeq K(\mathbb{Z}/2,q)$ and any CW complex has an open cover given by contractibles.
We show that the map (\[amap\]) is induced by a composite of maps: $$\label{1stm}
\mathbb{Z}/2(q)\xrightarrow{} tr _{\leq 2q}\mathbb{R}\epsilon _*\mathbb{Z}/2 \xrightarrow{3)} \mathbb{Z}/2(q)^{Bor} \xrightarrow{4)}
\mathbb{Z}/2(q)^{Bor} _{\mathbb{R}}\xrightarrow{5)} \phi _*\mathbb{Z}/2[-q].$$ of Zariski complexes of sheaves in $D^-(Shv _{Zar}(Sm/\mathbb{R}))$ which we now explain. Write $\epsilon :X _{et}\rightarrow X _{Zar}$ for the usual map of sites. The first unlabeled map is the cycle map from motivic cohomology to etale cohomology. The map 3) is obtained in Proposition \[CBE\] using Cox’s theorem [@Cox:real]. The map 4) is obtained by restriction to real points. The map 5) will be obtained from the adjoint of a map $\phi^*(\mathbb{Z}/2(q)^{Bor} _{\mathbb{R}})\rightarrow {\mathbb{Z}}/2[-q]$ as follows. Every CW-complex is locally contractible and so $$\phi^*(\mathbb{Z}/2(q)^{Bor} _{\mathbb{R}}) {\simeq}{\mathrm{Hom}}_{cts}(EG\times\Delta^{\bullet}_{top},{{\mathcal{Z}}/2_{0}({\mathbb{A}}^{q}_{{\mathbb{C}}})})^G[-2q]$$ is a quasi-isomorphism of complexes of Zariski sheaves where the right-hand side is the constant sheaf. In $D^-_{{\mathbb{Z}}/2}(Ab)$, any complex is quasi-isomorphic with the complex given by its cohomology. We have that $H^{p,q}(EG;\underline{{\mathbb{Z}}/2}) = H^{p+q}(BG;Z/2)$ and therefore $$H^k{\mathrm{Hom}}_{cts}(EG\times\Delta^{\bullet}_{top},{{\mathcal{Z}}/2_{0}({\mathbb{A}}^{q}_{{\mathbb{C}}})})^G[-2q] =
H^{q-k,q}(EG;\underline{{\mathbb{Z}}/2})={\mathbb{Z}}/2$$ for $0\leq k \leq 2q$ and is $0$ otherwise. This gives us the map $$\phi^*(\mathbb{Z}/2(q)^{Bor} _{\mathbb{R}}) \simeq \oplus _{0\leq i\leq 2q} \mathbb{Z}/2[-i]\rightarrow \mathbb{Z}/2[-i],$$ which induces the $i^{th}$ projection on cohomology $$H^n _G(X({\mathbb{R}}),{\mathbb{Z}}/2)=\oplus _{0\leq i\leq n}H^{n-i}(X({\mathbb{R}}),{\mathbb{Z}}/2)\rightarrow H^{n-i}(X({\mathbb{R}}),{\mathbb{Z}}/2).$$ In particular the adjoint of this map for $i=q$ gives us the map 5) $$\mathbb{Z}/2(q)^{Bor} _{\mathbb{R}}\rightarrow \phi _*{\mathbb{Z}}/2[-q].$$
Because both map \[1stm\] and map \[2ndm\] induce non-trivial maps in cohomology they have to coincide by Theorem \[MT\].
In the above proof, we used the following proposition which relies on Cox’s theorem identifying the etale cohomology of a real variety with Borel cohomology.
\[CBE\] There is a quasi-isomorphism $\rho: tr_{\leq 2q}{\mathbb{R}}\epsilon _*{\mathbb{Z}}/2\rightarrow {\mathbb{Z}}/2(q)^{Bor}$ of complexes of Zariski sheaves.
We show that the canonical map ${\mathbb{Z}}/2(q)^{Bor}\rightarrow {\mathbb{R}}\epsilon _*{\mathbb{Z}}/2$, constructed in [@HV:VT Proposition 5.5] induces a quasi-isomorphism ${\mathbb{Z}}/2(q)^{Bor}\rightarrow tr_{\leq 2q}{\mathbb{R}}\epsilon _*{\mathbb{Z}}/2$. The map $\rho$ is its inverse in the derived category. Its hypercohomology gives the cycle map $H^n _G(X,{\mathbb{Z}}/2)\rightarrow H^n _{et}(X,{\mathbb{Z}}/2)$, for every $n\geq 0$.
There is a quasi-isomorphism $\mathbb{R}\epsilon _*\mu^{\otimes q} _2 \xrightarrow{{\simeq}} \mathbb{R}\epsilon _*\mu^{\otimes q+i} _2$ and a commutative diagram $$\label{comm}
\xymatrix{
{{\mathbb{Z}}/2}(q)^{Bor}\ar[r]\ar[d]^{\simeq} & tr_{\leq 2q}\mathbb{R}\epsilon _*\mu^{\otimes q} _2 \ar[d] ^{\simeq}\\
tr_{\leq 2q}{{\mathbb{Z}}/2}(q+i)^{Bor}\ar[r] & tr_{\leq 2q}\mathbb{R}\epsilon _*\mu^{\otimes q+i} _2.
}$$ Take $i=q$. The result follows by showing the bottom map is a quasi-isomorphism. In [@HV:VT Section 5] it is shown that the composite $${\mathbb{Z}}/2(2q) \to tr_{\leq 2q}{\mathbb{Z}}/2(2q)^{Bor} \to tr_{\leq 2q}\mathbb{R}\epsilon _*\mu^{\otimes 2q} _2$$ is the usual cycle map ${\mathbb{Z}}/2(2q) \to tr_{\leq 2q}\mathbb{R}\epsilon _*\mu^{\otimes 2q} _2$. By Voevodsky’s resolution of the Milnor conjecture [@Voev:miln] this cycle map is a quasi-isomorphism. This implies that that $H^n _G(X,{\mathbb{Z}}/2)\rightarrow H^n _{et}(X,{\mathbb{Z}}/2)$ is a surjective map between finite-dimensional spaces for $n\leq 2q$. By [@Cox:real] both vector spaces have the same dimension and so the map is an isomorphism. Therefore ${\mathbb{Z}}/2(q)^{Bor}\simeq tr_{\leq 2q}{\mathbb{R}}\epsilon _*{\mathbb{Z}}/2$.
Let $a_{Zar}$ denote Zariski sheafification and define the following sheaves $$\begin{aligned}
\mathcal{H}^n_{{\mathcal{M}}}(q) =& a_{Zar}(U\mapsto H^{n}_{{\mathcal{M}}}(U(\mathbb{C})),\mathbb{Z}/2(q))) \\
\mathcal{H}^n _\mathbb{C}(G) =& a_{Zar}(U\mapsto H^{n}_{G}(U(\mathbb{C})),\mathbb{Z}/2)) \\
\mathcal{H}^n _\mathbb{R}(G) =& a_{Zar}(U\mapsto H^{n}_{G}(U(\mathbb{R}),\mathbb{Z}/2)), \\
\mathcal{H}^n _{et}(q) =& a_{Zar}(U\mapsto H^{n}_{et}(U,\mu_{2}^{\otimes q})), \\
\mathcal{H}^{n}_\mathbb{R} =& a_{Zar}(U\mapsto H^{n}(U(\mathbb{R}),\mathbb{Z}/2)). \end{aligned}$$
Sheafifying the diagram in Theorem \[thm:cycmaps\] for $p=q$ gives the commutative diagram $$\xymatrix{
{\mathcal{H}}^q _{{\mathcal{M}}}(q)\ar[r]^{{\cong}}\ar[d] & {\mathcal{H}}^q _{et}(q)\ar[r]^{{\cong}}\ar[d] & {\mathcal{H}}^q_{{\mathbb{C}}}(G)\ar[d]\\
{\mathcal{RL}}^q{\mathcal{H}}^{0}\ar[r]^{cyc_{q,0}} & {\mathcal{H}}^{0}_{{\mathbb{R}}} & \ar[l] {\mathcal{H}}^q_{{\mathbb{R}}}(G).\\
}$$
By Corollary \[dge\] and Corollary \[sm\] the sheafified cycle map $cyc_{q,0}$ induces the usual cycle map between reduced morphic cohomology and singular cohomology $$RL^{q}H^{n}(X)= H^{n}_{Zar}(X;{\mathcal{RL}}^{q}{\mathcal{H}}^{0}) \to H^{n}_{Zar}(X;{\mathcal{H}}^{0}_{{\mathbb{R}}})= H^{n}(X({\mathbb{R}});{\mathbb{Z}}/2).$$
The map ${\mathcal{H}}^q _{{\mathcal{M}}}(q)\to {\mathcal{H}}^q _{et}(q)$ induces the Bloch-Ogus isomorphism $$CH^{q}(X) = H^{q}_{Zar}(X; {\mathcal{H}}^q _{{\mathcal{M}}}(q)) {\cong}H^{q}_{Zar}(X; {\mathcal{H}}^q _{et}(q)).$$
The composite ${\mathcal{H}}^q _{et}(q) \to {\mathcal{H}}^q_{{\mathbb{C}}}(G)\to {\mathcal{H}}^q_{{\mathbb{R}}}(G) \to {\mathcal{H}}^{0}_{{\mathbb{R}}}$ induces a map $$CH^{q}(X) = {\mathcal{H}}^{q}_{Zar}(X;{\mathcal{H}}^{q}_{et}(q)) \to {\mathcal{H}}^{q}_{Zar}(X;{\mathcal{H}}^{0}_{{\mathbb{R}}}) = H^{q}(X({\mathbb{R}});{\mathbb{Z}}/2)$$ which by [@CTS:zero Remark 2.3.5] is just the Borel-Haeflinger cycle map sending a closed irreducible $Z\subseteq X$ to the Poincare dual of the fundamental class of $Z$ if $\dim Z({\mathbb{R}}) = \dim Z$ and zero otherwise. The above commutative diagram tells us that this agrees with the composite ${\mathcal{H}}^{q}(q)_{et} {\cong}{\mathcal{H}}^{q}_{{\mathcal{M}}}(q) \to {\mathcal{LR}}^{q}{\mathcal{H}}^{0} \to {\mathcal{H}}^{0}_{{\mathbb{R}}}$ and so we immediately obtain the following.
\[bh\] Let $X$ be a smooth quasi-projective real variety. For any $q\geq 0$, the Borel-Haeflinger cycle map factors as the composite $$CH^q(X)/2\rightarrow RL^qH^q(X)\xrightarrow{cyc_{q,n}} H^q(X(\mathbb{R}),\mathbb{Z}/2),$$ where the first map is the natural quotient.
Next we compare the $s$-map in reduced morphic cohomology with the operation $(-1)$ in étale cohomology. Recall that the operation $(-1):H^{i}_{et}(X;\mu_{2}^{\otimes q}) \to H^{i+1}_{et}(X;\mu_{2}^{\otimes q+1})$ is defined to be multiplication with the class $(-1)$ which is the image of $-1$ under the boundary map $H^{0}_{et}(X;\mathbb{G}_{m})\to H^{1}_{et}(X;\mu_{2})$ in the Kummer sequence. By naturality this is equal to the pullback of $(-1)\in H^{1}_{et}(\operatorname{\mathrm{Spec}}({\mathbb{R}});\mu_{2})$ under the structure map $X\to \operatorname{\mathrm{Spec}}({\mathbb{R}})$. Sheafifying gives the operation on Zariski sheaves $(-1):\mathcal{H}^q _{et}(q)\rightarrow \mathcal{H}^{q+1}_{et}(q+1)$.
\[cel\] Let $X$ be a smooth quasi-projective real variety of dimension $d$. The following square commutes for any $q\geq 0$ $$\begin{CD}
H^{i}_{Zar}(X;{\mathcal{H}}^{q}_{et}(q)) @>>> RL^qH^{i}(X)\\
@VV \cup (-1) V @VV\cup sV\\
H_{Zar}^{i}(X;{\mathcal{H}}_{et}^{ q+1}(q+1)) @>>> RL^{q+1}H^i(X).
\end{CD}$$ For any $q\geq d+1$ all maps are isomorphisms.
The $s$-operation is induced by multiplication with $s\in RL^1H^0(\operatorname{\mathrm{Spec}}({\mathbb{R}}))$, where $s$ is the generator. Sheafifying the $s$-map gives a map of Zariski sheaves and the composite $\mathcal{RL}^q{\mathcal{H}}^0\xrightarrow{} \mathcal{RL}^{q+1}{\mathcal{H}}^0\to\mathcal{H}_{{\mathbb{R}}}^0 $ induces the usual $s$-map on sheaf cohomology.
The class $(-1)\in H^1 _{et}(\operatorname{\mathrm{Spec}}({\mathbb{R}});\mu _2)$ and $(-1)$ maps to $s$ under the isomorphism $H^{1}_{et}(\operatorname{\mathrm{Spec}}({\mathbb{R}});\mu_{2}){\cong}RL^{1}H^{0}(\operatorname{\mathrm{Spec}}({\mathbb{R}}))$ because they both map to the generator of $H^{0}(pt;{\mathbb{Z}}/2)$. Therefore the above square commutes. When $q\geq d$ then the vertical maps are isomorphisms, both $H^{i}_{Zar}(X;{\mathcal{H}}^{q}_{et}(q)) \to H^{i}(X({\mathbb{R}});{\mathbb{Z}}/2)$ and $RL^qH^{i}(X) \to H^{i}(X({\mathbb{R}});{\mathbb{Z}}/2)$ are isomorphisms.
Let $X$ be a smooth quasi-projective real variety. We have the isomorphism of rings $$H^* _{et}(X,\mu^{\otimes *} _2)[s^{-1}]\simeq RL^*H^*(X)[s^{-1}]\simeq H^*(X({\mathbb{R}}),{\mathbb{Z}}/2),$$ where $s=(-1)$ under the left-hand isomorphism.
The map $\mathcal{H}^q _{et}(q)\rightarrow \mathcal{RL}^q{H}^0$ is not in general an isomorphism of sheaves as we can see for the case of a smooth projective variety of dimension $dim(X)=q$. In this case the first group surjects with non-trivial kernel in codimension $0,1,2$ (under some mild conditions on $X$) into cohomology of real points $X(\mathbb{R})$ by [@CTS:zero]. On the other hand the latter group is the cohomology of real points by [@Teh:real].
We now turn our attention to natural transformations from various cycle cohomology theories of interest to singular cohomology.
\[MT\] Let $k={\mathbb{R}}$ or ${\mathbb{C}}$ and write $\phi: (Top) _{an}\rightarrow (Sm/k) _{Zar}$ the map of sites that send $X\mapsto X(k)$ where $(Top)$ is the category of spaces homeomorphic to finite dimensional CW complexes equipped with the usual topology. Then we have
1. ${\mathrm{Hom}}_{D^-((Sm/{\mathbb{R}})_{Zar})}(\mathbb{Z}/2(q),\, \phi _*\mathbb{Z}/2[-q])=\mathbb{Z}/2$,
2. ${\mathrm{Hom}}_{D^-((Sm/{\mathbb{R}})_{Zar})}({\mathcal{R}}(q),\phi _*{\mathbb{Z}}/2[-q]) ={\mathbb{Z}}/2$,
3. ${\mathrm{Hom}}_{D^-((Sm/{\mathbb{C}})_{Zar})}(\mathbb{Z}(q)^{sst},\, \mathbb{R}\phi _*\mathbb{Z}/n)=\mathbb{Z}/n$, for any $n\geq 1$.
We have a quasi-isomorphism ${\mathbb{Z}}/2(q){\simeq}{\mathbb{Z}}/2(q)^{sst}$ and that $${\mathrm{Hom}}(\mathbb{Z}/2(q)^{sst},\, \phi _*\mathbb{Z}/2[-q])={\mathrm{Hom}}(\phi^*(\mathbb{Z}/2(q)^{sst}),\, \mathbb{Z}/2[-q]).$$ Because every CW complex is locally contractible in $D^-(Top)$ we have $$\phi^*(\mathbb{Z}/2(q)^{sst})[2q]\simeq {\mathrm{Hom}}_{cts}(- \times \Delta^{\bullet} _{top},\, {\mathcal{Z}}/2_0(\mathbb{A}^q _{{\mathbb{C}}})^{G}) {\simeq}\operatorname{\mathrm{Sing}}_{\bullet} {\mathcal{Z}}/2_0(\mathbb{A}^q _{{\mathbb{C}}})^{G}.$$ From [@DS:real (3.6)] it follows that ${\mathcal{Z}}/2_0(\mathbb{A}^q _{{\mathbb{C}}})^{G} {\simeq}\prod_{i=q}^{2q}K({\mathbb{Z}}/2,i)$. This yields the result because we then have $${\mathrm{Hom}}_{D^-(Top)}(\phi^*(\mathbb{Z}/2(q)),\, \mathbb{Z}/2[-q]) = \bigoplus_{i=q}^{2q}H^{q}(K({\mathbb{Z}}/2,i);{\mathbb{Z}}/2) = {\mathbb{Z}}/2.$$
In the proof of Theorem \[thm:cycmaps\] we observed that $\phi^*{\mathcal{R}}(q)\simeq {\mathbb{Z}}/2[-q]$ and so $$\begin{gathered}
{\mathrm{Hom}}_{D^-((Sm/{\mathbb{R}})_{Zar})}({\mathcal{R}}(q), \phi _*{\mathbb{Z}}/2[-q]) ={\mathrm{Hom}}_{D^-((Sm/{\mathbb{R}})_{Zar})}(\phi^*{\mathcal{R}}(q),{\mathbb{Z}}/2[-q]) \\
={\mathrm{Hom}}_{D^-((Sm/{\mathbb{R}})_{Zar})}({\mathbb{Z}}/2[-q],{\mathbb{Z}}/2[-q]) ={\mathbb{Z}}/2.\end{gathered}$$
The last item follows from the equivalence $\phi^{*}{\mathbb{Z}}(q)^{sst} {\simeq}{\mathbb{Z}}$. We have $${\mathrm{Hom}}_{D^-((Sm/{\mathbb{C}})_{Zar})}(\mathbb{Z}^{sst}(q), \mathbb{R}\phi _*\mathbb{Z}/n)= {\mathrm{Hom}}_{D^-((Sm/{\mathbb{R}})_{Zar})}(\mathbb{Z},\mathbb{Z}/n)=\mathbb{Z}/n,$$ for any $n\geq 1$.
Because $\mathbb{Z}^{sst}(q) = tr_{\leq q}\mathbb{Z}(q)^{sst}$ we have, according to Theorem \[MT\], 3), that $${\mathrm{Hom}}_{D^-(Sm/{\mathbb{C}})}(\mathbb{Z}(q)^{sst},\, tr _{\leq n}\mathbb{R}\phi_*\mathbb{Z}) = {\mathbb{Z}}.$$ Let $\alpha$ be a generator of this group. Recall that Suslin’s conjecture is the statement that $\alpha$ is a quasi-isomorphism for all $q\geq 0$. We end the section with the following corollary of Voevodsky’s resolution of the Beilinson-Lichtenbaum conjectures.
The map $\alpha \otimes \mathbb{Z}/n: \mathbb{Z}/n(q)^{sst} \rightarrow tr _{\leq q}\mathbb{R}\phi _*\mathbb{Z}/n$ is a quasi-isomorphism for any $n\geq 2$.
Theorem \[MT\] and the quasi-isomorphism ${\mathbb{Z}}(q)^{sst}\otimes{\mathbb{Z}}/p{\simeq}{\mathbb{Z}}/p(q)^{sst}$ gives $${\mathrm{Hom}}_{D^-}(\mathbb{Z}(q)^{sst},\, tr _{\leq q}\mathbb{R}\phi _*\mathbb{Z}/p)={\mathrm{Hom}}_{D^-}(\mathbb{Z}/p(q)^{sst},\, tr _{\leq q}\mathbb{R}\phi _*\mathbb{Z}/p)=\mathbb{Z}/p$$ f or any prime $p>1$. Let $\epsilon :(Sm/{\mathbb{C}}) _{et}\rightarrow (Sm/{\mathbb{C}}) _{Zar}$ be the usual map of sites. The cycle map $\beta: \mathbb{Z}/p(q) \rightarrow tr _{\leq q}\mathbb{R}\epsilon _*\mathbb{Z}/p$ can be seen as a generator of the group ${\mathrm{Hom}}_{D^{-}(Sm/{\mathbb{C}})}(\mathbb{Z}/p(q)^{sst}, tr _{\leq n}\mathbb{R}\phi _*\mathbb{Z}/p)$ because we have $\mathbb{Z}/p(q)^{sst}\simeq \mathbb{Z}/p(q)$ and ${\mathbb{R}}\phi _*\mathbb{Z}/p={\mathbb{R}}\epsilon _*{\mathbb{Z}}/p$ by the classical comparison theorem between étale and singular cohomology. This means that there is $0<N<p$ such that $\alpha\otimes {\mathbb{Z}}/p=N\beta$. Since $N$ is prime to $p$ it follows that $\alpha\otimes {\mathbb{Z}}/p$ is a quasi-isomorphism.
One easily obtains the result for powers of primes and then products of distinct primes using 5-lemma.
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---
abstract: 'Consider the generalized iterated wreath product $S_{r_1}\wr \ldots \wr S_{r_k}$ of symmetric groups. We find a bijection between the equivalence classes of irreducible representations of the generalized iterated wreath product with orbits of labels on certain rooted trees. We find a recursion for the number of these labels and the degrees of irreducible representations of the generalized iterated wreath product. We give upper bound estimates for fast Fourier transforms.'
address:
- 'Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA'
- 'Department of Mathematics, University of Chicago, Chicago, IL 60637 USA'
author:
- Mee Seong Im
- Angela Wu
bibliography:
- 'wreath-products-FFT.bib'
title: Generalized iterated wreath products of symmetric groups and generalized rooted trees correspondence
---
Introduction {#section:introduction}
============
Wreath products of symmetric groups arise as the automorphism groups of regular rooted trees, with applications ranging from functions on rooted trees, pixel blurring, nonrigid molecules, to choosing subcommittees from a set of committees. With motivation from [@MR2081042], we find a bijection between the equivalence classes of irreducible representations of the generalized iterated wreath product $W(\vec{r}|_k)$ and the orbits of families of labels on certain complete trees. We find a recursion for the number of equivalence classes and dimensions of the irreducible representations of the generalized iterated wreath products. We also give upper bound estimates for fast Fourier transforms of this chain of groups.
Background and Notation {#section: background}
-----------------------
Throughout this paper, we will fix $\vec{r} = (r_1,r_2,r_3,\ldots) \in \mathbb{N}^\omega$, a positive integral vector. We denote by $\vec{r}|_k :=(r_1,r_2,\ldots, r_k)$ the $k$-length vector found by truncating $\vec{r}$. For a group $G$, we denote by $\widehat{G}$ the set of irreducible representations of $G$. We say that $\mathcal{R}_G$ is a *traversal* for $G$ if $\mathcal{R}_G \subset \widehat{G}$ contains exactly one irreducible for each isomorphism class. A consequence of basic representation theory is that $\displaystyle{\sum_{\rho \in \mathcal{R}}} \dim(\rho)^2 = |G|$.
\[definition:chain-of-subgroups\]
Let the [*(generalized) $k$-th $\vec{r}$-symmetric wreath product $W(\vec{r}|_k)$*]{} be defined recursively by $$W(\vec{r}|_0) = 1 \text{ and } W(\vec{r}|_k) = W(\vec{r}|_{k-1}) \wr S_{r_k}.$$
Note that $W(\vec{r}|_1) = S_{r_1}$, $W(\vec{r}|_2) = S_{r_1} \wr S_{r_2}$, and $W(\vec{r}|_k) = S_{r_1} \wr \ldots \wr S_{r_k}$. Throughout this paper, we will be considering the chain of groups given in Definition \[definition:chain-of-subgroups\].
Irreducible Representations of Iterated Wreath Products {#section:irrep-iterated-wreath-products}
=======================================================
Let ${\mathop{\mathrm{Ind}}\nolimits}_H^K \sigma$ denote the representation of $K$ with dimension $[K:H]\cdot \dim \sigma$ induced from representation $\sigma \in \widehat{H}$ of subgroup $H \leq K$. Let $\alpha \vdash_h n$ denote that $\alpha = (\alpha_1,\ldots, \alpha_h) \in \left( \mathbb{Z}_{\geq 0}\right)^h$ is a weak composition of $n$ with $h$ parts so that $\alpha$ satisfies $\sum_{i=1}^h \alpha_i = n$.
If $\mathcal{R} = \{ \rho_1, \ldots , \rho_h \}$ is a traversal for $G \leq S_N$, then the irreducible representations given by $$\left\{ {\mathop{\mathrm{Ind}}\nolimits}_{G\wr S_\alpha}^{G \wr S_n} (\rho_1^{\alpha_1} \otimes\cdots \otimes \rho_h^{\alpha_h} \otimes \sigma) \bigg| \alpha \vdash_h n, \sigma \in \widehat{S}_{\alpha} \right\}$$ form a traversal of $G\wr S_n$, where $S_\alpha := S_{\alpha_1} \times S_{\alpha_2} \times \ldots \times S_{\alpha_h}$ with $S_0 = 1$. In particular, if $\mathcal{R} = \{ \rho_1 ,\ldots , \rho_h\}$ is a traversal for $W(\vec{r}|_{k-1})$, then the following set forms a traversal of $W(\vec{r}|_k)$: $$\label{eq_set-of-irreps}
\mathcal{R}_{W(\vec{r}|_{k})} := \left\{ {\mathop{\mathrm{Ind}}\nolimits}_{W(\vec{r}|_{k-1}) \wr S_\alpha}^{W(\vec{r}|_k)} (\rho_1^{\alpha_1} \otimes\cdots \otimes \rho_h^{\alpha_h} \otimes \sigma) \bigg| \alpha \vdash_h n, \sigma \in \widehat{S}_{\alpha} \right\}.$$
The set $\{ \rho^\alpha = \rho_1^{\alpha_1} \otimes \ldots \otimes \rho_h^{\alpha_h}: \alpha \vdash_h n \}$ forms a traversal for $G^n$. The theorem follows as a consequence of Clifford theory to the structure of irreducible representations of $G \wr S_n = G^n \rtimes S_n$.
Let $N(\vec{r}|_k)$ denote the number of irreducible representations of $W(\vec{r}|_k)$.
Define $P(n,h) : = \displaystyle{\sum_{\alpha \vdash_h n} \prod_{i=1}^h} \alpha_i!$. We find that $N(\vec{r}|_k)$ satisfies the following recursion: $$N(\vec{r}|_k) = P\left( r_k, N(\vec{r}|_{k-1}) \right) = \sum_{k=0}^h k! \cdot P(n-k, h-1).$$
Branching Diagram to $\vec{r}$-Label Correspondence
===================================================
We find a combinatorial structure describing the branching diagrams for the iterated wreath products of symmetric groups.
We define the complete $\vec{r}$-tree $T(\vec{r}|_k)$ of height $k$, or $\vec{r}|_k$-tree, recursively as follows. Let $T(r_1)$ be the tree consisting of a root node only. Let $T(\vec{r}|_k)$ consist of a root node with $r_k$ children, with each the root of a copy of the $k-1$-level tree $T(\vec{r}|_{k-1})$, which yields a tree with $k$ levels of nodes.
Notice that $T(\vec{r}|_k)$ has $\displaystyle{\prod_{i=2}^k } r_i$ leaves. We say a node $v$ is in the [*$j$-th layer*]{} of $T(\vec{r}|_k)$ if it is at distance $j$ from the root. The tree $T(\vec{r}|_k)$ has $\displaystyle{\prod_{i=k-j+1}^k r_i}$ nodes in the $j$-th layer. The subtree of $T$ rooted at some vertex $v$ denoted $T_v$ is the tree with root $v$ consisting of all the children and descendants of $v$. We call $T_v$ a maximal subtree of $T$ if $v$ is a child of the root, or equivalently if $v$ is in the first layer. Let $\deg(v)$ denote the number of leaves of the subtree $T_v$, and let $[n]:=\{ 1,2,\ldots, n\}$ be the set of integers from $1$ to $n$.
We denote by $\widehat{S}_* := \bigsqcup_{n \in \mathbb{N}} \bigsqcup_{\alpha \models n} \widehat{S}_\alpha$, where $\models$ denotes “is a partition of".
An [*$\vec{r}|_k$-label* ]{}is a function $\phi: V_{T(\vec{r}|_k)} \rightarrow \widehat{S}_*$ satisfying $\phi(v) \in \bigsqcup_{\alpha \vdash \deg(v)} \hat{S}_\alpha$.
We say that two labels $\phi$ and $\psi$ on $T$ are [*equivalent*]{}, or $\phi \sim \psi$, if there exists $\sigma \in \operatorname{Aut}(T)$ such that $\phi^\sigma = \psi$, where $\phi^\sigma(v) := \phi(v^\sigma)$.
An $\vec{r}|_k$-label $\phi: V_{T(\vec{r}|_k)} \rightarrow \widehat{S}_* $ is [*valid*]{} if it satisfies all of the following recursive conditions. We denote by $\mathcal{T}(\vec{r}|_k) := \left\{ \phi: \phi \text{ is a valid $\vec{r}|_k$-label }\right\} $ and $\mathcal{T} = \sqcup_k \mathcal{T}(\vec{r}|_k)$.
- Given an $\vec{r}|_1$-label $ \phi : V_{T(\vec{r}|_1)} = \{ \text{root node} \} \rightarrow \widehat{S}_*$: we require $\phi \in \widehat{S}_{r_1}$.
- Given for $k>1$ an $\vec{r}|_k$-label $\phi: V_{T(\vec{r}|_k)} \rightarrow \widehat{S}_*$: we require
1. for any child $v$ of the root, the $\vec{r}|_{k-1}$-label $\phi|_{T_v} \in \mathcal{T}_{k-1}$, and
2. $\phi(\text{root node}) \in \hat{S}_\alpha$, where $S_\alpha$ gives the stabilizer of the action by $S_{r_k}$ on $\vec{r}|_{k-1}$ sublabels of $\phi$, so that $\alpha \models r_k$ is the partition of $[r_k]$ given by the number of $\vec{r}|_{k-1}$-sublabels of $\phi$ in each non-empty equivalence class,
where $\phi|_{T_v}$ denotes the restriction of $\phi$ to the subtree $T_v$.
\[theorem:bijection-tree-iterated-wreath\] There is a bijection between equivalence classes of $\widehat{W}(\vec{r}|_k)$ and $W(\vec{r}|_k)$-orbits of $\mathcal{T}( \vec{r}|_k)$.
It suffices to define a map only on a traversal of $\widehat{W}(\vec{r}|_k)$ such as that given in . We will define $F: \mathcal{R}_{W(\vec{r}|_k)} \rightarrow \mathcal{T}( \vec{r}|_k)$ recursively and it suffices to prove that each orbit of $\mathcal{T}(\vec{r}|_k)$ under action by $W(\vec{r}|_k)$ has exactly one pre-image under $F$.
Let $k=1$. For any $\rho \in \widehat{W}(\vec{r}|_k) = \widehat{S}_{r_1} $, we define the $\vec{r}|_1$-label $F(\rho): V_{T(\vec{r}|_1)} = \{\text{root}\} \rightarrow \widehat{S}_{r_1}$ as $F(\rho) (\text{root}) := \rho$. This is clearly a bijection as desired.
Now let $k>1$. By the inductive hypothesis, $F: \mathcal{R}_{W(\vec{r}|_{k-1})} \rightarrow \mathcal{T}( \vec{r}|_{k-1})$ has exactly one pre-image per orbit of $\mathcal{T}(\vec{r}|_k)$. Suppose that a traversal for $W(\vec{r}|_{k-1})$ is given by the set $\{\rho_1, \ldots \rho_h \}$. We need to define $F$ on $\mathcal{R}_{W(\vec{r}|_k)}$ and show that orbits have exactly one pre-image as desired.
Pick an arbitrary element of $\rho_1^{\alpha_1} \otimes \ldots \otimes \rho_h^{\alpha_h} \otimes \sigma$ of $\mathcal{R}_{W(\vec{r}|_k)}$. Denote its image under $F$ by $\phi :=F(\rho_1^{\alpha_1} \otimes \ldots \otimes \rho_h^{\alpha_h} \otimes \sigma): V_{T(\vec{r}|_k)} \rightarrow S_{r_k}$. Let $U \subset V_{T(\vec{r}|_k)}$ be the $r_k$ children of the root. Assign an ordering to $U= \{u_1,\ldots, u_{r_k} \}$; then partition $U$ as $U = U_1 \sqcup \ldots \sqcup U_h$ satisfying $|U_i| = \alpha_i$ while preserving the ordering. For each $u_i \in U$, define the value of $\phi$ on all nodes in subtree $T_{u_i}$ to satisfy $\phi|_{u_i} = F(\rho_{j^i})$, where $j^i$ satisfies $U_{j^i} \ni u_i$ and where $\phi|_{T_{u_i}}$ denotes the restriction of $\phi$ to $T_{u_i} \subseteq T$. It remains to define the value of $\phi$ on the root node. We let $\phi(\text{root}) = \sigma$.
Notice that $\phi_{u_i} \in \mathcal{T}(\vec{r}|_{k-1})$ by definition and induction. Since $\sigma$ is in the stabilizer of the action by $S_{r_k}$ on $\rho^\alpha$, which is exactly $S_\alpha$, $\phi$ is a compatible label for $T(\vec{r}|_k)$. Thus, $F$ is well-defined and clearly each orbit of $\mathcal{T}(\vec{r}|_k)$ has exactly one pre-image.
Degrees of Irreducible Representations {#section:degrees}
--------------------------------------
Following the discussion in [@MR2081042], we define for any $\vec{r}|_k$-tree $T$ the companion tree $C_T$.
Fix $T(\vec{r}|_k)$ and $\vec{r}|_k$-label $\phi$. Let the [*companion label $C_\phi : V_{T(\vec{r}|_k)} \rightarrow \mathbb{N}$*]{} be defined by:
$$C_\phi(v) = \begin{cases}
\dim(\phi(v)) & \text{ if $v$ is a leaf of $T(\vec{r}|_k)$,} \\
|S_{r_i}/S_\alpha| = {r_i \choose \alpha} & \text{ otherwise, where $v$ is in the $(k-i)$-th layer of $T$ and $\phi(v) \in S_\alpha$}.
\end{cases}$$
If $\rho$ is an irreducible representation of $W(\vec{r}|_k)$ associated to $\vec{r}|_k$-label $\phi$, then the dimension $d_\rho$ of $\rho$ is given by
$$d_\rho = \prod_{v} C_\phi(v).$$
the product of the value of the companion label $C_\phi$ on all vertices.
Fast Fourier Transforms, Adapted Bases and Upper Bound Estimates {#section:FFT}
================================================================
We use the FFT estimates derived in [@MR1192969] and [@MR1339806] to find an overall upper bound on the running time of finding an FFT for $W(\vec{r}|_k)$. From [@MR1339806], we cite the result:
We have $$T(G \wr S_n) \leq n T(G) \cdot |G \wr S_{n-1}| + n T( G \wr S_{n-1}) \cdot |G| + n^3 2^{|\widehat{G}|} |G \wr S_n|.$$
Acknowledgment
--------------
The authors would like to acknowledge Mathematics Research Communities for providing the authors with an exceptional working environment at Snowbird, Utah and they would like to thank Michael Orrison for helpful discussions. This paper was written during the first author’s visit to the University of Chicago in 2014. She thanks their hospitality.
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---
abstract: 'A spin model that displays inverse melting and inverse glass transition is presented and analyzed. Strong degeneracy of the interacting states of an individual spin leads to entropic preference of the “ferromagnetic” phase, while lower energy associated with the non-interacting states yields a “paramagnetic” phase as temperature decreases. An infinite range model is solved analytically for constant paramagnetic exchange interaction, while for its random exchange, analogous results based on the replica symmetric solution are presented. The qualitative features of this model are shown to resemble a large class of inverse melting phenomena. First and second order transition regimes are identified.'
author:
- 'Nurith Schupper and Nadav M. Shnerb'
title: Spin Model for Inverse Melting and Inverse Glass Transition
---
We all tend to associate order parameter with order, namely, with less entropic microscopic realizations. This is indeed the general situation in nature: crystals are more ordered than liquids, ferromagnets have less entropy than paramagnets. Even the entropy associated with a glass, an out of equilibrium, frozen frustrated state, is less than that of a liquid phase of the same material.
There are, however, exceptions, where an “order parameter” does not reflect order, and the entropy growth during crystallization or freezing. The prototype of these phenomena is *inverse melting*, i.e., a reversible transition between a liquid phase at low temperatures to a high temperature crystalline phase, observed in $He^3$ and $He^4$ at extreme conditions (temperature below $1^\circ$K, pressure above 25 bar) [@helium]. A similar phenomenon was observed recently at room temperature and atmospheric pressure in P4MP1 polymer solutions [@greer]. Ferroelectricity in Rochelle salt is another example, where the spontaneous polarization is lost below the (lower) Curie temperature, this time the transition is second order in type [@rochelle]. The pinned-crystalline inverse transition of vortex lines in the presence of point disorder at high temperature superconductors [@pinning] is also considered as an example of inverse melting. However, in that system, the intensive order parameter (bulk magnetization) is lower in the crystalline phase, and the response functions are higher, i.e., the disordered phase is stiffer than the ordered phase.
Even if the crystalline state is the thermodynamically preferred one, the dynamics of the system may prevent its appearance. In glass forming materials ergodicity breaking takes place at a finite temperature and the system is trapped into a frozen disordered state. One expects that an “inverse” glass transition phenomenon, analogous to inverse melting, may also take place. An interesting example in polymeric systems is the reversible thermogelation of Methyl Cellulose solution in water [@Chevillard]. When a (soft and transparent) solution of Methyl Cellulose is heated (above $50^\circ$C, for a 10 gr/liter solution) it turns into a white, turbid and mechanically strong gel. Unlike the boiling of an egg that involves an irreversible transition from a metastable to a stable state, this transition is reversible upon cooling, and the polymer is redissolved on subsequent cooling. In its high temperature phase, the Methyl Cellulose gel exhibits, like many other gels [@gel], glassy features. Non monotonic temperature dependence of the glassy order parameter has been already reported for a random heteropolymer in a disordered medium [@randompoly]; this may be considered as the glassy analogue of the flux line crystallization [@pinning]. The liquid-liquid transition theory for polyamorphous materials predicts an inverse freezing transition even for the most known liquid, water. In the hypothesized phase diagram presented in [@stanley] a low density liquid (at about 150 bar, $-100^\circ$C) becomes a low density amorphous ice upon heating.
In many branches of statistical physics the presentation of a simple spin model (Ising, Potts, and SK models, for example) turns out to be a very beneficial step that yields both physical insight and quantitative predictions. In this paper, such a model for inverse melting is presented and analyzed for homogenous and heterogenous systems in the mean field level. The model exhibits both inverse melting and inverse glass transition, and allows first order and second order transitions. We believe that this generic model is applicable for the qualitative description of the above mentioned phase transitions (except for the inverse melting in superconductors which requires a different model).
Let us begin with an intuitive argument focusing on one of the above mentioned systems, namely, a single Methyl Cellulose polymer chain in water. In order to explain the inverse freezing it seems plausible to assume that its folded conformation is favored energetically while its unfolded conformation is favored entropicaly \[See figure (\[fig1\])\]. The entropy growth of the open conformation may be related to the number of possible microscopic configurations of the polymer itself, but it may be attributed also to the spatial arrangement of the water molecules in its vicinity [@haque].
![A sketch of the energy and entropy dependence on the linear size of a Methyl Cellulose polymer in water. The folded conformation costs less energy due to more favorable interactions between hydrophobic sequences along a single chain, but are less entropic as water molecules has to arrange in cage like structures around the hydrophobic constituents of the chain. The unfolded conformation admits much more microscopic configurations. The interaction with other polymers in the solution is suppressed in the folded state.[]{data-label="fig1"}](polymer1b.eps "fig:"){width="7.7cm"}\
The main cause for inverse freezing is that the “open” conformations of the polymer are also the *interacting* structures, as they allow for the formation of hydrophobic links with other polymers in the solution, a process that leads to gelation. This seems to be a general prescription to both inverse melting and inverse glass transitions: the noninteracting state is favored energetically, while the interacting state is favored by the entropy.
Let us now present a very simple model that incorporates these features. It is based on the Blume-Capel model [@Blume],[@Capel] for a spin one particle with “lattice field” that lower the energy of the “zero” (noninteracting) state. In contrast with the original Blume-Capel model, we consider the $\pm 1$ spin states (that interact with other spins) to be more degenerate. The system consists of a lattice of N sites and the Hamiltonian is given by $$\begin{aligned}
\label{eq:1} H=-J\sum_{<i,j>} S_{i}S_{j}+D\sum_{i=1}^N S_{i}^2\end{aligned}$$ where the spin variables are allowed to assume the values $S_i=0,\pm 1$. The summation over $<i,j>$ is over each distinct pair once. Turning back to our polymer analogy, spin $0$ represents schematically the compact non-interacting polymer coil, the stretched polymer (interacting with its neighbors) is represented by spin $\pm 1$. The positive constant $D$ measures the energy preference of the compact spatial configurations, and the “ferromagnetic” interaction between spins, $J$, is related to the concentration of polymers (or the pressure). The $0$ spin state is assumed to be n-fold degenerate, and the $\pm 1$ states are m-fold degenerate so that $r=m/n \geq 1$ is the degeneracy ratio that dictates the entropic advantage of the interacting states. It turns out that all the results presented here are independent of the absolute degeneracies $m$ and $n$, and depend only on their ratio $r$.
Using standard gaussian integral techniques one finds an expression for the free energy per spin in the infinite range limit: $$\begin{aligned}
\label{diff}
\label{eq:2} f \equiv F/N =\beta J m^2/2-ln[1+2 \ r \ cosh(\beta
J m)e^{-\beta D}]\end{aligned}$$ where m is the order parameter of the system (magnetization per spin), $m \equiv \langle \frac{1}{N}
\sum_{i=1}^N S_i \rangle$. The phase transition curves are obtained numerically by solving for the minimum of Eq. (\[diff\]) with respect to $m$. Scaling the temperature and $D$ with the interaction strength $J$, the phase diagram is shown in Figure (\[fig3\]). In the inset, results are presented for the original Blume-Capel model (i.e., the $r=1$ case): the line AB is a second order transition line, above it is a paramagnetic ($m=0$) phase and below it the system is Ferromagnetic ($m \neq 0$). Below the tricritical point (B) the phase transition is first order, and the three lines plotted are: the spinodal line of the ferromagnetic phase BE (above this line the $m\neq0$ solution ceases to exist), the spinodal line of the paramagnetic phase BC (below this line there is no $m=0$ minimum of the free energy) and the first order transition line BD. Along BD the free energy of the paramagnetic phase is equal to that of the ferromagnetic state. Clearly, the Blume-Capel model displays no inverse melting: an increase of the temperature induces smaller order parameter.
The situation is different as $r$ increases, as emphasized by the main part of Figure (\[fig3\]). The same phase diagram is presented, but now $r=6$, so the interacting states have larger entropy. The tricritical point is shifted to the left, leaving a region of second order inverse melting, and the orientation of the BD line also changes, establishing the possibility of first order inverse melting. Note that the $r=6$ transition lines converge to the $r=1$ lines as $T \to 0$, since the entropy has no effect on the free energy at that limit. The ferromagnetic phase also covers larger area of the phase diagram for $r=6$, a fact that reflects again its entropic advantage.
![Phase diagram and spinodal lines for the ordered model Eq. (\[diff\]) in the $D-T$ plane for $r=1$ (Blume-Capel model, inset) and for $r=6$. The value of $r=6$ has been chosen in order for the effect to be more pronounced, but inverse melting is seen for $r$ lower than $2$ []{data-label="fig3"}](fig2d.eps "fig:"){width="7.7cm"}\
![Phase diagram and the spinodal lines for the ordered Blume-Capel model in the interaction-temperature plane with $r=6$. The interaction J/D represents the concentration (“pressure”).[]{data-label="fig4"}](fig3e.eps "fig:"){width="7.7cm"}\
To allow qualitative comparison of our cartoon model with experimental results, the appropriate parameters should be identified. There are three parameters in the model as it stands: $D$ represents the energetic advantage of the noninteracting state, $r$ (if larger than 1) is the entropic gain of the interacting state, and $J$ is the strength of the interaction. In most of the physical systems that display inverse melting the controlled external parameter is the strength of the interaction: pressure (for $He^3$ and $He^4$) or concentration of the interacting objects (for polymeric systems and Rochelle salt - Ammonium Rochelle salt mixtures). As long as the only effect of the pressure is to increase the strength of the effective interaction among constituents, it may be modelled by changing $J$. The resulting phase diagram should be compared, though, with the $T-J$ plot of our model presented in Figure (\[fig4\]). The decrease of the transition temperature with the increase of interaction strength (pressure) is physically intuitive, as larger interaction favors energetically the ferromagnetic phase. As emphasized recently by [@stillinger], the slope of the first order transition line in the pressure-temperature plane is required by the corresponding Clausius-Clapeyron equation: $$\begin{aligned}
\label{eq:3} \frac{d P}{d T}=\frac{S_2-S_1}{V_2-V_1}\end{aligned}$$ where $V_2$, $V_1$ are the volume (the extensive parameter conjugate to the pressure) of the solid and liquid phases respectively, and $S_2$, $S_1$ are their entropies. Inverse melting is possible if the numerator of (\[eq:3\]) is negative, so for “normal” transitions ($V_2
> V_1$) one expects a negative slope of the transition line. In real magnetic or electric system the intensive-extensive pairs \[magnetization-magnetic field ($\textbf{M} \cdot d\textbf{H}$) or polarization-electric field ($\textbf{P}\cdot d\textbf{E}$)\], appear in the free energy function with inverse sign relative to $PdV$. If the order parameter vanishes, or takes smaller values, in the “liquid” (disordered) phase, this implies also negative slope of the first order transition line in the temperature-external field plane. An interesting exception is the inverse melting of vortex liquid in superconductors, where the magnetization of the crystalline phase is smaller than that of the liquid and the transition line slope is actually positive.
Inverse freezing, the (reversible) appearance of glassy features in a system upon raising the temperature, may be incorporated in our model by introducing random coupling $J_{ij}$, as in the standard spin-glass models [@binder]. This randomness may fit, in particular, to the gelation transition of Methyl Cellulose, as it occur *only* when the hydrophobic sequences are deposited at random along the chain. The random-exchange analogue of the Hamiltonian (\[eq:1\]) is: $$\begin{aligned}
\label{eq:4} H=\sum_{<i,j>} J_{ij}S_{i}S_{j}+\sum_{i=1}^N D
S_{i}^2\end{aligned}$$ where the exchange interaction between the $i$ and the $j$ spin is taken at random from some predetermined distribution. Following the paradigmatic Sherrington-Kirkpatrick (SK) analysis [@binder] of the infinite range spin glass, we assume gaussian distribution of the exchange term with zero mean: $$\begin{aligned}
\label{eq:5}
P(J_{ij})=\sqrt{\frac{N}{2\pi\sigma^{2}}}\exp-(\frac{NJ_{ij}^{2}}{2J^{2}}),\end{aligned}$$ where $\frac{J}{\sqrt{N}}$ is the width of the distribution. The replica trick is then implemented to get the free energy at the large $N$ limit.
The case $r=1$, namely the random exchange version of the Blume-Capel model, was first introduced and discussed by Ghatak and Sherrington (GS) [@Ghatak] who used symmetric replica to obtain the relevant phase diagram. The GS solution seems to display inverse freezing even for the $r=1$ case, but more detailed analysis by da Costa et. al. [@Costa] revealed that the glass order-parameter takes nonzero values (with a variety of stability features) in the area below the GS transition line, and the temperature dependence is monotonic. Recently, the full replica symmetry breaking analysis has been implemented for the GS model [@crisanti], and the results admit no inverse glass transition. Here we present a replica symmetric analysis of the same hamiltonian where the interacting states are highly degenerate, i.e., $r>1$. Following [@Costa], we obtain the phase transition and the spinodal lines, and the results support, again, both first and second order inverse glass transition.
The replica technique [@Edwards] relies on the identity $\overline{ln[Z]}=lim_{n \rightarrow 0}\frac{1}{n}(\overline{Z^n}
-1)$, where $Z$ is the partition function of the system and $Z^n$ is interpreted as the partition function of an n-fold replicated system $S_i \rightarrow S_{ia}, a=1...n$. The average free energy may be computed using $\beta f =-lim_{n\rightarrow
0}\frac{1}{n}(\overline{Z^n}-1)$. The disorder average is taken for $Z^n$ using the Gaussian distribution (\[eq:5\]) and gives: $$\begin{aligned}
\label{eq:6} \overline {Z^{n}}= Tr \exp \left [\frac{\beta^2
J^2}{N} \sum_{a>b} (\sum_iS_{ia} S_{ib})^2+ \right. \nonumber\\
\left. \frac{\beta^2 J^2}{2N}\sum_{a}(\sum_i S_{ia}^2)^2-\beta D
\sum_{i} S_{ia} ^2 \right]\end{aligned}$$ where $a,b=1...n$ denotes the replica. Implementing the Hubbard-Stratanovitch identity yields the free energy per spin: $$\begin{aligned}
\label{ff}
\label{eq:7} -\beta \frac{F}{N}=-\beta^2J^2\sum_{a>b}q_{ab}^2\
-\frac{\beta^2J^2}{2}\sum_{a}q_{aa}^2+lnTr e^{ \hat{L}}\end{aligned}$$ where $$\begin{aligned}
\label{eq:8}
\hat{L}=2\beta^2J^2\sum_{a>b}q_{ab}S_aS_b+\beta^2J^2\sum_{a}q_{aa}S_{a}^2-\beta
D\sum_{a}S_{a}^2 .\nonumber\\\end{aligned}$$ $q_{aa}$ and $q_{ab}$, the diagonal and the off diagonal entries of the “order parameter matrix”, are given self-consistently by the saddle-point condition: $$\begin{aligned}
\label{eq:9} q_{ab}&=&\langle{S_a S_b}\rangle\nonumber\\
q_{aa}&=&\langle{S_a ^2}\rangle\end{aligned}$$ where $\langle ... \rangle$ stands for thermal average. In order to solve this model it is necessary to make assumptions on the order parameter matrix elements $q_{ab}$, and the simplest ansatz, is symmetry with respect to permutations of any pair of the replicas: $q_{ab}=q, \ \ \forall a \neq b$, $q_{aa}=p, \ \
\forall a$. Using this *replica symmetric* assumption one obtains $$\begin{aligned}
\label{fff}
\label{eq:10} -\beta f&=&\frac{\beta^2 J^2}{2}(q^2-p^2)+
\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} dz \exp
(-\frac {z^2}{2})\cdot\nonumber\\ &ln&[1+2 \ r \ e^{\gamma}\cosh
(\beta J\sqrt{2q}z)\end{aligned}$$ with $$\begin{aligned}
\label{eq:11} \gamma=\beta^2 J^2 (p-q)-\beta D\end{aligned}$$
Extremizing the free energy with respect to q and p one gets by the following coupled equations: $$\begin{aligned}
\label{qqq}
\label{eq:12} q=\int_{-\infty}^{\infty}\frac{dz \ exp(-\frac
{z^2}{2})}{\sqrt{2\pi}}
\frac {4 r ^2 e^{2\gamma}\sinh ^2(\beta J
\sqrt{2q}z)}{[1+2 r e^{\gamma}\cosh (\beta J \sqrt{2q}z)]^2}\end{aligned}$$ $$\begin{aligned}
\label{ppp}
\label{eq:13} p=\int_{-\infty}^{\infty} \frac{dz \ exp(-\frac
{z^2}{2})}{\sqrt{2\pi}}\frac {2 r e^{\gamma} \cosh(\beta J
\sqrt{2q}z)}{1+2 r e^{\gamma}\cosh (\beta J \sqrt{2q}z)}\end{aligned}$$ The coupled equations (\[qqq\]) and (\[ppp\]) are numerically solved (with the possibility of multiple solutions if more than one stable state exists), and the location of the first order transition line is then determined by comparison of the free energy values (plugging $q$ and $p$ into (\[fff\])). The resulting phase diagram is shown in Fig. (\[fig5\]) for the case $r=6$, and displays all the essential features that exist in the ordered model, including a tricritical point and spinodal lines.
![Phase diagram and the spinodal lines for the disordered model in the D-T plane for a constant interaction J for $r=6$.[]{data-label="fig5"}](fig4b.eps "fig:"){width="7.7cm"}\
To conclude, the basic theoretical insight of Blume and Capel, to have a spin model with low energy non-interacting (zero) state, may yield an inverse melting transition once the model is enriched with an entropic advantage of the interacting phase. It should be emphasized that the higher entropy associated with the interaction do not unavoidably entail inverse melting; this property may be “buried” below energetic and other constraints that dominate the system, yet it may change the phase diagram predicted by the naive assumption that higher energy implies higher entropy.
The authors wish to acknowledge Prof. Y. Rabin for most helpful discussions of the subject.
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abstract: 'Nonlinear magnetotransport of a microwave-irradiated high mobility two-dimensional electron system under a finite direct current excitation is analyzed using a dc-controlled scheme with photon-assisted transition mechanism. The predicted amplitudes, extrema and nodes of the oscillatory differential resistance versus the magnetic field and the current density, are in excellent agreement with the recent experimental observation \[Hatke [*et al.*]{} Phys. Rev. B [**77**]{}, 201304(R) (2008)\].'
author:
- 'X. L. Lei and S. Y. Liu'
title: 'Nonlinear magnetoresistance of an irradiated two-dimensional electron system'
---
The prediction[@Ryz-1970] and detection[@Zud01; @Ye] of radiation induced magnetoresistance oscillation (RIMO) in two-dimensional (2D) electron systems (ES), especially the discovery of the zero-resistance state[@Mani02; @Zud03], have stimulated intensive experimental[@Dor03; @Yang03; @Zud04; @Mani04; @Willett; @Du; @Kovalev; @Mani05; @Dor05; @Stud; @Smet05; @Yang06; @Bykov05; @Bykov06] and theoretical[@Shi; @Durst; @Lei03; @Lei04; @Ryz03; @Vav04; @Dmitriev03; @DGHO05; @Torres05; @Ng05; @Ina-prl05; @Kashuba; @Andreev03; @Alicea05; @Auerbach05; @Mikhailov04] studies on this extraordinary transport phenomenon of electrons in very high Landau levels.
Despite the fact that basic features of RIMO have been established and the understanding that it stems from impurity scattering has been reached, so far there has been no common agreement as to the accurate microscopic origin of these giant resistance oscillations. Presented in different forms, many theoretical models[@Shi; @Durst; @Lei03; @Lei04; @Ryz03; @Vav04; @Torres05; @Ina-prl05; @Kashuba; @Ng05; @Lei07-2; @Auerbach07] consider RIMO to arise from electron transitions between different Landau states due to impurity scattering accompanied by absorbing and emitting microwave photons. This origin is called the “photon-assisted transition” or “displacement” mechanism. A different origin, called the “inelastic” or “distribution function” mechanism,[@Dor03; @Dmitriev03; @DGHO05] considers RIMO to arise from a microwave-induced nonequilibrium oscillation of the time-averaged isotropic electron distribution function in the density-of-states (DOS) modulated system. Both mechanisms exist in a real 2D semiconductor and have been shown to produce magentoresistance oscillations qualitatively having the observed period, phase and magnetic field damping. The “displacement” mechanism predicts a well-defined photoresistivity with given impurity scattering and Landau-level broadening, while the “inelastic” mechanism yields an additional factor proportional to the ratio of the inelastic scattering time $\tau_{\rm in}$ to the impurity-induced quantum scattering time $\tau_{q}$.[@Dmitriev03] Since the inelastic scattering time $\tau_{\rm in}$ or the thermalization time $\tau_{\rm th}$,[@Lei03] being the property of a nonequilibrium state and contributed by the direct Coulomb interactions between electrons and by all other possible impurity- and phonon-scattering mediated effective electron-electron scatterings,[@Lei03] is very hard to determine theoretically or to measure experimentally, the sharp controversy whether $\tau_{\rm in}/\tau_{q}\gg 1$ or $\tau_{\rm in}/\tau_{q}\ll 1$, i.e. which mechanism plays the dominant role in the experimental systems,[@Lei03; @Dmitriev03] has been an unsolved issue. The detailed comparison between theoretical predictions and experiments may provide a useful way to distinguish them.
Introducing additional parameters into microwave-illuminated 2DESs, such as dc excitations, can be of help to distinguish different models and mechanisms. It has been shown that a finite current alone, can also induce substantial magnetoresistance oscillation and zeroresistance without microwave radiation.[@Yang02; @WZhang07; @JZhang07; @Bykov07; @Lei07-1; @Vav07] Simultaneous application of a direct current and a microwave radiation leads to very interesting and complicated oscillatory behavior of resistance and differential resistance.[@WZhang07-2; @Lei07-2; @Auerbach07] Recent careful measurements[@WZhang08; @Hatke08] disclosed further details of such nonlinear magnetotransport in a high-mobility 2D semiconductor under both ac and dc exitations, allowing a careful comparison with theoretical predictions.
Our examination is based on a current-controlled scheme of photon-assisted transport,[@Lei03] which deals with a 2DES of short thermalization time having $N_{\rm s}$ electrons in a unit area of the $x$-$y$ plane and subject to a uniform magnetic field ${\bm B}=(0,0,B)$ in the $z$ direction. When an electromagnetic wave with incident electric field ${\bm E}_{{\rm i}s}\sin \omega t$ irradiates perpendicularly on the plane together with a dc electric field ${\bm E}_0$ inside, the steady transport state of this 2DES is described by the electron drift velocity ${\bm v}_0$ and an electron temperature $T_{\rm e}$, satisfying the force and energy balance equations[@Lei03] $$\begin{aligned}
N_{\rm s}e{\bm E}_{0}+N_{\rm s} e ({\bm v}_0 \times {\bm B})+
{\bm F}_0&=&0,\label{eqv0}\\
N_{\rm s}e{\bm E}_0\cdot {\bm v}_0+S_{\rm p}- W&=&0.
\label{eqsw}\end{aligned}$$ Here, the frictional force resisting electron drift motion, $${\bm F}_0=\sum_{{\bm q}_\|}\left| U({\bm q}_\|)\right| ^{2}
\sum_{n=-\infty }^{\infty }{\bm q}_\|{J}_{n}^{2}(\xi ){\Pi}_{2}
({\bm q}_\|,\omega_0+n\omega ),\label{ff0}
\label{eqf0}$$ is given in terms of the electron density correlation function ${\Pi}_2({\bm q}_{\|},{\Omega})$, the effective impurity potential $U({\bm q}_{\|})$, a radiation-related coupling parameter $\xi$ in the Bessel function $J_n(\xi)$, and $\omega_0\equiv{\bm q}_{\|}\cdot {\bm v}_0$. The electron energy absorption from the radiation field, $S_{\rm p}$, and the electron energy dissipation to the lattice, $W$, are given in Ref.. The nonlinear longitudinal resistivity and differential resistivity in the presence of a radiation field are obtained from Eq.(\[eqv0\]) by taking ${\bm v}_0$ and the current density ${\bm J}=N_{\rm s}e{\bm v}_0 $ in the $x$ direction, ${\bm v}_0=(v_{0},0,0)$ and ${\bm J}=(J,0,0)$, $$R_{xx}=-{F}_0/(N_{\rm s}^2 e^2 v_{0}),\,\,\,
r_{xx}=-({\partial F_0}/{\partial v_0})/(N_{\rm s}^2 e^2).\label{rrxx}$$
We have calculated the differential resistivity $r_{xx}$ from above equations (taking up to three-photon processes) under different magnetic fields $B$ and bias drift velocities $v_0$ for a GaAs-based heterosystem with carrier density $N_{\rm s}=3.7\times 10^{15}$/m$^{2}$ and low-temperature linear mobility $\mu_0=1200$m$^{2}$/Vs at lattice temperature $T=1.5$K, irradiated by a linearly $x$-polarized microwave of frequency $\omega/2\pi=69$GHz with incident amplitude $E_{{\rm i}s}=3.6$V/cm. The elastic scatterings are assumed due to a mixture of short-range and background impurities, and the Landau-level broadening $\Gamma$ is taken to be a Gaussian form with a broadening parameter $\alpha=7$.[@Lei03]
![(Color online) Differential magnetoresistivity $r_{xx}$ vs $\epsilon_{\omega}=\omega/\omega_c$ under fixed bias current densities from $J=0$ to $J=0.16$ A/m, in $0.01$A/m increments.[]{data-label="fig1"}](fig1.eps){width="45.00000%"}
![(Color online) (a) Differential resistivity $r_{xx}$ vs $\epsilon_j$ at fixed $\epsilon_{\omega}$ from 2 to 3.5 in steps of 0.0625. (b) Amplitude of $r_{xx}$ oscillations. (b) Positions of $r_{xx}$ maxima extracted.[]{data-label="fig2"}](fig2.eps){width="45.00000%"}
Figure 1 presents the calculated $r_{xx}$ versus $\epsilon_{\omega}\equiv \omega/\omega_{c}$ ($\omega_c=eB/m$ is the cyclotron frequency) at fixed bias drift velocities from $2v_0/v_{\rm F}=0$ ($v_{\rm F}$ is the Fermi velocity) to $2\times 10^{-3}$ in steps of $1.25\times 10^{-4}$, corresponding to current densities $J=0$ to $0.16$A/m in steps of $0.01$A/m. The $J=0$ case exhibits typical RIMO with a sequence of resistance maxima ($1^+,2^+,3^+,4^+$ and $5^+$) and negative values around the resistance minima $1^-$ and $2^-$. With increasing $J$ to 0.16A/m, the maxima $2^+,3^+$ and $4^+$ (minima $2^-,3^-$ and $4^-$) evolve into minima (maxima) having seemingly little change in the $B$ positions. Further, all the curves cross approximately at $\epsilon_{\omega}=1.5,2,2.5,3,3.5$ and 4.5, indicating that $J$ at this range has little effect on photoresistance there. These and other features of Fig.1 reproduce what was exactly observed in Ref..
Figure 2(a) shows the calculated $r_{xx}$ versus $\epsilon_j \equiv \omega_j/\omega_c$ ($\omega_j=2k_{\rm F}v_0$, $k_{\rm F}$ is the Fermi wave vector of the 2D electron system) at fixed $\epsilon_{\omega}$ from $2$ to $3.5$ in steps of 0.0625. Traces are vertically offset in increments of $0.625\,\Omega$ for clarity. The amplitudes $\Delta r$ of $r_{xx}$-$\epsilon_j$ oscillations are maximized around $\epsilon_{\omega}\simeq 2.2,2.8$ and 3.2 and strongly suppressed at $\epsilon_{\omega} \simeq 2.5$ and 3.5, as shown in Fig.2(b).[@note] The positions of $r_{xx}$ maxima extracted are plotted in Fig.2(c) as dots. All these are in excellent agreement with the experimental results \[Fig.2(a),(b) and (c) of Ref.\].
In the present current-controlled transport model the oscillations of $R_{xx}$ and $r_{xx}$ are referred to the behavior of function ${\Pi}_{2}({\bm q}_\|,\omega_0+n\omega )$ in Eq.(\[ff0\]). The electron density correlation function ${\Pi}_{2}({\bm q}_\|,\Omega )$ is essentially a multiplication of two energy-$\Omega$ shifted periodically modulated DOS functions of electrons in the magnetic field.[@Lei03] Its periodicity with changing frequency $\Omega\rightarrow \Omega+\omega_c$ at low temperatures and high Landau-level occupations, determines the main periodical behavior of magnetoresistance.[@Lei03] The previous examination[@Lei07-2] focused on the node positions of the oscillatory peak-valley pairs of $R_{xx}$, which appear periodically roughly along the lines $
\epsilon_\omega+\eta\, \epsilon_j=m=1,2,3,4,...
$ in the $\epsilon_\omega$-$\epsilon_j$ plane, where $\epsilon_\omega\equiv\omega/\omega_c$ is the control parameter of RIMOs, and $\epsilon_j \equiv \omega_j/\omega_c$ is the control parameter of current-induced magnetoresistance oscillations, and $\eta \lesssim 1$, dependent on the scattering potential.[@Lei07-1]
The maxima of differential resistivity $r_{xx}$ show up at lower values in the $\epsilon_j$ axis in comparison with the node positions of related valley-peak pairs of $R_{xx}$, and its appearance exhibits a periodicity $\Delta\epsilon_j \lesssim 1$.[@Lei07-1] In the $\epsilon_\omega$-$\epsilon_j$ plane, the differential resistance maxima are expected to show up roughly in the vicinity along the lines $$\epsilon_\omega+\lambda\, \epsilon_j=m=1,2,3,4,... \label{lambdam}$$ with $\lambda \gtrsim 1$, dependent on the scattering potential ($\lambda = 1.04$ for the system on discussion). Eq.(\[lambdam\]) qualitatively accounts for the periodical change of $r_{xx}$ in a large scale in steps of $\Delta(\epsilon_\omega+\lambda\, \epsilon_j)=1$.
Under strong microwave irradiation, as in the present case, the role of virtual photon process \[the $n=0$ term in the sum of Eq.(\[ff0\])\] is negligible due to samll $J_0^{2}(\xi)$,[@Lei03] and main contributions to resistivity come from $n=\pm1,\pm2,...$ terms (single- and multiple-photon processes). Noticing that the frequency differentiate ${\Pi}_{2}^{\prime}({\bm q}_\|,\Omega)$ is an even function of $\Omega$ and considering contributions from scatterings parallel and antiparallel to the drift velocity ${\bm v}_0$ and from $\pm |n|$ terms, we see that, in the case of finite bias current, the $r_{xx}$ behavior is determined by the sum of two terms: (a) ${\Pi}_{2}^{\prime}({\bm q}_\|,|n|\omega+q_{\|}v_0\cos{\theta})$ and (b) ${\Pi}_{2}^{\prime}({\bm q}_\|,|n|\omega-q_{\|}v_0\cos{\theta})$. Depending on the ${\Pi}_{2}^{\prime}({\bm q}_\|,\Omega)$ function behavior in the vicinity of $\Omega=|n|\omega$, effects of these two terms can be cancelled or added, completely or partly, at different locations of $\epsilon_{\omega}$. ${\Pi}_{2}^{\prime}({\bm q}_\|,\Omega )$ function reaches maxima (positive) at around $\Omega/\omega_c=N-\frac{1}{4}$, reaches minima (negative) at around $\Omega/\omega_c=N+\frac{1}{4}$, and passes through zero (changing sign) at around $\Omega/\omega_c=N$ and $N+\frac{1}{2}$ for all integers $N\geq 2$. Thus, at $\epsilon_{\omega}\simeq l$ or $l+\frac{1}{2}$ ($l=2,3,4,...$) with which all involved $|n|\omega$ frequencies are located around $N\omega_c$ or $(N+\frac{1}{2})\omega_c$, contributions from (a) and (b) are almost cancelled out for modest $v_0$. In the case of $\epsilon_{\omega}\simeq l-\frac{1}{4}$ \[$\epsilon_{\omega}\simeq l+\frac{1}{4}$\], there always exists a term of frequency $(N-\frac{1}{4})\omega_c$ \[($N+\frac{1}{4})\omega_c$\] in $|n|\omega$, and contributions from (a) and (b) are positively \[negatively\] additive. These clearly account for the suppression of the current effect at $\epsilon_{\omega}\simeq l$ and $l+\frac{1}{2}$, and the enhancement of it around $\epsilon_{\omega} \simeq l-\frac{1}{4}$ and $\epsilon_{\omega}\simeq l+\frac{1}{4}$.
Above discussions are general. The accurate behavior of resistivity $r_{xx}$ inside a period scale is relevant to the detailed shape of the DOS function. Figs.1 and 2 represent the result of a Gaussian-type DOS. The good quantitative agreement with experiment without adjusting parameters indicates that the present current-controlled scheme of photon-assisted transport captures the main physics of RIMOs in the discussed quasi-2D system.
This work was supported by the projects of the National Science Foundation of China, and the Shanghai Municipal Commission of Science and Technology.
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The amplitude maxima will be at $\epsilon_{\omega}\simeq 2.25,2.75$ and 3.25 if one considers only up to single-photon process ($|n|\leq 1$).
|
---
abstract: 'The puzzle of $\RDs$ associated with $B\to\Ds\tau\nu$ decay is addressed in the two-Higgs-doublet model. An anomalous coupling of $\tau$ to the charged Higgs is introduced to fit the data from BaBar, Belle, and LHCb. It is shown that all four types of the model yield similar values for the minimum $\chi^2$. We also show that the newly normalized $\RDs$ with the branching ratio of $B\to\tau\nu$ decay exhibits a much smaller minimum $\chi^2$.'
author:
- 'Jong-Phil Lee'
title: '$B\to D^{(*)}\tau\nu_\tau$ in the 2HDM with an anomalous $\tau$ coupling'
---
Introduction
============
One of the most interesting puzzles in flavor physics in recent years has been the excess of the semitaunic $B$ decays, $B\to\Ds\tau\nu_\tau$. The excess is well expressed in terms of the ratio $$\RDs\equiv\frac{\calB(B\to\Ds\tau\nu)}{\calB(B\to\Ds\ell\nu)}~,$$ where $\calB$ is the branching ratio. The standard model (SM) prediction is [@Na; @Fajfer] $$\begin{aligned}
\label{RSM}
R(D)_\SM&=&0.300\pm0.008~,\nn\\
R(D^*)_\SM&=&0.252\pm0.003~.\end{aligned}$$ The BaBar Collaboration has reported that the measured $R(D)$ exceeds the SM prediction by $2.0\sigma$, while $R(D^*)$ exceeds the SM prediction by $2.7\sigma$, and the combined significance of the disagreement is $3.4\sigma$ [@BaBar1; @BaBar_PRL]. BaBar analyzed the possible effect of a charged Higgs boson in the Type-II two-Higgs-doublet model (2HDM), and excluded the model at the 99.8% confidence level.
The Belle measurements of $\RDs$ are slightly smaller than those of BaBar, but still larger than the SM expectations [@Belle1; @Belle1607]. Interestingly, Belle’s results are compatible with the Type-II 2HDM in the $\tan\beta/m_{H^\pm}$ region around $0.45c^2/\text{GeV}$ (where $\beta$ is the ratio of the two vacuum expectation values of the 2HDM) and zero [@Belle1], and recent measurements of $R(D^*)$ are consistent with the SM predictions [@Belle1612] On the other hand, LHCb reported that $R(D^*)$ is larger than the SM predictions by $2.1\sigma$ [@LHCb1].
In this paper we try to fit the global data on $\RDs$ with the 2HDM of all types. The 2HDM is a natural extension of the SM Higgs sector, so it has been tested to fit the $\RDs$ puzzle [@Andreas; @Fazio; @Cline; @Koerner]. Out of all the types of 2HDM, the Type-II model is the most promising because the new physics (NP) effects are involved with the coupling of $\tan^2\beta$ while for other types the couplings are 1 or $1/\tan^2\beta$. As mentioned before, there is tension between BaBar and Belle regarding the compatibility of the Type-II 2HDM to the data. In this analysis we introduce an anomalous $\tau$ coupling to the charged Higgs [@Dhargyal]. Since the NP effects are enhanced by new couplings and suppressed by the charged Higgs mass, the new couplings should be large enough to allow a heavy charged Higgs to fit the data. We also investigate possible roles of leptonic decay $B\to\tau\nu$ to solve the $\RDs$ puzzle. It was suggested that the normalized $\RDs$ with $\calB(B\to\tau\nu)$, $\RtDs$ are consistent with the SM [@Nandi]. We implement the global $\chi^2$ fitting to $\RDs$ as well as $\RtDs$ with the anomalous $\tau$ coupling.
The paper is organized as follows. Section II introduces the 2HDM with the anomalous $\tau$ coupling to describe $B\to\Ds\tau\nu$ and $B\to\tau\nu$ transitions. In Sec. III $\RDs$ and $\RtDs$ are expressed in the 2HDM with the new coupling. Our results and discussions are given in Sec. IV, and conclusions follow in Sec. V.
2HDM with anomalous $\tau$ couplings
====================================
The Yukawa interaction in the 2HDM is given by [@Aoki] $$\begin{aligned}
\label{LY}
\calL_{\rm Yukawa}&=&
-\sum_{f=u,d,\ell}\frac{m_f}{v}\left(
\xi_h^f{\bar f}fh + \xi_H^f{\bar f}fH -i\xi_A^f{\bar f}\gamma_5 fA\right)\nn\\
&&\left[
\frac{\sqrt{2}V_{ud}}{v}{\bar u}\left(m_u\xi_A^uP_L\ + m_d\xi_A^dP_R\right)dH^+
+\frac{\sqrt{2}\xi_A^\ell m_\ell}{v}{\bar\nu}_L\ell_RH^+ +{\rm h.c.}\right]~,\end{aligned}$$ where $v=\sqrt{v_1^2+v_2^2}=246$ GeV, $v_{1,2}$ are the vacuum expectation values (VEVs) of the scalar fields $\Phi_{1,2}$ of the 2HDM with $\tan\beta\equiv v_2/v_1$, and the $\xi$s are the couplings defined in Table \[T1\].
$\xi_A^u$ $\xi_A^d$ $\xi_A^\ell$
--------- ------------- -------------- --------------
Type-I $\cot\beta$ $-\cot\beta$ $-\cot\beta$
Type-II $\cot\beta$ $ \tan\beta$ $ \tan\beta$
Type-X $\cot\beta$ $-\cot\beta$ $ \tan\beta$
Type-Y $\cot\beta$ $ \tan\beta$ $-\cot\beta$
: $\xi$s for each type of 2HDM.[]{data-label="T1"}
Here we introduce an anomalous factor $\eta$ to enhance $\xi_A^\tau$ [@Dhargyal]. The motivation is that $\tau$ is screened from the second Higgs VEV $v_2$ and the neutral component of $\Phi_2$ by a factor of $\eta$. In this case the tau mass is $\sim Y_{\rm Yukawa}v_2/\eta$, effectively enhancing the Yukawa coupling of $\tau$ to $H^\pm$, while that of $\tau$ to neutral Higgses remains unchanged. This kind of model can be easily constructed within extra dimensions. For example, as in Refs. [@Archer; @Agashe], the overlappings between the wave functions of $\tau$ and the neutral component of $\Phi_2$ over the extra dimension would determine the strength of the $\tau$ coupling to the neutral part of $\Phi_2$. We could simply assume that the overlapping of $\tau$ and the neutral $\Phi_2$ is rather weak compared to other cases. The enhancement occurs for Type-I and Type-Y models because in these models leptons couple only to $\Phi_2$. The same thing could happen for $\tau$ and the neutral component of $\Phi_1$ to screen $\tau$ from $v_1$, resulting in $\eta$ enhancement for $\tau$-$H^\pm$ couplings in Type-II and X models. In this work we assume that phenomenologically $\tau$ couplings to $H^\pm$ are enhanced by a factor of $\eta$ for all types of the model, $$\xi_A^\tau \to \eta\xi_A^\tau~.$$ Now the effective Lagrangian for the $b\to c\ell\nu$ transition is $$\begin{aligned}
\label{Lb2c}
\calL(b\to c\ell\nu)&=&\calL(b\to c\ell\nu)_{\rm SM}+\calL(b\to c\ell\nu)_{\rm 2HDM}\\\nn
&=&
\frac{G_FV_{cb}}{\sqrt{2}}\left[{\bar c}\gamma^\mu\left(1-\gamma_5\right)b\
{\bar\ell}\gamma_\mu\left(1-\gamma_5\right)\nu_\ell\right]\\\nn
&&
+\frac{V_{cb}}{m_{H^\pm}^2}\left[{\bar c}\left(g_s^c+g_p^c\gamma_5\right)b\
{\bar\ell}\left(f_s^\ell - f_p^\ell\gamma_5\right)\nu_\ell\right]~,\end{aligned}$$ where $$\begin{aligned}
\label{gcfl}
g_s^c &=& \frac{m_c\xi_A^u +m_b\xi_A^d}{\sqrt{2}v}~,~~~
g_p^c = \frac{-m_c\xi_A^u +m_b\xi_A^d}{\sqrt{2}v}~,\\
f_s^\ell &=& f_p^\ell = -\frac{m_\ell\xi_A^\ell}{\sqrt{2}v}~.\end{aligned}$$ For $B\to\tau\nu$ decay, $$\begin{aligned}
\label{Lb2u}
\calL(B\to\tau\nu)&=&\calL(B\to\tau\nu)_{\rm SM}+\calL(B\to\tau\nu)_{\rm 2HDM}\\\nn
&=&
\frac{G_FV_{ub}}{\sqrt{2}}\left[{\bar u}\gamma^\mu\left(1-\gamma_5\right)b\
{\bar\tau}\gamma_\mu\left(1-\gamma_5\right)\nu_\tau\right]\\\nn
&&
+\frac{V_{ub}}{m_{H^\pm}^2}\left[{\bar u}\left(g_s^u+g_p^u\gamma_5\right)b\
{\bar\tau}\left(f_s^\tau - f_p^\tau\gamma_5\right)\nu_\tau\right]~,\end{aligned}$$ where $$\begin{aligned}
\label{gcfl}
g_s^u &=& \frac{m_u\xi_A^u +m_b\xi_A^d}{\sqrt{2}v}~,~~~
g_p^u = \frac{-m_u\xi_A^u +m_b\xi_A^d}{\sqrt{2}v}~,\\
f_s^\tau &=& f_p^\tau = -\frac{m_\tau\xi_A^\tau}{\sqrt{2}v}~.\end{aligned}$$ Note that $\xi_A^\tau$ contains the enhancement factor $\eta$, $\xi_A^\tau=\eta\xi_A^{\ell=e,\mu}$.
$B\to\Ds\tau\nu$ and $B\to\tau\nu$ decays
=========================================
The decay rates of $B\to\Ds\ell\nu$ in the 2HDM can be expressed as $$\Gamma^\Ds = \Gamma^\Ds_\SM +\Gamma^\Ds_{\rm mix} +\Gamma^\Ds_{H^\pm}~.$$ The differential decay rates for $B\to D\ell\nu$ are given by $$\begin{aligned}
\label{dGDdsSM}
\frac{d\Gamma^D_\SM}{ds}&=&
\frac{G_F^2 |V_{cb}|^2}{96\pi^3m_B^2}\left\{
4m_B^2P_D^2\left(1+\frac{m_\ell^2}{2s}\right)|F_1|^2 \right. \\\nn
&&
\left. +m_B^4\left(1-\frac{m_D^2}{m_B^2}\right)^2\frac{3m_\ell^2}{2s}|F_0|^2\right\}
\left(1-\frac{m_\ell^2}{s}\right)^2P_D~,
\\
%
\label{dGDdsmix}
\frac{d\Gamma^D_{\rm mix}}{ds}&=&
\frac{G_F}{\sqrt{2}m_{H^\pm}^2}\frac{g_s^c|V_{cb}|^2}{32\pi^3}\left(f_s^\ell+f_p^\ell\right)m_\ell\\\nn
&&\times
\left(1-\frac{m_D^2}{m_B^2}\right)\left(\frac{m_B^2-m_D^2}{m_b-m_c}\right)|F_0|^2
\left(1-\frac{m_\ell^2}{s}\right)^2 P_D~,
\\
%
\label{dGDdsHpm}
\frac{d\Gamma^D_{H^\pm}}{ds}&=&
\frac{(g_s^c)^2|V_{cb}|^2}{64\pi^3 m_{H^\pm}^4 m_B^2}\left[(f_s^\ell)^2+(f_p^\ell)^2\right]
|F_0|^2 s\left(1-\frac{m_\ell^2}{s}\right)^2 P_D~,\end{aligned}$$ where $s=(p_B-p_D)^2$ is the momentum-transfer squared, and $$\label{PD}
P_D\equiv\frac{\sqrt{s^2+m_B^4+m_D^4-2(sm_B^2+sm_D^2+m_B^2m_D^2)}}{2m_B}~,$$ is the momentum of $D$ in the $B$ rest frame. The form factors $F_0$ and $F_1$ are given by $$\begin{aligned}
\label{F01}
F_0 &=& \frac{\sqrt{m_Bm_D}}{m_B+m_D} (w+1)S_1~,\\
F_1 &=& \frac{\sqrt{m_Bm_D}(m_B+m_D)}{2m_BP_D}\sqrt{w^2-1}V_1~,\end{aligned}$$ where $$\begin{aligned}
V_1(w)&=&
V(1)\left[1-8\rho_D^2z(w)+(51\rho_D^2-10)z(w)^2-(252\rho_D^2-84)z(w)^3\right]~,\\
S_1(w)&=&
V_1(w)\left\{
1+\Delta\left[-0.019+0.041(w-1)-0.015(w-1)^2\right]\right\}~,\end{aligned}$$ with $$\begin{aligned}
w&=&\frac{m_B^2+m_D^2-s}{2m_Bm_D}~,~~~
z(w)=\frac{\sqrt{w+1}-\sqrt{2}}{\sqrt{w+1}+\sqrt{2}}~,\\
\rho_D^2&=&1.186\pm0.055~,~~~
\Delta=1\pm1~.\end{aligned}$$
For $B\to D^*\ell\nu$ decay, $$\begin{aligned}
\label{dGDsds}
\frac{d\Gamma^{D^*}_\SM}{ds}
&=&
\frac{G_F^2|V_{cb}|^2}{96\pi^3m_B^2}\left[
\left( |H_+|^2+|H_-|^2+|H_0|^2\right)\left(1+\frac{m_\ell^2}{2s}\right)
+\frac{3m_\ell^2}{2s}|H_s|^2\right] \\\nn
&&\times
s\left(1-\frac{m_\ell^2}{s}\right)^2P_{D^*}~,\\
%
\frac{d\Gamma^{D^*}_{\rm mix}}{ds}
&=&
\frac{G_F}{\sqrt{2}}\frac{m_\ell g_p^c|V_{cb}|^2}{8\pi^3 m_{H^\pm}^2}
\frac{f_s^\ell+f_p^\ell}{m_b+m_c}A_0^2
\left(1-\frac{m_\ell^2}{s}\right)^2P_{D^*}^3~,\\
%
\frac{d\Gamma^{D^*}_{H^\pm}}{ds}
&=&
\frac{(g_p^c)^2|V_{cb}|^2}{16\pi^3 m_{H^\pm}^4}
\frac{(f_s^\ell)^2+(f_p^\ell)^2}{(m_b+m_c)^2} A_0^2
s\left(1-\frac{m_\ell^2}{s}\right)^2P_{D^*}^3~,\end{aligned}$$ where $P_{D^*}=P_D(m_D\to m_{D^*})$. The form factors are given by $$\begin{aligned}
\label{Hpm0s}
H_\pm(s) &=&
(m_B+m_{D^*})A_1(s)\mp\frac{2m_B}{m_B+m_{D^*}}P_{D^*} V(s)~,\\
H_0(s) &=&
\frac{-1}{2m_{D^*}\sqrt{s}}\left[
\frac{4m_B^2P_{D^*}^2}{m_B+m_{D^*}}A_2(s)
-(m_B^2-m_{D^*}^2-s)(m_B+m_{D^*})A_1(s)\right]~,\\
H_s(s) &=&
\frac{2m_B P_{D^*}}{\sqrt{s}} A_0(s)~,\end{aligned}$$ where $$\begin{aligned}
\label{A012V}
A_1(w^*) &=& \frac{w^*+1}{2}r_{D^*}h_{A_1}(w^*)~,\\
A_0(w^*) &=& \frac{R_0(w^*)}{r_{D^*}}h_{A_1}(w^*)~,\\
A_2(w^*) &=& \frac{R_2(w^*)}{r_{D^*}}h_{A_1}(w^*)~,\\
V(w^*) &=& \frac{R_1(w^*)}{r_{D^*}}h_{A_1}(w^*)~,\\\end{aligned}$$ with $$\label{wsrDs}
w^* = \frac{m_B^2+m_{D^*}^2-s}{2m_Bm_{D^*}}~,~~~
r_{D^*} = \frac{2\sqrt{m_Bm_{D^*}}}{m_B+m_{D^*}}~,$$ and $$\begin{aligned}
\label{hR}
h_{A_1}(w^*)&=&
h_{A_1}(1)\left[1-8\rho_{D^*}^2z(w^*)+(53\rho_{D^*}^2-15)z(w^*)^2
-(231\rho_D{^*}^2-91)z(w^*)^3\right] ~,\\
R_0(w^*) &=&
R_0(1)-0.11(w^*-1)+0.01(w^*-1)^2 ~,\\
R_1(w^*) &=&
R_1(1)-0.12(w^*-1)+0.05(w^*-1)^2 ~,\\
R_2(w^*) &=&
R_2(1)+0.11(w^*-1)-0.01(w^*-1)^2 ~.\end{aligned}$$ Here[@Dhargyal] $$\begin{aligned}
\rho_{D^*}^2 &=& 1.207\pm0.028~,~~~ R_0(1)=1.14\pm 0.07~,\\
R_1(1)&=&1.401\pm0.033~,~~~ R_2(1)=0.854\pm 0.020~.\end{aligned}$$ For the leptonic two-body decay $B\to\tau\nu$, the branching ratio is $$\calB(B\to\tau\nu)=
\calB(B\to\tau\nu)_\SM\left(1+ r_{H^\pm}\right)^2~,$$ where $$\begin{aligned}
\calB(B\to\tau\nu)_\SM &=&
\frac{G_F^2 |V_{ub}|^2m_\tau^2 m_B}{8\pi}f_B^2
\left(1-\frac{m\tau^2}{m_B^2}\right)^2\tau_B~,\\
%
r_{H^\pm}&=&
\frac{(m_u/m_b)\xi_A^u-\xi_A^d}{1+m_u/m_b}\xi_A^\tau\left(\frac{m_B}{m_{H^\pm}}\right)^2
~.\end{aligned}$$ Here $f_B$ and $\tau_B$ are the decay constant and lifetime of $B$, respectively.
The experimental data are summarized in Table \[T2\] [@Nandi].
$R(D)$ $R(D^*)$ $\calB(B\to\tau\nu)$
------------- -------------------------- ------------------------------------------------- -------------------------------------------------
BABAR $0.440\pm0.058\pm0.042$ $0.332\pm0.024\pm0.018$ [@BaBar1] $1.83^{+0.53}_{-0.49}\times 10^{-4}$ [@BaBar2]
Belle(2015) $0.375\pm0.064\pm0.026$ $0.293\pm0.038\pm0.015$ [@Belle1] $(1.25\pm0.28)\times 10^{-4}$ [@Belle3]
Belle(1607) $-$ $0.302\pm0.030\pm0.011$ [@Belle1607] $-$
Belle(1612) $-$ $0.270\pm0.035^{+0.028}_{-0.025}$ [@Belle1612] $-$
LHCb $-$ $0.336\pm0.027\pm0.030$ [@LHCb1] $-$
: Experimental data for $R(\Ds)$ and $\calB(B\to\tau\nu)$. For $\RDs$ measurements the uncertainties are $\pm$(statistical)$\pm$(systematic).[]{data-label="T2"}
At first we try to fit the data of Table \[T2\] by minimizing $\chi^2$. BABAR results [@BaBar1] already ruled out the Type-II 2HDM. We introduce an anomalous $\tau$ coupling for all types of 2HDM, which will be shown to significantly reduce the $\chi^2$ minimum.
In addition, it was suggested that the ratio $$R_\tau(\Ds)\equiv\frac{R(\Ds)}{\calB(B\to\tau\nu)}~,
\label{RtDs}$$ has some advantages in this analysis [@Nandi]. First of all the $\tau$ detection systematics is canceled in the ratio. But it should be noted that the ratio of Eq. (\[RtDs\]) introduces the theoretical error on $V_{ub}$.
$R_\tau(D)\times 10^3$ $R_\tau(D^*)\times 10^3$
------------------- ------------------------- ---------------------------
BABAR $2.404 \pm 0.838$ $1.814 \pm 0.582$
BABAR($\tau$ tag) $5.96 \pm 2.26$ $4.49 \pm 3.54$
Belle(2015) $3.0 \pm 1.1$ $2.344 \pm 0.799$
Belle($\tau$ tag) $5.7 \pm 3.3$ $4.49 \pm 3.54$
Belle(1607) $-$ $2.416\pm 0.794$
Belle(1612) $-$ $2.160\pm 0.835$
: $R_\tau(\Ds)$ values for different experiments [@Nandi]. The value of “Belle(1612)” is a new one.[]{data-label="T3"}
We use the values of $R_\tau(\Ds)$ in Table \[T3\] for the fit.
Results and Discussions
=======================
In our analysis $\tan\beta$ and $M_{H^\pm}$ are by default the fitting parameters to minimize $\chi^2$, defined by $$\chi^2=\sum_i\frac{(x_i-\mu_i)^2}{(\delta\mu_i)^2}~,$$ where the $x_i$s are model predictions and the $(\mu_i\pm\delta\mu_i)$s are experimental data. Figure \[RD1\] shows the $R(D)$ values vs $\chi^2$ with the anomalous $\tau$ coupling $\eta$. In Fig. \[RD1\](a), $\eta$ is set to be an additional fitting parameter, $-1000\le\eta\le 1000$. Plots for the Type-I and Type-X models are overlapped.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[RD1\] $\chi^2$ vs $R(D)$ for (a) free anomalous couplings and (b) $\eta=\tan^2\beta$ at the $1\sigma$ level. In panel (b) Type-I with $\eta=-\tan^4\beta$ and Type-II with $\eta=-\tan^2\beta$ are also shown. In panel (a), Type-I and Type-X are overlapped; in panel (b), Type-I($\eta=-\tan^4\beta$) and Type-X($\eta=\tan^2\beta$) are overlapped.](RD-chisq-1.jpg "fig:") ![\[RD1\] $\chi^2$ vs $R(D)$ for (a) free anomalous couplings and (b) $\eta=\tan^2\beta$ at the $1\sigma$ level. In panel (b) Type-I with $\eta=-\tan^4\beta$ and Type-II with $\eta=-\tan^2\beta$ are also shown. In panel (a), Type-I and Type-X are overlapped; in panel (b), Type-I($\eta=-\tan^4\beta$) and Type-X($\eta=\tan^2\beta$) are overlapped.](RD-chisq-2.jpg "fig:")
(a) (b)
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
As can be seen from Eqs. (\[dGDdsmix\]) and (\[dGDdsHpm\]), the 2HDM contributes to $R(D)$ as $$\RDs_{H^\pm}\sim
\frac{g_{s,p}^c f_s^\tau}{m_{H^\pm}^2}
+\left(\frac{g_{s,p}^c f_s^\tau}{m_{H^\pm}^2}\right)^2 ~,
\label{RDHpm}$$ where the coefficients are omitted for simplicity. For free $\eta$, Types I and X behave similarly because $\xi_A^u$ and $\xi_A^d$ are the same (see Table \[T1\]). This is also true for Types II and Y. We also consider the case of fixed $\eta\equiv\tan^2\beta$ as in Ref. [@Dhargyal] in Fig. \[RD1\](b). The dominant contribution to Eq. (\[RDHpm\]) comes from $$\begin{aligned}
\RDs_{H^\pm}&\sim&
-\frac{(m_b\xi_A^d)(m_\tau\eta\xi_A^\ell)}{2v^2m_{H^\pm}^2}
+\left[\frac{(m_b\xi_A^d)(m_\tau\eta\xi_A^\ell)}{2v^2m_{H^\pm}^2}\right]^2 \nn\\
&=&
\label{RDHpm2}
\begin{cases}
-\left(\frac{m_b m_\tau}{2v^2m_{H^\pm}^2}\right)(\eta\cot^2\beta)
+ \left[\frac{m_b m_\tau}{2v^2m_{H^\pm}^2}\right]^2(\eta\cot^2\beta)^2 & \text{for Type-I} \\
-\left(\frac{m_b m_\tau}{2v^2m_{H^\pm}^2}\right)(\eta\tan^2\beta)
+ \left[\frac{m_b m_\tau}{2v^2m_{H^\pm}^2}\right]^2(\eta\tan^2\beta)^2 & \text{for Type-II} \\
-\left(\frac{m_b m_\tau}{2v^2m_{H^\pm}^2}\right)(-\eta)
+ \left[\frac{m_b m_\tau}{2v^2m_{H^\pm}^2}\right]^2(-\eta)^2 & \text{for Type-X} \\
-\left(\frac{m_b m_\tau}{2v^2m_{H^\pm}^2}\right)(-\eta)
+ \left[\frac{m_b m_\tau}{2v^2m_{H^\pm}^2}\right]^2(-\eta)^2 & \text{for Type-Y}~. \\
\end{cases}\end{aligned}$$ Since the first term is negative for Types I and II for $\eta\equiv\tan^2\beta>0$, the $\chi^2$ values are very poor compared to those for Types X and Y, as shown in Fig. \[RD1\](b). If we allow $\eta= -\tan^4\beta$ for the Type-I model, the $\chi^2$ distribution over $R(D)$ overlaps with that for the Type-X model. Similar things happen for the Type-II model with $\eta=-\tan^2\beta$ and the Type-Y model. In this case, Eq. (\[RDHpm\]) is not the same for Types II and Y; the sign of $\eta$ is more relevant to the $\chi^2$ distribution than the power of $\eta$. We can see that introducing the anomalous $\tau$ coupling improves the $\chi^2$ fitting, and any Type of 2HDM model is as good (or bad) as another. The best-fit values of $\RDs$ and the corresponding minimum $\chi^2$ per degree of freedom (d.o.f.) are given in Table \[T4\],
Types I II X Y
----------------------------------- --------- --------- --------- ---------
$R(D)$ $0.342$ $0.362$ $0.342$ $0.362$
$R(D^*)$ $0.255$ $0.253$ $0.255$ $0.254$
$\chi^2_\text{min}/\text{d.o.f.}$ $2.881$ $2.813$ $2.881$ $2.861$
: The best-fit $\RDs$ values with free $\eta$ for different Types of the model.[]{data-label="T4"}
and the allowed region of $R(D)$ and $R(D^*)$ at the $1\sigma$ level is given in Fig. \[RDRDs\]
![\[RDRDs\] Allowed region in the $R(D)$-$R(D^*)$ plane at the $1\sigma$ level with free $\eta$. Vertical and horizontal lines are the best-fit points. ](RD-RDs.jpg)
Figure \[RD2\] shows the allowed region of $m_{H^\pm}$ vs $\tan\beta$.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[RD2\] $\RDs$-fitting results of $\tan\beta$ vs $m_{H^\pm}$ for (a) free anomalous couplings within $-1000\le\eta\le 1000$ and (b) $\eta=(\pm)\tan^\alpha\beta$ for some $\alpha$ for different Types of the model, at the $1\sigma$ level. Regions for Type-I with $\eta=-\tan^4\beta$, Type-X with $\eta=\tan^2\beta$, and Type-Y with $\eta=\tan^2\beta$ are overlapped; regions for Type-X with $\eta=\tan^3\beta$ and Type-Y with $\eta=\tan^3\beta$ are also overlapped. ](RD-tanb-1.jpg "fig:") ![\[RD2\] $\RDs$-fitting results of $\tan\beta$ vs $m_{H^\pm}$ for (a) free anomalous couplings within $-1000\le\eta\le 1000$ and (b) $\eta=(\pm)\tan^\alpha\beta$ for some $\alpha$ for different Types of the model, at the $1\sigma$ level. Regions for Type-I with $\eta=-\tan^4\beta$, Type-X with $\eta=\tan^2\beta$, and Type-Y with $\eta=\tan^2\beta$ are overlapped; regions for Type-X with $\eta=\tan^3\beta$ and Type-Y with $\eta=\tan^3\beta$ are also overlapped. ](RD-tanb.jpg "fig:")
(a) (b)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
In Fig. \[RD2\] (a), $\eta$ is a free parameter within $-1000\le\eta\le 1000$. In this case $m_{H^\pm}$ cannot be large enough because the $R(D)_{H^\pm}$ term of Eq. (\[RDHpm2\]) gets smaller and cannot fit the data. One exception is the Type-II model. As shown in Eq. (\[RDHpm2\]), there is a $\tan^2\beta$ enhancement for $R(D)_{H^\pm}$, which allows $m_{H^\pm}$ to be large. If we require that the charged Higgs mass is $m_{H^\pm}\gtrsim500$ GeV, only the Type-II model survives in Fig. \[RD2\](a). In Fig. \[RD2\] (b) we fix $\eta\equiv\pm\tan^\alpha\beta$ for some $\alpha$. For Types X and Y, the allowed stripe stretches to larger $m_{H^\pm}$ with smaller $\tan\beta$ as $\alpha$ goes from 2 to 3. This is because $R(D)_{H^\pm}\sim
\eta m_bm_\tau/m_{H^\pm}^2+\left(\eta m_bm_\tau/m_{H^\pm}^2\right)^2$. Also shown in Fig. \[RD2\](b) are the Type-I model with $\eta=-\tan^4\beta$ and the Type-II model with $\eta=-\tan^2\beta$ for comparison. It would be expected from Eq. (\[RDHpm2\]) that stripes for the Type-X and Y models are coincident up to $\sim\calO(m_c/m_b)$. They also overlap with the stripe of the Type-I model with $\eta=-\tan^4\beta$. The stripe for the Type-II model with $\eta=-\tan^2\beta$ lies in the lowest region of $\tan\beta$ since there is already a $\tan^2\beta$ term in $R(D)_{H^\pm}$.
Now we turn to the $\RtDs$. Figure \[RtD1\] shows $R_\tau(D)$ vs $\chi^2$.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[RtD1\] $\chi^2$ vs $R_\tau(D)$ for (a) free anomalous couplings and (b) $\eta=\tan^2\beta$ at the $1\sigma$ level. In panel (b) Type-I with $\eta=-\tan^4\beta$ and Type-II with $\eta=-\tan^2\beta$ are also shown. In panel (a), Type-I is overlapped with Type-X; in panel (b), grey-green, cyan-blue, and magenta-red curves are overlapped, respectively.](RtauD-chisq-1.jpg "fig:") ![\[RtD1\] $\chi^2$ vs $R_\tau(D)$ for (a) free anomalous couplings and (b) $\eta=\tan^2\beta$ at the $1\sigma$ level. In panel (b) Type-I with $\eta=-\tan^4\beta$ and Type-II with $\eta=-\tan^2\beta$ are also shown. In panel (a), Type-I is overlapped with Type-X; in panel (b), grey-green, cyan-blue, and magenta-red curves are overlapped, respectively.](RtauD-chisq-2.jpg "fig:")
(a) (b)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Note that the minimum $\chi^2$ reduces significantly compared to Fig. \[RD1\]; $\chi^2_\text{min}/\text{d.o.f.}=$ 0.623 (Type-I, X), 0.614 (Type-II) 0.615 (Type-Y) for free $\eta$ in Fig. \[RtD1\](a). As discussed in Ref. [@Nandi], $\RtDs$ values from the BABAR and Belle results are consistent with each other and not so far from the SM predictions [@Nandi], $$\begin{aligned}
R_\tau(D)_\SM&=&(3.136\pm0.628)\times 10^3~,\\
R_\tau(D^*)_\SM&=&(2.661\pm0.512)\times 10^3~.\end{aligned}$$
In Fig. \[RtD1\](b) we fix $\eta=\pm\tan^\alpha\beta$ for some $\alpha$. Any Type of the model is as good as another. $R_\tau(D^*)$ vs $\chi^2$ shows similar behavior. The new contribution to $\calB(B\to\tau\nu)$ is $$r_{H^\pm} \sim
-\xi_A^d\xi_A^\tau\left(\frac{m_B}{m_{H^\pm}}\right)^2 =
\label{rHpm}
\begin{cases}
(-\eta\cot^2\beta)\left(\frac{m_B}{m_{H^\pm}}\right)^2 & \text{for Type-I} \\
(-\eta\tan^2\beta)\left(\frac{m_B}{m_{H^\pm}}\right)^2 & \text{for Type-II} \\
\eta\left(\frac{m_B}{m_{H^\pm}}\right)^2 & \text{for Type-X} \\
\eta\left(\frac{m_B}{m_{H^\pm}}\right)^2 & \text{for Type-Y} ~,\\
\end{cases}$$ where terms of $\calO(m_u/m_b)$ are neglected. As in Eq. (\[RDHpm2\]), only the combination of $\xi_A^d\xi_A^\tau$ is relevant, and thus the Type-I model with $\eta=-\tan^4\beta$ looks much like the Type-X models with $\eta=\tan^2\beta$, and so on.
Table \[T5\] shows the best-fit values of $\RtDs$ and $\chi^2_\text{min}/\text{d.o.f}$,
Types I II X Y
----------------------------------- --------- --------- --------- ---------
$R_\tau(D)\times 10^{-3}$ $2.828$ $2.885$ $2.828$ $2.915$
$R_\tau(D^*)\times 10^{-3}$ $2.223$ $2.188$ $2.223$ $2.223$
$\chi^2_\text{min}/\text{d.o.f.}$ $0.623$ $0.614$ $0.623$ $0.615$
: The best-fit $\RtDs$ values with free $\eta$ for different Types of the model.[]{data-label="T5"}
and Fig. \[RtDRtDs\] shows the allowed region of $R_\tau(D)$ and $R_\tau(D^*)$ at the $1\sigma$ level.
![\[RtDRtDs\] Allowed region in the $R_\tau(D)$-$R_\tau(D^*)$ plane at the $1\sigma$ level with free $\eta$. Vertical and horizontal lines are the best-fit points. ](RtD-RtDs.jpg)
Figure \[RtD2\] shows the allowed region in the $m_{H^\pm}$-$\tan\beta$ plane to fit the $\RtDs$.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[RtD2\] $\RtDs$-fitting results of $\tan\beta$ vs $m_{H^\pm}$ for (a) free anomalous couplings within $-1000\le\eta\le 1000$ and (b) $\eta=(\pm)\tan^\alpha\beta$ for some powers of $\alpha$ for different Types of the model, at the $1\sigma$ level. Regions for Type-I with $\eta=-\tan^4\beta$, Type-X with $\eta=\tan^2\beta$, and Type-Y with $\eta=\tan^2\beta$ are overlapped; regions for Type-X with $\eta=\tan^3\beta$ and Type-Y with $\eta=\tan^3\beta$ are also overlapped. ](RtauD-tanb-1.jpg "fig:") ![\[RtD2\] $\RtDs$-fitting results of $\tan\beta$ vs $m_{H^\pm}$ for (a) free anomalous couplings within $-1000\le\eta\le 1000$ and (b) $\eta=(\pm)\tan^\alpha\beta$ for some powers of $\alpha$ for different Types of the model, at the $1\sigma$ level. Regions for Type-I with $\eta=-\tan^4\beta$, Type-X with $\eta=\tan^2\beta$, and Type-Y with $\eta=\tan^2\beta$ are overlapped; regions for Type-X with $\eta=\tan^3\beta$ and Type-Y with $\eta=\tan^3\beta$ are also overlapped. ](RtauD-tanb.jpg "fig:")
(a) (b)
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
In Fig. \[RtD2\](a), $\eta$ is a free parameter. For Types X and Y, almost the entire region is allowed. The different behaviors of the Type I and II models are due to the factors of $\eta/\tan^2\beta$ (Type-I) and $\eta\tan^2\beta$ (Type-II) in Eq. (\[RDHpm2\]).
In Fig. \[RtD2\](b), $\eta\equiv\pm\tan^\alpha\beta$ for some $\alpha$. Compared to Fig. \[RD2\](b), each Type shows similar behavior, but with broader bands. The reason is that the $\RtDs$ values are more consistent with each other than $\RDs$ ones, and thus more points in the $m_{H^\pm}$-$\tan\beta$ plane are allowed around $\chi^2_\text{min}$. And the bands for Types X and Y with $\eta=\tan^2\beta$ stretch to the region of $m_{H^\pm}=1000$ GeV.
Conclusions
===========
In this work we tried to solve the puzzle of $\RDs$ in the 2HDM. We introduced $\eta$ as an anomalous $\tau$ coupling to $H^+$ to fit the data through minimizing $\chi^2$. To fit the excess of the data over the SM predictions it needs to enhance the charged Higgs contributions, which come in the form of $\RDs_{H^\pm}\sim\eta m_bm_\tau/m_{H^\pm}^2+(\eta m_bm_\tau/m_{H^\pm}^2)^2$. Thus, for small values of $\eta$, $m_{H^\pm}$ cannot be large enough to avoid detection. For the Type-II the situation is alleviated because there is already a factor of $\tan^2\beta$ (but with opposite sign) in $\RDs_{H^\pm}$. As shown in Fig. \[RD2\](b), a large $m_{H^\pm}\sim1000$ GeV is allowed if $\RDs_{H^\pm}\sim (m_bm_\tau/m_{H^\pm}^2)\tan^3\beta
+(m_bm_\tau/m_{H^\pm}^2)^2\tan^6\beta$ in any Type of 2HDM model.
The new ratios $\RtDs$ fit much better. Contributions of the form $\sim(m_bm_\tau/m_{H^\pm}^2)\tan^2\beta
+(m_bm_\tau/m_{H^\pm}^2)^2\tan^4\beta$ allow a large $m_{H^\pm}\sim 1000$ GeV as in Fig. \[RtD2\](b), which is not true for the $\RDs$ fitting. In both cases of $\RDs$ and $\RtDs$ fitting, any type of 2HDM is as good as another with an appropriate $\eta$. For a sufficiently large $m_{H^\pm}\gtrsim 1000$ GeV, new contributions of the form $\sim (m_bm_\tau/m_{H^\pm}^2)\tan^k\beta$ with $k=2$ fit the data well for $\RtDs$, while $k\ge 3$ for $\RDs$. It should be noted that the errors in $\RtDs$ are still large.
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---
abstract: 'Given a positive integer $n$, we let ${{\rm sfp}}(n)$ denote the squarefree part of $n$. We determine all positive integers $n$ for which $\max \{ {{\rm sfp}}(n), {{\rm sfp}}(n+1), {{\rm sfp}}(n+2) \} \leq 150$ by relating the problem to finding integral points on elliptic curves. We also prove that there are infinitely many $n$ for which $$\max \{ {{\rm sfp}}(n), {{\rm sfp}}(n+1), {{\rm sfp}}(n+2) \} < n^{1/3}.$$'
address:
- 'Department of Mathematics, Wake Forest University, Winston-Salem, North Carolina 27109 USA'
- 'Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109 USA'
author:
- Jeremy Rouse
- Yilin Yang
bibliography:
- 'refs.bib'
title: Three consecutive almost squares
---
Introduction
============
The positive integers $48$, $49$ and $50$ are consecutive, and are “almost squares”, namely $48 = 3 \cdot 4^{2}$, $49 = 7^{2}$ and $50 = 2 \cdot 5^{2}$. Does this phenomenon ever occur again? That is, is there a positive integer $n > 48$ for which $$\begin{aligned}
n &= 3x^{2}\\
n+1 &= y^{2}\\
n+2 &= 2z^{2}\end{aligned}$$ has an integer solution? The answer is no. One perspective on this is given by Cohen in Section 12.8.2 of [@Cohen], where this problem is solved using linear forms in logarithms. Another approach is to recognize that the pair of equations $2z^{2} - y^{2} = y^{2} - 3x^{2} = 1$ define an intersection of two quadrics in ${\mathbb{P}}^{3}$ and hence define a curve of genus $1$. Siegel proved that there are only finitely many integer points on a genus $1$ curve, and hence the system of equations above has only finitely many solutions. This method of solving simultaneous Pell equations is considered in [@Tzanakis].
We consider more generally the problem of, given integers $a$, $b$ and $c$, finding integers $n$ for which $n = ax^{2}$, $n+1 = by^{2}$ and $n+2 = cz^{2}$. Multiplying these equations gives $n(n+1)(n+2) =
abc(xyz)^{2}$ and is related to the problem of finding consecutive integers that multiply to an “almost” perfect square. There is an extensive literature on related problems. On one hand, Erdős and Selfridge proved in [@ErdSelf] that no product of consecutive integers is a perfect power (a result generalized to arithmetic progressions by others, see [@Ben1] and [@GHP]). In another direction, Győry proved [@Gyory] that the equation $n(n+1)\cdots(n+k-1) = bx^{l}$ has no integer solutions with $n > 0$ if the greatest prime factor of $b$ does not exceed $k$.
For a positive integer $n$, define ${{\rm sfp}}(n)$ to be the “squarefree part” of $n$, the smallest positive integer $a$ so that $a | n$ and $\frac{n}{a}$ is a perfect square. In this paper, we consider positive integers $n$ for which ${{\rm sfp}}(n)$, ${{\rm sfp}}(n+1)$ and ${{\rm sfp}}(n+2)$ are all small. Our first result is a classification of values of $n$ for which ${{\rm sfp}}(n)$, ${{\rm sfp}}(n+1)$ and ${{\rm sfp}}(n+2)$ are all $\leq 150$. We say that such an $n$ is “non-trivial” if ${{\rm sfp}}(n) < n$, ${{\rm sfp}}(n+1) < n+1$ and ${{\rm sfp}}(n+2) < n+2$.
\[upto150\] There are exactly $25$ non-trivial $n$ for which ${{\rm sfp}}(n) \leq 150$, ${{\rm sfp}}(n+1) \leq 150$ and ${{\rm sfp}}(n+2) \leq 150$, the largest of which is $n = 9841094$.
A table of all $25$ values of $n$ is given in Section \[table\].
After this manuscript was complete, the authors were informed that the essence of this result follows from work of Michael Bennett and Gary Walsh (see [@BenWal]). Given positive integers $a$, $b$ and $c$, a positive integer solution to $n = ax^{2}$, $n+1 = by^{2}$, $n+2 =
cz^{2}$ yields $(n+1)^{2} - n(n+2) = b^{2} y^{4} - (ac) (xz)^{2} =
1$. Bennett and Walsh prove that the system $b^{2} Y^{4} - d X^{2} =
1$ has at most one solution in positive integers. Moreover, if $T + U
\sqrt{d}$ is a fundamental solution of Pell’s equation $X^{2} - dY^{2}
= 1$ define for $k \geq 1$, $T_{k} + U_{k} \sqrt{d} = (T + U
\sqrt{d})^{k}$. Then $T_{k} = bx^{2}$ for some integer $x \in {\mathbb{Z}}$ for at most one $k$, and if any such $k$ exists, then $k$ is the smallest positive integer for which $T_{k}$ is a multiple of $b$. Bennett and Walsh’s approach uses linear forms in logarithms, while our approach relies heavily on the theory of elliptic curves. In particular, if $C$ is the curve defined by the two equations $by^{2} - ax^{2} = 1, cz^{2}
- by^{2} = 1$, then the Jacobian of $C$ is isomorphic to $E : y^{2} =
x^{3} - (abc)^{2} x$.
We rule out many of the $778688$ candidates for the triple $(a,b,c)$ by checking to see whether there are integer solutions to each of the three equations $by^{2} - ax^{2} =
1$, $cz^{2} - b^{2} = 1$ and $cz^{2} - ax^{2} = 2$. We also test $C$ for local solvability, and use Tunnell’s theorem (see [@Tunnell]) to determine if the rank of $E$ is positive. Finally, we use the surprising property that the natural map from $C$ to $E$ sends an integral solution on $C$ to an integral point on $E$. It suffices therefore to compute all the integral points on $E$ (using Magma [@Magma]), which requires computing generators of the Mordell-Weil group. In many cases this is straightforward, but a number of cases require more involved methods (12-descent, computing the analytic rank, and the use of Heegner points).
Given the existence of large solutions, it is natural to ask how large\
$\max \{ {{\rm sfp}}(n), {{\rm sfp}}(n+1), {{\rm sfp}}(n+2) \}$ can be as a function of $n$. This is the subject of our next result.
\[family\] There are infinitely many positive integers $n$ for which $$\max \{ {{\rm sfp}}(n), {{\rm sfp}}(n+1), {{\rm sfp}}(n+2) \} < n^{1/3}.$$
The following heuristic suggests that the exponent $1/3$ above is optimal. A partial summation argument shows that the number of positive integers $n \leq x$ with ${{\rm sfp}}(n) \leq z$ is $\frac{12
\sqrt{xz}}{\pi^{2}} + O(\sqrt{x} \log(z))$. Assuming that $n$, $n+1$ and $n+2$ are “random” integers it follows that ${{\rm sfp}}(n),
{{\rm sfp}}(n+1)$ and ${{\rm sfp}}(n+2)$ are all $\leq n^{\alpha}$ with probability about $n^{3 (\alpha - 1)/2}$. Therefore, the “expected” number of $n$ for which $\max \{ {{\rm sfp}}(n), {{\rm sfp}}(n+1),
{{\rm sfp}}(n+2) \} \leq n^{\alpha}$ is infinite if $3 (\alpha - 1)/2 \geq
-1$ and finite otherwise.
This work arose out of the second author’s undergraduate thesis at Wake Forest University. The authors used Magma [@Magma] V2.20-9 for computations. The authors would also like to thank the anonymous referee for a number of very helpful comments.
Background
==========
We denote by $\mathbb{Q}_{p}$ the field of $p$-adic numbers. A necessary condition for a curve $X/\mathbb{Q}$ to have a rational solution is for it to have such a solution in $\mathbb{Q}_{p}$ for all primes $p$.
If $d$ is an integer which is not a perfect square, let ${\mathbb{Z}}[\sqrt{d}] = \{ a + b \sqrt{d} : a, b \in {\mathbb{Z}}\}$. This is a (not necessarily maximal) order in the quadratic field ${\mathbb{Q}}[\sqrt{d}]$. Let $N : {\mathbb{Z}}[\sqrt{d}] \to {\mathbb{Z}}$ be the norm map given by $N(a + b \sqrt{d}) = a^{2} - db^{2}$.
We now describe some background about elliptic curves. For our purposes an elliptic curve is a curve of the shape $$E : y^{2} = x^{3} + ax + b$$ where $a, b \in {\mathbb{Q}}$. Let $E({\mathbb{Q}})$ be the set of pairs $(x,y)$ of rationals numbers that solve the equation, together with the “point at infinity”. This set has a binary operation on it: given $P, Q$ in $E({\mathbb{Q}})$, the line $L$ through $P$ and $Q$ intersects $E$ in a third point $R = (x,y)$. The point $P+Q$ is defined to be $(x,-y)$. This binary operation endows $E({\mathbb{Q}})$ with the structure of an abelian group.
The Mordell-Weil theorem (see [@Silverman], Theorem VIII.4.1, for example) states the following.
\[TZ\] For any elliptic curve $E/{\mathbb{Q}}$, the group $E({\mathbb{Q}})$ is finitely generated. More, precisely, $$E( \mathbb{Q} ) \cong E(\mathbb{Q})_{\rm tors} \times \mathbb{Z}^{{{\rm rank}}(E({\mathbb{Q}}))},$$ where $E(\mathbb{Q})_{tors}$ is the (finite) torsion subgroup.
There is in general no algorithm which is proven to compute the rank of $E$, (see Section 3 of Rubin and Silverberg’s paper [@Rubin]) but there are a number of procedures which work well in practice for relatively simple curves $E$. We will reduce the problem of solving $n = ax^{2}$, $n+1 =
by^{2}$ and $n+2 = cz^{2}$ to finding points $(X,Y)$ on the curve $$E : Y^{2} = X^{3} - (abc)^{2} X,$$ with $X, Y \in {\mathbb{Z}}$. A theorem of Siegel (see Theorem IX.4.3 of [@Silverman]) states that there are only finitely many points in $E({\mathbb{Q}})$ with both coordinates integral. There are effective and practical algorithms (see [@ST], [@IntegralPoints], [@Sm] and [@Tz]) to determine the set of integral points, provided the rank $r$ can be computed and a set of generators for $E({\mathbb{Q}})$ found. Given a point $P = (x,y) \in E({\mathbb{Q}})$, the “naive height” of $P$ is defined by writing $x = \frac{a}{b}$ with $a, b \in {\mathbb{Z}}$ with $\gcd(a,b) = 1$ and defining $h(P) = \log \max \{
|a|, |b| \}$. The “canonical height” of $P$ is defined to be $$\hat{h}(P) = \lim_{n \to \infty} \frac{h(2^{n} P)}{4^{n}}.$$
The $L$-function of $E/{\mathbb{Q}}$ is defined to be $$L(E,s) = \sum_{n=1}^{\infty} \frac{a_{n}(E)}{n^{s}}
= \prod_{p} \left(1 - a_{p}(E) p^{-s} + \varepsilon(p) p^{1-2s}\right)^{-1}$$ where $a_{p}(E) = p + 1 - \# E({\mathbb{F}}_{p})$, and $$\varepsilon(p) = \begin{cases}
1 & \text{ if } p \text{ does not divide the conductor of } E,\\
0 & \text{ otherwise. }
\end{cases}$$
The Birch and Swinnerton-Dyer conjecture states that ${{\rm ord}}_{s=1} L(E,s) = {{\rm rank}}(E({\mathbb{Q}}))$, and moreover that $$\lim_{s \to 1} \frac{L(E,s)}{(s-1)^{r}} = \frac{\Omega R(E/{\mathbb{Q}}) \Sha(E/{\mathbb{Q}}) \prod_{p} c_{p}}{|T|^{2}}.$$ In our case, $\Omega$ is twice the real period, $R(E/{\mathbb{Q}})$ is the regulator of $E({\mathbb{Q}})$ computed using the function $\hat{h}$ above, the $c_{p}$ are the Tamagawa numbers, and $\Sha(E/{\mathbb{Q}})$ is the Shafarevich-Tate group. The completed $L$-function $\Lambda(E,s) = N^{s/2} (2 \pi)^{-s} \Gamma(s) L(E,s)$ satisfies the function equation $\Lambda(E,s) = w_{E} \Lambda(E,2-s)$, where $w_{E}$ is the root number of $E$. Note that $w_{E} = 1$ implies that ${{\rm ord}}_{s=1} L(E,s)$ is even, and $w_{E} = -1$ if ${{\rm ord}}_{s=1} L(E,s)$ is odd.
The best partial result in the direction of the Birch and Swinnerton-Dyer conjecture is the following.
\[bsdpartial\] Suppose that $E/{\mathbb{Q}}$ is an elliptic curve and ${{\rm ord}}_{s=1} L(E,s) = 0$ or $1$. Then, ${{\rm ord}}_{s=1} L(E,s) = {{\rm rank}}(E({\mathbb{Q}}))$.
The work of Bump-Friedberg-Hoffstein [@BFH] or Murty-Murty [@MM] is necessary to remove a condition imposed in the work of Gross-Zagier and Kolyvagin.
Proof of Theorem \[family\]
===========================
Motivated by the observation that $8388223 = 127 \cdot 257^{2}$ and $8388225 = 129 \cdot 255^{2}$, we find a parametric family of solutions where $n = a \cdot b^{2}$ and $n+2 = (a+2) \cdot (b-2)^2$.
If we write $(4 + \sqrt{13}) (649+180 \sqrt{13})^{m} = x_{m} + y_{m} \sqrt{13}$ where $x_{m}$ and $y_{m}$ are integers, then $x_{m}^{2} - 13 y_{m}^{2} = 3$ for all $m \geq 0$. We have that $x_{0} = 4$, $x_{1} = 4936$, $y_{0} = 1$, $y_{1} = 1369$ and $$x_{m} = 1298 x_{m-1} - x_{m-2}, y_{m} = 1298 y_{m-1} - y_{m-2}.$$ It is easy to see that $x_{m}$ is periodic modulo $32$ with period $8$ and from this it follows that $x_{8m+7} \equiv 0 \pmod{32}$ for all $m \geq 0$. Set $a_{m} = x_{8m+7}/2$ and $n = 4 a_{m}^{3} - 3 a_{m} - 1$. Then we have $$\begin{aligned}
n &= (a_{m} - 1) (2 a_{m} + 1)^{2},\\
n+1 &= a_{m} (4 a_{m}^{2} - 3) = 13 a_{m} y_{8m+7}^{2}, \text{ and }\\
n+2 &= (a_{m} + 1) (2 a_{m} - 1)^{2}.\\\end{aligned}$$ Since $16 | a_{m}$, $\max \{ {{\rm sfp}}(n), {{\rm sfp}}(n+1), {{\rm sfp}}(n+2) \} \leq \max \{ a_{m} - 1, \frac{13 a_{m}}{16}, a_{m} + 1 \} = a_{m} + 1 < n^{1/3}$.
The polynomial $4x^{3} - 3x$ used in the proof above is the Chebyshev polynomial $T_{3}(x)$. This explains why $T_{3}(x) - 1$ and $T_{3}(x) + 1$ both have a double zero.
Proof of Theorem \[upto150\]
============================
Since there are $92$ squarefree integers $\leq 150$, there are $778688 = 92^{3}$ possibilities for the triple $(a,b,c)$. We first test four things before searching for integral points on $E : y^{2} = x^{3} - (abc)^{2} x$. Suppose that $n = ax^{2}$, $n+1 = by^{2}$ and $n+2 = cz^{2}$ is an integral solution to the system of equations $$\begin{aligned}
\label{eq1} by^{2} - ax^{2} &= 1,\\
\label{eq2} cz^{2} - by^{2} &= 1,\\
\label{eq3} cz^{2} - ax^{2} &= 2.\end{aligned}$$
Greatest common divisor conditions
----------------------------------
If $(x,y,z)$ is an integer solution to , and , then $\gcd(a,b) = 1$, $\gcd(b,c) = 1$ and $\gcd(a,c) = 1 \text{ or } 2$. This reduces the number of triples to consider to $425639$.
Norm equations
--------------
We next check that each of the equations $by^{2} - ax^{2} = 1$, $cz^{2} - by^{2} = 1$, and $cz^{2} - ax^{2} = 2$ has an integer solution. The equation $by^{2} - ax^{2} = 1$ has an integer solution if and only if $Y^{2} - abx^{2} = b$ has an integer solution. This equation has a solution if and only if ${\mathbb{Z}}[\sqrt{ab}]$ has an element of norm $b$. We use Magma’s routine [NormEquation]{} to test this. After these tests have been made, there are $2188$ possibilities for $(a,b,c)$ that remain.
Local solvability of $C$
------------------------
Let $C \subseteq {\mathbb{P}}^{3}$ be the curve defined by the two equations $by^{2} - ax^{2} = w^{2}$, $cz^{2} - by^{2} = w^{2}$. This curve must have a rational point on it in order for there to be a non-trivial solution. We check whether $C$ has points $(x : y : z : w)$ in ${\mathbb{Q}}_{p}$ for all primes $p$ dividing $2abc$. This is done via Magma’s [ IsLocallySolvable]{} routine. This eliminates $244$ possibilities.
Rank of the elliptic curve $E$
------------------------------
Let $E : y^{2} = x^{3} - (abc)^{2} x$. Define a map $M$ from non-trivial solutions to $C$, represented in the form $n = ax^{2}$, $n+1 = by^{2}$ and $n+2 = cz^{2}$ to $E$, given by $$M(x,y,z,n) = ((n+1)(abc), (abc)^{2} xyz).$$ If $n > 0$, then $xyz > 0$ and so $M(x,y,z,n) \in E({\mathbb{Q}})$ is an integral point, and one with a non-zero $y$-coordinate. We now use the following results about the family of elliptic curves $E$ we consider.
A natural number $N$ is called *congruent* if there exists a right triangle with all three sides rational and area $N$.
Consider the elliptic curve $E$ over $\mathbb{Q}$ given by: $$E(\mathbb{Q}): y^{2}=x^{3}-N^{2} x.$$ We have the following result.
\[congthm\] The number $N$ is congruent if and only if the rank of $E$ is positive.
Proposition I.9.17 of [@Koblitz] implies that the only points of finite order on $E$ are $(0,0)$, $(\pm N, 0)$ and the point at infinity. If for some $n > 0$ we have $n = ax^{2}$, $n+1 = by^{2}$ and $n+2 = cz^{2}$, then the point $M(x,y,z,n)$ on $E : Y^{2} = X^{3} - (abc)^{2} X$ is, according to the above, a non-torsion point, and hence $E$ has positive rank, and consequently, by Theorem \[congthm\], $abc$ is a congruent number.
\[Tunnell 1983\] If $N$ is squarefree and odd, and $$\# \left\{ x,y,z \in \mathbb{Z} \vert N =2x^{2}+y^{2}+32z^2 \right\} \ne
\frac{1}{2} \# \left\{ x,y,z \in \mathbb{Z} \vert
N=2x^{2}+y^{2}+8z^2 \right\},$$ then $N$ is not congruent. If $N$ is squarefree and even, and $$\# \left\{ x,y,z \in \mathbb{Z} : \frac{N}{2}=4x^{2}+y^{2}+32z^2 \right\} \ne
\frac{1}{2} \# \left\{ x,y,z \in \mathbb{Z} : \frac{N}{2}=4x^{2}+y^{2}+8z^2 \right\} ,$$ then $N$ is not congruent.
To use Tunnell’s theorem, we compute the generating function for the number of representations of $N$ by $2x^{2} + y^{2} + 8z^{2}$ as $$\sum_{x, y, z \in {\mathbb{Z}}} q^{2x^{2} + y^{2} + 8z^{2}}
= \left(\sum_{x \in {\mathbb{Z}}} q^{2x^{2}}\right) \left(\sum_{y \in {\mathbb{Z}}} q^{y^{2}}\right) \left(\sum_{z \in {\mathbb{Z}}} q^{8z^{2}}\right),$$ as well as the representations of $N$ by $2x^{2}+y^{2}+32z^{2}$, $4x^{2}+y^{2}+8z^{2}$ and $4x^{2}+y^{2}+32z^{2}$. Actually, for our purposes, it suffices to compute $$\sum_{x=-X}^{X} \sum_{y=-Y}^{Y} \sum_{z=-Z}^{Z}
q^{2x^{2} + y^{2} + 8z^{2}},$$ where $X = \lfloor \sqrt{150^{3}/2} \rfloor$, $Y = \lfloor
\sqrt{150^{3}} \rfloor$, and $Z = \lfloor \sqrt{150^{3}/8} \rfloor$.
The use of Tunnell’s theorem rules out $530$ of the $1944$ remaining cases. In the case that the hypothesis of Tunnell’s theorem is false, the Birch and Swinnerton-Dyer conjecture predicts that $E({\mathbb{Q}})$ does have positive rank. In this case, we proceed to the next step.
Computing integral points
-------------------------
Once the program determines that the elliptic curve $E(\mathbb{Q}):
Y^{2} = X^{3} - (abc)^{2} X$ has positive rank, we let Magma attempt to compute the integral points on the curve, using the routine [IntegralPoints]{}, which is based on the method developed in [@IntegralPoints] and [@ST]. If this routine does not succeed within $15$ minutes, which is sufficient time to perform a 4-descent, we abort the computation.
In $1377$ of the $1414$ cases that remain, Magma is able to determine the Mordell-Weil group and determine all of the integral points within $15$ minutes. Once the integral points are determined, we check to see if they are in the image of the map $M(x,y,z,n)$ and if so, whether they correspond to a non-trivial solution.
There are $37$ more difficult cases that remain, and in each case we are able to use other methods to determine the Mordell-Weil group. A table of these cases and the generators of the Mordell-Weil group are given at the page [http://users.wfu.edu/rouseja/MWgens.html]{}.
Of the $37$ cases, there are four cases with root number $-1$ and rank $\leq 3$ for which one point of infinite order has low height. In these, we numerically compute $L'(E,1)$ and show that it is nonzero. Theorem \[bsdpartial\] proves the rank is one in these cases. One of these cases is $a =
139$, $b = 89$ and $c = 109$. In this case, $E$ has conductor $\approx
5.82 \cdot 10^{13}$ and computing $L'(E,1)$ takes about $5$ hours.
In the remaining $33$ cases, rather than use the $8$-descent (the default if the [IntegralPoints]{} command were to keep running), we use the 12-descent routines in Magma (due to Tom Fisher [@Fisher]) to find points of large height. The curve with $a = 137$, $b = 109$ and $c = 101$ has a generator with canonical height $1234$, which is found after searching on the $12$-covers for about $90$ minutes. This approach is successful in all but one case.
Heegner points
--------------
All but one of the $778688$ original cases are handled by the methods of the previous sections. The remaining case is $a = 67$, $b = 131$ and $c = 109$. The curve $E$ has root number $-1$, rank $\leq 1$ and conductor $\approx 2.9 \cdot 10^{13}$. A long computation shows that $L'(E,1) \approx 72.604$. This suggests, assuming the Birch and Swinnerton-Dyer conjecture is true and $\Sha(E/{\mathbb{Q}})$ is trivial, that a generator of the Mordell-Weil group has canonical height about $1692$. Searching for points on the 12-covers up to a height of $3^{42} \cdot 10^{5} \approx 10^{25}$ does not succeed in finding points.
For this final case, we use the method of Elkies described in [@ElkiesHeegner]. This is a variant of the usual Heegner point method and is quite fast on quadratic twists of curves with low conductor. The modular curve $X_{0}(32)$ parametrizes pairs $(E,C)$, where $E$ is an elliptic curve, and $C \subseteq E$ is a cyclic subgroup of order $32$. It is well-known that $X_{0}(32)$ is isomorphic to $y^{2} = x^{3} + 4x$, which in turn has a degree $2$ map to $y^{2} = x^{3} - x$. Finding a rational point on $y^{2} = x^{3} - D^{2} x$ is equivalent to finding a point on $y^{2} = x^{3} - x$ with $x$ and $y/\sqrt{-D}$ both rational. If $E$ is an elliptic curve with complex multiplication whose endomorphism ring $\mathcal{O}$ contains an ideal $I$ so that $\mathcal{O}/I \cong {\mathbb{Z}}/32 {\mathbb{Z}}$, this naturally gives rise to a point on $X_{0}(32)$, which is defined in the ring class field of $\mathcal{O}$. Taking the trace of such a point (in the Mordell-Weil group) gives rise to a point on $X_{0}(32)$ over an imaginary quadratic field.
In [@ElkiesHeegner], Elkies gives a procedure for constructing rational points on $y^{2} = x^{3} - D^{2} x$ with $D \equiv 7 \pmod{8}$. We must handle one case with $D \equiv 5 \pmod{8}$. We let $\mathcal{O} = {\mathbb{Z}}[4 \sqrt{-D}]$. This ring has an ideal $I = \langle 32, 4 + 4 \sqrt{-D} \rangle$ and $\mathcal{O}/I \cong {\mathbb{Z}}/32 {\mathbb{Z}}$. We take a representative ideal $J$ for each element of the class group of $\mathcal{O}$. Thinking of $J$ as a lattice, $\mathbb{C}/IJ$ is an elliptic curve, and $I/IJ$ is a subgroup of order $32$ on the curve. Thus, $(\mathbb{C}/IJ, I/IJ)$ is a Heegner point on $X_{0}(32)$. Via the isomorphism with $E : y^{2} = x^{3} + 4x$, we obtain a collection of points. Adding these points together on $E$ gives a point defined in ${\mathbb{Q}}(\sqrt{-D})$, and eventually a rational point on $y^{2} = x^{3} - D^{2} x$. Running this computation in several simpler cases suggests that the resulting point $P$ on $y^{2} = x^{3} - D^{2} x$ satisfies $mQ = P$ for some rational point $Q$, where $m$ is equal to the number of divisors of $D$. Any point $Q$ must have an $x$-coordinate of the form $x = em^{2}/n^{2}$, where $e$ is a divisor of $2D$. We use this method to minimize the amount of decimal precision needed to compute $P$.
For $D = 67 \cdot 131 \cdot 109$, the class number of $\mathcal{O}$ is $3712$. Given that we expect a point with canonical height $1692$, we compute the Heegner points and their trace in the manner described in [@ElkiesHeegner] using $850$ digits of decimal precision. The result is a point $Q$ with canonical height $1692.698$, and the computation takes about 18 minutes. This point is a generator of the Mordell-Weil group of $E : y^{2} = x^{3} - D^{2} x$. Once the Mordell-Weil group is found, we check that there are no integral points on $E$. This completes the proof of Theorem \[upto150\].
Table of $n$ {#table}
============
The following is a table of all $25$ positive integers $n$ with ${{\rm sfp}}(k) < \max \{ k, 150 \}$ for $k = n$, $n+1$ and $n+2$.
$n$ ${{\rm sfp}}(n)$ ${{\rm sfp}}(n+1)$ ${{\rm sfp}}(n+2)$
----------- ------------------ -------------------- --------------------
$48$ $3$ $1$ $2$
$98$ $2$ $11$ $1$
$124$ $31$ $5$ $14$
$242$ $2$ $3$ $61$
$243$ $3$ $61$ $5$
$342$ $38$ $7$ $86$
$350$ $14$ $39$ $22$
$423$ $47$ $106$ $17$
$475$ $19$ $119$ $53$
$548$ $137$ $61$ $22$
$845$ $5$ $94$ $7$
$846$ $94$ $7$ $53$
$1024$ $1$ $41$ $114$
$1375$ $55$ $86$ $17$
$1519$ $31$ $95$ $1$
$1680$ $105$ $1$ $2$
$3724$ $19$ $149$ $46$
$9800$ $2$ $1$ $58$
$31211$ $59$ $3$ $13$
$32798$ $62$ $39$ $82$
$118579$ $19$ $5$ $141$
$629693$ $53$ $46$ $55$
$1294298$ $122$ $19$ $7$
$8388223$ $127$ $26$ $129$
$9841094$ $134$ $55$ $34$
|
---
author:
- 'H.J. Hoeijmakers'
- 'R.J. de Kok'
- 'I.A.G. Snellen'
- 'M. Brogi [^1]'
- 'J.L. Birkby [^2]'
- 'H. Schwarz'
bibliography:
- 'MMP.bib'
date: 'Received August 12, 2014; accepted October 28, 2014'
title: 'A search for TiO in the optical high-resolution transmission spectrum of HD 209458b: Hindrance due to inaccuracies in the line database'
---
[The spectral signature of an exoplanet can be separated from the spectrum of its host star using high-resolution spectroscopy. During such observations, the radial component of the planet’s orbital velocity changes, resulting in a significant Doppler shift which allows its spectral features to be extracted.]{} [In this work, we aim to detect TiO in the optical transmission spectrum of HD 209458b. Gaseous TiO has been suggested as the cause of the thermal inversion layer invoked to explain the dayside spectrum of this planet.]{} [We used archival data from the 8.2m Subaru Telescope taken with the High Dispersion Spectrograph of a transit of HD209458b in 2002. We created model transmission spectra which include absorption by TiO, and cross-correlated them with the residual spectral data after removal of the dominating stellar absorption features. We subsequently co-added the correlation signal in time, taking into account the change in Doppler shift due to the orbit of the planet.]{} [We detect no significant cross-correlation signal due to TiO, though artificial injection of our template spectra into the data indicates a sensitivity down to a volume mixing ratio of $\sim 10^{-10}$. However, cross-correlating the template spectra with a HARPS spectrum of Barnard’s star yields only a weak wavelength-dependent correlation, even though Barnard’s star is an M4V dwarf which exhibits clear TiO absorption. We infer that the TiO line list poorly match the real positions of TiO lines at spectral resolutions of $\sim 100,000$. Similar line lists are also used in the PHOENIX and Kurucz stellar atmosphere suites and we show that their synthetic M-dwarf spectra also correlate poorly with the HARPS spectra of Barnard’s star and five other M-dwarfs. We conclude that the lack of an accurate TiO line list is currently critically hampering this high-resolution retrieval technique.]{}
Introduction
============
High Dispersion Spectroscopy
----------------------------
Recently, carbon monoxide was identified in the transmission spectrum of the exoplanet HD209458b [@Snellen2010]. During transit, the radial velocity of the hot-Jupiter changes by a few tens of ${\textrm{ km } \textrm{s}^{-1}}$, resulting in a changing Doppler shift which is measurable at a resolution of $R \sim 10^5$. This time-variation of the planet’s spectral signature provides a powerful tool to discriminate between the planet and the overwhelmingly bright host star, because the absorption features in the stellar spectrum are quasi-invariant on these time scales (see Figure \[toymodel\]). Futhermore, because the orbital period of the planet is well known, its radial velocity is known at all times and the Doppler shift of the planets’ absorption features can be targeted specifically.
This technique has also been successfully applied during other parts of of the orbit, where direct thermal emission of the planet is probed. Absorption in the dayside spectra of both transiting and non-transiting hot-Jupiters has been observed: [$\textrm{CO}$ ]{}and [$\textrm{H}_2 \textrm{O}$ ]{}absorption in the atmospheres of $\tau$ Boötes b, HD 189733 b, 51 Peg b and HD 179733 b [@Birkby2013; @Brogi2012; @Brogi2013; @Brogi2014; @DeKok2013; @Lockwood2014; @Rodler2013]. In case of the non-transiting planets, this has allowed for their previously unknown orbital inclinations to be determined: $44.5^{\circ} \pm 1.5^{\circ}$ for $\tau$ Boötes b [@Brogi2012], consistent with the measurement of @Rodler2012, $79.6^{\circ} - 82.2^{\circ}$ for 51 Peg b [@Brogi2013], and $67.7^{\circ} \pm 4.3^{\circ}$ for HD 179733 b [@Brogi2014], which in turn lead to a determination of their masses.
![Toy model of the phase-dependent Doppler-shift of TiO lines along the orbit of HD209458b. The white curves represent TiO-emission features due to the inversion layer (greatly enhanced for visual purposes). During transit around $\phi = 0$, TiO produces slanted absorption lines (black). The black vertical lines are stellar absorption lines which are stable in time. This difference in the behaviour of stellar and planetary features provides a means of contrast between star and planet.[]{data-label="toymodel"}](gfx/toymodel2.png){width="9cm"}
Inversion layer in HD 209458b
-----------------------------
HD 209458b is a well-studied exoplanet. It orbits a solar-type star once every $3.5 {\textrm{ days}}$, located $47 {\textrm{ pc}}$ from the Sun (see Table \[tab:systemparameters\] for the physical properties of the system). Using the Spitzer Space Telescope @Knutson2008 found evidence for $\textrm{H}_2 \textrm{O}$ emission in the IRAC bands at $4.5 {\textrm{ } \upmu \textrm{m}}$ and $5.8 {\textrm{ } \upmu \textrm{m}}$, indicative of a thermal inversion layer in the atmosphere of the planet. A temperature inversion is caused by the absorption of incident starlight in a high-altitude layer of the atmosphere. On Earth, solar UV radiation is absorbed in the oxygen-ozone cycle, effectively absorbing all sunlight short of $350 {\textrm{ nm}}$, heating the atmosphere and forming the stratosphere between $10$ and $50 {\textrm{ km}}$ altitude [see e.g. @Portmann2007 and references therein]. In the atmospheres of hot-Jupiters, similar processes may occur, albeit in different chemical environments.
A number of compounds have been proposed to be capable of causing the inversion layer in the atmosphere of HD 209458b, one of which is TiO [@Hubeny2003; @Burrows2007], a diatomic molecule which is known to cause major absorption features throughout the optical and near-infrared spectra of M-dwarfs. An atmosphere containing a strong optical absorber like TiO would furthermore be consistent with HD 209458b’s low albedo [@Rowe2006]. @Desert2008 reported a tentative detection of TiO absorption in the transmission spectrum of HD 209458b, though more recent broadband studies seem to support a lack of TiO absorption in the transmission spectra of other hot-Jupiters [see e.g. @Huitson2013; @Sing2013; @Gibson2013; @Bento2014]. Also, models of hot-Jupiter atmospheres suggest that maintaining a significant gaseous TiO concentration at high altitudes is difficult due to condensation of TiO either at depth or at the night-side [@Spiegel2009; @Burrows2009; @Fortney2010; @Parmentier2013]. More recent observations of the dayside emission spectrum of HD 209458b, call into question the existence of the inversion layer altogether [@Zellem2014; @Diamond-Lowe2014]. Interestingly, @Stevenson2014 observed molecular absorption features in the near-infrared dayside spectrum of Wasp-12b, possibly attributable to TiO. This contrasts with the optical transmission spectra of the same planet, put forth by @Sing2013. Thus, it is as yet unclear whether TiO is an important chemical component of hot-Jupiter atmospheres.
In this paper we apply a cross-correlation-based retrieval method to search for TiO in the transmission spectrum of HD 209458b at high spectral resolution. Section 2 describes our dataset and the application of the method. Section 3 discusses our results, followed by our conclusions in Section 4.
[0.5]{}[|l|l|l|]{} **Parameter** & **Symbol** & **Value**\
Visible magnitude & $V$ & $7.67 \pm 0.01$\
Distance (pc) & $d$ & $47.4 \pm 1.6$\
Effective temperature (K) & $T_{\textrm{eff}}$ & $6065 \pm 50$\
Luminosity ($L_{\odot}$) & $L_*$ & $1.622^{+0.097}_{-0.10}$\
Mass ($M_{\odot}$) & $M_*$ & $1.119 \pm 0.033$\
Radius ($R_{\odot}$) & $R_*$ & $1.155^{+0.014}_{-0.016}$\
$\textrm{Systemic velocity } (\textrm{km}\textrm{s}^{-1})^{a}$ & $\gamma$ & $-14.7652 \pm 0.0016$\
Metallicity (dex) & $\left[ \textrm{F}/ \textrm{H} \right]$ & $0.00 \pm 0.05$\
Age (Gyr)& & $3.1^{+0.8}_{-0.7}$\
**$\textrm{Orbital period (days)}^{b}$** & $P$ & $3.52474859$\
& & $\pm 0.00000038$\
Semi-major axis (AU) & $a$ & $0.04707^{+0.00046}_{-0.00047}$\
Inclination (deg) & $i$ & $86.71 \pm 0.05$\
$\textrm{Eccentricity}^{c}$ & $e \cos \omega$ & $0.00004 \pm 0.00033$\
Impact parameter & $b$ & $ 0.507 \pm 0.005 $\
$\textrm{Transit central time (HJD)}^{b}$ & $t_c$ & $2,452,826.628521$\
& & $\pm 0.000087$\
$\textrm{Transit duration (min)}^{d}$ & $t_T$ & $183.89 \pm 3.17$\
Mass ($M_J$) & $M_p$ & $0.685^{+0.015}_{-0.014}$\
Radius ($R_J$) & $R_p$ & $1.359^{+0.016}_{-0.019}$\
Density ($\textrm{g }\textrm{cm}^{-3}$) & $\rho_p$ & $0.338^{+0.016}_{-0.014}$\
Equilibrium temperature (K) & $T_{\textrm{eq}}$ & $1449 \pm 12$\
Observations and data reduction
===============================
Subaru data of HD 209458 {#sec:HDS}
------------------------
The data used in this analysis were taken on October 25, 2002 using the High Dispersion Spectrograph (HDS) on the Subaru 8.2m telescope. The observations were performed by @Narita2005 and were originally aimed at detecting absorption of sodium and a number of other atomic species in the transmission spectrum of HD 209458b. These were later used by @Snellen2008 who did indeed detect sodium. The data consist of 31 exposures with a mean exposure time of $500 \sec$ which cover one 3h transit, plus 1.5h and 0.5h of baseline before and after, respectively. The spectrum is observed over 20 echelle orders, covering a wavelength range between $554 {\textrm{ nm}}$ and $682 {\textrm{ nm}}$ at a spectral resolution of $R \sim 45,000$ ($6.7 {\textrm{ km } \textrm{s}^{-1}}$ resolution). Each spectral order is sampled at $0.9 {\textrm{ km } \textrm{s}^{-1}}$ per pixel over 4100 pixels. The basic data reduction (such as bias subtraction, flat fielding and 1D spectral extraction) were performed by @Snellen2008 and are described therein in detail.
HARPS data of M-dwarfs {#sec:HARPS}
----------------------
Our analysis relies on cross-correlating the transmission spectra of HD 209458b with high-resolution template spectra of TiO-bearing models of HD 209458b’s absorption spectrum which rely on line lists published by @Freedman2008 [see sections \[Postproc\] and \[sec:models\]]. The optical spectra of M-dwarfs are dominated by TiO absorption features and are therefore natural benchmarks against which we tested the accuracy of these TiO templates. Many nearby M-dwarfs have been observed extensively at high spectral resolution in order to detect the radial velocity signature induced by potential orbiting companions. We downloaded high resolution spectra of 5 different M-dwarfs from the ESO data archive. These spectra were obtained by the HARPS spectrograph at ESO’s 3.6m La Silla telescope, between $500{\textrm{ nm}}$ and $691{\textrm{ nm}}$ at resolutions of $R=115,000$. Information about the 5 stars is shown in Table \[tab:Mdwarfs\]. Barnard’s star is used throughout this work as the primary example, and is shown in the top panel of Figure \[templates\] before the removal of all broadband features using a high-pass filter (the spectra of the other M-dwarfs were filtered in the same way), and after correcting for its radial velocity of $-110.51 {\textrm{ km } \textrm{s}^{-1}}$ [@Nidever2002].
[0.5]{}[l|l|l|l]{} **Name** & **Spectral type** & **PID (ESO)** &**PI**\
Proxima Centauri & M6.0V & 183.C-0437(A) & Bonfils\
GJ402 & M5.0V & 183.C-0437(A) & Bonfils\
GJ9066 & M4.5V & 072.C-0488(E) & Mayor\
GJ876 & M5.0V & 183.C-0437(A) & Bonfils\
Barnard’s star & M4.0V & 072.C-0488(E) & Mayor\
{width="\linewidth"}
Post-processing and cross-correlation {#Postproc}
-------------------------------------
The analysis of the high dispersion transmission spectra of HD 209458b differs at several points from that of infrared data used in previous high dispersion spectroscopy studies by our group [see @Birkby2013; @Brogi2012; @Brogi2013; @Brogi2014; @DeKok2013], which are dominated by time-dependent telluric absorption features, mainly due to variations in airmass. In the optical, the atmosphere presents no significant absorption features except in some specific wavelength ranges. Rather, the spectrum is dominated by stellar absorption lines. Assuming that the stellar spectrum is stable during the transit, it can be removed by dividing each of the individual spectra in the time-series with the time-averaged stellar spectrum. In principle, this operation does not affect the planet signal because it significantly shifts in wavelength during the observations due to the change in the radial component of the orbital velocity during transit (from $-15 {\textrm{ km } \textrm{s}^{-1}}$ to $+15 {\textrm{ km } \textrm{s}^{-1}}$).
The full sequence of processing steps is summarized below. Intermediate data products are shown in Figure \[arrays\].
1. **Broadband removal and normalization:** To remove strong broadband components such as the stellar continuum and the spectrograph’s blaze function, the brightest exposure of each spectral order was divided into 50 wavelength regions. A 7-th order polynomial was fit through the maximum of each part to obtain the continuum level. This fit was removed from all other exposures through division, removing the strongest broadband features. Subsequently, the continuum of each of the exposures was obtained by dividing each exposure into the same 50 regions as above and again finding the maximum of each part. These continua were interpolated, and removed through division. This acts to remove residual broad-band variations not removed by the 7th-order polynomial fit, and normalizes the continuum of each exposure to unity. At this stage there were still significant residual broad-band variations, which were removed at a later stage (see step 7).
2. **Alignment of spectra:** The spectra slowly drift in wavelength due to the change in radial velocity of the observatory during the observations, and instrumental instability. These misalignments are of the order of a pixel and would adversely affect the removal of the time-average stellar spectrum if left untreated. In each spectral order a dozen strong stellar absorption lines were identified by eye. To these lines in each of the 31 exposures, Gaussian profiles were fitted to obtain the position of each line center. A linear fit through the misalignments of the different absorption lines was used to align the spectra to a common reference frame (a higher order fit was not warranted because the scatter in the determination of the line centroids was too high to identify higher order components).
3. **Wavelength solution:** As HD 209458 is a close solar analogue, we matched our time-averaged spectrum (see next step) of the HD 209458 system to a model solar spectrum (adopted from the Kurucz stellar atmosphere atlas) in a step-wise fashion. The strongest stellar absorption lines were identified and matched by eye. This produced a crude wavelength solution which was subsequently used to identify 30 - 40 less prominent lines in each order. We subsequently fitted Gaussian profiles to the cores of these lines in both the data and the solar template, producing a wavelength solution with a standard deviation of $1.7 \times 10^{-3} {\textrm{ nm}}\pm 0.7 \times 10^{-3}$, corresponding to $0.83 \pm 0.36 {\textrm{ km } \textrm{s}^{-1}}$ or $ \sim 12 \%$ of the spectral resolution. Because the solar spectrum is synthesized at rest, the wavelength solution is calibrated to the rest frame of HD 209458 and the mean radial velocity with respect to the observatory on Earth is thus automatically corrected for. Any time-dependent Doppler shifts had already been corrected in step 2. Therefore, each exposure is now in the rest frame of the host star.
4. **Removal of stellar spectrum:** Each pixel in each spectral order was time-averaged by taking the median of the aligned spectra, which corresponds to taking the median of each of the columns of the array in panel 2 of Figure \[arrays\]. This average contains all time-constant stellar features, but only a small part of the time-varying planet signal, since it is shifted to a different column in each exposure. It is removed from each exposure through division. The width of planetary absorption lines is limited to $6.7{\textrm{ km } \textrm{s}^{-1}}$, which is the resolution of the instrument. Between exposures, the planet signal is shifted by $\sim1.8 {\textrm{ km } \textrm{s}^{-1}}$. Therefore, the planet signal is present in no more than 4 exposures per pixel column. These 4 exposures are part of a median with 27 other exposures which don’t contain the planet signal. Therefore, we expect that up to $10\% - 15\%$ of the planet signal is removed along with the median stellar spectrum. This has no importance for the rest of the analysis, because we measure the significance of any TiO detection by our ability to retrieve injected model spectra, which suffer in the same way from this degradation (see section \[sec:Results\]).
5. **Removal of bad pixels and cosmics:** The resulting residuals contain bad pixels and cosmic rays. All pixels deviating by more than $4 \sigma$ from the mean were reset to the mean, removing all significant bad pixels and cosmic rays. 0.21% of all pixel values were affected.
6. **Normalization by signal to noise:** The noise in the residual spectrum varies with pixel position due to the presence of deep stellar absorption lines up until step 4. We divide each pixel in the residual spectrum by the variance of that pixel value (variance of each column of panel 3, Figure \[Postproc\]) to weigh down such pixels. Low SNR regions would otherwise add a significant amount of noise to the cross-correlation function.
7. **Suppression of broadband residuals:** The residuals feature broadband variations due to imperfect removal of instrumental effects and broad stellar lines. Latent broadband variations were characterized by smoothing each exposure with a Gaussian with a width of 40px ($36 {\textrm{ km } \textrm{s}^{-1}},$) and were divided out of the data. The last panel of Figure \[arrays\] shows the residual of this final processing step, on which the cross-correlation is performed.
8. **Cross correlation:** The planet’s potential absorption features were extracted by cross-correlating the residual spectra with template spectra of TiO (see section \[sec:models\]) across the full wavelength range of the data i.e. between $\sim 554$ and $\sim 682 {\textrm{ nm}}$. The templates were Doppler-shifted to velocities ranging between $\pm 150 {\textrm{ km } \textrm{s}^{-1}}$ relative to the star, in steps of $1{\textrm{ km } \textrm{s}^{-1}}$ (which roughly corresponds to the sampling rate of the data, $0.9{\textrm{ km } \textrm{s}^{-1}}$ per pixel). At the Doppler shift equal to the radial velocity of the planet, the cross-correlation effectively co-adds all of the individual absorption lines of the molecule.
{width="\linewidth"}
Model spectra {#sec:models}
-------------
For the purpose of applying the cross-correlation method to the case of TiO in the atmosphere of HD 209458b (see below), we generated several model transmission spectra of atmospheres containing various concentrations of TiO, at a sampling resolution of $R=240,000$. The line database stems from @Freedman2008, who presented a modified line list based on the one published by @Schwenke1998. The model atmosphere has a temperature-pressure profile as shown in Figure \[models\], and includes [$\textrm{H}_2$ ]{}scattering and molecular absorption due to TiO with volume mixing ratios (VMR) of $10^{-7}$, $10^{-8}$, $10^{-9}$, $10^{-10}$ and $10^{-11}$. The solar photospheric abundance of titanium is $10^{-7.1}$ [@Asplund2006], so the transmission spectrum of TiO is modelled between $\sim 1$ and $\sim 10^{-4}$ times the solar abundance of Ti. These mixing ratios were assumed to be uniform throughout the atmosphere. As such, no chemistry, phase-changes or dynamics were taken into account. The atmosphere was assumed to have a temperature inversion as displayed in the right panel of Figure \[models\] [adopted from @Burrows2007], but because the background continuum originates from a star with an effective temperature of $6065 {\textrm{ K}}$, the assumption of a particular T-P-profile does not significantly influence the template spectrum. To confirm the robustness of the template to a temperature mismatch, we also modelled a TiO-bearing atmosphere at a constant temperature of $3000 {\textrm{ K}}$ and cross-correlated it with the template shown in Figure \[models\]. The result is shown in Figure \[xcorcompare\], and discussed in more detail in section \[sec:evaluation\].
For benchmark purposes (see section \[sec:HARPS\]), we also cross-correlated with synthetic M-dwarf spectra originating from the Kurucz and PHOENIX stellar atmosphere model suites which necessarily include models of the TiO molecule. Both models use a line list calculated by @Plez1998. The Kurucz model is a template spectrum of the M2V dwarf GL411 at a resolution of $R=100,000$ [@Castelli2003] and the PHOENIX model, a template of a generic M4V dwarf ($T_{\textrm{eff}}=3100\textrm{K}$, $\log g = 5.0$, $\textrm{Fe/H} = +0.5$, obtained from the PHOENIX online database as presented by @Husser2013) at a resolution of $R=125,000$. They are shown together with the HARPS spectrum of Barnard’s star in Figure \[templates\].
{width="\linewidth"}
Results and discussion {#sec:Results}
======================
The left panels of Figure \[xcorresult\] show the correlations between the HDS spectra of HD 209458b and our TiO template at five modeled VMRs (see section \[sec:models\]). The correlation coefficients are plotted as a function of the radial velocity to which the template is shifted (horizontal axis) for every individual exposure (i.e. time, vertical axis).
{width="\linewidth"}
To assess the ability of the cross-correlation procedure to retrieve the template spectra from real data, we inject the templates into the data prior to cross-correlation using the known orbital velocity of HD 209458b, between steps 3 and 4 in section \[Postproc\], similar to @Snellen2010. Prior to injection, we convolve the template with a Gaussian to match the spectral resolution of the data $(45,000)$, and convolve again with a box-function to account for the widening of the planetary absorption lines due to the changing Doppler-shift during the $500 \textrm{s}$ exposures. This gives rise to the middle panels in Figure \[xcorresult\]. Injection of a TiO-bearing template at a VMR greater than $10^{-10}$ clearly produces a slanted feature, the gradient of which is a measure of the orbital velocity of the planet at which the template was injected.
Given that the radial velocity of the planet is known at all times, each exposure (i.e. row in Figure \[xcorresult\]) can be shifted to the rest-frame of the planet. This allows for co-addition of all exposures, the result of which is shown in the right panels in Figure \[xcorresult\]. Here, the solid lines represents the correlation of the templates with the residuals of the data, and the dashed line represents the correlation after injection of the template. The signal-to-noise of the observed correlation peak is indicative of the significance at which the template would be retrieved, had it been identically present in the residuals. Evidently, the peak correlation strength decreases with decreasing VMR, hence the VMR at which the correlation peaks at the $3 \sigma$ level, we consider to be the theoretical limiting VMR at which TiO would be retrievable by applying the current analysis on this data.
It is clear from the left panels of Figure \[xcorresult\] that the residuals do not correlate with our TiO templates. This non-detection is significant down to VMRs of $10^{-10}$, as evidenced by the successful retrieval of our templates after having artificially injected them into the data. This sensitivity limit corresponds to $10^{-3}$ times the solar abundance of Ti [@Asplund2006]. The models of the atmosphere of HD209458b are greatly simplified approximates to the real physical environment. Especially the assumption of a uniform VMR across the entire atmosphere is questionable, as the atmosphere is not expected to be homogeneous. However, by extending the absorbing TiO layer uniformly throughout the atmosphere, the TiO absorption at that VMR is maximized. The limiting VMR of $10^{-10}$ is therefore a true lower limit: If TiO would only be present at a VMR of $10^{-10}$ in parts of the atmosphere, it would not be retrieved in this analysis.
Before invoking a plethora of reasons for this non-detection of TiO such as hazes, obscuring clouds, screening by other species, cold-traps or destructive chemistry, it should be noted that the sensitivity of the cross-correlation procedure critically depends on the accuracy of the line positions in the template spectrum. If the wavelength positions of the lines are inaccurate, the cross-correlation procedure may not add their contributions constructively. As shown below, this is a real concern.
Evaluation of template spectra {#sec:evaluation}
------------------------------
Because TiO absorption visibly dominates M-dwarf optical spectra, a high-resolution optical spectrum of Barnard’s star can be used to test the accuracy of our TiO template. Their mutual correlation should be very clear.
![Correlation of our TiO template (at $T\sim1500{\textrm{ K}}$), as shown in Figure \[models\], with the PHOENIX M4V model (green), the Kurucz model of GL411 (red), Barnard’s star’s optical spectrum (light blue) and the TiO template modelled at a temperature of $3000{\textrm{ K}}$ (dark blue). All modelled spectra correlate strongly with our TiO template, while Barnard’s star’s spectrum does not. The high peak correlation between the two templates at different temperatures suggests that the assumption of an inaccurate T-P profile has little effect on the templates.[]{data-label="xcorcompare"}](gfx/template_correlation.png){width="9cm"}
Figure \[xcorcompare\] shows the cross-correlation between our TiO template at different temperatures, the stellar atmosphere models and the HARPS-spectrum of Barnard’s star. The TiO-bearing model of HD 209458b (VMR=$10^{-7}$, T-P profile as shown in Figure \[models\]) shows a correlation peak of 0.35 with the stellar atmosphere models. The cross-correlation between our TiO templates at $3000{\textrm{ K}}$ and $\sim 1500{\textrm{ K}}$ peaks at 0.9. M-dwarf transmission spectra feature a host of other optical absorbers, which decrease the correlation between an M-dwarf spectrum and a pure-TiO template. We regard this as the main reason for the non-prefect correlation between our TiO template and the stellar atmosphere models, rather than a mismatch of the T-P profile associated with the template. In contrast with the stellar atmosphere models, a marginal correlation peak of only $0.1$ is observed between the TiO template and the HARPS spectrum. From this we may conclude that our TiO-bearing model spectra do not accurately reproduce real-world TiO absorption at high resolution.
To test the extent at which stellar atmosphere models and real M-dwarf spectra match each other at high resolution, we cross-correlate Barnard’s star’s spectrum with the spectra of the other 5 M-dwarfs, and with the PHOENIX M4V model. This is shown in Figure \[Mdwarf\_mutual\]. Regardless of differences in effective temperature, metallicity and signal-to-noise, real M-dwarf spectra correlate strongly with each other; with a peak of >0.75 in all cases. However, as is the case with the TiO template, the PHOENIX model correlates at less than 0.15 with all of the M-dwarf spectra. This strengthens our conclusion that the modelled spectrum of TiO is not representative of real TiO. The high correlation between the TiO template and the PHOENIX and Kurucz models, suggests that the TiO line lists have a common source, and the same uncertainties in the line positions seem to have persisted in the works of both @Plez1998 and @Schwenke1998. A visual comparison of the models and the spectrum of Barnard’s star, shows great similarity between all three models at high resolution, but a severe mismatch with the HARPS data (Figure \[resolution-compare\]).
![Correlation of several M-dwarf spectra (see Table \[tab:Mdwarfs\]) with Barnard’s star (solid lines), and with the PHOENIX M4V model (dashed lines). The cross-correlation procedure retrieves the Doppler shifts of the other M-dwarfs, as we did not shift these to their own rest frames, for clarity. All M-dwarfs correlate by more than 0.75 with Barnard’s star, while correlation with the model absorption spectrum never exceeds 0.15.[]{data-label="Mdwarf_mutual"}](gfx/Mdwarf_mutual.pdf){width="\linewidth"}
{width="0.75\linewidth"}
Although the TiO models do not match the HARPS data well, our TiO template is able to partially retrieve the presence of TiO in Barnard’s star spectrum. The extent of the inaccuracies of the models as shown in Figure \[resolution-compare\] near $530{\textrm{ nm}}$, raises the suspicion that the marginal correlation with Barnard’s star is not due to the haphazard alignment of model absorption lines. Instead, it is possible that certain absorption bands are + more accurately than others. To test the hypothesis that inaccuracies are associated with certain absorption bands only, we divide the available $130 {\textrm{ nm}}$ range of our HARPS data into smaller sections and again cross-correlate our model of HD 209458b with each section separately. The result is shown in Figure \[xcor-ladder\].
{width="12cm"}
Over most of the wavelength range, our model does not correlate with the TiO spectrum. However above $630 {\textrm{ nm}}$, the TiO template appears to better match the data, increasing in correlation towards longer wavelengths (a trend which may continue further towards the near infra-red). The four sections shown in in Figure \[xcor-ladder\] correlate maximally at slightly different radial velocity shifts (few km/s), meaning that ensembles of lines are collectively offset. This specifically indicates inaccuracies in the calculation of the energy levels of the TiO molecule.
Alternative template spectra
----------------------------
As shown above, the cross-correlation procedure is very sensitive to species with rich optical absorption spectra in the HDS residuals. As for TiO, the sensitivity reaches down to VMRs below the ppb level. Although we have also shown that the line-lists which lie at the basis of our TiO (and other’s) model spectra display extensive inaccuracies, the template is able to partially retrieve TiO features from Barnard’s star’s optical spectrum at wavelengths longwards of $\sim 640 {\textrm{ nm}}$ and is thus to a certain extent useful to probe the atmosphere of HD 209458b for the presence of TiO. We therefore cross-correlate the residuals with the TiO template at a VMR of $10^{-7}$, using only wavelengths between $640{\textrm{ nm}}- 682{\textrm{ nm}}$, the result of which is shown in the top row of figure \[fig:alternative\_correlation\].
{width="\linewidth"}
Injecting the template into the residuals like before, reveals a theoretical sensitivity of $7.6 \sigma$ (Top right panel of Figure \[fig:alternative\_correlation\]). However, because the line list is only partially accurate at these wavelengths, this is only an upper limit: I.e. if line list was perfect, the presence of TiO in the atmosphere of HD 209458b at a volume-mixing ratio of $10^{-7}$ would have been retrieved with a significance of $7.6 \sigma$. Though since the line-list suffers from uncharacterised inaccuracies, the true limiting VMR is not determined.
Earlier it has been shown (Figure \[xcorcompare\]) that if the temperature-pressure environment of the TiO template does not match that of the data, the correlation is not severely affected. Therefore, although the temperature of the photosphere of Barnard’s star is some $1500\textrm{K}$ higher than the atmosphere of HD 209458b at the terminator region, the TiO absorption present in the spectra of M-dwarfs may also be useful to identify TiO absorption in exoplanet atmosphere at high resolution. As shown before (Figure \[Mdwarf\_mutual\]), the presence of noise in a cross-correlation template does not destroy the cross-correlation at high-resolution, and thus we propose to use the HARPS spectrum of Barnard’s star as a template for cross-correlation with the HDS data of HD 209458b.
Injecting the absorption spectrum of Barnard’s star into the residuals of the HDS data does not provide a physically meaningful sensitivity. Although our template at $\sim 1500\textrm{K}$ correlates significantly with the stellar atmosphere models, the absorption spectrum of an M-dwarf is not a good model for the absorption spectrum of the upper atmosphere of HD 209458b during transit, because the underlying physical parameters are very different: In addition to the different temperature-pressure environment, the presence of other species in an M-dwarf’s photosphere and the different continuum level (HD 209458 is a solar type star) make it difficult to adjust Barnard’s star’s absorption spectrum to form a template from which parameters such as the limiting VMR can be derived as before. Extracting a physically meaningful TiO absorption spectrum from real-world absorption spectra of M-dwarfs would require more work and is beyond the scope of this paper. For now, we only use Barnard’s star’s spectrum to qualitatively probe for the presence of TiO absorption in the HDS dataset. The resulting cross-correlation is shown in Figure \[fig:alternative\_correlation\]. Again, no significant correlation is detected.
Conclusion
==========
We have probed the optical transmission spectrum of HD 209458b for TiO absorption using the time-dependent radial velocity of the planet, which can be spectrally resolved at high resolution. At first glance our null-result seems to be in contrast with the possible detection of TiO by @Desert2008, and in line with more recent observations of hot-Jupiters and models. Injection of model TiO spectra into the data shows that our cross-correlation retrieval method is sensitive to volume mixing ratios down to $10^{-10}$, and is thus theoretically capable of ruling out the presence of TiO at a stringent level.
However, this method relies on the accuracy of template spectra of TiO at resolutions in the order of $R \sim 10^5$. Although such templates are available in the literature, we observe that the line databases used to synthesise these spectra are based on model calculations of the TiO molecule which display widespread inaccuracies. We infer that the energy levels of the TiO molecule are not determined well enough to accurately synthesise absorption spectra at these resolutions. Nonetheless, we have found that the TiO line list used to generate our template spectrum is accurate to some extent at wavelengths between $640 {\textrm{ nm}}$ and $670 {\textrm{ nm}}$, and that the spectra of M-dwarfs can also serve as cross-correlation templates. However, using a high-resolution spectrum of Barnard’s star and restricting our template to the aforementioned wavelength range, do not result in a significant correlation. It would therefore seem that the presence of a large concentration of TiO in the upper atmosphere of HD 209458b at the terminator region is unlikely. However, due to the uncertainties regarding the physical meaning of these template models, such a conclusion is tentative.
The lack of an accurate TiO line list is thus a critical hindrance in the application of this retrieval technique to search for the presence of TiO in the optical transmission spectrum of HD 209458b. Also, since the TiO molecule is among the best characterized molecules in astrophysics, we suspect that the absorption spectra of less common and more complicated species are likely to be even less accurately characterized. Furthermore, inaccuracies in the calculations of line positions do not only severely hinder the application of high-resolution retrieval techniques like the one employed in this work, but they also pervade widely used stellar atmosphere models. Modern-day stellar atmosphere codes employ highly advanced modelling techniques to poor line databases, which in the case of TiO is over a decade old.
Therefore we conclude that an increasing community of observational and theoretical astronomers would benefit from an immediate and extensive effort in improving the quality of absorption spectra of molecules like TiO, either through more sophisticated theoretical calculations or by means of direct measurements in the laboratory.
This work is part of the research programmes PEPSci and VICI 639.043.107, which are financed by the Netherlands Organisation for Scientific Research (NWO), and was also supported by the Leiden Observatory Huygens Fellowship. This work was performed in part under contract with the California Institute of Technology (Caltech)/Jet Propulsion Laboratory (JPL) funded by NASA through the Sagan Fellowship Program executed by the NASA Exoplanet Science Institute. Support for this work was provided in part by NASA through Hubble Fellowship grant HST-HF2-51336 awarded by the Space Telescope Science Institute. Also, this research has made use of the Exoplanet Orbit Database and the Exoplanet Data Explorer at exoplanets.org. Finally, we would like to thank Leigh Fetcher, Jaemin Lee and Patrick Irwin for providing the TiO line-list necessary to perform this analysis.
[^1]: NASA Hubble Fellow
[^2]: NASA Sagan Fellow
|
---
bibliography:
- 'mabib.bib'
---
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---
abstract: 'We consider compound multi-input multi-output (MIMO) wiretap channels where minimal channel state information at the transmitter (CSIT) is assumed. Code construction is given for the special case of isotropic mutual information, which serves as a conservative strategy for general cases. Using the flatness factor for MIMO channels, we propose lattice codes universally achieving the secrecy capacity of compound MIMO wiretap channels up to a constant gap (measured in nats) that is equal to the number of transmit antennas. The proposed approach improves upon existing works on secrecy coding for MIMO wiretap channels from an error probability perspective, and establishes information theoretic security (in fact semantic security). We also give an algebraic construction to reduce the code design complexity, as well as the decoding complexity of the legitimate receiver. Thanks to the algebraic structures of number fields and division algebras, our code construction for compound MIMO wiretap channels can be reduced to that for Gaussian wiretap channels, up to some additional gap to secrecy capacity.'
author:
- 'Antonio Campello, Cong Ling and Jean-Claude Belfiore[^1]'
bibliography:
- 'block\_fading.bib'
title: |
Semantically Secure Lattice Codes\
for Compound MIMO Channels
---
Introduction
============
Due to the open nature of the wireless medium, wireless communications are inherently vulnerable to eavesdropping attacks. Information theoretic security offers additional protection for wireless data, since it only relies on the physical properties of wireless channels, thus representing a competitive/complementary approach to security compared to traditional cryptography.
The fundamental wiretap channel model was first introduced by Wyner [@Wyner75]. In this seminal paper, Wyner defined the secrecy capacity and presented the idea of coset coding to encode both data and random bits to mitigate eavesdropping. In recent years, the quest for the secrecy capacity of many classes of channels has been one of the central topics in wireless communications [@BBRM08; @Bloch_Barros_2011; @Liang_Poor_Shamai_2009; @polarsecrecy; @BargWiretap; @GeneralWiretap; @Tyagi15].
In the information theory community, a commonly used secrecy notion is *strong secrecy*: the mutual information $\mathbb{I}(M;Z^T)$ between the confidential message $M$ and the channel output $Z^T$ should vanish when the code length $T \to \infty$. This common assumption of uniformly distributed messages was relaxed in [@Bellare2012], which considered the concept of *semantic security*: for *any* message distribution, the advantage obtained by an eavesdropper from its received signal vanishes for large block lengths. This notion is motivated by the fact that the plaintext can be fixed and arbitrary.
For the Gaussian wiretap channel, [@OSB] introduced the *secrecy gain* of lattice codes while [@LLBS_12] proposed semantically secure lattice codes based on the lattice Gaussian distribution. To obtain semantic security, the *flatness factor* of a lattice was introduced in [@LLBS_12] as a fundamental criterion which implies that conditional outputs are indistinguishable for different input messages. Using a random coding argument, it was shown that there exist families of lattice codes which are *good for secrecy*, meaning that their flatness factor vanishes. Such families achieve semantic security for rates up to $1/2$ nat from the secrecy capacity.
Compared to the Gaussian wiretap channel, the cases of fading and multi-input multi-output (MIMO) wiretap channels are more technically challenging. The fundamental limits of fading wireless channels with secrecy constraints have been investigated in [@BR06; @LPS07; @BBRM08], where the achievable rates and the secrecy outage probability were given. The secrecy capacity of the MIMO wiretap channel was derived in [@OH11; @Khisti10; @LiuShamai09; @Loyka16Wiretap], assuming full channel state information at the transmitter (CSIT). A code design in this setting was given in [@KhinaKK15] by reducing to scalar Gaussian codes. Although CSIT is sometimes available for the legitimate channel, it is hardly possible that it would be available for the eavesdropping channel. An achievability result was given in [@He14] for varying MIMO wiretap channels with no CSI about the wiretapper, under the condition that the wiretapper has less antennas than the legitimate receiver. Schaefer and Loyka [@SL15] studied the secrecy capacity of the *compound* MIMO wiretap channel, where a transmitter has no knowledge of the realization of the eavesdropping channel, except that it remains fixed during the transmission block and belongs to a given set (the *compound set*). The compound model represents a well-accepted reasonable approach to information theoretic security, which assumes minimal CSIT of the eavesdropping channel [@Liang09; @Bjelakovic2013; @Khisti11]. It can also model a multicast channel with several eavesdroppers, where the transmitter sends information to all legitimate receivers while keeping it secret from all eavesdroppers [@Liang09].
When it comes to code design for fading and MIMO wiretap channels, an error probability criterion was used in several prior works [@BO_TComm; @KHHV; @KOO15], while information theoretic security was only addressed recently with the help of flatness factors [@Hamed; @LVL16]. In particular, [@LVL16] established strong secrecy over MIMO wiretap channels for secrecy rates that are within a constant gap from the secrecy capacity.
Main Contributions
------------------
In this paper, we propose universal codes for compound Gaussian MIMO wiretap channels that complement the recent work reported in [@LVL16]. The key method is discrete Gaussian shaping and a “direct" proof of the universal flatness of the eavesdropper’s lattice. This method is similar to that used in [@Our] to approach the capacity of compound MIMO channels so that the present paper can be considered a companion paper of [@Our] for wiretap channels. Note that [@LVL16] used an “indirect" proof, which was based on an upper bound on the smoothing parameter in terms of the minimum distance of the dual lattice. Besides considering different channel models ([@LVL16] is focused on ergodic stationary channels although it also briefly addresses compound channels), the code constructions of this paper and [@LVL16] are also different: the construction of [@LVL16] is based on a particular sequence of algebraic number fields with increasing degrees, while the algebraic construction of this work combines algebraic number fields of fixed degree and random error correcting codes of increasing lengths. The proposed construction enjoys a significantly smaller gap to secrecy capacity, as well as lower decoding complexity, than [@LVL16], over compound MIMO wiretap channels.
We focus on a compound channel formed by the set of all matrices with the same white-input capacity (see for the precise model). Our lattice coding scheme universally achieves rates (in nats) up to $(C_b - C_e - n_a)^{+}$, where $C_b$ is the capacity of the legitimate channel, $C_e$ is the capacity of the eavesdropper channel, $n_a$ is the number of transmit antennas and $(x)^+ = \max\left\{x,0\right\}$. We believe the $n_a$-nat gap is an artifact of our proof technique based on the flatness factor, which may be removed by improving the flatness-factor method. This is left as an open problem for future research.
For this special compound model, we also show how to extend the analysis in order to accommodate number-of-antenna mismatch, *i.e.*, security is valid *regardless* of the number of antennas at the eavesdropper[^2]. This is a very appealing property, since the number of receive antennas of an eavesdropper may be unknown to the transmitter.
We present two techniques to prove universality of the proposed lattice codes. The first technique is based on Construction A (see Sect. \[ConstructionA\] for the definition) and the usual argument for compound channels [@RootVarayia1968; @Loyka16Compound], which combines fine quantization of the channel space with mismatch encoding for quantized states. This method is a generic proof of the existence of good codes which potentially incurs large blocklengths and performance loss. The second technique is based on algebraic lattices and assumes that the codes admit an “algebraic reduction” and can absorb the channel state. In fact, any code which is good for the *Gaussian* wiretap channel can be coupled with this second technique, as long as it also possesses an additional algebraic structure (for precise terms see Definition \[def:algRed\]). It is inspired by previous works on algebraic reduction for fading and MIMO channels [@GhayaViterboJC], [@LuzziGoldenCode], which are revisited here in terms of secrecy.
Relation to Previous Works
--------------------------
An idea of approaching the secrecy capacity of fading wiretap channels using nested lattice codes was outlined in [@LingISTC16]. Code construction for compound wiretap channels has been further developed in [@CLB-IZS18], which leads to the current work where proof details are given.
The technique for establishing universality of the codes in [@SL15] over the compound MIMO channel with (uncountably) infinite uncertainty sets consists of quantizing the channel space and designing a (random Gaussian) codebook for the quantized channels. This method is similar to the proof of Theorem \[thm:achievableRatesFinal\] in the present paper.
Compound MIMO channels *without* secrecy constraints have been considered earlier in [@RootVarayia1968; @Loyka16Compound; @ShiWesel07] for random codebooks. Lattice codes are shown to achieve the optimal diversity-multiplexing tradeoff for MIMO channels in [@HeshamElGamal04]. More recently it was proven that precoded integer forcing [@Ordentlich15] achieves the compound capacity up to a gap, while algebraic lattice codes [@Our] achieve the compound capacity with ML decoding and a gap to the compound capacity of MIMO channels with reduced decoding complexity. As mentioned above, some techniques (generalized Construction A and channel quantization) of this paper are similar to those used in [@Our].
Organization
------------
The technical content of this paper is organized as follows. In Section \[sec:preliminaries\] we discuss the main problem and notions of security. In Section \[sec:correlated\], we introduce the main notation on lattices and discrete Gaussians, stating generalized versions of known results for correlated Gaussian distributions. In Section \[sec:IV\] we give an overview of the main coding scheme and analyze the information leakage and reliability. The proof of universality, however, is postponed until Section \[sec:universallyFlat\], where we show that lattice codes can achieve vanishing information leakage under semantic security through the two aforementioned techniques. Section \[sec:discussion\] concludes the paper with a discussion of other compound models and future work.
Notation
--------
Matrices and column vectors are denoted by upper and lowercase boldface letters, respectively. For a matrix $\mathbf{A}$, its Hermitian transpose, inverse, determinant and trace are denoted by $\mathbf{A}^{\dag}$, $\mathbf{A}^{-1}$, $|\mathbf{A}|$ and $\mathrm{tr}(\mathbf{A})$, respectively. We denote the Frobenius norm of a matrix by $\left\|\mathbf{A} \right\|_F \triangleq \sqrt{\mbox{tr}(\mathbf{A}^\dagger \mathbf{A})}$ and the spectral norm (*i.e.*, $2$-norm) by $\left\|\mathbf{A} \right\| \triangleq \sqrt{\lambda_1}$, where $\lambda_1$ is the largest eigenvalue of $\mathbf{A}^{\dagger} \mathbf{A}$. $\mathbf{I}$ denotes the identity matrix. We write $\mathbf{A}\succeq\mathbf{0}$ for a symmetric matrix $\mathbf{A}$ if it is positive semi-definite. Similarly, we write $\mathbf{A}\succeq\mathbf{B}$ if $(\mathbf{A}-\mathbf{B})\succeq\mathbf{0}$. We use the standard asymptotic notation $f\left( x\right) =O\left( g\left( x\right) \right) $ when $\lim\sup_{x\rightarrow
\infty}|f(x)|/g(x) < \infty$ , $f\left( x\right) =o\left( g\left( x\right) \right) $ when $\lim_{x\rightarrow
\infty}f(x)/g(x) =0$, $f\left( x\right) =\Omega\left( g\left( x\right) \right) $ when $\lim\inf_{x\rightarrow
\infty}f(x)/g(x) > 0$, and $f\left( x\right) =\omega\left( g\left( x\right) \right) $ when $\lim_{x\rightarrow
\infty}f(x)/g(x) =\infty$. Finally, in this paper, the logarithm is taken with respect to base $e$ (where $e$ is the Neper number) and information is measured in nats.
Problem Statement {#sec:preliminaries}
=================
Consider the following wiretap model. A transmitter (Alice) sends information through a MIMO channel to a legitimate receiver (Bob) and is eavesdropped by an illegitimate user (Eve). The channel equations for Bob and Eve read: $$\label{eq:block-fading}\begin{split}
\underbrace{\mathbf{Y}_b}_{n_b\times T}&=\underbrace{\mathbf{H}_b}_{n_b\times n_a}\underbrace{\mathbf{X}}_{n_a \times T}+\underbrace{\mathbf{W}_b}_{n_b\times T} \\ \underbrace{\mathbf{Y}_e}_{n_e\times T}&=\underbrace{\mathbf{H}_e}_{n_e\times n_a}\underbrace{\mathbf{X}}_{n_a \times T}+\underbrace{\mathbf{W}_e}_{n_e\times T},
\end{split}$$ where $n_a$ is the number of transmit antennas, $n_{b}$ ($n_{e}$, resp.) is the number of receive antennas for Bob (Eve, resp.), $T$ is the coherence time, and $\mathbf{W}_b$ ($\mathbf{W}_e$, resp.) has circularly symmetric complex Gaussian i.i.d. entries with variance $\sigma_b^2$ ($\sigma_e^2$, resp.) per complex dimension. We can vectorize in a natural way: $$\label{eq:block-fading-vec}\begin{split}
\underbrace{\mathbf{y}_b}_{n_bT \times 1}&=\underbrace{\mathcal{H}_b}_{n_bT \times n_aT}\underbrace{\mathbf{x}}_{n_aT \times 1}+\underbrace{\mathbf{w}_b}_{n_bT\times1} \\ \underbrace{\mathbf{y}_e}_{n_eT \times 1}&=\underbrace{\mathcal{H}_e}_{n_eT \times n_aT}\underbrace{\mathbf{x}}_{n_aT \times 1}+\underbrace{\mathbf{w}_e}_{n_eT\times1},
\end{split}$$ where $\mathcal{H}_b$ and $\mathcal{H}_e$ are the block diagonal matrices $$\mathcal{H}_{b} = \mathbf{I}_{T} \otimes \mathbf{H}_b = \left(\begin{array}{cccc} {\mathbf{H}}_b & & & \\ & {\mathbf{H}}_b & & \\ & & \ddots & \\ & & & {\mathbf{H}}_b \end{array}\right),$$ $$\mathcal{H}_{e} = \mathbf{I}_{T} \otimes \mathbf{H}_e = \left(\begin{array}{cccc} {\mathbf{H}}_e & & & \\ & {\mathbf{H}}_e & & \\ & & \ddots & \\ & & & {\mathbf{H}}_e \end{array}\right).$$ For convenience, we denote the transmit signal-to-noise ratio (SNR) in Bob and Eve’s channels by $$\rho_b \triangleq \frac{P}{\sigma_b^2} \mbox{ and } \rho_e \triangleq \frac{P}{\sigma_e^2},$$ respectively, where $P$ is the power constraint, *i.e.*, the transmitted signal satisfies $\mathbb{E}[\mathbf{x}^{\dagger}\mathbf{x}]\leq n_a T P$.
We assume that the channel realizations $(\mathbf{H}_b,\mathbf{H}_e)$ are *unknown* to Alice but belong to a compound set $\mathcal{S}=\mathcal{S}_b \times \mathcal{S}_e \in \mathbb{C}^{n_b \times n_a} \times \mathbb{C}^{n_e \times n_a}$. From the security perspective, we further make the conservative assumption that Eve knows both $\mathbf{H}_b$ and $\mathbf{H}_e$. Under this general scenario the (strong) secrecy capacity is bounded by [@SL15]: $$C_s \geq \max_{{\mathbf{R}}} \min_{{\mathbf{H}}_b,{\mathbf{H}}_e} \left(\log |{\mathbf{I}}+\sigma_b^{-1}{\mathbf{H}}_b^\dagger{\mathbf{H}}_b {\mathbf{R}}| - \log \left|{\mathbf{I}}+\sigma_e^{-1}{\mathbf{H}}_e^\dagger{\mathbf{H}}_e {\mathbf{R}}\right| \right)^{+},$$ where the minimum is over all realizations in $\mathcal{S}$ and the maximum over the matrices $\mathbf{R} \succeq 0$ such that $\text{tr}(\mathbf{R}) \leq n_a P$. Suppose that $\mathcal{S}_b$ and $\mathcal{S}_e$ are the set of channels with the same isotropic mutual information, *i.e.*, $$\begin{split}
\mathcal{S}_b &= \left\{{\mathbf{H}}_b \in \mathbb{C}^{n_b \times n_a } : |{\mathbf{I}}+\rho_b {\mathbf{H}}_b^\dagger{\mathbf{H}}_b |= e^{C_b} \right\}, \\ \mathcal{S}_e &= \left\{{\mathbf{H}}_e \in \mathbb{C}^{n_e \times n_a } :\left|{\mathbf{I}}+\rho_e{\mathbf{H}}_e^\dagger{\mathbf{H}}_e \right|= e^{C_e} \right\},
\end{split}
\label{eq:BandE}$$ for fixed $C_b,C_e\geq0$. In this case, the bound gives $C_s \geq (C_b - C_e)^{+}$. The worst case is achieved by taking a specific “isotropic” realization $\mathbf{H}_b^\dagger \mathbf{H}_b = \alpha_b {\mathbf{I}}$, $\mathbf{H}_e^\dagger \mathbf{H}_e = \alpha_e {\mathbf{I}}$, where $\alpha_b$ and $\alpha_e$ are such that $\mathbf{H}_b$ and $\mathbf{H}_e$ belong to $\mathcal{S}_b$ and $\mathcal{S}_e$, respectively. From this we conclude that $C_s = C_b-C_e$. The goal of this paper is to construct universal lattice codes that approach the secrecy capacity $C_s$ with *semantic* security. As a corollary, the semantic security capacity and the strong secrecy capacity of the compound set $\mathcal{S}_b \times \mathcal{S}_e$ coincide.
A practical motivation to consider the compound model is the following. Firstly, notice that the secrecy capacity is the same if we replace the equality in the definition of $\mathcal{S}_b$ and $\mathcal{S}_e$ with upper/lower bounds; more precisely the secrecy capacity of the channel with compound set $\overline{\mathcal{S}}_e \times \overline{\mathcal{S}}_b$, where $$\begin{split}
\overline{\mathcal{S}}_b &= \left\{{\mathbf{H}}_b \in \mathbb{C}^{n_b \times n_a } :|{\mathbf{I}}+\rho_b {\mathbf{H}}_b^\dagger{\mathbf{H}}_b |\geq e^{C_b} \right\},\\ \overline{\mathcal{S}}_e &= \left\{{\mathbf{H}}_e \in \mathbb{C}^{n_e \times n_a } :|{\mathbf{I}}+\rho_e{\mathbf{H}}_e^\dagger{\mathbf{H}}_e |\leq e^{C_e} \right\},
\end{split}
\label{eq:BandEbar}$$ is the same as for $\mathcal{S}_e \times \mathcal{S}_b$. Note that the sets $\mathcal{S}_b$, $\mathcal{S}_e$ and $\overline{\mathcal{S}}_e$ are compact whereas $\overline{\mathcal{S}}_b$ is not. In other words, universal codes are robust, in the sense that only a lower bound on the legitimate channel capacity and an upper bound on the eavesdropper channel are needed. From the security perspective, this is a safe strategy in the scenario where the capacities are not known precisely. Even if Bob and Eve’s channels are random, an acceptable secrecy-outage probability can be guaranteed by setting $C_b$ and $C_e$ properly. Then, the problem still boils down to the design of universal codes for the compound model .
Notions of Security
-------------------
A secrecy code for the compound MIMO channel can be formally defined as follows.
An $(R,R^\prime,T)$-secrecy code for a compound MIMO channel with set $\mathcal{S} = \mathcal{S}_b \times \mathcal{S}_e$ consists of
- A set of messages $\mathcal{M}_T = \left\{1,\ldots,e^{TR}\right\}$ (the secret message rate $R$ is measured in nats and $e^{TR}$ is assumed to be an integer for convenience).
- An auxiliary (not necessarily uniform) source $U$ taking values in $\mathcal{U}_T$ with entropy $R^\prime=H(U)$.
- A stochastic encoding function $f_T: \mathcal{M}_T \times \mathcal{U}_T \to \mathbb{C}^{n_a\times T}$ satisfying the power constraint $$\frac{1}{T}\text{{\upshape tr}}\left(\mathbb{E}\left[f_T(m,U)^{\dagger} f_T(m,U)]\right]\right) \leq n_a P,$$ for any $m \in \mathcal{M}_T$.
- A decoding function $g_T: \mathcal{S}_b \times \mathbb{R}^{n_b \times T} \to \mathcal{M}_T$ with output $\hat{m} = g_T(s_b,\mathbf{Y}_b)$.
A pair $(s_b,s_e) \in\ \mathcal{S}_b \times \mathcal{S}_e$ is referred to as a *channel state* (or *channel realization*). To ensure reliability for all channel states we require a sequence of codes whose error probability for message $M$ vanishes uniformly: $$\mathbb{P}_{\text{err}|M} \triangleq \mathbb{P}(\hat{M} \neq M) \to 0, \forall s_b \in \mathcal{S}_b\mbox{, as } T \to \infty.
\label{eq:Reliability1}$$ Let $p_M$ be a message distribution over $\mathcal{M}_T$. For strong secrecy, $p_M$ is usually assumed to be uniform; however, this assumption is not sufficient from the viewpoint of semantic security, which is the standard notion of security in modern cryptography. Let $\mathbf{Y}_{e}$ be the output of the channel to the eavesdropper, who is omniscient. The following security notions are adapted from [@Bellare2012; @LLBS_12] and should hold in the limit $T \to \infty$:
- *Mutual Information Security (<span style="font-variant:small-caps;">MIS</span>)*: Unnormalized mutual information $$\label{eq:MIS}
\mathbb{I}(M; \mathbf{Y}_{e}) \to 0$$ for any message distribution $p_M$ and $\mbox{for \textit{all} }s_e \in \mathcal{S}_e$.
- *Semantic Security (<span style="font-variant:small-caps;">SemanticS</span>)*: Adversary’s advantage $$\sup_{f,p_M}\left\{ \max_{m'} \mathbb{P}(f(M) = f(m') | \mathbf{Y}_e) - \max_{m''} \mathbb{P}(f(M) = f(m'')) \right\} \to 0$$ for any function $f$ from $M$ to finite sequences of bits in $\left\{0,1\right\}^*$, and *all* $s_e \in \mathcal{S}_e$.
- *Distinguishing Security (<span style="font-variant:small-caps;">DistS</span>)*: The maximum variational distance $$\max_{m',m''\in \mathcal{M}_T} \mathbb{V}(p_{\mathbf{Y}_e | m'},p_{\mathbf{Y}_e | m''}) \to 0 \mbox{ for all } s_e \in \mathcal{S}_e.$$
We stress that all three notions require a sequence of codes to be *universally* secure for all channel states. Treating these notions as classes, we have the inclusions $\textsc{MIS}\subseteq\textsc{SemanticS}=\textsc{DistS}$, *i.e.*, the sequences of codes satisfying $\textsc{DistS}$ are the same as the ones satisfying $\textsc{SemanticS}$ and also include those satisfying $\textsc{MIS}$ [@LLBS_12 Prop. 1]. Moreover, if in the above notions we require that the convergence rate is $o(1/T)$, the three sets coincide. We thus define universally secure codes as follows.
\[def:universal\] A sequence of codes of rate $R$ is universally secure for the MIMO wiretap channel if for all $(s_b,s_e) \in \mathcal{S}$, it satisfies the reliability condition and mutual information security uniformly.
Then, semantic security follows as a corollary, which is a direct consequence of established relations between $\textsc{MIS}$ and $\textsc{SemanticS}$ [@Bellare2012]:
The sequence of codes given in Definition \[def:universal\] is semantically secure for the compound MIMO wiretap channel.
In what follows we proceed to construct universally secure codes for the MIMO wiretap channel using lattice coset codes.
Correlated Discrete Gaussian Distributions {#sec:correlated}
==========================================
In this subsection, we exhibit essential results and concepts for the definition and analysis of our lattice coding scheme.
Preliminary Lattice Definitions
-------------------------------
A (complex) lattice $\Lambda$ with generator matrix $\mathbf{B}_c \in \mathbb{C}^{n \times 2n}$ is a discrete additive subgroup of $\mathbb{C}^n$ given by $$\Lambda = \mathcal{L}(\mathbf{B}_c) = \left\{ \mathbf{B}_c {\mathbf{x}}: {\mathbf{x}}\in \mathbb{Z}^{2n} \right\}.
\label{eq:complexLattice}$$ A complex lattice has an equivalent real lattice generated by the matrix obtained by stacking real and imaginary parts of matrix $\mathbf{B}_c$: $$\mathbf{B}_r = \left(\begin{array}{c} \Re(\mathbf{B}_c) \\ \Im(\mathbf{B}_c)\end{array} \right) \in \mathbb{R}^{2n \times 2n}.$$
A *fundamental region* $\mathcal{R}(\Lambda)$ for $\Lambda$ is any interior-disjoint region that tiles $\mathbb{C}^n$ through translates by vectors of $\Lambda$. For any ${\mathbf{y}}, {\mathbf{x}}\in \mathbb{C}^n$ we say that ${\mathbf{y}}= {\mathbf{x}}\pmod \Lambda$ iff ${\mathbf{y}}- {\mathbf{x}}\in \Lambda$. By convention, we fix a fundamental region and denote by ${\mathbf{y}}\pmod \Lambda$ the unique representative ${\mathbf{x}}\in \mathcal{R}(\Lambda)$ such that ${\mathbf{y}}= {\mathbf{x}}\pmod \Lambda$. The volume of $\Lambda$ is defined as the volume of a fundamental region for the equivalent real lattice, given by $V(\Lambda) = |{\mathbf{B}}_r|.$
Throughout this text, for convenience, we also use the matrix-notation of lattice points. If $\Lambda \subset \mathbb{C}^{nT}$ is a full-rank lattice, the matrix form representation of ${\mathbf{x}}=(x_1,\ldots,x_{nT}) \in \Lambda$ is $$\mathbf{X}= \left( \begin{array}{cccc} x_1 & x_2 & \cdots & x_T \\ x_{T+1} & x_{T+2} & \cdots & x_{2T} \\
x_{2T+1} & x_{2T+2} & \cdots & x_{3T} \\ \vdots & \vdots & \ddots & \vdots \\ x_{(n-1)T+1} & x_{(n-1)T+2} & \cdots & x_{nT}\end{array} \right).$$ The *dual* $\Lambda^*$ of a complex lattice is defined as $$\Lambda^* = \left\{{\mathbf{x}}\in \mathbb{C}^n : \Re\left\langle {\mathbf{x}}, {\mathbf{y}}\right\rangle \in \mathbb{Z} \mbox{ for all } {\mathbf{y}}\in \Lambda\right\}.$$
The Flatness Factor
-------------------
The flatness factor has been introduced in [@LLBS_12], and will be used here to bound the information leakage of information transmission of our coding scheme.
The p.d.f. of the complex Gaussian centered at $\mathbf{c} \in \mathbb{C}^n$ is defined as $$f_{\sigma,\mathbf{c}}({\mathbf{x}}) = \frac{1}{(\pi \sigma^2)^n} e^{-({\mathbf{x}}-\mathbf{c})^{\dagger} ({\mathbf{x}}-\mathbf{c})/\sigma^2}.$$ We write $f_{\sigma,\Lambda}({\mathbf{x}})$ for the sum of $f_{\sigma,\mathbf{c}}({\mathbf{x}})$ over $\mathbf{c} \in \Lambda$. The *flatness factor* of a lattice quantifies the distance between $f_{\sigma,\Lambda}({\mathbf{x}})$ and the uniform distribution over $\mathcal{R}(\Lambda)$ and, as we will see, bounds the amount of leaked information in a lattice coding scheme.
\[Flatness factor for spherical Gaussian distributions\] For a lattice $\Lambda$ and a parameter $\sigma$, the flatness factor is defined by: $$\epsilon_{\Lambda}(\sigma) \triangleq \max_{\mathbf{x} \in
\mathcal{R}(\Lambda)}|{
V(\Lambda)f_{\sigma,\Lambda}(\mathbf{x})-1}|$$ where $\mathcal{R}(\Lambda)$ is a fundamental region of $\Lambda$.
For a complex lattice $\Lambda \subset \mathbb{C}^n$, let $\gamma_{\Lambda}(\sigma) = \frac{
V(\Lambda)^{\frac{1}{n}}}{\sigma^2}$ be the volume-to-noise ratio (VNR). We recall the formulas of the flatness factor and smoothing parameter, adapted to complex lattices. The flatness factor can be written as [@LLBS_12 Prop. 2]: $$\epsilon_{\Lambda}(\sigma) = \left(\frac{\gamma_{\Lambda}(\sigma)}{{\pi}}\right)^{{n}}{
\Theta_{\Lambda}\left({\frac{1}{\pi\sigma^2}}\right)}-1
=\Theta_{\Lambda^*}\left({{\pi\sigma^2}}\right)-1,
\label{flatness-dual-lattice}$$ where $\Theta_{\Lambda}$ is the *theta series* of the lattice $\Lambda$.
\[Smoothing parameter [@MR07]\] \[def:smooth\] For a lattice $\Lambda$ and $\varepsilon > 0$, the smoothing parameter is defined by the function $\eta_{\varepsilon}(\Lambda)=\sqrt{2\pi}\sigma$, for the smallest $\sigma>0$ such that $\sum_{{\bm{\lambda}^*}\in \Lambda^* \setminus \{\mathbf{0}\}} e^{-\pi^2
\sigma^2\|{ \bm{\lambda}^*}\|^2}\leq \varepsilon$.
When we have a correlated Gaussian distribution with covariance matrix ${\mathbf{\Sigma}}$ $$\label{eq:corr-Gauss}
f_{\sqrt{{\mathbf{\Sigma}}}, \mathbf{c}}(\mathbf{x}) = \frac{1}{\pi^{n}|{\mathbf{\Sigma}}|} \exp\left\{-(\mathbf{x}-\mathbf{c})^T{\mathbf{\Sigma}}^{-1}(\mathbf{x}-\mathbf{c})\right\},$$ the flatness factor is similarly defined.
\[Flatness factor for correlated Gaussian distributions\] $$\epsilon_{\Lambda}(\sqrt{{\mathbf{\Sigma}}}) \triangleq \max_{\mathbf{x} \in
\mathcal{R}(\Lambda)}|{
V(\Lambda)f_{\sqrt{{\mathbf{\Sigma}}},\Lambda}(\mathbf{x})-1}|$$ where $\mathcal{R}(\Lambda)$ is a fundamental region of $\Lambda$.
The usual smoothing parameter in Definition \[def:smooth\] is a scalar. To extend its definition to matrices, we say $\sqrt{2\pi{\mathbf{\Sigma}}} \succeq \eta_{\varepsilon}(\Lambda)$ if $\epsilon_{\Lambda}(\sqrt{{\mathbf{\Sigma}}}) \leq \varepsilon$. This induces a partial order because $\epsilon_{\Lambda}(\sqrt{{\mathbf{\Sigma}}_1}) \leq \epsilon_{\Lambda}(\sqrt{{\mathbf{\Sigma}}_2})$ if ${\mathbf{\Sigma}}_1 \succeq {\mathbf{\Sigma}}_2$.
When $\mathbf{c} = 0$ we ignore the index and write $f_{\sqrt{{\mathbf{\Sigma}}}, \mathbf{0}}(\mathbf{x}) = f_{\sqrt{{\mathbf{\Sigma}}}}(\mathbf{x})$. For a covariance matrix ${\mathbf{\Sigma}}$ we define the generalized-volume-to-noise ratio as $$\gamma_{\Lambda}(\sqrt{{\mathbf{\Sigma}}}) = \frac{V(\Lambda)^{1/n}}{|{\mathbf{\Sigma}}|^{1/n}}.$$ Clearly, the effect of correlation on the flatness factor may be absorbed if we use a new lattice $\frac{ \sqrt{{\mathbf{\Sigma}}}} {\sigma} \cdot \Lambda$, *i.e.*, $\epsilon_{\Lambda}({\sigma}) = \epsilon_{\frac{ \sqrt{{\mathbf{\Sigma}}}}{\sigma} \cdot \Lambda}(\sqrt{{\mathbf{\Sigma}}})$. From this, and from the expression of the flatness factor, we have $$\begin{aligned}
\epsilon_{\Lambda}(\sqrt{\mathbf{{\mathbf{\Sigma}}}}) &=& \frac{V(\Lambda)}{\pi^n |{\mathbf{\Sigma}}|} \sum_{{\bm{\lambda}} \in \mathbf{\Lambda}} e^{-\bm{\lambda}^\dagger {\mathbf{\Sigma}}^{-1} \bm{\lambda}} - 1 \\
&=& \left(\frac{\gamma_{\sqrt{{\mathbf{\Sigma}}^{-1}} \Lambda}(\sigma^2)}{\pi}\right)^{n} \Theta_{\sqrt{{\mathbf{\Sigma}}^{-1}} \Lambda}\left(\frac{1}{\pi\sigma^2}\right)-1.\end{aligned}$$ In our applications, the matrix ${\mathbf{\Sigma}}$ will be determined by the channel realization . Figure \[fig:Flatness\] shows the effect of fading on the lattice Gaussian function. A function which is flat over the Gaussian channel (corresponding to ${\mathbf{\Sigma}}= {\mathbf{I}}$) need not be flat for a channel in deep fading (corresponding to an ill-conditioned ${\mathbf{\Sigma}}$), in which case an eavesdropper could clearly distinguish one dimension of the signal.
The Discrete Gaussian Distribution
----------------------------------
In order to define our coding scheme, we need a last element, which is the distribution of the sent signals. To this end, we define the *discrete Gaussian distribution* $\mathcal{D}_{\Lambda+\mathbf{c},\sqrt{{\mathbf{\Sigma}}}}$ as the distribution assuming values on $\Lambda+ \mathbf{c}$, such that the probability of each point $\bm{\lambda} + \mathbf{c}$ is given by $$\mathcal{D}_{\Lambda+\mathbf{c},\sqrt{{\mathbf{\Sigma}}}}(\bm{\lambda} + \mathbf{c}) = \frac{f_{\sqrt{{\mathbf{\Sigma}}}}({\bm\lambda} + \mathbf{c})}{f_{\sqrt{{\mathbf{\Sigma}}},\Lambda}(\mathbf{c})}.$$
Its relation to the continuous Gaussian distribution can be shown via the smoothing parameter or the flatness factor. For instance, a vanishing flatness factor guarantees that the power per dimension of $\mathcal{D}_{\Lambda+\mathbf{c},\sigma {\mathbf{I}}}$ is approximately $\sigma^2$ [@LLBS_12 Lemma 6].
The next proposition says that the sum of a continuous Gaussian and a discrete Gaussian is approximately a continuous Gaussian, provided that the flatness factor is small. The proof can be found in [@LVL16 Appendix I-A]:
\[lem:product\] Given $\mathbf{x}_1$ sampled from the discrete Gaussian distribution $D_{\Lambda+\mathbf{c},\sqrt{{\mathbf{\Sigma}}_1}}$ and $\mathbf{x}_2$ sampled from the continuous Gaussian distribution $f_{\sqrt{{\mathbf{\Sigma}}_2}}$. Let ${\mathbf{\Sigma}}_0 = {\mathbf{\Sigma}}_1 + {\mathbf{\Sigma}}_2$ and let ${\mathbf{\Sigma}}_3^{-1} = {\mathbf{\Sigma}}_1^{-1} +{\mathbf{\Sigma}}_2^{-1}$. If $\sqrt{{\mathbf{\Sigma}}_3} \succeq \eta_{\varepsilon}(\Lambda)$ for $\varepsilon \leq \frac{1}{2}$, then the distribution $g$ of $\mathbf{x}=\mathbf{x}_1+\mathbf{x}_2$ is close to $f_{\sqrt{{\mathbf{\Sigma}}_0}}$: $$g(\mathbf{x}) \in f_{\sqrt{{\mathbf{\Sigma}}_0}}(\mathbf{x})\left[ {1-4\varepsilon}, 1+4\varepsilon \right].$$
Coding Scheme and Analysis {#sec:IV}
==========================
Overview {#sec:overview}
--------
Given a pair of nested lattices $\Lambda_e^T \subset \Lambda_{b}^T \subset \mathbb{C}^{n_aT}$ such that $$\frac{1}{T} \log |\Lambda_b^T/\Lambda_e^T| = R,$$ the transmitter maps a message $m$ to a coset of $\Lambda_e^T$ in quotient $\Lambda_b^T/\Lambda_e^T$, then samples a point from that coset. Concretely, one can use a a one-to-one map $\phi$ such that $\phi(m) = \bm{\lambda}_m$, where $\bm{\lambda}_m$ is a representative of the coset and then samples the signal $\mathbf{x} \sim \mathcal{D}_{\Lambda_e^T+\bm{\lambda}_m,\sigma_s},$ broadcasting it to the channels. A block diagram for the transmission until the front-end receivers Bob and Eve is depicted in Figure \[fig:wiretapa\].
In order to find pairs of sequences of nested lattices $\Lambda_b^T$ and $\Lambda_e^T$ we employ constructions of lattices from error-correcting codes. The analysis and full construction are explained in Section \[sec:universallyFlat\]. Essentially, the lattice $\Lambda_b^T$ controls reliability and has to be chosen in such a way that it is *universally good* for the legitimate compound channel. The lattice $\Lambda_e^T$ controls the information leakage to the eavesdropper, and has to be chosen in such a way that the flatness factor vanishes universally for any eavesdropper realization (universally good for secrecy). The main result of this section is the following theorem, stating the existence of schemes with vanishing probability of error and vanishing information leakage for all pairs of realizations in the compound set $\mathcal{S}_b \times \mathcal{S}_e$.
\
There exists a sequence of pairs of nested lattices $(\Lambda_b^T,\Lambda_e^T)_{T=1}^{\infty}$, $\Lambda_b^T \subset\Lambda_e^T \subset \mathbb{C}^{n_aT}$ such that as $T\to \infty$, the lattice coding scheme *universally* achieves any secrecy rate $$R < (C_b - C_e - n_a)^{+}.$$ \[thm:achievableRatesFinal\]
Moreover, we show that both the probability of error and information leakage in Theorem \[thm:achievableRatesFinal\] vanishes uniformly for all realizations.
The Eavesdropper Channel: Security
----------------------------------
For a *fixed* realization $\mathbf{H}_e$, the key element for bounding the information leakage is the following lemma [@LLBS_12 Lem 2]:
Suppose that there exists a probability density function $q$ taking values in $\mathbb{C}^{n_e\times T}$ such that $\mathbb{V}(p_{\mathbf{Y}_e|m}, q_{\mathbf{Y}_e}) \leq \varepsilon_T$ for all $m \in \mathcal{M}_T$. Then, for all message distributions, the information leakage is bounded as: $$\mathbb{I}(M; \mathbf{Y}_e) \leq 2 n_e T \varepsilon_T R - 2 \varepsilon_T \log 2 \varepsilon_T.
\label{eq:leakage}$$
We will show that if the distribution is sufficiently flat, then $\mathbf{Y}_e| m$ is statistically close to a multivariate Gaussian for any $m \in \mathcal{M}_T$. Let us assume for now that ${\mathbf{H}}_e$ is an invertible square matrix (we next show how to reduce the other cases to this one). In this case, given a message $m$, we have $$\mathcal{H}_e \mathbf{x} \sim \mathcal{D}_{\mathcal{H}_e(\Lambda_e^T + \bm{\lambda}_m), \sqrt{(\mathcal{H}_e\mathcal{H}_e^{\dagger}) \sigma_s^2}}.$$
According to Lemma \[lem:product\], the distribution of $\mathcal{H}_e \mathbf{x} + \mathbf{w}_e$ is within variational distance $4 \varepsilon_T$ from the normal distribution $\mathcal{N}(0,\sqrt{{\mathbf{\Sigma}}_0})$, where $\varepsilon_T = \varepsilon_{\mathcal{H}_e \Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}_3})$ and $${\mathbf{\Sigma}}_0 = (\mathcal{H}_e\mathcal{H}_e^{\dagger}) \sigma_s^2 + \sigma_e^2 \mathbf{I}, \ \ {\mathbf{\Sigma}}_3^{-1} = (\mathcal{H}_e\mathcal{H}_e^{\dagger})^{-1} \sigma_s^{-2} + \sigma_e^{-2} \mathbf{I}.
\label{eq:sigma3}$$
We thus have the following bound for the information leakage ( with $\varepsilon_T$ replaced by $4\varepsilon_T$): $$\mathbb{I}(M; \mathbf{Y}_e) \leq 8 n_e T \varepsilon_T R - 8 \varepsilon_T \log 8 \varepsilon_T.
\label{eq:infoLeakage}$$
Therefore, if $\varepsilon_T=\varepsilon_{\mathcal{H}_e \Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}_3}) = o(1/T)$, the leakage vanishes as $T$ increases *for the specific realization $\mathcal{H}_e$.* To achieve strong secrecy universally, we must, however, ensure the existence of a lattice with vanishing flatness factor for *all* possible $\mathbf{{\mathbf{\Sigma}}}_3$. We postpone the universality discussion to Section \[sec:universallyFlat\] where it is proven that a vanishing flatness factor is possible simultaneously for all ${\mathbf{H}}_e \in \mathcal{S}_e$ and $\gamma_{\mathcal{H}_e \Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}_3}) < \pi$. This condition implies that semantic security is possible for any VNR, $$\gamma_{\mathcal{H}_e \Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}_3}) = \frac{|\mathcal{H}_ e^\dagger \mathcal{H}_e|^{1/n_a T} {V(\Lambda_e^T)^{1/n_a T}}}{ |{\mathbf{\Sigma}}_3|^{1/n_a T}} < \pi,
\label{eq:Secrecy}$$ $$\label{eq:vole}
V(\Lambda_e^T)^{1/n_aT} < \left| {\mathbf{I}}+ \rho_e {\mathbf{H}}_e^{\dagger} {\mathbf{H}}_e\right|^{-1/n_a} \pi\sigma_s^2 = (\pi \sigma_s^2)e^{-C_e/n_a}.$$\
**Number-of-Antenna Mismatch.** The above analysis assumed that $n_e = n_a$, *i.e.*, the number of eavesdropper receive antennas is *equal* to the the number of transmit antennas. Although analytically simpler, this assumption is not reasonable in practice, since we expect a compound scheme to perform well for any number of eavesdropper antennas. We show next how to reduce the other cases to the square case.
\(i) $n_e < n_a$: Recall that the signal received by the eavesdropper is given in matrix form by $$\mathbf{Y}_e = \mathbf{H}_e \mathbf{X}+ \mathbf{W}_e.$$ Let $\tilde{\mathbf{H}}_e \in \mathbb{C}^{(n_a-n_e) \times n_a}$ be a completion of ${\mathbf{H}}_e$ such that $$\overline{{\mathbf{H}}}_e = \left(\begin{array}{c} \mathbf{H}_e \\ \beta \tilde{\mathbf{H}}_e \end{array}\right),$$ is a full-rank sqaure matrix and $\beta > 0$ is some small number. Let $\tilde{\mathbf{W}}_e \in \mathbb{C}^{(n_a-n_e)\times T}$ be a matrix corresponding to circularly symmetric Gaussian noise. Consider the following surrogate MIMO channel: $$\left(\begin{array}{c} \mathbf{Y}_e \\ \tilde{\mathbf{Y}}_e \end{array}\right) = \left(\begin{array}{c} \mathbf{H}_e \\ \beta \tilde{\mathbf{H}}_e \end{array}\right) \mathbf{X}+ \ \left(\begin{array}{c} \mathbf{W}_e \\ \tilde{\mathbf{W}}_e \end{array}\right),$$ where $\tilde{{\mathbf{H}}}_e$ is scaled so that the capacity of the new channel is arbitrarily close to the original one. Indeed for any full rank completion $\tilde{{\mathbf{H}}}_e$, from the matrix determinant lemma, we have $$|{\mathbf{I}}+ \rho_e \overline{{\mathbf{H}}}_e^\dagger \overline{{\mathbf{H}}}_e| = |{\mathbf{I}}+ \rho_e {{\mathbf{H}}}_e^\dagger {{\mathbf{H}}}_e| \times |{\mathbf{I}}+ \beta^2 \tilde{{\mathbf{H}}}_e (|{\mathbf{I}}+ \rho_e {{\mathbf{H}}}_e^\dagger {{\mathbf{H}}}_e|)^{-1}\tilde{{\mathbf{H}}}_e^{\dagger}| \geq e^{C_e}.$$ Therefore, by letting $\beta \to 0$, the left-hand side tends to $e^{C_e}$. For any signal ${\mathbf{X}}$, the information leakage of the surrogate channel is strictly greater than the original one. Indeed, the the eavesdropper’s original channel is stochastically degraded with respect to the augmented one, thus $\mathbb{I}(M; (\mathbf{Y}_e,\tilde{\mathbf{Y}}_e)) \geq \mathbb{I}(M; \mathbf{Y}_e).$ A universally secure code for the $n_a \times n_a$ MIMO compound channel will have vanishing information leakage for the surrogate $n_a \times n_a$ channel (for *any* completion) and therefore will also be secure for the original $n_e \times n_a$ channel.
\(ii) $n_e > n_a$: Performing a rectangular $QR$ factorization of ${\mathbf{H}}_{e}$ we have: $${\mathbf{H}}_{e} = \mathbf{Q} \left(\begin{array}{c} \hat{{\mathbf{R}}} \\ \mathbf{0} \end{array}\right),$$ where ${\mathbf{Q}} \in \mathbb{C}^{n_e \times n_e}$ and $\hat{\mathbf{R}} \in \mathbb{C}^{n_a \times n_a}$ are square matrices. Therefore the eavesdropper’s received signal is equivalent to $$\begin{aligned}
{\mathbf{Y}}_e &=& \mathbf{Q}\left(\begin{array}{c} \hat{{\mathbf{R}}} \\ \mathbf{0} \end{array}\right)\mathbf{X} +\left(\begin{array}{c} {\mathbf{W}}_e^{(1)} \\ {\mathbf{W}}_e^{(2)} \end{array}\right) \\
\iff \mathbf{Q}^{\dagger} {\mathbf{Y}}_e & = & \left(\begin{array}{c} \hat{{\mathbf{R}}} \\ \mathbf{0} \end{array}\right)\mathbf{X} +\left(\begin{array}{c} \tilde{{\mathbf{W}}}_e^{(1)} \\ \tilde{{\mathbf{W}}}_e^{(2)} \end{array}\right),\end{aligned}$$ where the components of the noise matrices $\tilde{{\mathbf{W}}}_e^{(1)}, \tilde{{\mathbf{W}}}_e^{(2)}$ are i.i.d. Gaussian. The leakage is therefore the same as for the square channel $\hat{{\mathbf{R}}}$ and a universal code will also achieve vanishing leakage for the non-square channel.
The Legitimate Channel: Reliability
-----------------------------------
It was shown in [@Our] that if ${\mathbf{X}}\sim \mathcal{D}_{\Lambda_b^T,\sigma_s}$, then the maximum-a-posteriori (MAP) decoder for the signal ${\mathbf{Y}}_b$ is equivalent to lattice decoding of $\mathbf{F}_b {\mathbf{Y}}_b$, where $\mathbf{F}_b$ is the MMSE-GDFE matrix to be defined in the sequel. We cannot claim directly that ${\mathbf{X}}\sim \mathcal{D}_{\Lambda_b^T,\sigma_s}$, since the message distribution in $\mathcal{M}_T$ need not be uniform. Nonetheless, we show that reliability is still possible for all individual messages.
The full decoding process is depicted in Figure \[fig:wiretapb\]. Bob first applies a filtering matrix $\mathbf{F}_b$ so that $$\tilde{{\mathbf{Y}}}_b = \mathbf{F}_b {\mathbf{Y}}_b = \mathbf{R}_b {\mathbf{X}}+ \mathbf{W}_{b,{\text{eff}}},$$ where ${\mathbf{R}}_b^{\dagger} {\mathbf{R}}_b = {\mathbf{H}}_b^{\dagger} {\mathbf{H}}_b + \rho_b^{-1} {\mathbf{I}}$ and ${\mathbf{F}}_b^{\dagger} {\mathbf{R}}_b = \rho_b^{-1} {\mathbf{H}}_b$, and the effective noise is $$\mathbf{W}_{b,{\text{eff}}} = ({\mathbf{F}}_b {\mathbf{H}}_b - {\mathbf{R}}_b){\mathbf{X}}+ {\mathbf{F}}_b \mathbf{W}_b.$$ The next step is to decode $\tilde{{\mathbf{Y}}}_b$ in ${\mathbf{R}}_b \Lambda_b^T$, in order to obtain $Q_{{\mathbf{R}}_b \Lambda_b^T}(\tilde{{\mathbf{Y}}}_b),$ which is then remapped into the element of the coset ${\mathbf{R}}_b\Lambda_b^T/{\mathbf{R}}_b\Lambda_e^T$ through the operation $\mbox{mod }{\mathbf{R}}_b \Lambda_e^T$. We can then invert the linear transformation associated to ${\mathbf{R}}_b$ (notice that ${\mathbf{R}}_b$ has full rank) in order to obtain the coset in $\Lambda_b^T/\Lambda_e^T$ and re-map it to the message space $\mathcal{M}_T$ through $\phi^{-1}$.
In the first step, from Lemma \[lem:product\], the effective noise ${\mathbf{W}}_{b,{\text{eff}}}$ is statistically close to a Gaussian noise with covariance: $$\begin{aligned}
{\mathbf{\Sigma}}_{b,{\text{eff}}} &=& \sigma_s^2 (\mathbf{F}_b\mathbf{H}_b-\mathbf{R}_b)(\mathbf{F}_b\mathbf{H}_b-\mathbf{R}_b)^\dagger + \sigma_b^2\mathbf{F}_b{\mathbf{F}}_b^\dagger \\ \nonumber
&=& \rho_b^{-2} \sigma_s^2 \mathbf{R}_b^{-\dagger}\mathbf{R}_b^{-1} + \sigma_b^2\rho_b^{-2}\mathbf{R}_b^{-\dagger}\mathbf{H}_b^\dagger\mathbf{H}_b\mathbf{R}_b^{-1}\\ \nonumber
&=& \sigma_b^2 \mathbf{R}_b^{-\dagger} (\rho_b^{-1} \mathbf{I} + \mathbf{H}_b^\dagger\mathbf{H}_b)\mathbf{R}_b^{-1}= \sigma_b^2\mathbf{I}.
\label{eq:Variance}\end{aligned}$$ provided that $\varepsilon_{(F_b{\mathbf{H}}_b-{\mathbf{R}}_b)\Lambda_e^T}({\mathbf{\Sigma}}_{b,\text{inv}})$ is small, where $${\mathbf{\Sigma}}_{b,\text{inv}}^{-1} = (\sigma_s^2 (\mathbf{F}_b\mathbf{H}_b-\mathbf{R}_b)(\mathbf{F}_b\mathbf{H}_b-\mathbf{R}_b)^\dagger)^{-1} + (\sigma_b^2\mathbf{F}_b{\mathbf{F}}_b^\dagger)^{-1}.$$ The probability of error given *any* message $m$ is thus bounded by $$\mathbb{P}_{\text{err}|m} \leq \left(1+4\varepsilon_{(F_b{\mathbf{H}}_b-{\mathbf{R}}_b)\Lambda_e^T}({\mathbf{\Sigma}}_{b,\text{inv}})\right) P(\tilde{{\mathbf{W}}}_{b,{\text{eff}}} \notin \mathcal{V}(\mathbf{R}_b \Lambda_b^T)),$$ where each entry of $\tilde{{\mathbf{W}}}_{b,{\text{eff}}}$ is i.i.d. normal with variance $\sigma_b^2$. Therefore, if we guarantee that $\varepsilon_{(F_b{\mathbf{H}}_b-{\mathbf{R}}_b)\Lambda_e^T}({\mathbf{\Sigma}}_{b,\text{inv}})$ is bounded and if we choose a universally good lattice, the probability vanishes for all possible ${\mathbf{R}}_b$. This is possible [@Our] provided that $$\gamma_{{\mathbf{R}}_b\Lambda_b^T}(\sigma_b) > \pi e,
\label{eq:Reliability}$$ namely, $$\label{eq:volb}
V(\Lambda_b^T)^{1/n_aT} > \left| {\mathbf{I}}+ \rho_b {\mathbf{H}}_b^{\dagger} {\mathbf{H}}_b\right|^{-1/n_a} \pi e\sigma_s^2 = (\pi e \sigma_s^2)e^{-C_b/n_a}.$$
However, the evaluation of ${\mathbf{\Sigma}}_{b,\text{inv}}$ is cumbersome and implies an extra condition for the flatness of $\Lambda_e^T$. Next we show, instead, how to circumvent this problem by using the fact that that the effective noise is “asymptotically” sub-Gaussian with covariance matrix $\sigma_b^2 {\mathbf{I}}$. We say that a centred random vector ${\mathbf{w}}\in \mathbb{R}^n$ is sub-Gaussian with (proxy) parameter $\sigma$ if $$\log E[e^{t \left\langle{\mathbf{w}},{\mathbf{u}}\right\rangle}] \leq \frac{t^2 \sigma^2}{2}$$ for all $t \in \mathbb{R}$ and all unit norm vectors ${\mathbf{u}}\in \mathbb{R}^n$.
Let ${\mathbf{x}}$ be a random vector with distribution $\mathcal{D}_{\Lambda_e^T+\bm{\lambda}_m,\sigma_s}$, and let $\varepsilon^\prime = \varepsilon_{\Lambda_e^T}\left(\sigma_s\right).$ For any matrix $\mathbf{A}$ and any vector ${\mathbf{u}}\in \mathbb{C}^{n_b T}$, we have: $$E[e^{\Re\left\{{\mathbf{x}}^\dagger \mathbf{A} {\mathbf{u}}\right\}}] \leq \left(\frac{1+\varepsilon^{\prime}}{1-\varepsilon^{\prime}}\right) e^{\frac{\sigma_s^2}{4} \left\| \mathbf{A}{\mathbf{u}}\right\|^2}.$$
Notice that the average power per dimension of a sub-Gaussian random variable is always less than or equal to its parameter $\sigma_s^2$. Moreover, the sum of two sub-Gaussians is also a sub-Gaussian (for more properties, the reader is referred to [@LVL16]). The above lemma, along with , allows us to establish that ${\mathbf{W}}_{b,{\text{eff}}}$ is almost sub-Gaussian with parameter $\sigma_b^2$. Therefore, as long as $\varepsilon^{\prime} \approx 0$ the probability of error tends to zero if we choose $\Lambda_b^T$ to be universally AWGN-good.
Proof of Theorem \[thm:achievableRatesFinal\]: Achievable Secrecy Rates
-----------------------------------------------------------------------
From the previous subsections, semantic security is achievable if $\Lambda_b^T$ and $\Lambda_e^T$ satisfy:
1. Reliability : $\gamma_{{\mathbf{R}}_b\Lambda_b^T}(\sigma_b) >\pi e$
2. Secrecy : $\gamma_{\mathcal{H}_e \Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}_3}) < \pi$
3. Sub-Gaussianity of equivalent noise and power constraint: $\varepsilon_{\Lambda_e^T}(\sigma_s) \to 0$.
From \[eq:vole\] and , the first two conditions can be satisfied for rates up to $$\log |{\mathbf{I}}+ \rho_b {\mathbf{H}}_b^{\dagger} {\mathbf{H}}_b| - \log |{\mathbf{I}}+ \rho_e {\mathbf{H}}_e^{\dagger} {\mathbf{H}}_e| - n_a$$ nats per channel use, but the last conditions may, *a priori*, limit these rates to certain SNR regimes. Fortunately, if condition $2)$ is satisfied, we automatically satisfy the condition for $\varepsilon_{\Lambda_e^T}(\sigma_s) \to 0$, since $$\frac{V(\Lambda_e^T)^{1/n_aT}}{\sigma_s^2} \leq \frac{V(\Lambda_e^T)^{1/n_aT}}{e^{-C_e/n_a} \sigma_s^2} < \pi.$$ Therefore, if $(\Lambda_b^T, \Lambda_e^T)$ is a sequence of nested lattices, where
1. $\Lambda_b^T$ is universally good for the compound channel with set $\mathcal{S}_b$,
2. $\Lambda_e^T$ is universally secure for the compound channel with set $\mathcal{S}_e$,
then nested lattice Gaussian coding achieves any secrecy rate up to $$R \leq (C_b - C_e - n_a)^{+}.$$
The *existence* of such nested pairs is proved subsequently in Section \[sec:universallyFlat\] and Appendix \[ref:appSimultaneous\], which concludes the proof of Theorem \[thm:achievableRatesFinal\].
In fact using a method in [@HB14] we can further reduce the gap to approximately $n_a \log (e/2)$. We conjecture that this gap can be completely removed with tighter bounds for the variational distance between the discrete and continuous Gaussians. This is left as an open question.
Theorem \[thm:achievableRatesFinal\] is also a slight improvement on the main result of [@LLBS_12 Theormm 5] in the sense that one of the conditions on the SNR of Bob ($\rho_b > e$) is not needed any longer. Indeed, for the Gaussian channel, $n_a = 1$ and the SNR condition for non-zero secrecy rates is $C_b > C_e + 1$, which is equivalent to $$\frac{1+\rho_b}{1+\rho_e} > e.$$
Universally Flat Gaussians {#sec:universallyFlat}
==========================
The results in the previous section require the existence of sequences of lattices which are universally good for the wiretap channel. More specifically, we need a sequence $\Lambda_b^T$ which is universally AWGN-good and a sequence $\Lambda_e^{T}$ whose leakage vanishes for *all* channel realizations of the eavesdropper. The first condition was studied in [@Our], where it was shown, through a compactness argument, that random lattices are universal. In this section we deal with the second condition and prove the existence of lattices $\Lambda_e^{T}$ which are universally good for secrecy of the MIMO channel.
Two methods are provided to establish the main result. The first method relies solely on random lattice coding arguments and achieves secrecy capacity up to a gap of $n_a$ nats per channel use. The second method is based on algebraic reductions and exhibits a larger gap (by a factor of $\omega(n_a\log n_a)$) to capacity, but has the appealing property of reducing the problem to the one of constructing secrecy-good lattices for the *AWGN channel*, making it potentially more useful in practice.
Construction A
--------------
[\[ConstructionA\]]{} Construction A (or “mod-$p$”) lattices are certainly the simplest choice for constructing pairs of nested lattices, however generalizations based on algebraic lattices may offer greater flexibility in the code design, which could be leveraged to obtain better decoding complexity, diversity, or other parameters. Moreover, the coding scheme in Section \[sec:algebraicApproach\] entails an extra condition on the ensemble, which can be satisfied by assuming an algebraic structure. A general “flexible” construction can be defined via “generalized reductions”. Let $\psi: \Lambda_{\text{base}} \to \mathbb{F}_p^T$ be a surjective homomorphism from a base lattice $\Lambda_{\text{base}}$ of complex dimension $N \geq T$ to the vector space $\mathbb{F}_p^T$ (also referred to as a *reduction*). Define the lattice $\Lambda(\mathcal{C})$ as the pre-image of a linear code $\mathcal{C}$, $$\Lambda(\mathcal{C}) = \psi^{-1}(\mathcal{C}).$$ If $\mathcal{C}$ has length $T$ and dimension $k$, the volume of $\Lambda(\mathcal{C})$ equals to $p^{T-k} V(\Lambda_{\text{base}})$. For instance if $N=T$, $\Lambda_{\text{base}} = \mathbb{Z}[i]^T$ the mapping $\psi$ is the reduction modulo $p$: $$\begin{split}
&\psi(a_1+b_1 i, a_2+b_2 i, \ldots, a_T + b_T i) =\\
& (a_1 \,\, (\text{mod } p), b_1 \,\, (\text{mod } p), a_2 \,\, (\text{mod } p), b_2 \,\, (\text{mod } p), \cdots, \\
& a_T \,\, (\text{mod } p), b_T \,\, (\text{mod } p) ),
\end{split}$$ we recover an analogue of Loeliger’s (mod-$p$) Construction A [@Loeliger]. In this case we obtain a nested lattice beween $\mathbb{Z}[i]^T$ and $p\mathbb{Z}[i]^T$. More refined “direct” constructions can be obtained by using number theory and prime ideals of $\mathbb{Z}[i]$. For instance, if $\Lambda_{\text{base}}$ is the embedding of the ring of integers of a number field and $\psi$ is the reduction modulo a prime ideal we can recover the constructions in [@Our]. Notice that, for this construction, if $\mathcal{C}_1 \subset \mathcal{C}_2$, we obtain two nested lattices $\Lambda(\mathcal{C}_1) \subset \Lambda(\mathcal{C}_2)$.
It was shown in [@Campello17] that if $\{\psi\}$ is an infinite sequence of mappings, under mild conditions[^3] the ensemble of lattices averaged over all linear codes $\mathcal{C}$ of same dimension $k$ satisfies the *Minkowski-Hlawka theorem*, namely: $$\lim_{p \to \infty} \mathbb{E}_{\mathcal{C}}\left[\sum_{{\mathbf{x}}\in \beta \Lambda(\mathcal{C}) {\backslash\left\{\mathbf{0}\right\}}} f({\mathbf{x}})\right] = V^{-1} \int_{\mathbb{C}^{N}} f({\mathbf{x}}),$$ where $\beta = V^{1/2N} (p^{T-k} V(\Lambda))^{1/2N}$ is a constant so that all lattices have volume $V$. The result holds for any integrable function $f$ which decays sufficiently fast (in particular any function upper bounded by a constant times $1/(\left\|\mathbf{x}\right\| + 1)^{2N+\delta}$ for some $\delta > 0$). Clearly the Gaussian probability density function satisfies this restriction.
Lattices Which Are Good for Secrecy
-----------------------------------
In what follows we will apply the generalized version of Construction A to construct a sequence of lattices $\Lambda_e^T$ which is good for secrecy, *i.e.*, which has vanishing flatness factor for all eavesdropper channel realizations. As usual, $T$ will denote the blocklength (cf. Equation ), $N$ will be set to $n_aT$ (the complex dimension of the coding lattice) and $k < T$ is any positive integer.
Using the above Minkowski-Hlawka theorem, there exists an ensemble of lattices $\mathbb{L}$ of volume $V$ such that $$\mathbb{E}_{\mathbb{L}}\left[\sum_{{\mathbf{x}}\in \Lambda {\backslash\left\{\mathbf{0}\right\}}}f({\mathbf{x}})\right] \leq V^{-1} \int_{\mathbb{C}^{n_a T}} f({\mathbf{x}})\mathbf{d} {\mathbf{x}}+ \varepsilon,
\label{eq:MH}$$ for any $\varepsilon > 0$ . Equation implies that $$\mathbb{E}_{\mathbb{L}}\left[\sum_{{\mathbf{x}}\in \Lambda{\backslash\left\{\mathbf{0}\right\}}} e^{-{\mathbf{x}}^\dagger {\mathbf{\Sigma}}^{-1} {\mathbf{x}}} \right] \leq V^{-1}\pi^{n_a T} |{\mathbf{\Sigma}}| + \varepsilon,$$ therefore $$\mathbb{E}_{\mathbb{L}}\left[\varepsilon_{\Lambda}(\sqrt{{\mathbf{\Sigma}}}) \right] \leq \frac{V(1+\varepsilon)}{\pi^{n_a T} |{\mathbf{\Sigma}}| }.$$ Hence as long as $\varepsilon$ is bounded and $V^{1/T}/\pi |{\mathbf{\Sigma}}|^{1/T}$ is bounded by a constant less than $1$, the flatness factor tends to zero exponentially in the proposed lattice coding scheme. The condition for $\varepsilon$ can be achieved, for instance, by choosing $p$ sufficiently large in Construction A.
Let $\mathcal{H}_e = {\mathbf{H}}_e \otimes {\mathbf{I}}$ and ${\mathbf{\Sigma}}_3^{-1} = (\mathcal{H}_e\mathcal{H}_e^{\dagger})^{-1} \sigma_s^{-2} + \sigma_e^{-2} \mathbf{I},$ as in Equation . For any $\gamma < \pi$, there exists a sequence of lattices $\Lambda_e^T \subset \mathbb{C}^{n_aT}$ with $\gamma_{\mathcal{H}_e \Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}_3}) \leq \gamma $ and universally vanishing flatness factor, *i.e.*, $$\lim_{T \to \infty} \varepsilon_{\mathcal{H}_e \Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}_3})= 0 \mbox{ for all } {\mathbf{H}}_e \in \mathcal{S}_e.$$ Moreover, the convergence rate is exponential, *i.e.*, for all ${\mathbf{H}}_e \in \mathcal{S}_e$, $\varepsilon_{\mathcal{H}_e \Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}_3}) = e^{-\Omega(T)}$. \[lem:universallyFlatGaussians\]
The proof is analogous to the quantization argument for the probability of error in [@Our], which, in turn follows [@ShiWesel07].
\(i) *Fixed ${\mathbf{H}}_e$*. If $\mathbb{L}$ is a Minkowski-Hlawka ensemble with volume $V$, then
$$\mathbb{E}_{\mathbb{L}}\left[\varepsilon_{\mathcal{H}_e \Lambda^{(T)}}({\mathbf{\Sigma}}_3) \right] \leq (1+ \varepsilon) \left(\frac{\gamma_{\mathcal{H}_e \Lambda^{(T)}}(\sqrt{{\mathbf{\Sigma}}_3})}{\pi}\right)^{n_aT}$$ which guarantees a sequence $\Lambda_e^{(T)}$ (at this point, possibly depending on ${\mathbf{H}}_e$) with vanishing flatness factor as long as $\gamma < \pi$.
\(ii) *Finite set*. Let $\mathcal{S}_{Q} \subset \mathcal{S}_e$ be a *finite* subset of $\mathcal{S}_e$ with cardinality $Q$. We have $$\begin{aligned}
\mathbb{E}_{\mathbb{L}}\left[\max_{{\mathbf{H}}_e \in \mathcal{S}_Q} \varepsilon_{\mathcal{H}_e \Lambda_e^{(T)}}({\mathbf{\Sigma}}_3) \right] &\leq& \mathbb{E}_{\mathbb{L}}\left[\sum_{{\mathbf{H}}_e\in \mathcal{S}_Q} \varepsilon_{\mathcal{H}_e \Lambda_e^{(T)}}({\mathbf{\Sigma}}_3) \right] \\
=Q(1+\varepsilon)\left(\frac{\gamma_{\mathcal{H}_e \Lambda_e^{(T)}}(\sqrt{{\mathbf{\Sigma}}_3})}{\pi}\right)^{n_aT} &\to& 0\end{aligned}$$ which guarantees a sequence $\Lambda_e^{(T)}$ with exponentially vanishing flatness factor for any ${\mathbf{H}}_e \in \mathcal{S}_e$.
\(iii) *Quantization step*. By quantizing the channel space, we can extend step (ii) into a universal code for any channel in $\mathcal{S}_e$. This analysis is described in Appendix \[app:2\]. Here we provide a sketch of the argument. Suppose $\mathcal{S}_Q$ is a $\delta$-covering for $\mathcal{S}_e$, *i.e.*, for all ${\mathbf{H}}_{e}\in \mathcal{S}_e$, there exists ${\mathbf{H}}_{q} \in \mathcal{S}_Q$ such that $\left\| {\mathbf{H}}_e- {\mathbf{H}}_q \right\| \leq \delta$. From the compactness of $\mathcal{S}_e$, such a covering exists for any arbitrarily small $\delta > 0$, and the size of the covering depends only on $n_a$, which is fixed for the whole transmission. Furthermore, the theta series is a continuous function of ${\mathbf{H}}_e$, which implies that the flatness factor in different channel realizations are also close. From this, we can choose $\delta$ independently of $T$ that guarantees that the total exponent is negative. Therefore, the flatness factor tends to zero uniformly as $T \to \infty$.
The above proof does not rely on a specific realization but rather on the knowledge of the compact *compound* set $\mathcal{S}_e$. It is reminiscent of a widely used technique in coding for compound channels (*e.g.*, [@ShiWesel07]). Essentially, an encoder develops a code for $Q_{\delta}$ channels, where $Q_{\delta}$ is the cardinality of a good quantizer of the channel space. However the quantization $Q_{\delta}$ may increase the effective blocklength for a target information leakage. Moreover, the proof does not give us insights on how to effectively quantize $\mathcal{S}_e$, making algebraic approaches appealing in practice.
Lemma \[lem:universallyFlatGaussians\] shows the existence of universally flat Gaussians or, in other words, the existence of a sequence of lattices $\Lambda_e^{T}$ which are good for secrecy. Recall that in our construction \[sec:overview\], we required $\Lambda_e^{T}$ to be nested with $\Lambda_b^{T} \supset \Lambda_e^{T}$, where $\Lambda_b^T$ is a sequence of lattices which are good for the legitimate compound channel. The existence of $\Lambda_b^T$ was proven in [@Our]. In Appendix \[ref:appSimultaneous\] we argue that both conditions can be achieved by a nested pair $(\Lambda_b^T, \Lambda_e^T)$ which is the last missing part of the proof of Theorem \[thm:achievableRatesFinal\].
Algebraic Approach {#sec:algebraicApproach}
------------------
Following [@LuzziGoldenCode], we now define a lattice admitting algebraic reduction.
We say that $\Lambda$ admits algebraic reduction if for any unit determinant matrix $\mathbf{A} \in \mathbb{C}^{n_a \times n_a}$ there exists a matrix decomposition of the form $\mathbf{A} = {\mathbf{E}}{\mathbf{U}}$, where ${\mathbf{E}}$ and ${\mathbf{U}}$ are also unit-determinant satisfying the following properties:
1. ${\mathbf{U}}\Lambda = \Lambda$,
2. $\left\| {\mathbf{E}}^{-1} \right\|_F \leq \alpha$ for some absolute constant $\alpha$ that does not depend on $\mathbf{A}$.
\[def:algRed\]
The Golden Code is one example of a lattice that admits algebraic reduction [@LuzziGoldenCode]. Lattices built from generalized versions of Construction A based on number fields and division algebras also admit a similar reduction (if necessary we may relax requirement 1) to include equivalence instead of equality). This property was used in [@Our] to achieve capacity of the infinite compound MIMO channel. Note that $\alpha$ grows with $n_a$. See [@Our Theorem 3] for an upper bound on $\alpha$ in the case of number fields, and [@LuzziGoldenCode] in the case of division algebras. Next, we show that an ensemble of lattices satisfying Definition \[def:algRed\] achieves the secrecy capacity of the compound MIMO channel up to a constant gap.
Recall the following relation between the spectral norm and the Frobenius norm: $$\left\|{\mathbf{I}}\otimes \mathbf{A} \right\| = \left\|\mathbf{A} \right\| \leq \left\|\mathbf{A} \right\|_F,$$ for the identity matrix ${\mathbf{I}}$ of any dimension.
Suppose that $\Lambda \subset \mathbb{C}^{n_a T}$ is such that its dual lattice $\Lambda^*$ admits algebraic reduction. Then for $\mathbf{A} \in \mathbb{C}^{n_a \times n_a}$, $$\varepsilon_{\Lambda}(\sqrt{{\mathbf{I}}_T \otimes \mathbf{A}}) \leq \varepsilon_{\Lambda}\left(|{\mathbf{A}}|^{1/2n_a}/\alpha \right).$$
From the Poisson summation formula and the expression for the flatness factor : $$\begin{split}
\varepsilon_{\Lambda}(\sqrt{{\mathbf{I}}_T \otimes \mathbf{A}}) &= \sum_{\bm{\lambda} \in \Lambda^{*}{\backslash\left\{\mathbf{0}\right\}}} e^{-\pi^2 \bm{\lambda}^\dagger ({\mathbf{I}}_T \otimes \mathbf{A}) \bm{\lambda} } \\ &=\sum_{\bm{\lambda} \in \Lambda^{*}{\backslash\left\{\mathbf{0}\right\}}} e^{-\pi^2 \left\| \sqrt{{\mathbf{I}}_T \otimes \mathbf{A}} \bm{\lambda}\right\|^{2} } \\
&= \sum_{\bm{\lambda} \in \Lambda^{*}{\backslash\left\{\mathbf{0}\right\}}} e^{-\pi^2 |\mathbf{A}|^{1/n_a} \left\| \frac{\sqrt{{\mathbf{I}}_T \otimes \mathbf{A}}}{\left(\sqrt{| \mathbf{A}|}\right)^{1/n_a}} \,\, \bm{\lambda}\right\|^{2} }.
\end{split}$$
Upon decomposing $\frac{\sqrt{\mathbf{A}}}{\left(\sqrt{|\mathbf{A}|}\right)^{1/n_a}} = {\mathbf{E}}{\mathbf{U}}$ as in Definition \[def:algRed\], the last equation becomes $$\begin{split}
&\sum_{\bm{\lambda} \in \Lambda^{*}{\backslash\left\{\mathbf{0}\right\}}} e^{-\pi^2 |\mathbf{A}|^{1/n_a} \left\|({\mathbf{I}}\otimes {\mathbf{E}}) \bm{\lambda}\right\|^2} \stackrel{(a)}{\leq} \sum_{\bm{\lambda} \in \Lambda^{*}{\backslash\left\{\mathbf{0}\right\}}} e^{-\pi^2 |\mathbf{A}|^{1/n_a} \frac{\left\|\bm{\lambda}\right\|^2}{\left\|{\mathbf{E}}^{-1}\right\|^2}} \\
&\stackrel{(b)}{\leq} \sum_{\bm{\lambda} \in \Lambda^{*}{\backslash\left\{\mathbf{0}\right\}}} e^{-\pi^2 |\mathbf{A}|^{1/n_a} \frac{\left\|\bm{\lambda}\right\|^2}{\left\|{\mathbf{E}}^{-1}\right\|_F^2}} \stackrel{(c)}{\leq} \sum_{\bm{\lambda} \in \Lambda^{*}{\backslash\left\{\mathbf{0}\right\}}} e^{-\pi^2 |\mathbf{A}|^{1/n_a} \frac{\left\|\bm{\lambda}\right\|^2}{\alpha^2}} \\
&= \varepsilon_{\Lambda}\left(|{\mathbf{A}}|^{1/2n_a}/\alpha \right),
\end{split}$$ where $(a)$ is due to the bound $\left\| \bm{\lambda}\right\| \leq \left\|({\mathbf{I}}\otimes {\mathbf{E}})^{-1}\right\| \left\|({\mathbf{I}}\otimes {\mathbf{E}})\bm{\lambda}\right\|$ and the fact that $\left\|({\mathbf{I}}\otimes {\mathbf{E}})^{-1}\right\| = \left\|({\mathbf{I}}\otimes {\mathbf{E}}^{-1})\right\| = \left\|{\mathbf{E}}^{-1}\right\| $, $(b)$ is due to the inequality between the $2$-norm and the Frobenius norm and $(c)$ follows from Definition \[def:algRed\].
Since $$\varepsilon_{\mathcal{H}_e\Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}_3}) = \varepsilon_{\Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}}),$$ where ${\mathbf{\Sigma}}^{-1} = \mathcal{H}_e^\dagger {\mathbf{\Sigma}}_3^{-1} \mathcal{H}_e$ and ${\mathbf{\Sigma}}_3^{-1} = (\mathcal{H}_e\mathcal{H}_e^{\dagger})^{-1} \sigma_s^{-2} + \sigma_e^{-2} \mathbf{I}$ is block-diagonal, we can apply the above lemma. Therefore, if we construct an ensemble of lattices such that their duals admit algebraic reduction for some constant $\alpha > 0$, then there exist lattices with vanishing flatness factor $\varepsilon_{\mathcal{H}_e \Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}_3})$ provided that $$\varepsilon_{\Lambda_e^T}\left(\sqrt{\sigma_s^2 \alpha^{-2} e^{-C_e/n_a}}\right) \to 0.
\label{eq:secrecyGoodGaussian}$$ This can be achieved if: $$V(\Lambda_e^T)^{1/n_aT} < \pi e^{-C_e/n_a} \alpha^{-2} \sigma_s^2.
\label{eq:lastexpression}$$ Notice that the right-hand side of depends only on the determinant of ${\mathbf{\Sigma}}_3$ or on the capacity of the eavesdropper channel, not on any individual realization. For this condition to hold, we only need a sequence of secrecy-good lattices for a surrogate eavesdropper channel with smaller noise variance (by a factor ${\alpha^{-2}}$). Therefore, by combining and , we arrive at the the following result:
Let $(\Lambda_b^{T}, \Lambda_e^{T})$ be a sequence of nested lattices where: (i) $\Lambda_b^{T}$ is universally good for the compound MIMO channel and (ii) $\Lambda_e^{T}$ satisfies Definition \[def:algRed\] and is secrecy good **for the AWGN channel** (Condition ). Then nested lattice Gaussian coding achieves any secrecy rate up to $$R \leq (C_b - C_e - n_a -2n_a\log (\alpha))^{+}.$$ \[thm:acheivableRatesAlgebraic\]
Notice that the gap has a different nature than the one in the previous subsection. It consists of two parts: $n_a$ due to the same restriction on the flatness factor in Theorem \[thm:achievableRatesFinal\], and $\log \alpha$ due to algebraic reduction. Although we have conjectured that the gap in Theorem \[thm:achievableRatesFinal\] can be essentially removed, this is not the case for $\log \alpha$ in Theorem \[thm:acheivableRatesAlgebraic\]. Indeed, since $\alpha$ cannot be smaller than $\sqrt{n_a}$ [@Our Theorem 3], this gap is always larger than $n_a\log{n_a}$. However, the code construction can be reduced to the problem of finding good lattices for the Gaussian wiretap channel (with some additional algebraic structure), making the design potentially more practical.
Notice also that this strategy is closely related to the “decoupled design" for compound MIMO channels [@Our Sect. VI]. Both strategies can indeed be combined, *i.e.*, Bob’s code can also benefit from algebraic reduction. In this case both the original channel decoder and the code design can be greatly simplified, at the cost of an extra gap (*i.e.*,, an extra factor $2n_a\log (\alpha)$) to the compound capacity.
Discussion {#sec:discussion}
==========
In this paper, we have presented a construction of nested lattice codes to achieve the secrecy capacity of compound MIMO wiretap channels, up to a gap equal to the number of transmit antennas. Compared to [@LVL16], the construction in this work is not only more practical, but also enjoys a smaller gap. With algebraic reduction, further simplification has been made, at the cost of an extra gap to the secrecy capacity. Interestingly, the algebraic approach reaffirms the important role of the dual lattice of $\Lambda_e^T$ in wiretap channels, firstly discovered in [@LVL16].
#### Encoding and decoding
Encoding and decoding are not much different from those of lattice codes for compound MIMO channels in [@Our]. The generalized Construction A employed in this paper may be viewed as a concatenated code, where the inner code is a lattice with some desired properties, while the outer code is an error correction code. Therefore, decoding can be run successively, which greatly reduces the decoding complexity. As for encoding, the discrete Gaussian shaping can be facilitated by choosing a nice base lattice, *e.g.*, a rotated $\mathbb{Z}^{n_a}$ lattice whose Gaussian shaping is easy. There are highly efficient algorithms for Gaussian shaping over specific lattices [@Antonio16], but more research is needed for Gaussian shaping over generalized Construction A. Practical implementation of the proposed codes is left as future work.
#### Comparison to other compound models
When the channel ${\mathbf{H}}_b$ is known and the eavesdropper channel has bounded norm, [@SL15] has shown that the eavesdropper’s worst channel is also isotropic. In this case the capacity can be achieved by decomposing the channels into different independent substreams with appropriate power, and applying independent coding for the Gaussian channel. This is also the case when ${\mathbf{H}}_b$ has a linear uncertainty. In these cases, a combination of correct power allocation and a similar argument to Lemma \[lem:universallyFlatGaussians\] shows that semantic secrecy is also achievable by random lattice codes. On the other hand, the algebraic approach (Theorem \[thm:acheivableRatesAlgebraic\]) heavily relies on the fact that the channels in $\mathcal{S}_e$ have the same white-input mutual information.
#### Finite-length performance
The results of this work are based on asymptotic analysis as $T\to \infty$. The practical performance of the proposed universal codes at finite block lengths warrants an investigation. In particular, how large $T$ is required to approach the promised gap in practice? For given $T$, how far do practical codes perform from secrecy capacity? It may be a challenging problem to design good, practical universal codes.
As a further perspective, one may consider an “outage” analysis of the MIMO wiretap channel in a finite blocklength regime, where the channel matrices ${\mathbf{H}}_b$ and ${\mathbf{H}}_e$ may be random. In other words, one may analyze the probability that the code rate $R$ exceeds the secrecy capacity. In such scenarios, we believe that lattices with the non-vanishing determinant property will be able to provide universal bounds for the outage probability. We leave it as an open problem.
Quantization of Channel Space {#app:2}
=============================
In this appendix we show bounds on the flatness factor in the quantized channel space, formalizing part (iii) in the proof of Lemma \[lem:universallyFlatGaussians\]. Instead of performing the quantization directly in the eavesdropper space $\mathcal{S}_b$, we will consider the corresponding covariance matrices. Following the notation of Lemma \[lem:universallyFlatGaussians\], we have: $$\varepsilon_{\mathcal{H}_e\Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}_3}) = \varepsilon_{\Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}}),$$ where ${\mathbf{\Sigma}}^{-1} = \mathcal{H}_e^\dagger {\mathbf{\Sigma}}_3^{-1} \mathcal{H}_e$ and ${\mathbf{\Sigma}}_3^{-1} = (\mathcal{H}_e\mathcal{H}_e^{\dagger})^{-1} \sigma_s^{-2} + \sigma_e^{-2} \mathbf{I}$. Let $\Omega_e$ be the space of co-variance matrices of the form ${\mathbf{\Sigma}}$, where ${\mathbf{H}}_e$ can be any matrix in the space of eavesdropper matrices $\mathcal{S}_b$: $$\Omega_e = \left\{ {\mathbf{\Sigma}}= (\mathcal{H}_e^\dagger {\mathbf{\Sigma}}_3^{-1} \mathcal{H}_e)^{-1} : \mathcal{H}_e = {\mathbf{I}}_T\otimes{\mathbf{H}}_e, {\mathbf{H}}_e \in \mathcal{S}_e \right\}.$$ By using the definition of the flatness factor, we can show the following:
Let ${\mathbf{\Sigma}}, \bar{{\mathbf{\Sigma}}} \in \Omega_e$ be two matrices satisfying $\left\|\mathbf{{\mathbf{\Sigma}}}-\overline{\mathbf{{\mathbf{\Sigma}}}}\right\| \leq \delta$. If $\delta$ is sufficiently small, then $\overline{{\mathbf{\Sigma}}} - \delta {\mathbf{I}}$ is positive-definite and $$\varepsilon_{\Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}}) \leq \varepsilon_{\Lambda_e^T}(\sqrt{\overline{{\mathbf{\Sigma}}} - \delta {\mathbf{I}}}).$$
For any $\bm{\lambda} \in \mathbb{C}^{n_aT}$ we have $|\bm{\lambda}^\dagger ({\mathbf{\Sigma}}-\overline{{\mathbf{\Sigma}}}) \bm{\lambda}| \leq \left\|\bm{\lambda}\right\|^2 \delta.$ Therefore $$\begin{split}
\varepsilon_{\Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}}) &= \sum_{\bm{\lambda} \in \Lambda^{*}{\backslash\left\{\mathbf{0}\right\}}} e^{-\pi^2 \bm{\lambda}^\dagger {\mathbf{\Sigma}}\bm{\lambda} } \\ &= \sum_{\bm{\lambda} \in \Lambda^{*}{\backslash\left\{\mathbf{0}\right\}}} e^{-\pi^2 \bm{\lambda}^\dagger\left({\mathbf{\Sigma}}-\overline{{\mathbf{\Sigma}}}\right) \bm{\lambda} }e^{-\pi^2 \bm{\lambda}^\dagger\overline{{\mathbf{\Sigma}}} \bm{\lambda} }
\\ &\leq \sum_{\bm{\lambda} \in \Lambda^{*}{\backslash\left\{\mathbf{0}\right\}}} e^{\pi^2 \delta \bm{\lambda}^\dagger \bm{\lambda} }e^{-\pi^2 \bm{\lambda}^\dagger\overline{{\mathbf{\Sigma}}} \bm{\lambda} } = \varepsilon_{\Lambda_e^T}(\sqrt{\overline{{\mathbf{\Sigma}}} - \delta {\mathbf{I}}}).
\end{split}$$
Suppose now that $\mathcal{S}_{\delta}$ is a $\delta$-quantizer for $\Omega_e$ with cardinality $Q_{\delta}$, *i.e.*, for all ${\mathbf{\Sigma}}$ there exists $\overline{{\mathbf{\Sigma}}} \in \mathcal{S}_{\delta}$ such that $\left\|\mathbf{{\mathbf{\Sigma}}}-\overline{\mathbf{{\mathbf{\Sigma}}}}\right\| \leq \delta$. For any ${\mathbf{\Sigma}}$ we have: $$\begin{split}
\mathbb{E}[\varepsilon_{\Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}})] &\leq \mathbb{E}[\varepsilon_{\Lambda_e^T}(\sqrt{\overline{{\mathbf{\Sigma}}}-\delta{\mathbf{I}}})] \leq
\mathbb{E}[\max_{\overline{{\mathbf{\Sigma}}} \in \mathcal{S}_{\delta}}\varepsilon_{\Lambda_e^T}(\sqrt{\overline{{\mathbf{\Sigma}}}-\delta {\mathbf{I}}})] \\ &\leq
\mathbb{E}[\sum_{\overline{{\mathbf{\Sigma}}} \in \mathcal{S}_{\delta}}\varepsilon_{\Lambda_e^T}(\sqrt{\overline{{\mathbf{\Sigma}}}-\delta {\mathbf{I}}})] \\
&\leq Q_{\delta} (1+\varepsilon_T)\left(\frac{\gamma_{\Lambda_e^T}(\sqrt{\overline{{\mathbf{\Sigma}}}-\delta{\mathbf{I}}})}{\pi}\right)^{n_aT} \\ &= Q_{\delta}(1+\varepsilon_T) f(\delta) \left(\frac{\gamma_{\Lambda_e^T}(\sqrt{{{\mathbf{\Sigma}}}})}{\pi}\right)^{n_aT},
\end{split}$$ where $$f(\delta) = \frac{|{\mathbf{\Sigma}}|}{|{\mathbf{\Sigma}}-\delta {\mathbf{I}}|} = \frac{1}{|{\mathbf{I}}- \delta {\mathbf{\Sigma}}^{-1}|}.$$
The last upper bound is universal, in the sense that it does not depend on the specific realization ${\mathbf{H}}_e$. Note that if the VNR condition is satisfied, namely $\gamma_{\Lambda_e^T}(\sqrt{{{\mathbf{\Sigma}}}}) < \pi$, then the term $(\gamma_{\Lambda_e^T}(\sqrt{{{\mathbf{\Sigma}}}})/\pi)^{n_aT}$ decays exponentially in $T$ with exponent given by $$c_1 = -n_a \log (\gamma_{\Lambda_e^T}(\sqrt{{{\mathbf{\Sigma}}}})/\pi).$$ From this, we obtain the bound $$\begin{split}
\mathbb{E}[\varepsilon_{\Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}})] &\leq (1+\varepsilon_T)Q_{\delta} e^{-T \left(c_1 - \frac{\log f(\delta)}{T}\right)} \\ &= (1+\varepsilon_T)Q_{\delta} e^{-T \left(c_1 - {\log |{\mathbf{I}}-\delta(\sigma_{e}^{-2}{\mathbf{H}}_e^\dagger {\mathbf{H}}_e + \sigma_s^{-2} {\mathbf{I}}) | }\right)},
\end{split}
\label{eq:boundAverageFlatnessFactor}$$ which holds for any ${\mathbf{\Sigma}}\in \Omega_e$. We can therefore choose a small $\delta$ (independently of $T$) such that the total exponent is negative. Since $Q_{\delta}$ does not depend on $T$, and $\varepsilon_T$ can be made arbitrarily small, we obtain an exponential decay of the flatness factor.
Simultaneous Goodness {#ref:appSimultaneous}
=====================
From Section \[sec:universallyFlat\], the construction of universally secure codes boils down to finding a sequence of pairs of nested lattices $\Lambda_{b}^{T} \subset \Lambda_{e}^{T}$ such that
- $\Lambda_{b}^{T}$ has vanishing probability of error: $\mathbb{P}_{\Lambda_b}({\mathbf{R}}_b) \triangleq \mathbb{P}(\tilde{{\mathbf{W}}}_{b,{\text{eff}}} \notin \mathcal{V}({\mathbf{R}}_b\Lambda_b^T)) \to 0$ as $T \to \infty$;
- $\Lambda_{e}^T$ has vanishing flatness factor: $\varepsilon_{\Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}}) \to 0$ as $T \to \infty$,
where we recall that $\tilde{{\mathbf{W}}}_{b,{\text{eff}}}$ is the effective noise, sub-Gaussian with co-variance matrix $\sigma_b^2 {\mathbf{I}}$, ${\mathbf{R}}_b^\dagger {\mathbf{R}}_b = {\mathbf{H}}_b^\dagger {\mathbf{H}}_b + \rho_b^{-1} {\mathbf{I}}$, and $${\mathbf{\Sigma}}^{-1} = \mathcal{H}_e^\dagger {\mathbf{\Sigma}}_3^{-1} \mathcal{H}_e, \mbox{with } {\mathbf{\Sigma}}_3^{-1} = (\mathcal{H}_e\mathcal{H}_e^{\dagger})^{-1} \sigma_s^{-2} + \sigma_e^{-2} \mathbf{I}.$$ First suppose that $\mathbf{R}_b$ and ${\mathbf{\Sigma}}$ are fixed. Let $\Lambda_b^T = \Lambda(\mathcal{C}_b)$ be obtained by choosing $\mathcal{C}_b$ uniformly in the set of all codes with parameters $(T,k_b,p)$. Let $\Lambda_e^T = \Lambda(\mathcal{C}_e)$ be obtained by expurgating $k_b-k_e$ columns from $\mathcal{C}_b$. With this process $\mathcal{C}_e$ will be also chosen uniformly from all $(T,k_e,p)$ codes. We have: $$\label{eq:simult-good}
\begin{split}&\mathbb{E}_{\mathcal{C}_b}[\max\left\{\mathbb{P}_{\Lambda_b^T}({\mathbf{R}}_b),\varepsilon_{\Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}})\right\}] \\ &\leq \mathbb{E}_{\mathcal{C}_b}[\mathbb{P}_{\Lambda_b^T}({\mathbf{R}}_b) +\varepsilon_{\Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}}) ]\\ &= \mathbb{E}_{\mathcal{C}_b}[\mathbb{P}_{\Lambda_b^T}({\mathbf{R}}_b)] +\mathbb{E}_{\mathcal{C}_e}[\varepsilon_{\Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}}) ] \to 0.
\end{split}$$ Convergence of both terms in the last equation is guaranteed to be exponentially fast. Indeed:
- The term $\mathbb{E}_{\mathcal{C}_b}[\mathbb{P}_{\Lambda_b^T}({\mathbf{R}}_b)]$ tends to zero exponentially provided that $\gamma_{{\mathbf{R}}_b}(\Lambda_b) > \pi e$, due to AWGN-goodness of $\Lambda_b^T$.
- The term $\mathbb{E}_{\mathcal{C}_e}[\varepsilon_{\Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}}) ]$ tends to zero exponentially provided that $\gamma_{\Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}}) \to 0$, due to Appendix \[app:2\], Equation .
Furthermore, by considering the quantized channel spaces, similarly to Appendix \[app:2\], we conclude that the convergence is universal. Therefore, there exists a pair of lattices ${\Lambda_b^T,\Lambda_e^T}$ where $\Lambda_b^T$ is universally AWGN-good and $\Lambda_e^T$ is universally secrecy-good, and Theorem \[thm:achievableRatesFinal\] follows.
Although the above argument only demonstrates the existence of a pair of good lattices, it is possible to show a concentration result on the performance of the ensemble of nested lattices. Suppose some exponential bound $e^{-c T}$ on for some $c > 0$. Then, using Markov’s inequality, we have that for the ensemble of nested lattices considered, $$\mathbb{P}(\mathbb{P}_{\Lambda_b^T}({\mathbf{R}}_b) +\varepsilon_{\Lambda_e^T}(\sqrt{{\mathbf{\Sigma}}}) > e^{-(c-c') T}) < e^{-c' T}, \ \ \forall \ 0<c'<c.$$ That is, with probability higher than $1-e^{-c' T}$ over the choice of $\mathcal{C}_b$, stays below $e^{-(c-c') T}$. In other words, most of these nested lattices have a performance concentrating around $e^{-c T}$.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to thank Laura Luzzi and Roope Vehkalahti for helpful discussions.
[^1]: This work was presented in part at the International Zurich Seminar on Communications (IZS) 2018 and in part at the International Symposium on Turbo Codes and Iterative Information Processing (ISTC) 2016.
A. Campello is with the Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, U.K. (e-mail: a.campello@imperial.ac.uk).
C. Ling is with the Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, U.K. (e-mail: cling@ieee.org).
J.-C. Belfiore is with the Mathematical and Algorithmic Sciences Lab, France Research Center, Huawei Technologies (e-mail: belfiore@telecom-paristech.fr).
[^2]: Previous works [@BO_TComm; @LVL16] required that the number of the eavesdropper’s antennas be greater than or equal to $n_a$.
[^3]: More specifically, it is required that that the sequence of lattices corresponding to the kernels of $\psi$ has a non-vanishing Hermite parameter.
|
---
abstract: 'We consider a spin system containing pure two spin Sherrington-Kirkpatrick Hamiltonian with Curie-Weiss interaction. The model where the spins are spherically symmetric was considered by @Baiklee16 and @Baikleewu18 which shows a two dimensional phase transition with respect to temperature and the coupling constant. In this paper we prove a result analogous to @Baiklee16 in the “paramagnetic regime" when the spins are i.i.d. Rademacher. We prove the free energy in this case is asymptotically Gaussian and can be approximated by a suitable linear spectral statistics. Unlike the spherical symmetric case the free energy here can not be written as a function of the eigenvalues of the corresponding interaction matrix. The method in this paper relies on a dense sub-graph conditioning technique introduced by @Ban16. The proof of the approximation by the linear spectral statistics part is close to @Banerjee2017.'
author:
- |
Debapratim Banerjee\
Dept. of Statistics\
University of Pennsylvania\
dban@wharton.upenn.edu\
bibliography:
- 'PAR\_SPI.bib'
title: 'Fluctuation of the free energy of Sherrington-Kirkpatrick model with Curie-Weiss interaction: the paramagnetic regime'
---
Introduction
============
The model description
---------------------
We at first give the description of the model. We start with a symmetric matrix $A=\left( A_{i,j} \right)_{i,j=1}^{n}$ where the entries in the strict upper triangular part of $A$ are i.i.d. standard Gaussian and for simplicity one might take $A_{i,i}=0$. The Hamiltonian corresponding to the Sherrington-Kirkpatrick model without any external field is given by $$\label{SK:hamiltonian}
H_{n}^{SK}(\sigma):= \frac{1}{\sqrt{n}}\langle \sigma, A\sigma \rangle =\frac{1}{\sqrt{n}} \sum_{i,j} A_{i,j} \sigma_{i} \sigma_{j}= \frac{2}{\sqrt{n}} \sum_{1\le i< j \le n} A_{i,j} \sigma_{i}\sigma_{j}.$$ Here $\sigma_{i}$’s are called spins and in this paper we shall only consider the case when $\sigma_{i} \in \{ -1, 1 \}$ for each $i$. In particular, one might consider the case when the spins $\sigma_{i}$’s are i.i.d. Rademacher random variables. This is known as the classical Sherrington- Kirkpatrick model. This model has got significant amount of interest in the study of spin glasses over the last few decades. Celebrated result like the proof of Parisi formula is considered one of the major advancements in this field. One might look at @Panbook, @Tal05 for some information in this regard.
However the main focus of this paper is the following Hamiltonian $$\label{our:hamiltonian}
H_{n}(\sigma):= H_{n}^{SK}(\sigma)+ H_{n}^{CW}(\sigma)$$ where the Curie-Weiss Hamiltonian with coupling constant $J$ is defined by $$\label{cw:hamiltonian}
H_{n}^{CW}(\sigma):= \frac{J}{n} \sum_{i,j=1}^{n} \sigma_{i}\sigma_{j} = \frac{J}{n}\left( \sum_{i=1}^{n} \sigma_{i} \right)^2.$$ Note that model corresponding to the Hamiltonian $H_{n}^{CW}(\sigma)$ is a simple model which can be studied in considerable details. One might look at @54 for a reference.
This Hamiltonian in was defined in @Talbook(see (4.22)) as a model which have difficulties of the ferromagnetic interactions however with a familiar disorder. This model was introduced as a prelude to study the Hopfield model in @Talbook.
The main result of this paper is a limit theorem for the free energy corresponding to the Hamiltonian $H_{n}(\sigma)$ when $\beta< \frac{1}{2}$ and $\beta J < \frac{1}{2}$ whenever $\sigma_{i}$’s are i.i.d. Rademacher variables. If the spins $\mathbf{\sigma}=(\sigma_{1},\ldots,\sigma_{n})$ are distributed according to the uniform measure on the sphere $S_{n-1}$ where $S_{n-1}:= \left\{ \sigma \in \mathbb{R}^{n}~|~ ||\sigma||^2=n \right\}$, then the analogous Hamiltonian was considered in @Baiklee16 and @Baikleewu18. However the results in @Baiklee16 are much more general than the current paper in the sense they are able to consider any $\beta>0,J>0$. Depending on the values of $\beta, J$, there are three distinct regimes where the free energy shows different behaviors. In particular, the regime $\beta< \frac{1}{2}$ and $\beta J < \frac{1}{2}$ is known as the para-magnetic regime where the result analogous to this paper was obtained in @Baiklee16. The regime when $\beta> \frac{1}{2}$ and $J<1$ is known as the spin glass regime and the other case ($\beta J > \frac{1}{2} $ and $J>1$) is known as the ferromagnetic regime. Although the results in @Baiklee16 are much more general than the current paper in terms of possible choices of $(\beta,J)$, the technique of that paper is restricted to the case when the spins $\mathbf{\sigma}=(\sigma_{1},\ldots,\sigma_{n})$ are distributed according to the uniform measure on the sphere $S_{n-1}$ which does not cover the case when $\sigma_{i}$’s are i.i.d. Rademacher random variables. This is the problem we consider in this paper.
We now give a very brief overview of the literature for the fluctuation of free energy of classical Sherrington-Kirkpatrick model in with absence of any external field.
The classical Sherrington-Kirkpatrick model with no external field ($h=0$) under goes a phase transition at $\beta= \frac{1}{2}$. When the spins $\sigma_{i}$’s are i.i.d. Rademacher and $\beta< \frac{1}{2}$ the free energy has a Gaussian limiting distribution. One might look at @ALR87 and @CoNe95 for some references. The case $\beta> \frac{1}{2}$ is known as the low temperature regime. To the best of our limited knowledge, very few things are known about the fluctuations of the free energy in this regime. One might look at @Cha17 where it is proved that the fluctuation of the free energy of the Sherrington-Kirkpatrick model is at least $O(1)$. When the spins are uniformly distributed on $S_{n-1}$, the free energy analogously undergoes a phase transition at $\beta=\frac{1}{2}$. When $\beta< \frac{1}{2}$, the free energy has a Gaussian limiting distribution and can be approximated by a linear spectral statistics of the eigenvalues. The low temperature case ($\beta> \frac{1}{2}$) is also well-known. Here the free energy has a limiting GOE Tracy-Widom distribution with $O\left( n^{-\frac{2}{3}} \right)$ fluctuations. One might look at @Baikleestat for a reference.
Finally, The model considered in this paper (Hamiltonian defined in ) was also considered in @Chen12 from the point of view of thermodynamic limit of the free energy. One might also look at @CadelR where this model was studied using cavity method. However to the best of our limited knowledge the problem of fluctuation of free energy remained open.
Preliminary definitions
-----------------------
We now give some preliminary definitions. We start with defining a Hamiltonian which generalizes the one defined in .
(interactions)\[def:ourham\] Suppose $A_{i,j}$, $1\le i \le j \le n$ be i.i.d. standard Gaussian random variables. Set $A_{j,i}=A_{i,j}$ for $i<j$. Let $M_{i,j}= \frac{1}{\sqrt{n}}A_{i,j}+ \frac{J}{n}$ and $M_{i,i}= \frac{1}{\sqrt{n}}A_{i,i}+ \frac{J'}{n} $ for some $n$ independent non negative fixed constants $J$ and $J'$. One considers the Hamiltonian $H_{n}(\sigma)= \langle \sigma, M\sigma \rangle$. The defined Hamiltonian is more general than the one defined in in the following sense. Definition \[def:ourham\] allows $A_{i,i}$ to be non zero random variables with $J'$ being any arbitrary constant.
(Partition function and Free energy) Given any Hamiltonian $H_{n}(\sigma)$ where $\sigma=(\sigma_{1},\ldots, \sigma_{n})$ are distributed according to a measure $\Psi_{n}$, the partition function and free energy at an inverse temperature $\beta$ are denoted by $Z_{n}(\beta)$ and $F_{n}(\beta)$ respectively and defined as follows. $$\label{eq:partition}
Z_{n}(\beta) := \int \exp\left\{\beta H_{n}(\sigma) \right\} d \Psi_{n}(\sigma)$$ and $$F_{n}(\beta) : =\frac{1}{n} \log\left(Z_{n}(\beta)\right).$$ In our case we take $\Psi_{n}$ to be the uniform probability measure on the Hypercube $\{-1, +1 \}^n$.
In our case $Z_{n}(\beta)$ is as follows: $$\begin{split}
Z_{n}(\beta)&= \sum_{\sigma \in \{ -1, +1 \}^{n}}\frac{1}{2^{n}}\exp\left\{ \sum_{i,j=1}^{n}\frac{\beta}{\sqrt{n}} A_{i,j}\sigma_{i}\sigma_{j} + \frac{\beta J}{n} \sum_{i,j=1}^{n} \sigma_{i}\sigma_{j}+ \beta(J'-J)\right\}\\
&= \sum_{\sigma \in \{ -1, +1 \}^{n}}\frac{1}{2^{n}} \exp\left\{ \frac{2\beta}{\sqrt{n}}\sum_{1\le i<j\le n}^{n}A_{i,j}\sigma_{i}\sigma_{j} + \frac{2\beta J}{n} \sum_{1\le i<j\le n}\sigma_{i}\sigma_{j} + \frac{\beta}{\sqrt{n}}\sum_{i=1}^{n}\left(A_{i,i}+\frac{J'}{\sqrt{n}}\right)\right\}.
\end{split}$$ Finally we define the Wasserstein distance between two distribution functions. This distance is crucially used at many places of the proofs.
\[Wass\] We at first fix $p \ge 1$. Suppose $F^{1}$ and $F^{2}$ are two distribution functions such that $\int_{x \in \mathbb{R}} |x|^{p}d F^{1}(x)< \infty$ and $\int_{x \in \mathbb{R}} |x|^{p}dF^{2}(x)< \infty$. Then the Wasserstein distance for $p$ between $F^{1}$ and $F^{2}$ is is denoted by $W_{p}$ and defined to be $$W_{p}\left( F^{1}, F^{2} \right):= \left[\inf_{X \sim F^{1}; Y \sim F^{2}}\operatorname{E}\left[ \left| X- Y \right|^{p} \right]\right]^{\frac{1}{p}}.$$
Observe that the Wasserstein distance is defined for two distribution functions. However when we write $\left[\inf_{X \sim F^{1}; Y \sim F^{2}}\operatorname{E}\left[ \left| X- Y \right|^{p} \right]\right]^{\frac{1}{p}}$, we consider two random variables $X\sim F^{1}$ and $Y \sim F^{2}$ such that $X$ and $Y$ are defined on the same measure space such that $\operatorname{E}\left[ \left| X- Y \right|^{p}\right]$ takes the lowest possible value.
The following result on the Wasserstein distance is well known.
\[prop:wass\] Suppose $\left\{X_{n}\right\}_{n=1}^{\infty}$ be a sequence of random variables and $X$ be a random variable. If $W_{2}\left( F^{X_{n}}, F^{X}\right) \to 0$, then $X_{n} \stackrel{d}{\to} X$ and $\operatorname{E}[X_{n}^2] \to \operatorname{E}[X^2]$.
One might see @Mal72 for a reference.
Main result
===========
We are ready to state the main result of this paper.
\[thm:asymptotic\]
1. (Asymptotic normality) Consider the Hamiltonian $H_{n}(\sigma)$ as defined in Definition \[def:ourham\]. Let $F_{n}(\beta)$ be the free energy corresponding to the Hamiltonian $H_{n}(\sigma)$. When $\beta < \frac{1}{2}$ and $\beta J< \frac{1}{2}$ the following result holds: $$n\left(F_{n}(\beta)- F(\beta)\right) \stackrel{d}{\to} N(f_{1},\alpha_{1})$$ where $F(\beta)= \beta^2$, $$\alpha_{1}= -\beta^2- \frac{1}{2}\log\left(1- 4\beta^2\right)$$ and $$f_{1}= -\frac{1}{2} \log\left(1- 2\beta J\right) + \beta(J'- J) + \frac{1}{4}\log\left(1- 4\beta^2\right).$$
2. (Approximation by signed cycle counts) For any sequence $m_{n}$ diverging to infinity such that $m_{n}=o\left( \sqrt{\log n} \right)$, one also has the following approximation result for the log partition function $\log\left(Z_{n}(\beta)\right)$. $$\begin{split}
&\log\left(Z_{n}(\beta)\right) + \frac{1}{2} \log\left(1- 2\beta J\right)- (n-1)\beta^2 + \beta (J -J') - \beta C_{n,1} -\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \sum_{k=2}^{m_{n}}\frac{2(2\beta)^{k}\left(C_{n,k}- (n-1)\mathbb{I}_{k=2}\right)- (2\beta)^{2k}}{4k} \stackrel{p}{\to} 0.
\end{split}$$ Here the statistics $C_{n,k}$’s are taken according to Definition \[def:signedcycles\].
Our next result is Theorem \[thm:asymptotic\] where the approximations of $C_{n,k}$’s by linear spectral statistics are stated. Before going to Theorem \[Thm:approximation\], we need some important definitions. We now introduce an important generating function. Given any $r \in \mathbb{N}$, let $$\label{gen:f}
\left( \frac{1-\sqrt{1-4z^2}}{2z} \right)^r= \sum_{m=r}^{\infty} f(m,r) z^{m}.$$ The coefficients $f(m,r)$’s are key quantities for defining the variances and covariances of linear spectral statistics constructed from different power functions. For any $k \in \mathbb{N}$ denote $$\label{def:catalan}
\psi_{k}=\left\{
\begin{array}{ll}
0 & \text{if $k$ is odd}\\
\frac{1}{\frac{k}{2}+1} \binom{k}{\frac{k}{2}} & \text{if $k$ is even}.
\end{array}
\right.$$ So $\psi_k$ is the $\frac{k}{2}$-th Catalan number for every even $k$. Finally, we define a set of rescaled Chebyshev polynomials. These polynomials are important for drawing the connection between signed cycles $C_{n,k}$’s and the spectrum of adjacency matrix. The standard Chebyshev polynomial of degree $m$ is denoted by $S_{m}(x)$ and can be defined by the identity $$\label{def:ChebyshevI}
S_{m}\left(\cos(\theta)\right) = \cos(m\theta).$$ In this paper we use a slight variant of $S_m$, denoted by $P_{m}$ and defined as $$\label{def:ChebyshevII}
P_{m}(x)= 2S_{m}\left(\frac{x}{2}\right).$$ In particular, $P_{m}(2\cos(\theta)) = 2 \cos(m\theta)$. It is easy to note that $P_{m}\left( z+ z^{-1}\right)= z^{m} + z^{-m}$ for all $z \in \mathbb{C}$. One also notes that $P_{m}(\cdot)$ is even and odd whenever $m$ is even or odd respectively.
\[Thm:approximation\](Approximation of cycles by linear spectral statistics) Let $\tilde{A}$ be the matrix obtained by putting $0$ on the diagonal of the matrix $A$. Let $P_{k}$ be as defined in . Then to following is true for any $3\le k= o\left( \sqrt{\log n}\right)$ under $\mathbb{P}_{n}$. $$C_{n,k}- \left\{\operatorname{Tr}\left( P_{k}\left( \frac{1}{\sqrt{n}} \tilde{A} \right) \right)- \operatorname{E}\left[ \operatorname{Tr}\left( P_{k}\left( \frac{1}{\sqrt{n}}\tilde{A} \right) \right) \right]\right\} \stackrel{p}{\to} 0.$$ Here for any function $f$ and a matrix $A$ $$\operatorname{Tr}\left[ f(A) \right]=\sum_{i=1}^{n} f(\lambda_{i})$$ where $\lambda_{1},\ldots,\lambda_{n}$ are the eigenvalues of the matrix $A$.
The proof of Theorem \[Thm:approximation\] is given in Section \[sec:thmapp\].
Proof techniques and related definitions
========================================
As mentioned earlier, the fundamental technique of the proof of Theorem \[thm:asymptotic\] is completely different from that of @Baiklee16. The proof in the current paper is based on the dense sub graph conditioning technique introduced in @Ban16. The fundamental idea is to view the free energy as the log of the Radon-Nikodym derivative $\left(\log\frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}}\right)$ of two suitably defined sequences of measures $\mathbb{P}_{n}$ and $\mathbb{Q}_{n}$. Now one introduces a class of random variables called the signed cycles (Definition \[def:signedcycles\]) and prove that these variables asymptotically determine the full Radon-Nikodym derivative. This is done by a fine second moment argument. The argument in this part is highly motivated from a paper by @Jan where it is proved that a similar kind of argument holds for random regular graphs where the signed cycle counts are replaced by standard cycle counts. The technique of cycle conditioning was also used in @MNS12 in their proof of contiguity of the probability measures induced by a planted partition model and the Erdős- Rényi model in the sparse regime.
We now start with defining the signed cycles random variables.
\[def:signedcycles\] Let $A$ be a $n \times n$ symmetric matrix with with the strict upper triangular part being i.i.d. mean $0$ and variance $1$. For $k\ge 2$, we define the signed cycles random variables $C_{n,k}$ as follows: $$C_{n,k}:= \left( \frac{1}{\sqrt{n}} \right)^{k}\sum_{i_0,i_1,\ldots,i_{k-1}}A_{i_{0},i_{1}} A_{i_{1},i_{2}} \ldots A_{i_{k-1},i_{0}}.$$ Here $i_{0},\ldots,i_{k-1}$ are taken to be all distinct. For $k=1$, $C_{n,k}$ is simply defined as follows: $$C_{n,1} := \left( \frac{1}{\sqrt{n}} \right) \sum_{i} A_{i,i}.$$
In this paper we require the concept of mutual contiguity of two sequences of measures heavily. Now we define these concepts. If someone is interested one might have a look at @LeCam and @LeCam00 for general discussions on contiguity.
(Contiguity) For two sequences of probability measures $\left\{\mathbb{P}_n\right\}_{n=1}^{\infty}$ and $\left\{\mathbb{Q}_n\right\}_{n=1}^{\infty}$ defined on $\sigma$-fields $(\Omega_n,\mathcal{F}_n)$, we say that $\mathbb{Q}_n$ is contiguous with respect to $\mathbb{P}_n$, denoted by $\mathbb{Q}_n \triangleleft \mathbb{P}_n$, if for any event sequence $A_n$, $\mathbb{P}_n(A_n)\to 0$ implies $\mathbb{Q}_n(A_n)\to 0$. We say that they are (asymptotically) mutually contiguous, denoted by $\mathbb{P}_n \triangleleft\triangleright \mathbb{Q}_n$, if both $\mathbb{Q}_n\triangleleft \mathbb{P}_n$ and $\mathbb{P}_n\triangleleft \mathbb{Q}_n$ hold.
The following result gives an useful way to study mutual contiguity:
\[prop:useI\] Suppose that $L_n=\frac{d\mathbb{Q}_n}{d\mathbb{P}_n}$, regarded as a random variable on $(\Omega_n,\mathcal{F}_n,\mathbb{P}_n)$, converges in distribution to some random variable $L$ as $n \to \infty$. Then $\mathbb{P}_n$ and $\mathbb{Q}_n$ are mutually contiguous if and only if $L > 0$ a.s. and $\operatorname{E}[L] = 1$.
One might look at Proposition 3 of @Jan for a proof.
We now state a result on mutual contiguity of measures.
\[prop:norcont\](Janson’s second moment method): Let $\mathbb{P}_n$ and $\mathbb{Q}_n$ be two sequences of probability measures such that for each $n$, both are defined on the common $\sigma$-algebra $(\Omega_n, \mathcal{F}_n)$. Suppose that for each $i\geq 1$, $W_{n,i}$ are random variables defined on $(\Omega_n,\mathcal{F}_n)$. Then the probability measures $\mathbb{P}_n$ and $\mathbb{Q}_n$ are asymptotically mutually contiguous if the following conditions hold simultaneously:
(i) $\mathbb{Q}_n$ is absolutely continuous with respect to $\mathbb{P}_n$ for each $n$;
(ii) The likelihood ratio statistic $Y_n = \frac{\mathrm{d}\mathbb{Q}_n}{\mathrm{d}\mathbb{P}_n}$ satisfies $$\label{eq:lr-square}
\limsup_{n\to\infty}\operatorname{E}_{\mathbb{P}_n}\left[Y_n^2\right] \leq
\exp\left\{\sum_{i=1}^\infty \frac{\mu_i^2}{\sigma_i^2}\right\} < \infty.$$
(iii) For any fixed $k\ge 1$, one has $\left( W_{n,1},\ldots, W_{n,k} \right)|\mathbb{P}_{n} \stackrel{d}{\to} \left(Z_{1}, \ldots, Z_{k}\right) $ and $\left( W_{n,1},\ldots, W_{n,k} \right)|\mathbb{Q}_{n} \stackrel{d}{\to} \left(Z'_{1}, \ldots, Z'_{k}\right)$. Further $Z_{i}\sim N(0,\sigma_{i}^{2})$ and $Z'_{i} \sim N(\mu_{i},\sigma_{i}^{2})$ are sequences of independent random variables.
(iv) Under $\mathbb{P}_{n}$, $W_{n,i}$’s are uncorrelated and there exists a sequence $m_{n} \to \infty$ such that $$\operatorname{Var}\left[ \sum_{i=1}^{m_{n}} \frac{\mu_{i}}{\sigma_{i}^2} W_{n,i} \right] \to C < \infty$$ Here the $\operatorname{Var}$ is considered with respect to the measure $\mathbb{P}_{n}$.
In addition, we have that under $\mathbb{P}_n$, $$\label{eq:lr-limit}
Y_n \stackrel{d}{\to} \exp\left\{\sum_{i=1}^\infty \frac{\mu_i Z_i - \frac{1}{2}\mu_i^2}{\sigma_i^2}\right\}.$$ Furthermore, given any $\epsilon,\delta>0$ there exists a natural number $K=K(\delta,\epsilon)$ such that for any sequence $n_l$ there is a further subsequence $n_{l_m}$ such that $$\label{eq:janson-decomp}
\limsup_{m\to\infty} \mathbb{P}_{n_{l_m}}\left( \left| \log(Y_{n_{l_m}}) -
\sum_{k=1}^{K} \frac{2\mu_{k} W_{n_{l_m},k} -\mu_{k}^{2}}{2\sigma_{k}^{2}}
\right|
\ge \epsilon \right) \le \delta.$$
Proposition \[prop:norcont\] is one of the most important results required for the proof of Theorem \[thm:asymptotic\]. In particular, the rest of the proof relies on defining the measures $\mathbb{P}_{n}$ and $\mathbb{Q}_{n}$ and $W_{n,i}$’s properly. It is worth noting that in this context the statistics $C_{n,i}$’s serve as $W_{n,i}$’s.
We now give the proof of Proposition \[prop:norcont\].
**Proof of Proposition \[prop:norcont\]:**
#### Proof of mutual contiguity and
This proof is broken into two steps. We focus on proving . Given , mutual contiguity is a direct consequence of Proposition \[prop:useI\].
**Step 1.** We first prove the random variable on the right hand side of is almost surely positive and has mean $1$. Let us define $$L:=\exp\left\{ \sum_{i=1}^{\infty}\frac{2\mu_{i}Z_{i}-\mu_{i}^{2}}{2\sigma_{i}^{2}} \right\},\qquad
L^{(m)}:= \exp\left\{ \sum_{i=1}^{m}\frac{2\mu_{i}Z_{i}-\mu_{i}^{2}}{2\sigma_{i}^{2}} \right\},\quad
m\in \mathbb{N}.$$ As $Z_i \sim N(0,\sigma_{i}^{2})$, for any $i\in\mathbb{N}$, and so $$\operatorname{E}\left[{\exp}
\left\{\frac{2\mu_{i}Z_{i}-\mu_{i}^{2}}{2\sigma_{i}^{2}} \right\}\right]=1.$$ So $\{L^{(m)}\}_{m=1}^{\infty}$ is a martingale sequence and $$\operatorname{E}\left[ \big(L^{(m)}\big)^2 \right]=\prod_{i=1}^{m} \exp\left\{ \frac{\mu_{i}^{2}}{\sigma_{i}^{2}} \right\}=\exp\left\{ \sum_{i=1}^{m} \frac{\mu_{i}^{2}}{\sigma_{i}^{2}} \right\}.$$ Now by the righthand side of , $L^{(m)}$ is a $L^2$ bounded martingale. Hence, $L$ is a well defined random variable with $$\operatorname{E}[L] = 1,\qquad
\operatorname{E}[L^2]= \exp\left\{ \sum_{i=1}^{\infty} \frac{\mu_{i}^{2}}{\sigma_{i}^{2}} \right\}.$$ On the other hand $\log(L)$ is a limit of Gaussian random variables, hence $\log(L)$ is Gaussian with $$\operatorname{E}[\log(L)]= -\frac{1}{2} \sum_{i=1}^{\infty} \frac{\mu_{i}^{2}}{\sigma_{i}^{2}}, \qquad
\operatorname{Var}( \log(L) ) = \sum_{i=1}^{\infty} \frac{\mu_{i}^{2}}{\sigma_{i}^{2}}.$$ Hence $\mathbb{P}(L=0)= \mathbb{P}(\log(L)=-\infty)=0$.
**Step 2.** Now we prove $Y_n \stackrel{d}{\to} L$. Since $$\limsup_{n \to \infty}\operatorname{E}_{\mathbb{P}_n}\left[ Y_n^2\right]<\infty,$$ condition (iv) implies that the sequence $Y_n$ is tight. Prokhorov’s theorem further implies that there is a subsequence $\{ n_{k} \}_{k=1}^{\infty}$ such that $Y_{n_k}$ converge in distribution to some random variable $L(\{ n_{k} \})$. In what follows, we prove that the distribution of $L(\{ n_{k} \})$ does not depend on the subsequence $\{ n_{k} \}$. In particular, $L(\{ n_{k} \})\stackrel{d}{=} L$. To start with, note that since $Y_{n_k}$ converges in distribution to $L(\{ n_{k} \})$, for any further subsequence $\{ n_{k_l} \}$ of $\{ n_{k}\}$, $Y_{n_{k_l}}$ also converges in distribution to $L(\{ n_{k} \})$.
Given any fixed $\epsilon>0$ take $m$ large enough such that $$\exp\left\{ \sum_{i=1}^{\infty} \frac{\mu_{i}^{2}}{\sigma_{i}^{2}} \right\}-\exp\left\{ \sum_{i=1}^{m} \frac{\mu_{i}^{2}}{\sigma_{i}^{2}} \right\} < \epsilon.$$ For this fixed number $m$, consider the joint distribution of $(Y_{n_{k}},W_{n_k,1},\ldots,W_{n_k,m})$. This sequence of $m+1$ dimensional random vectors with respect to $\mathbb{P}_{n_k}$ is tight by condition (ii). So it has a further subsequence such that $$(Y_{n_{k_l}},W_{n_{k_l},1},\ldots,W_{n_{k_l},m})|\mathbb{P}_{n_{k_l}}\stackrel{d}{\to}\left((H_1,\ldots,H_{m+1})\in (\Omega(\{ n_{k_l} \}),\mathcal{F}(\{ n_{k_l} \}),P(\{ n_{k_l} \}))(say).\right).$$ where $H_{1}\stackrel{d}{=} L(\{ n_{k} \})$ and $\left(H_{2},\ldots, H_{m+1}\right)\stackrel{d}{=}\left( Z_{1},\ldots,Z_{m}\right)$ We are to show that we can define the random variables $L^{(m)}$ and $L(\{ n_{k} \})$ in such a way that there exist suitable $\sigma$-algebras $\mathcal{F}_1 \subset\mathcal{F}_2$ such that $L^{(m)} \in \mathcal{F}_1$, $L(\{ n_{k} \}) \in \mathcal{F}_2$, and $\operatorname{E}\left[ L(\{ n_{k} \})\left|\right. \mathcal{F}_{1} \right]= L^{(m)}$.
Since $\limsup_{n \to \infty}\operatorname{E}_{\mathbb{P}_n}\left[ Y_n^2\right]< \infty$, the sequence $Y_{n_{k_l}}$ is uniformly integrable. This, together with condition (i), leads to $$\label{eqn_expder}
\operatorname{E}[L(\{ n_{k} \})] = \lim_{l\to\infty}
\operatorname{E}_{\mathbb{P}_{n_{k_l}}} [ Y_{n_{k_l}} ] = 1.
$$ Now take any positive bounded continuous function $f:\mathbb{R}^m \to \mathbb{R}$. By Fatou’s lemma $$\label{eqn_ineq}
\liminf_{l\to\infty}
\operatorname{E}_{\mathbb{P}_{n_{k_l}}} \left[f (W_{n_{k_l},1},\ldots, W_{n_{k_l},m} )Y_{n_{k_l}} \right] \ge \operatorname{E}\left[ f\left(Z_1,\ldots,Z_{m}\right)L(\{ n_{k} \})\right].$$ However for any constant $\xi$, implies $\xi=\xi\operatorname{E}_{\mathbb{P}_{n_{k_l}}}[ Y_{n_{k_l}} ] \to \xi\operatorname{E}[L(\{ n_{k} \})]= \xi$. Observe that given any bounded continuous function $f$ we can find $\xi$ large enough so that $f+ \xi$ is a positive bounded continuous function. So is indeed implied by Fatou’s lemma.
Now $$\begin{split}
&\liminf \operatorname{E}_{\mathbb{P}_{n_{k_l}}} \left[\left(f (W_{n_{k_l},1},\ldots,W_{n_{k_l},m} )+ \xi\right)Y_{n_{k_l}} \right]\\
&=\liminf \operatorname{E}_{\mathbb{P}_{n_{k_l}}}\left[ f(W_{n_{k_l},1},\ldots,W_{n_{k_l},m} ) Y_{n_{k_l}} \right] +\xi\\
& \ge \operatorname{E}\left[ \left(f(Z_{1},\ldots,Z_{m} )+\xi\right) L(\{ n_{k} \}) \right]
\end{split}$$ So holds for any bounded continuous function $f$. On the other hand, replacing $f$ by $-f$ we have $$\label{eqn_ineqII}
\lim_{l\to\infty}
\operatorname{E}_{\mathbb{P}_{n_{k_l}}}
\left[f(W_{n_{k_l},1},\ldots,W_{n_{k_l},m})Y_{n_{k_l}} \right]
= \operatorname{E}\left[ f(Z_1,\ldots,Z_{m})L(\{ n_{k} \})\right].$$ Now condition (ii) leads to $$\int f(W_{n_{k_l},1},\ldots,W_{n_{k_l},m})Y_{n_{k_l}} \mathrm{d}\mathbb{P}_{n_{k_l}}= \int f(W_{n_{k_l},1},\ldots,W_{n_{k_l},m})\mathrm{d}\mathbb{Q}_{n_{k_l}} \to \int f(Z_1',\ldots,Z_{m}') \mathrm{d}Q.$$ Here $Q$ is the measure induced by $(Z_1',\ldots,Z_{m}')$. In particular, one can take the measure $Q$ such that $(Z_1,\ldots, Z_{m})$ themselves are distributed as $(Z_1',\ldots,Z_{m}')$ under the measure $Q$. This is true since $$\int f(Z_1',\ldots, Z_{m}') \mathrm{d}Q= \operatorname{E}\left[ f(Z_1,\ldots, Z_{m}) L^{(m)}\right].$$ for any bounded continuous function $f$, and so $\int_{A} \mathrm{d}Q= \operatorname{E}[ \mathbf{1}_{A} L^{(m)} ]$ for any $A \in \sigma(Z_1,\ldots, Z_{m})$. Now looking back into , we have for any $A \in \sigma(Z_1,\ldots, Z_{m})$, $\operatorname{E}[ \mathbf{1}_{A} L^{(m)} ]= \operatorname{E}\left[ \mathbf{1}_{A} L(\{ n_{k} \}) \right]$. Since by definition $L^{(m)}$ is $\sigma(Z_1,\ldots, Z_{m})$ measurable, we have $$L^{(m)}=\operatorname{E}\left[ L(\{ n_{k} \}) \left|\right. \sigma(Z_1,\ldots, Z_{m}) \right].$$
From Fatou’s lemma $$\operatorname{E}[ L(\{ n_{k} \})^2]\le \liminf_{n \to \infty} \operatorname{E}_{\mathbb{P}_n}[Y_n^2]= \exp\left\{ \sum_{i=1}^{\infty} \frac{\mu_i^{2}}{\sigma_{i}^{2}} \right\}.$$ As a consequence, we have $$0 \le \operatorname{E}|L(\{ n_{k} \})-L^{(m)}|^2 = \operatorname{E}[L(\{ n_{k} \})^2]-\operatorname{E}[L^{(m)2}]< \epsilon.$$ So $W_2(F^{L^{(m)}},F^{L(\{ n_{k} \})})< \sqrt{\epsilon}$. Here $F^{L^{(m)}}$ and $F^{L(\{ n_{k} \})}$ denote the distribution functions corresponding to $L^{(m)}$ and $L(\{ n_{k} \})$ respectively. As a consequence, $W_2(F^{L^{(m)}},F^{L(\{ n_{k} \})}) \to 0$ as $m \to \infty.$ Hence $L^{(m)} \stackrel{d}{\to} L(\{ n_{k} \})$ by the result stated after Definition \[Wass\]. On the other hand, we have already proved $L^{(m)}$ converges to $L$ in $L^2$. So $L(\{ n_{k} \})\stackrel{d}{=}L$.
#### Proof of
We start with a sub sequence $\{ n_{l} \}$. We shall choose $k$ large enough which shall be specified later. We also know that both the random variables $\log\left(Y_{n_{l}}\right)$ and $ \left\{ \sum_{i=1}^{k} \frac{2\mu_{i} W_{n_{l},i}- \mu_{i}^2}{2\sigma_{i}^2} \right\}$ are tight.
We now prove that there is a $M$ invariant of $k$ such that both the probabilities $$\begin{split}
&\mathbb{P}_{n_{l}}\left[ -M \le \log\left(Y_{n_{l}}\right) \le M \right] \ge 1- \frac{\delta}{100}\\
&\mathbb{P}_{n_{l}} \left[ -M \le \left\{ \sum_{i=1}^{k} \frac{2\mu_{i}W_{n_{l},i}- \mu_{i}^2 }{2 \sigma_{i}^2} \right\} \le M \right] \ge 1 -\frac{\delta}{100}
\end{split}$$ for all $n_{l}$. Since the random variable $Y_{n_{l}}$ do not depend on $k$ the first inequality is obvious. For the second inequality observe that $$\operatorname{Var}\left[ \left\{ \sum_{i=1}^{k} \frac{2\mu_{i}W_{n_{l},i}- \mu_{i}^2 }{2 \sigma_{i}^2} \right\} \right]\le \operatorname{Var}\left[ \sum_{i=1}^{m_{n}} \frac{2\mu_{i}W_{n_{l},i}- \mu_{i}^2 }{2 \sigma_{i}^2} \right]$$ where $m_{n}$ is a sequence increasing to infinity as mentioned in Proposition \[prop:norcont\]. Now $$\begin{split}
& \operatorname{Var}\left[ \sum_{i=1}^{m_{n}} \frac{2\mu_{i}W_{n_{l},i}- \mu_{i}^2 }{2 \sigma_{i}^2} \right] < C'
\end{split}$$ for all $n_{l}$. for a deterministic constant $C'$. As a consequence, $$\begin{split}
\mathbb{P}_{n_{l}}\left[ \left| \sum_{i=1}^{k} \frac{2\mu_{i}W_{n_{l},i}- \mu_{i}^2 }{2 \sigma_{i}^2} \right|> M \right]\le \frac{C'}{M^2}\le \frac{\delta}{100}
\end{split}$$ where $M^2= \frac{100C'}{\delta}$. $$\mathbb{P}_{n_{l}}\left[ -M \le \log\left(Y_{n_{l}}\right) \le M \cap -M \le \left\{ \sum_{i=1}^{k} \frac{2\mu_{i}W_{n_{l},i}- \mu_{i}^2 }{2 \sigma_{i}^2} \right\} \le M \right] \ge 1- \frac{\delta}{50}.$$ Now $\log(\cdot)$ is an uniformly continuous function on $[e^{-M}, e^{M}]$. So given $\epsilon>0$, there exists $\tilde{\epsilon}$ such that for any $x,y\in [e^{-M}, e^{M}]$, $$\begin{split}
\left| x-y \right|\le \tilde{\epsilon} &\Rightarrow \left| \log(x)- \log(y) \right| \le \epsilon\\
\Leftrightarrow \left| x- y \right| > \tilde{\epsilon} & \Leftarrow \left| \log(x)- \log(y) \right| > \epsilon.
\end{split}$$ We know that there is a further sub-sequence $n_{l_{m}}$ such that $(Y_{n_{l_{m}}}, W_{n_{l_{m}},1}, \ldots, W_{n_{l_{m}},k})$ converges jointly in distribution to $$(Y_{n_{l_{m}}}, W_{n_{l_{m}},1}, \ldots, W_{n_{l_{m}},k}) \stackrel{d}{\to} (H_{1},H_{2},\ldots, H_{k+1})\in (\Omega\{ n_{l_{m}} \}, \mathcal{F}\{ n_{l_{m}} \},\mathbb{P}\{ n_{l_{m}} \}).$$ Let $\mathcal{F}\{ n_{l_{m}},1 \}\subset \mathcal{F}\{ n_{l_{m}} \}$ be the sigma algebra generated by $(H_{2},\ldots, H_{k+1})$. Here $H_{1} \stackrel{d}{=} L$ and $\left(H_{2},\ldots, H_{k+1}\right)\stackrel{d}{=}\left( Z_{1},\ldots, Z_{k} \right)$. Using the arguments same as the previous proof we see that $$\operatorname{E}\left[ H_{1} \left| \mathcal{F}_{n_{l_{m}},1} \right. \right]=\exp \left\{ \sum_{i=1}^{k} \frac{2\mu_{i}H_{i+1}- \mu_{i}^2 }{2 \sigma_{i}^2} \right\}.$$ As a consequence, we have $$0\le \operatorname{E}\left( H_{1} - \exp \left\{ \sum_{i=1}^{k} \frac{2\mu_{i}H_{i+1}- \mu_{i}^2 }{2 \sigma_{i}^2} \right\} \right)^2\le \exp\left\{ \sum_{i=1}^{\infty} \frac{\mu_{i}^2}{\sigma_{i}^2} \right\} - \exp\left\{ \sum_{i=1}^{k}\frac{\mu_{i}^2}{\sigma_{i}^2}\right\}.$$ We shall choose this $k$ large enough so that $$\exp\left\{ \sum_{i=1}^{\infty} \frac{\mu_{i}^2}{\sigma_{i}^2} \right\} - \exp\left\{ \sum_{i=1}^{k}\frac{\mu_{i}^2}{\sigma_{i}^2}\right\} < \frac{\delta \tilde{\epsilon}^2}{100}.$$ Now by Chebyshev’s inequality $$\mathbb{P}\left[\left| H_{1} - \exp \left\{ \sum_{i=1}^{k} \frac{2\mu_{i}H_{i+1}- \mu_{i}^2 }{2 \sigma_{i}^2} \right\} \right|\ge \frac{\tilde{\epsilon}}{2}\right]\le \frac{\delta \tilde{\epsilon}^2}{25 \tilde{\epsilon}^2}=\frac{\delta}{25}.$$ Since $$\left( Y_{n_{l_{m}}}, W_{n_{l_{m}},1},\ldots, W_{n_{l_{m}},k} \right) \stackrel{d}{\to} \left( H_{1},H_{2},\ldots, H_{k+1} \right)$$ by continuous mapping theorem for in distributional convergence, we have $$Y_{n_{l_{m}}} - \exp\left\{ \sum_{i=1}^{k} \frac{2\mu_{i} W_{n_{l_{m}},i}- \mu_{i}^2}{2\sigma_{i}^2} \right\} \stackrel{d}{\to} H_{1} - \exp \left\{ \sum_{i=1}^{k} \frac{2\mu_{i}H_{i+1}- \mu_{i}^2 }{2 \sigma_{i}^2} \right\}.$$ Since the set $[\frac{\tilde{\epsilon}}{2},\infty)$ is closed, we have by Portmanteau theorem, $$\begin{split}
&\limsup_{n_{l_{m}}} \mathbb{P}_{n_{l_{m}}}\left[ \left| Y_{n_{l_{m}}} - \exp\left\{ \sum_{i=1}^{k} \frac{2\mu_{i} W_{n_{l_{m}},i}- \mu_{i}^2}{2\sigma_{i}^2} \right\} \right| > \tilde{\epsilon}\right]\\
&\le \limsup_{n_{l_{m}}}\mathbb{P}_{n_{l_{m}}}\left[ \left| Y_{n_{l_{m}}} - \exp\left\{ \sum_{i=1}^{k} \frac{2\mu_{i} W_{n_{l_{m}},i}- \mu_{i}^2}{2\sigma_{i}^2} \right\} \right|\ge \frac{\tilde{\epsilon}}{2}\right]\\
& \le \frac{\delta}{25}.
\end{split}$$ As a consequence, $$\begin{split}
&\frac{\delta}{25}
\ge \mathbb{P}_{n_{l_{m}}}\left[ \left| Y_{n_{l_{m}}} - \exp\left\{ \sum_{i=1}^{k} \frac{2\mu_{i} W_{n_{l_{m}},i}- \mu_{i}^2}{2\sigma_{i}^2} \right\} \right|> \tilde{\epsilon}\right]\\
&\ge \mathbb{P}_{n_{l_{m}}}\left[ Y_{n_{l_{m}}} \in [e^{-M}, e^{M}] \cap \exp \left\{ \sum_{i=1}^{k} \frac{2\mu_{i}H_{i+1}- \mu_{i}^2 }{2 \sigma_{i}^2} \right\} \in [e^{-M}, e^{M}] \cap \left| Y_{n_{l_{m}}} - \exp \left\{ \sum_{i=1}^{k} \frac{2\mu_{i}H_{i+1}- \mu_{i}^2 }{2 \sigma_{i}^2} \right\} \right| > \tilde{\epsilon} \right]\\
& \ge \mathbb{P}_{n_{l_{m}}}\left[ Y_{n_{l_{m}}} \in [e^{-M}, e^{M}] \cap \exp \left\{ \sum_{i=1}^{k} \frac{2\mu_{i}H_{i+1}- \mu_{i}^2 }{2 \sigma_{i}^2} \right\} \in [e^{-M}, e^{M}] \cap \left| \log\left(Y_{n_{l_{m}}}\right) - \left\{ \sum_{i=1}^{k} \frac{2\mu_{i}H_{i+1}- \mu_{i}^2 }{2 \sigma_{i}^2} \right\} \right|> \epsilon \right]\\
& \ge 1- \mathbb{P}_{n_{l_{m}}}\left[\left( Y_{n_{l_{m}}} \in [e^{-M}, e^{M}] \cap \exp \left\{ \sum_{i=1}^{k} \frac{2\mu_{i}H_{i+1}- \mu_{i}^2 }{2 \sigma_{i}^2} \right\} \in [e^{-M}, e^{M}] \right)^{c} \right]\\
&~~~~~~~~~~ -\mathbb{P}_{n_{l_{m}}}\left[\left| \log\left(Y_{n_{l_{m}}}\right) - \left\{ \sum_{i=1}^{k} \frac{2\mu_{i}H_{i+1}- \mu_{i}^2 }{2 \sigma_{i}^2} \right\} \right|\le \epsilon \right]\\
& \ge \mathbb{P}_{n_{l_{m}}}\left[\left| \log\left(Y_{n_{l_{m}}}\right) - \left\{ \sum_{i=1}^{k} \frac{2\mu_{i}H_{i+1}- \mu_{i}^2 }{2 \sigma_{i}^2} \right\} \right|> \epsilon \right]- \frac{\delta}{100}\\
&\Rightarrow \mathbb{P}_{n_{l_{m}}}\left[\left| \log\left(Y_{n_{l_{m}}}\right) - \left\{ \sum_{i=1}^{k} \frac{2\mu_{i}H_{i+1}- \mu_{i}^2 }{2 \sigma_{i}^2} \right\} \right|> \epsilon \right] \le \frac{\delta}{25} + \frac{\delta}{100}< \delta.
\end{split}$$
Construction of $\mathbb{P}_{n}$ and $\mathbb{Q}_{n}$ and asymptotic distribution of signed cycles
==================================================================================================
Construction of the measure $\mathbb{Q}_{n}$ {#subsec:meas}
--------------------------------------------
We at first give the construction of measures $\mathbb{P}_{n}$ and $\mathbb{Q}_{n}$. We assume that the random variables $\left( A_{i,j} \right)_{1\le i <j \le n}$ are defined on $\left(\Omega_{n},\mathcal{F}_{n}\right)$.
In this paper $\mathbb{P}_{n}$ is simply taken to be the measure induced by $\left( A_{i,j} \right)_{1\le i <j \le n}$. We now define the measure $\mathbb{Q}_{n}$ in the following way: At first for any given $\sigma \in \{ -1, +1 \}^n$, we define the measure $\mathbb{Q}_{n,\sigma}$ by $$\label{def:qsigma}
\frac{d\mathbb{Q}_{n,\sigma}}{d\mathbb{P}_{n}} := \exp \left\{ \sum_{i<j} \left(\frac{2\beta}{\sqrt{n}}\sigma_{i}\sigma_{j} A_{i,j}- \frac{2\beta^2}{n}\right) + \frac{\beta J}{n} \left( \sum_{i=1}^{n}\sigma_{i} \right)^2 \right\}.$$ Observe that $\mathbb{Q}_{n,\sigma}$ is not in general a probability measure. In particular, $$\int_{\Omega_{n}} d \mathbb{Q}_{n,\sigma} = \exp\left\{ \frac{\beta J}{n} \left( \sum_{i=1}^{n}\sigma_{i} \right)^2 \right\}.$$ Here we have used the fact that $\operatorname{E}\left[ \exp{tX} \right]=\exp\left\{\frac{t^2}{2}\right\}$ whenever $X \sim N(0,1)$. Here $\Omega_{n}$ is the sample space on which the random variables $A_{i,j}$’s are defined. Finally, we define $$\label{def:q}
\mathbb{Q}_{n} := \frac{1}{\operatorname{E}_{\Psi_{n}}\left[ \exp\left\{ \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2\right\} \right]}\sum_{\sigma \in \{-1, +1 \}^n} \frac{1}{2^n} \mathbb{Q}_{n,\sigma}.$$ Observe that $\mathbb{Q}_{n}$ is a valid probability measure on $\Omega_{n}$. Let $$\tau_{n}:=\operatorname{E}_{\Psi_{n}}\left[ \exp\left\{ \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2\right\} \right].$$ One might observe that $\tau_{n}$ is the partition function of Curie-Weiss model. From Hoeffding’s inequality we have $$\begin{split}
&\mathbb{P}\left[ \frac{1}{\sqrt{n}}\left| \sum_{i=1}^{n} \sigma_{i} \right|>t \right]\le 2 \exp\left\{ - \frac{t^2}{2} \right\}\\
& \Rightarrow \mathbb{P} \left[ \exp\left\{\frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2 \right\} \ge t \right] \le \exp\left\{ - \left( \frac{\log t}{2\beta J} \right) \right\}= \left( \frac{1}{t} \right)^{\alpha_{0}}
\end{split}$$ for some $\alpha_{0}>1$. This makes the random variable $\exp\left\{\frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2 \right\}$ uniformly integrable. Hence $$\tau_{n}\to \frac{1}{\sqrt{1-2\beta J}}.$$ Plugging in the definition of partition function in , it is worth noting that: $$\begin{split}
\frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}}&= \frac{1}{\tau_{n}} \sum_{\sigma \in \{-1, +1 \}^n} \frac{1}{2^n}\exp \left\{ \sum_{i<j} \left(\frac{2\beta}{\sqrt{n}}\sigma_{i}\sigma_{j} A_{i,j}- \frac{2\beta^2}{n}\right) + \frac{\beta J}{n} \left( \sum_{i=1}^{n}\sigma_{i} \right)^2 \right\}\\
&= \frac{1}{\tau_{n}} \exp\left\{-(n-1)\beta^2 + \beta J\right\} \exp\left\{ - \frac{\beta}{\sqrt{n}} \sum_{i=1}^{n} A_{i,i} -\beta J'\right\}Z_{n}(\beta).
\end{split}$$ So in order to prove Theorem \[thm:asymptotic\] it is enough to prove a central limit theorem for $\log\left(\frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}}\right)$ and to prove that $\log\left(\frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}}\right)$ is asymptotically independent of $\frac{1}{\sqrt{n}}\sum_{i=1}^{n} A_{i,i}$.
Asymptotic distribution of $C_{n,i}$’s under $\mathbb{P}_{n}$ and $\mathbb{Q}_{n}$
----------------------------------------------------------------------------------
In order to derive the limiting distribution of $C_{n,i}$’s under $\mathbb{Q}_{n}$ we at first need to define another sequence of measure $\mathbb{Q}_{n}'$. We shall at first derive the limiting distribution of $C_{n,i}$’s under $\mathbb{Q}_{n}'$ and then we shall find the limiting distribution of $C_{n,i}$’s under $\mathbb{Q}_{n}$.\
Let for any given $\sigma \in \{-1,+1 \}^n$, $\mathbb{Q}_{n,\sigma}'$ be defined as $$\frac{d\mathbb{Q}_{n,\sigma}'}{d\mathbb{P}_{n}}= \exp \left\{ \sum_{i<j} \left(\frac{2\beta}{\sqrt{n}}\sigma_{i}\sigma_{j} A_{i,j}- \frac{2\beta^2}{n}\right) \right\}.$$ Observe that $\mathbb{Q}_{n,\sigma}'$ is a probability measure. In fact $\left(A_{i,j}\right)_{1 \le i < j \le n }\left|_{\mathbb{Q}_{n,\sigma}'} \right.$ are independent normal random variables with $A_{i,j}\left|_{\mathbb{Q}_{n,\sigma}'} \right. \sim N\left( \frac{2\beta}{\sqrt{n}}\sigma_{i}\sigma_{j},1 \right)$. Here $\left(A_{i,j}\right)_{1 \le i < j \le n }\left|_{\mathbb{Q}_{n,\sigma}'} \right.$ denote the joint distribution of the random variables $\left(A_{i,j}\right)_{1\le i,j \le n}$ under the measure $\mathbb{Q}_{n,\sigma}'$. Finally $$\mathbb{Q}_{n}':= \frac{1}{2^{n}}\sum_{\sigma\in \{-1, 1 \}^{n}} \mathbb{Q}_{n,\sigma}'.$$
The first result in this section gives the asymptotic distribution of $C_{n,i}$’s under $\mathbb{P}_{n}$ and $\mathbb{Q}_{n}$.
\[prop:signdistr\]
1. Under $\mathbb{P}_{n}$, we have for any $2 \le k_{1}< k_{2}\ldots < k_{l} = o\left( \sqrt{\log(n)} \right)$ with $l$ fixed, $$\left( \frac{C_{n,k_{1}}- (n-1)\mathbb{I}_{k_{1}=2}}{\sqrt{2k_{1}}}, \ldots, \frac{C_{n,k_{l}}}{\sqrt{2k_{l}}} \right) \stackrel{d}{\to} N_{l}(0, I_{l}).$$
2. Let $\Psi_{n}$ be the uniform probability measure on the hyper cube $\{ -1, +1\}^{n}$. Then there exists a set $S_{n}$ with $\Psi_{n}\left(S_{n}\right) \to 0$, we have for all $\sigma \in S_{n}^{c}$, under $\mathbb{Q}_{n,\sigma}'$ $$\label{res:qnsig}
\left( \frac{C_{n,k_{1}}- (n-1)\mathbb{I}_{k_{1}=2}- \mu_{k_{1}}}{\sqrt{2k_{1}}}, \ldots, \frac{C_{n,k_{l}}-\mu_{k_{l}}}{\sqrt{2k_{l}}} \right) \stackrel{d}{\to} N_{l}(0, I_{l})$$ where $\mu_{i}:= \left( 2 \beta \right)^{i}$. This implies under $\mathbb{Q}_{n}'$, $$\left( \frac{C_{n,k_{1}}- (n-1)\mathbb{I}_{k_{1}=2}- \mu_{k_{1}}}{\sqrt{2k_{1}}}, \ldots, \frac{C_{n,k_{l}}-\mu_{k_{l}}}{\sqrt{2k_{l}}} \right) \stackrel{d}{\to} N_{l}(0, I_{l}).$$
3. Finally, $C_{n,1}\stackrel{d}{\to} N(0,1)$ under $\mathbb{P}_{n}$ and is asymptotically independent of the process $\{ C_{n,k}- (n-1)\mathbb{I}_{k=2} \}_{k\ge 2}$.
Here $N_{l}(\mu,\Sigma)$ denotes an $l$ dimensional normal random vector with mean parameter $\mu$ and variance parameter $\Sigma$. The proof of Proposition \[prop:signdistr\] is given in Section \[sec:propsign\]. With Proposition \[prop:signdistr\], we now give the asymptotic distribution of $C_{n,i}$’s under $\mathbb{Q}_{n}$.
\[prop:signdistrnew\] Under $\mathbb{Q}_{n}$, we have for any $2 \le k_{1}< k_{2}\ldots < k_{l} = o\left( \sqrt{\log(n)} \right)$ with $l$ fixed, $$\left( \frac{C_{n,k_{1}}- (n-1)\mathbb{I}_{k_{1}=2}-\mu_{k_1}}{\sqrt{2k_{1}}}, \ldots, \frac{C_{n,k_{l}}-\mu_{k_{l}}}{\sqrt{2k_{l}}} \right) \stackrel{d}{\to} N_{l}(0, I_{l}).$$
We assume Proposition \[prop:signdistr\] and give the proof. We need to prove for any bounded continuous function $f: \mathbb{R}^{l} \to \mathbb{R}$, $$\int f\left(\frac{C_{n,k_{1}}- (n-1)\mathbb{I}_{k_{1}=2}-\mu_{k_1}}{\sqrt{2k_{1}}}, \ldots, \frac{C_{n,k_{l}}-\mu_{k_{l}}}{\sqrt{2k_{l}}}\right) d \mathbb{Q}_{n} \to \operatorname{E}\left[ f(Z_{k_{1}},\ldots, Z_{k_{l}}) \right]$$ where $Z_{k_{1}},\ldots, Z_{k_{l}}$ are independent standard Gaussian random variables. Now $$\begin{split}
&\int_{\Omega_{n}} f\left(\frac{C_{n,k_{1}}- (n-1)\mathbb{I}_{k_{1}=2}-\mu_{k_1}}{\sqrt{2k_{1}}}, \ldots, \frac{C_{n,k_{l}}-\mu_{k_{l}}}{\sqrt{2k_{l}}}\right) d \mathbb{Q}_{n}\\
& = \frac{1}{2^n}\sum_{\sigma \in \{ -1,+1 \}^{n}} \int_{\Omega_{n}} f\left(\frac{C_{n,k_{1}}- (n-1)\mathbb{I}_{k_{1}=2}-\mu_{k_1}}{\sqrt{2k_{1}}}, \ldots, \frac{C_{n,k_{l}}-\mu_{k_{l}}}{\sqrt{2k_{l}}}\right) d \mathbb{Q}_{n,\sigma}\\
&= \frac{1}{2^n}\sum_{\sigma \in \{ -1,+1 \}^{n}}\int_{\Omega_{n}} f\left(\frac{C_{n,k_{1}}- (n-1)\mathbb{I}_{k_{1}=2}-\mu_{k_1}}{\sqrt{2k_{1}}}, \ldots, \frac{C_{n,k_{l}}-\mu_{k_{l}}}{\sqrt{2k_{l}}}\right) \frac{d\mathbb{Q}_{n,\sigma}}{d\mathbb{P}_{n}} d\mathbb{P}_{n}\\
&= \frac{1}{\tau_{n}}\frac{1}{2^n}\sum_{\sigma \in \{ -1,+1 \}^{n}}\int_{\Omega_{n}} f\left(\frac{C_{n,k_{1}}- (n-1)\mathbb{I}_{k_{1}=2}-\mu_{k_1}}{\sqrt{2k_{1}}}, \ldots, \frac{C_{n,k_{l}}-\mu_{k_{l}}}{\sqrt{2k_{l}}}\right)\exp\left\{ \frac{\beta J}{n}\left(\sum \sigma_{i}\right)^2\right\} \frac{d\mathbb{Q}_{n,\sigma}'}{d\mathbb{P}_{n}} d\mathbb{P}_{n}\\
&= \frac{1}{\tau_{n}} \frac{1}{2^n}\sum_{\sigma \in \{ -1,+1 \}^{n}} \exp\left\{ \frac{\beta J}{n}\left(\sum \sigma_{i}\right)^2\right\} F_{n}(\sigma) \\
&= \frac{1}{\tau_{n}} \operatorname{E}_{\Psi_{n}} \left[ \exp\left\{ \frac{\beta J}{n}\left(\sum \sigma_{i}\right)^2\right\} F_{n}(\sigma) \right]
\end{split}$$ Here $F_{n}(\sigma)= \int_{\Omega_{n}} f\left(\frac{C_{n,k_{1}}- (n-1)\mathbb{I}_{k_{1}=2}-\mu_{k_1}}{\sqrt{2k_{1}}}, \ldots, \frac{C_{n,k_{l}}-\mu_{k_{l}}}{\sqrt{2k_{l}}}\right) \frac{d\mathbb{Q}_{n,\sigma}'}{d\mathbb{P}_{n}} d\mathbb{P}_{n}.$ From of Proposition \[prop:signdistr\], we know that under the measure $\Psi_{n}(\cdot)$, $F_{n}(\sigma) \stackrel{p}{\to} \operatorname{E}\left[ f(Z_{k_{1}},\ldots, Z_{k_{l}}) \right]$. Now from central limit theorem, $$\frac{1}{n}\left(\sum \sigma_{i}\right)^2 \stackrel{d}{\to} Y$$ where $Y$ is a Chi-squared random variable with $1$ degree of freedom. So by Slutsky’s theorem we have under the measure $\Psi_{n}$ $$F_{n}(\sigma)\exp\left\{ \frac{\beta J}{n}\left(\sum \sigma_{i}\right)^2\right\} \stackrel{d}{\to} \operatorname{E}\left[ f(Z_{k_{1}},\ldots, Z_{k_{l}}) \right]\exp\left\{ \beta J Y\right\}.$$ Further, from Hoeffding’s inequality we also have when $\beta J <\frac{1}{2}$, the sequence $\exp\left\{ \frac{\beta J}{n}\left(\sum \sigma_{i}\right)^2\right\}$ is uniformly integrable. Since the random variables $F_{n}(\sigma)$’s are uniformly bounded, the sequence $ F_{n}(\sigma)\exp\left\{ \frac{\beta J}{n}\left(\sum \sigma_{i}\right)^2\right\} $ is also uniformly integrable. As a consequence, $$\operatorname{E}_{\Psi_{n}} \left[ \exp\left\{ \frac{\beta J}{n}\left(\sum \sigma_{i}\right)^2\right\} F_{n}(\sigma) \right] \to \operatorname{E}\left[ f(Z_{k_{1}},\ldots, Z_{k_{l}}) \right] \frac{1}{\sqrt{1- 2\beta J}}.$$
Along with Proposition \[prop:norcont\], Proposition \[prop:signdistrnew\] is another important Result to prove Theorem \[thm:asymptotic\]. In particular, Proposition \[prop:signdistrnew\] allows us to verify condition $(iii)$ of Proposition \[prop:norcont\] in the context of Theorem \[thm:asymptotic\]. One might observe that in proof of Proposition \[prop:signdistrnew\] there are two important facts. First of all, part (2) of Proposition \[prop:signdistr\] where one proves the asymptotic normality of the signed cycles holds with same parameter for almost all $\sigma$’s. This makes $F_{n}(\sigma)$ converge to $\operatorname{E}\left[ f(Z_{k_{1}},\ldots, Z_{k_{l}}) \right]$ for almost all $\sigma$’s. Secondly, it is also important that the partition function of the Curie-Weiss model in high temperature has a limit $\frac{1}{\sqrt{1-2\beta J}}$. Hence $\frac{1}{\tau_{n}}$ cancels out the $\frac{1}{\sqrt{1-2\beta J}}$ factor giving us the needed result.
Proof of Theorem \[thm:asymptotic\]
===================================
As mentioned in subsection \[subsec:meas\], we at first prove a central limit theorem for $\log\left( \frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}}\right)\left| \mathbb{P}_{n} \right.$ and finally proving $\log\left( \frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}} \right)\left| \mathbb{P}_{n} \right.$ is asymptotically independent of $C_{n,1}$. The main idea is to use Proposition \[prop:norcont\] to a class of measure $\tilde{\mathbb{Q}}_{n}$ which is close to $\mathbb{Q}_{n}$ in total variation distance. We now give a formal proof of Theorem \[thm:asymptotic\].\
**Proof of Theorem \[thm:asymptotic\]:**\
We at first prove the central limit theorem for $\log\left(\frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}}\right)\left| \mathbb{P}_{n} \right.$. The proof is broken into two steps as follows.
**Step 1 (Construction of the measure $\tilde{\mathbb{Q}}_{n}$) :** To begin with we shall consider a set $\Omega(\sigma)_{n} \subset \{ -1,+1 \}^{n}$ such that $\Psi_{n}\left(\Omega(\sigma)_{n}\right)\to 1$. The precise definition of $\Omega(\sigma)_{n}$ will be provided later. Now we consider the measure $\tilde{\mathbb{Q}}_{n}$ as follows $$\tilde{\mathbb{Q}}_{n}= \frac{1}{\operatorname{E}_{\Psi_{n}}\left[ \mathbb{I}_{\Omega(\sigma)_{n}} \exp\left\{ \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2\right\} \right]}\sum_{\sigma \in \Omega(\sigma)_{n}} \frac{1}{2^n}\mathbb{Q}_{n,\sigma}= \frac{1}{\tilde{\tau}_{n}}\sum_{\sigma \in \Omega(\sigma)_{n}} \frac{1}{2^n}\mathbb{Q}_{n,\sigma}$$ where we define $$\tilde{\tau}_{n}:= \operatorname{E}_{\Psi_{n}}\left[ \mathbb{I}_{\Omega(\sigma)_{n}} \exp\left\{ \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2\right\} \right].$$ Since the sequence of random variables $\exp\left\{ \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2\right\}$ is uniformly integrable it follows that for any sequence of sets $\Omega_{n}(\sigma)$ such that $\Psi_{n}\left[ \Omega_{n}(\sigma) \right] \to 1,$ $$\tilde{\tau}_{n} \to \frac{1}{\sqrt{1-2\beta J}}.$$ Now we prove the sequences of measures $\mathbb{Q}_{n}$ and $\tilde{\mathbb{Q}}_{n}$ are close in the total variation sense. Let $A_{n}\in \mathcal{F}_{n}$ be a sequence of measurable sets. We have $$\label{eq:tvcal}
\begin{split}
&\left|\mathbb{Q}_{n}(A_{n})- \tilde{\mathbb{Q}}_{n}(A_{n}) \right|\\
&=\left| \frac{1}{\tau_{n}}\sum_{\sigma \in \{ -1,+1 \}^n} \frac{1}{2^n} \int_{A_{n}}\frac{d\mathbb{Q}_{n,\sigma}}{d\mathbb{P}_{n}}d\mathbb{P}_{n}- \frac{1}{\tilde{\tau}_{n}} \sum_{\sigma \in \Omega_{n}(\sigma) }\frac{1}{2^n} \int_{A_{n}}\frac{d\mathbb{Q}_{n,\sigma}}{d\mathbb{P}_{n}}d\mathbb{P}_{n}\right|\\
&\le \left| \frac{1}{\tau_{n}} \sum_{\sigma \in \Omega_{n}(\sigma)^{c}} \frac{1}{2^n} \int_{A_{n}}\frac{d\mathbb{Q}_{n,\sigma}}{d\mathbb{P}_{n}}d\mathbb{P}_{n} \right| + \left|\left( \frac{1}{\tau_{n}} -\frac{1}{\tilde{\tau}_{n}}\right)\sum_{\sigma \in \Omega_{n}(\sigma)}\frac{1}{2^n} \int_{A_{n}} \frac{d\mathbb{Q}_{n,\sigma}}{d\mathbb{P}_{n}}d\mathbb{P}_{n}\right|\\
&\le \left| \frac{1}{\tau_{n}} \sum_{\sigma \in \Omega_{n}(\sigma)^{c}} \frac{1}{2^n} \int_{\Omega_{n}}\frac{d\mathbb{Q}_{n,\sigma}}{d\mathbb{P}_{n}}d\mathbb{P}_{n} \right| + \left| \left( \frac{1}{\tau_{n}} -\frac{1}{\tilde{\tau}_{n}}\right) \right|\left| \sum_{\sigma \in \Omega_{n}(\sigma)}\frac{1}{2^n} \int_{\Omega_{n}} \frac{d\mathbb{Q}_{n,\sigma}}{d\mathbb{P}_{n}}d\mathbb{P}_{n} \right|\\
& \le \left|\frac{1}{\tau_{n}} \operatorname{E}_{\Psi_{n}}\left[ \mathbb{I}_{\Omega(\sigma)_{n}^{c}} \exp\left\{ \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2\right\} \right] \right|+ \left| \left( \frac{1}{\tau_{n}} -\frac{1}{\tilde{\tau}_{n}}\right) \right|\operatorname{E}_{\Psi_{n}}\left[ \mathbb{I}_{\Omega(\sigma)_{n}} \exp\left\{ \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2\right\} \right]
\end{split}$$ Observe that the final expression in does not depend on the set $A_{n}$ and also it has been argued earlier that the final expression in converges to $0$. By Proposition \[prop:signdistrnew\], under the measure $\tilde{\mathbb{Q}}_{n}$ the random variables for any $2 \le k_{1}< k_{2}\ldots < k_{l} = o\left( \sqrt{\log(n)} \right)$ with $l$ fixed, $$\left( \frac{C_{n,k_{1}}- (n-1)\mathbb{I}_{k_{1}=2}-\mu_{k_1}}{\sqrt{2k_{1}}}, \ldots, \frac{C_{n,k_{l}}-\mu_{k_{l}}}{\sqrt{2k_{l}}} \right) \stackrel{d}{\to} N_{l}(0, I_{l}).$$ Now we prove that $\limsup_{n \to \infty}\operatorname{E}_{\mathbb{P}_{n}}\left[\left(\frac{d\tilde{\mathbb{Q}}_{n}}{d\mathbb{P}_{n}}\right)^2\right]\le \exp\left\{ \sum_{k=2}^{\infty} \frac{\mu_{k}^2}{\sigma_{k}^2} \right\}$ where $\mu_{k}= (2\beta)^{k}$. This will allow us to use Proposition \[prop:norcont\] for $\tilde{\mathbb{Q}}_{n}$. In particular, we shall get $\left( \frac{d\tilde{\mathbb{Q}}_{n}}{d\mathbb{P}_{n}} \right)\left|\mathbb{P}_{n}\right.$ has a normal limiting distribution. Once this is done, the limiting distribution of $\frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}}\left| \mathbb{P}_{n} \right.$ can be derived by the following arguments which proves $$\frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}}- \frac{d\tilde{\mathbb{Q}}_{n}}{d\mathbb{P}_{n}}\left|\mathbb{P}_{n} \right. \stackrel{p}{\to} 0.$$
Since both $\tau_{n}$ and $\tilde{\tau}_{n}$ have the same finite limit, the random variable $$\tilde{Y}_{n}:=\frac{\tilde{\tau_{n}}}{\tau_{n}}\frac{d\tilde{\mathbb{Q}}_{n}}{d\mathbb{P}_{n}}\left|\mathbb{P}_{n} \right.$$ has the same limiting distribution as $\frac{d\tilde{\mathbb{Q}}_{n}}{d\mathbb{P}_{n}}\left| \mathbb{P}_{n} \right.$. In particular, $$\left(\tilde{Y}_{n}- \frac{d\tilde{\mathbb{Q}}_{n}}{d\mathbb{P}_{n}}\right)\left| \mathbb{P}_{n} \right. \stackrel{p}{\to} 0.$$ So it is enough to prove $$\left(\frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}}- \tilde{Y}_{n} \right)\left|\mathbb{P}_{n}\right. \stackrel{p}{\to} 0.$$ However, $$\begin{split}
0\le \frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}}- \tilde{Y}_{n} &= \frac{1}{\tau_{n}}\left(\sum_{\sigma \in \Omega_{n}(\sigma)^{c}}\frac{1}{2^n} \frac{d\mathbb{Q}_{n,\sigma}}{d\mathbb{P}_{n}}\right)\\
\Rightarrow \operatorname{E}_{\mathbb{P}_{n}}\left[\frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}}- \tilde{Y}_{n} \right] &= \frac{1}{\tau_{n}}\operatorname{E}_{\Psi_{n}}\left[ \mathbb{I}_{\Omega_{n}(\sigma)^{c}}\exp\left\{ \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2\right\} \right]\to 0.
\end{split}$$ This completes the proof of $$\left(\frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}}- \frac{d\tilde{\mathbb{Q}}_{n}}{d\mathbb{P}_{n}}\right)\left| \mathbb{P}_{n} \right. \stackrel{p}{\to} 0.$$ **Step 2 (Upper bounding $\operatorname{E}_{\mathbb{P}_{n}}\left[ \left( \frac{d\tilde{\mathbb{Q}}_{n}}{d\mathbb{P}_{n}} \right)^2 \right]$ ):**
We know that $$\label{eq:loglikemombound}
\begin{split}
&\left( \frac{d\tilde{\mathbb{Q}}_{n}}{d\mathbb{P}_{n}} \right)^2 = \left(\frac{1}{\tilde{\tau_{n}}}\right)^2\frac{1}{4^n}\sum_{\sigma \in \Omega(\sigma)_{n}} \sum_{\sigma' \in \Omega(\sigma)_{n}}\frac{d\mathbb{Q}_{n,\sigma}}{d\mathbb{P}_{n}}\frac{d\mathbb{Q}_{n,\sigma'}}{d\mathbb{P}_{n}}\\
&= \left(\frac{1}{\tilde{\tau_{n}}}\right)^2\frac{1}{4^n}\sum_{\sigma \in \Omega(\sigma)_{n}} \sum_{\sigma' \in \Omega(\sigma)_{n}}\exp\left\{ \sum_{i<j} \left(\frac{2\beta}{\sqrt{n}}A_{i,j}\left( \sigma_{i}\sigma_{j} + \sigma_{i}' \sigma_{j}'\right)- \frac{4\beta^2}{n}\right) + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2 + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma'_{i} \right)^2 \right\}\\
& \Rightarrow \operatorname{E}_{\mathbb{P}_{n}}\left[ \left( \frac{d\tilde{\mathbb{Q}}_{n}}{d\mathbb{P}_{n}} \right)^2 \right]\\
&= \left(\frac{1}{\tilde{\tau_{n}}}\right)^2\frac{1}{4^n}\sum_{\sigma \in \Omega(\sigma)_{n}} \sum_{\sigma' \in \Omega(\sigma)_{n}}\exp\left\{\sum_{i<j}\left( \frac{2\beta^2}{n}\left( \sigma_{i}\sigma_{j}+ \sigma'_{i}\sigma'_{j} \right)^2- \frac{4\beta^2}{n} \right)+ \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2 + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma'_{i} \right)^2 \right\}\\
&= \left(\frac{1}{\tilde{\tau_{n}}}\right)^2\frac{1}{4^n}\sum_{\sigma \in \Omega(\sigma)_{n}} \sum_{\sigma' \in \Omega(\sigma)_{n}}\exp\left\{\sum_{i<j}\left( \frac{4\beta^2}{n} \sigma_{i}\sigma_{j} \sigma'_{i}\sigma'_{j} \right)+ \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2 + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma'_{i} \right)^2 \right\}\\
&= \left(\frac{1}{\tilde{\tau_{n}}}\right)^2\frac{1}{4^n}\sum_{\sigma \in \Omega(\sigma)_{n}} \sum_{\sigma' \in \Omega(\sigma)_{n}}\exp\left\{ \frac{2\beta^2}{n}\left( \sum_{i=1}^{n} \sigma_{i}\sigma'_{i} \right)^2 -2\beta^2\ + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2 + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma'_{i} \right)^2 \right\}\\
&= \exp\left\{ -2\beta^2 \right\}\left(\frac{1}{\tilde{\tau_{n}}}\right)^2 \operatorname{E}_{\Psi_{n}\otimes \Psi_{n}}\left[ \mathbb{I}_{\sigma \in \Omega(\sigma)_{n}} \mathbb{I}_{\sigma' \in \Omega(\sigma)_{n}}\exp\left\{ \frac{2\beta^2}{n}\left( \sum_{i=1}^{n} \sigma_{i}\sigma'_{i} \right)^2 + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2 + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma'_{i} \right)^2 \right\} \right]
\end{split}$$ Here $\Psi_{n}\otimes \Psi_{n}$ denote the two fold product of the uniform probability measure on $\{-1 , 1\}^{n} \times \{ -1, 1 \}^{n}$.
Observe that the random variable $$\label{def:exp}
\mathbb{I}_{\sigma \in \Omega(\sigma)_{n}} \mathbb{I}_{\sigma' \in \Omega(\sigma)_{n}}\exp\left\{ \frac{2\beta^2}{n}\left( \sum_{i=1}^{n} \sigma_{i}\sigma'_{i} \right)^2 + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2 + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma'_{i} \right)^2 \right\} \stackrel{d}{\to} \exp\left\{ 2\beta^2 Y_{1} + \beta J Y_{2} + \beta J Y_{3} \right\}$$ where $Y_{1},Y_{2},Y_{3}$ are three independent chi-square random variables each with one degree of freedom. Our target is to prove the random variable in the l.h.s. of is uniformly integrable. This done by proving $$\limsup_{n \to \infty}\operatorname{E}_{\Psi_{n}\otimes \Psi_{n}}\left[ \mathbb{I}_{\sigma \in \Omega(\sigma)_{n}} \mathbb{I}_{\sigma' \in \Omega(\sigma)_{n}}\exp\left\{(1+\eta) \left(\frac{2\beta^2}{n}\left( \sum_{i=1}^{n} \sigma_{i}\sigma'_{i} \right)^2 + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2 + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma'_{i} \right)^2 \right\}\right) \right]<\infty$$ for sufficiently small $\eta$. We at first write $$\begin{split}
&=\operatorname{E}_{\Psi_{n}\otimes \Psi_{n}}\left[ \mathbb{I}_{\sigma \in \Omega(\sigma)_{n}} \mathbb{I}_{\sigma' \in \Omega(\sigma)_{n}}\exp\left\{ (1+\eta)\left(\frac{2\beta^2}{n}\left( \sum_{i=1}^{n} \sigma_{i}\sigma'_{i} \right)^2 + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2 + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma'_{i} \right)^2\right) \right\} \right]\\
&= \operatorname{E}\left[\operatorname{E}\left[ \mathbb{I}_{\sigma \in \Omega(\sigma)_{n}} \mathbb{I}_{\sigma' \in \Omega(\sigma)_{n}}\exp\left\{ (1+\eta)\left(\frac{2\beta^2}{n}\left( \sum_{i=1}^{n} \sigma_{i}\sigma'_{i} \right)^2 + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2 + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma'_{i} \right)^2 \right)\right\} \left| \sigma \right.\right]\right]\\
&= \operatorname{E}\left[ \mathbb{I}_{\sigma \in \Omega(\sigma)_{n}} \exp\left\{ (1+\eta)\frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2\right\}\operatorname{E}\left[ \mathbb{I}_{\sigma' \in \Omega(\sigma)_{n}}\exp\left\{(1+\eta)\left(\frac{2\beta^2}{n}\left( \sum_{i=1}^{n} \sigma_{i}\sigma'_{i} \right)^2 + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma'_{i} \right)^2\right) \right\}\left| \sigma \right. \right]\right]\\
&= \operatorname{E}\left[ \mathbb{I}_{\sigma \in \Omega(\sigma)_{n}} \exp\left\{ (1+\eta)\frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2 \right\}\operatorname{E}\left[ \mathbb{I}_{\sigma' \in \Omega(\sigma)_{n}}\exp\left\{(1+\eta)\frac{1}{n}\left(\sigma'\right)^{\mathrm{T}} A'A \left(\sigma'\right) \right\}\left| \sigma \right. \right] \right]\\
&\le \operatorname{E}\left[ \mathbb{I}_{\sigma \in \Omega(\sigma)_{n}} \exp\left\{ (1+\eta)\frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2 \right\}\operatorname{E}\left[ \exp\left\{(1+\eta)\frac{1}{n}\left(\sigma'\right)^{\mathrm{T}} A^{\mathrm{T}}A \left(\sigma'\right) \right\}\left| \sigma \right. \right] \right].
\end{split}$$ Here $\mathrm{T}$ denotes the transpose of a matrix and the matrix $A_{2\times n}$ is given by $$A=\left(
\begin{array}{llll}
\beta J& \beta J & \ldots & \beta J\\
2\beta^2\sigma_{1}& 2\beta^2\sigma_{2} & \ldots & 2\beta^2\sigma_{n}
\end{array}
\right).$$ Since $\operatorname{E}\left[\exp\left\{ \alpha^{\mathrm{T}}\sigma' \right\}\right]\le \exp\left\{ \frac{1}{2} || \alpha ||^{2} \right\}$ for any $\alpha \in \mathbb{R}^{n}$, we have the following tail estimate by Theorem 1 and Remark 1 of @HSU: $$\mathbb{P}\left[ \frac{1}{n}\left(\sigma'\right)^{\mathrm{T}} A^{\mathrm{T}}A \left(\sigma'\right)\ge \mathrm{tr}(\Sigma) +2 \sqrt{\mathrm{tr}(\Sigma^2)t} + 2 || \Sigma ||t\left| \sigma \right. \right]\le e^{-t}$$ where $\Sigma= \frac{1}{\sqrt{n}}A $. Observe that the nonzero eigenvalues of $\Sigma$ are same as the nonzero eigenvalues of $\frac{1}{n}AA^{\mathrm{T}}$. Now $$\frac{1}{n}AA^{\mathrm{T}}=\left(
\begin{array}{ll}
\beta J& 2\beta^3J \left( \frac{1}{n}\sum_{i=1}^{n}\sigma_{i} \right)\\
2\beta^3J\left( \frac{1}{n} \sum_{i=1}^{n}\sigma_{i} \right) & 2\beta^2
\end{array}
\right).$$ We now choose the set $$\Omega(\sigma)_{n}:= \left\{ \frac{1}{n}\sum_{i=1}^{n}\sigma_{i} \le \delta_{n}\right\}$$ for some $\delta_{n} \to 0$ as $n \to \infty$. The existence of such $\Omega(\sigma)_{n}$ is ensured by weak law of large numbers. Now by Weyl’s interlacing inequality, we have the eigenvalues of $\frac{1}{n}AA^{\mathrm{T}}$ are given by $\left\{\beta J+ O(\delta_{n}), 2\beta^2 +O(\delta_{n}) \right\} $. Also note that on $\Omega(\sigma)_{n}$, $\mathrm{tr}(\Sigma)$ and $\mathrm{tr}(\Sigma^2)$ remain uniformly bounded. So given any $\epsilon>0$ we can find a $t_{0}$ large enough such that $$\mathrm{tr}(\Sigma) +2 \sqrt{\mathrm{tr}(\Sigma^2)t} < \epsilon 2 || \Sigma ||t$$ for all $t>t_{0}.$ As a consequence, for all $t>t_{0}$ $$\begin{split}
&\mathbb{P}\left[ \frac{1}{n}\left(\sigma'\right)^{\mathrm{T}} A^{\mathrm{T}}A \left(\sigma'\right)\ge (1+\epsilon) 2|| \Sigma ||t\left| \sigma \right. \right]\\
&\le \mathbb{P}\left[ \frac{1}{n}\left(\sigma'\right)^{\mathrm{T}} A^{\mathrm{T}}A \left(\sigma'\right)\ge \mathrm{tr}(\Sigma) +2 \sqrt{\mathrm{tr}(\Sigma^2)t} + 2 || \Sigma ||t\left| \sigma \right. \right]< e^{-t}\\
& \Rightarrow \mathbb{P}\left[ (1+\eta)\frac{1}{n}\left(\sigma'\right)^{\mathrm{T}} A^{\mathrm{T}}A \left(\sigma'\right) \ge \log(t) \left| \sigma \right.\right]\le t^\frac{-1}{2(1+\epsilon)(1+\eta)||\Sigma||} ~ \forall ~ t> \tilde{t}_{0}.
\end{split}$$ where $\tilde{t}_{0}$ is another deterministic constant. Here the last step comes from replacing $(1+\eta)(1+\epsilon) 2|| \Sigma ||t$ by $\log(t)$. Since $\max\left\{ \beta J, 2 \beta^2 \right\}< \frac{1}{2}$, we can choose $\epsilon $ and $\eta$ small enough such that $$\frac{1}{2(1+\epsilon)(1+\eta)||\Sigma||} >\alpha_{0}>1.$$ As a consequence, $$\label{eqn:argjoint}
\begin{split}
\mathbb{I}_{\sigma \in \Omega(\sigma)_{n}}\operatorname{E}\left[ \exp\left\{(1+\eta)\frac{1}{n}\left(\sigma'\right)^{\mathrm{T}} A^{\mathrm{T}}A \left(\sigma'\right) \right\}\left| \sigma \right. \right]\le \tilde{t}_{0} + \int_{t > \tilde{t}_{0}} \frac{1}{t^{\alpha_0}} dt= \tilde{t}_{0} + \frac{1}{\alpha_0-1} \frac{1}{t^{\alpha_{0}-1}}
\end{split}$$ On the other hand we can choose $\eta$ small enough such that $\beta J (1+\eta) < \gamma_{0}< \frac{1}{2}$. Now it is enough to prove that $$\limsup\operatorname{E}\left[ \exp\left\{ (1+\eta)\frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2\right\} \right]<\infty.$$ However we know that for any $t>0$, $$\begin{split}
&\operatorname{E}\left[ \exp\left\{ \frac{t}{\sqrt{n}} \sum_{i=1}^{n} \sigma_{i}\right\} \right]\le \exp\left\{ \frac{t^2}{2} \right\}\\
& \Rightarrow
\mathbb{P}\left[ \left|\frac{1}{\sqrt{n}} \sum_{i=1}^{n} \sigma_{i} \right| >t \right] = 2 \mathbb{P}\left[ \frac{1}{\sqrt{n}} \sum_{i=1}^{n} \sigma_{i} > t \right]=2\mathbb{P}\left[ \exp \left\{ \frac{t}{\sqrt{n}} \sum_{i=1}^{n} \sigma_{i}\right\} > \exp\left\{ t^2 \right\}\right]\le 2 \exp\left\{ -\frac{t^2}{2} \right\}\\
\end{split}$$ Here the last inequality is a straight forward application of Markov’s inequality. Now $$\begin{split}
\mathbb{P}\left[ \exp\left\{\frac{\beta J(1+\eta)}{n} \left( \sum_{i=1}^{n} \sigma_{i} \right)^2 \right\} >t \right]&= \mathbb{P}\left[ \frac{\beta J (1+\eta)}{n}\left( \sum_{i=1}^{n} \sigma_{i}\right)^2 > \log (t) \right]\\
= \mathbb{P}\left[ \left| \frac{1}{\sqrt{n}} \sum_{i=1}^{n} \sigma_{i}\right| > \sqrt{\frac{\log t}{\beta J (1+\eta)}} \right]&\le 2\exp\left\{ - \frac{\log t}{2 \beta J (1+\eta)} \right\} \le 2 \left( \frac{1}{t} \right)^{\frac{1}{2 \beta J (1+\eta)}}< 2 \left( \frac{1}{t} \right)^{\frac{1}{2\gamma_{0}}}.
\end{split}$$ Observe that $\frac{1}{2\gamma_{0}}>1$. Hence by argument similar to we have $$\limsup \operatorname{E}\left[ \exp\left\{ (1+\eta)\frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2\right\} \right]< \infty.$$ This completes the proof of uniform integrability of the random variable in the l.h.s. of . As a consequence, $$\begin{split}
&\lim_{n\to \infty}\operatorname{E}\left[\mathbb{I}_{\sigma \in \Omega(\sigma)_{n}} \mathbb{I}_{\sigma' \in \Omega(\sigma)_{n}}\exp\left\{ \frac{2\beta^2}{n}\left( \sum_{i=1}^{n} \sigma_{i}\sigma'_{i} \right)^2 + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i} \right)^2 + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma'_{i} \right)^2 \right\}\right]\\
& = \operatorname{E}\left[ 2\beta^2 Y_{1} + \beta J Y_{2} + \beta J Y_{3} \right]= \frac{1}{\sqrt{1-4\beta^2}}\frac{1}{1-2\beta J}.
\end{split}$$ Plugging this into we have $$\begin{split}
\lim_{n \to \infty}\operatorname{E}_{\mathbb{P}_{n}}\left[ \left( \frac{d\tilde{\mathbb{Q}}_{n}}{d\mathbb{P}_{n}} \right)^2 \right] &= \exp\left\{-2\beta^2\right\} \left( 1- 2\beta J \right)\frac{1}{\sqrt{1-4\beta^2}}\frac{1}{1-2\beta J}\\
&= \exp\left\{-2\beta^2\right\} \frac{1}{\sqrt{1-4\beta^2}}\\
&= \exp \left\{ -2 \beta^2 \right\} \exp \left\{ -\frac{1}{2} \log\left(1-4\beta^2\right) \right\}\\
&= \exp \left\{ -2 \beta^2 \right\} \exp \left\{ \frac{1}{2} \sum_{k=1}^{\infty} \frac{\left(4\beta^2\right)^{k}}{k} \right\}= \exp\left\{ \sum_{k=2}^{\infty} \frac{\mu_{k}^2}{2k} \right\}
\end{split}$$ where $\mu_{k}=(2\beta)^{k}$. Now using Proposition \[prop:norcont\] with $W_{n,k}= C_{n,k+1}-(n-1)\mathbb{I}_{k=1}$, we have for the sequences of measures $\tilde{\mathbb{Q}}_{n}$ and $\mathbb{P}_{n}$ $$\frac{d\mathbb{\tilde{Q}}_{n}}{d\mathbb{P}_{n}}\left|\mathbb{P}_{n} \right. \stackrel{d}{\to} \exp\left\{ \sum_{k=1}^{\infty} \frac{2\mu_{k+1} Z_{k}-\mu_{k+1}^2}{4(k+1)} \right\}$$ where $Z_{k}\sim N\left(0,2(k+1)\right)$. Hence $$\frac{d\mathbb{\tilde{Q}}_{n}}{d\mathbb{P}_{n}}\left|\mathbb{P}_{n} \right. \stackrel{d}{\to} \exp\left\{ \sum_{k=1}^{\infty} \frac{2\mu_{k+1} Z_{k}-\mu_{k+1}^2}{4(k+1)} \right\}.$$ As we have proved earlier that $\frac{d\mathbb{\tilde{Q}}_{n}}{d\mathbb{P}_{n}}- \frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}} \left| \mathbb{P}_{n} \right. \stackrel{p}{\to} 0$, this completes the proof of the asymptotic normality of $\log\left( \frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}} \right)\left| \mathbb{P}_{n}. \right.$
**Proof of part (2) of Theorem \[thm:asymptotic\]:** Before proving part $(1)$ of Theorem \[thm:asymptotic\], we prove part $(2)$. Since $$\frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}}
= \frac{1}{\tau_{n}} \exp\left\{-(n-1)\beta^2 + \beta J\right\} \exp\left\{ - \frac{\beta}{\sqrt{n}} \sum_{i=1}^{n} A_{i,i} -\beta J'\right\}Z_{n}(\beta),$$ in order to prove part $(2)$ of Theorem \[thm:asymptotic\], we need to prove that $$\label{eqn:toprove}
\log \left( \frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}} \right)- \sum_{k=2}^{m_{n}} \frac{2\mu_{k} \left( C_{n,k}- (n-1)\mathbb{I}_{k=2} \right)-\mu_{k}^2}{4k} \left| \mathbb{P}_{n} \right. \stackrel{p}{\to} 0.$$ We at first prove the result analogous to for $\log\left(\frac{d\tilde{\mathbb{Q}}_{n}}{d\mathbb{P}_{n}}\right)$. then follows from the fact that $\frac{d\mathbb{Q}_{n}}{d\mathbb{P}_{n}} - \frac{d\tilde{\mathbb{Q}}_{n}}{d\mathbb{P}_{n}}\left| \mathbb{P}_{n} \right.\stackrel{p}{\to} 0$ and an application of continuous mapping theorem.
By , for any given $\epsilon,\delta >0$ there exists $K=K(\epsilon,\delta)$ and for any subsequence $n_l$ there exists a further subsequence $n_{l_q}$ such that $$\label{bdd:liketoseriesIII}
\mathbb{P}_{n_{l_q}}\left( \left| \log(\frac{d\tilde{\mathbb{Q}}_{n_{l_{q}}}}{d\mathbb{P}_{n_{l_{q}}}}) -
\sum_{k=2}^{K} \frac{2\mu_{k}(C_{n_{l_q},k}-(n-1)\mathbb{I}_{k=2})-\mu_{k}^{2}}{4k} \right|\ge \frac{\epsilon}{2} \right) \le \frac{\delta}{2}.$$ Now choose $K'\ge K$ such that $$\sum_{K'+1}^{\infty}\frac{\mu_{k}^{2}}{2k}\le \max\left\{\frac{\delta\epsilon^{2}}{100},\frac{\epsilon}{100}\right\}.$$ For any $K'< k_{1}<k_{2}<m_{n}= o(\sqrt{\log n})$, the proof of Proposition \[prop:signdistr\] implies that $\operatorname{E}_{\mathbb{P}_{n}}\left[ C_{n,k_{1}} \right]=0$, ${\text{Cov}}(C_{n,k_{1}},C_{n,k_{2}})=0$ and $\operatorname{Var}(C_{n,k_{i}})= 2k_{i}(1+ O({k_{i}^{2}}/{n}))$ for $i \in \{1,2 \}.$ So $$\operatorname{Var}\left(\sum_{k=K'+1}^{m_{n_{l_q}}} \frac{2\mu_{k} C_{n_{l_q},k}-\mu_{k}^{2}}{4k}\right)
= (1+o(1))\sum_{k=K'+1}^{m_{n_{l_q}}}\frac{\mu_{k}^{2}}{2k}\le \frac{\delta\epsilon^{2}}{100}.$$ Now for large values of $n_{l_q}$, $$\label{bdd:series}
\begin{split}
&\mathbb{P}_{n_{l_q}}\left( \left|\sum_{k=K+1}^{m_{n_{l_q}}} \frac{2\mu_{k}C_{n_{l_q},k}}{4k}\right| \ge \frac{\epsilon}{4}\right) \le \frac{16 \delta\epsilon^{2}}{100\epsilon^{2}},
\qquad \mbox{and so} \\
& \mathbb{P}_{n_{l_q}} \left( \left|\sum_{k=K+1}^{m_{n_{l_q}}} \frac{2\mu_{k} C_{n_{l_q},k} -\mu_{k}^{2}}{4k} \right| \ge \frac{\epsilon}{4}+ \frac{\epsilon}{100} \right)
\le\frac{16 \delta\epsilon^{2}}{100\epsilon^{2}}.
\end{split}$$
Plugging in the estimates of and we have for all large values of $n_{l_q}$, $$\label{eq:subseq-conv}
\mathbb{P}_{n_{l_q}}\left( \left| \log\left(\frac{d\tilde{\mathbb{Q}}_{n_{l_{q}}}}{d\mathbb{P}_{n_{l_{q}}}}\right) - \sum_{k=1}^{m_{n_{l_q}}} \frac{2\mu_{k}(C_{n_{l_q},k}- (n-1)\mathbb{I}_{k=2})-\mu_{k}^{2}}{4k}
\right| \ge \epsilon \right)
\le \delta.
$$ Since occurs to any subsequence and any $(\epsilon,\delta)$ pair, this completes the proof.
**Proof of part (1) of Theorem \[thm:asymptotic\]:** Consider the random variable $$M:= W+ \sum_{k=1}^{\infty}\frac{2\mu_{k+1}Z_{k}-\mu_{k+1}^2}{4(k+1)}$$ where $W\sim N(0, \beta^2)$ and is independent of the random variable $$\sum_{k=1}^{\infty}\frac{2\mu_{k+1}Z_{k}-\mu_{k+1}^2}{4(k+1)}.$$ Observe that from the proof of part $(2)$ we have $$\begin{split}
&\log\left(Z_{n}(\beta)\right) + \frac{1}{2} \log\left(1- 2\beta J\right)- (n-1)\beta^2 + \beta (J -J') - \beta C_{n,1}\\
& ~~~~~~~~~~ - \sum_{k=2}^{m_{n}}\frac{2\mu_{k}\left(C_{n,k}- (n-1)\mathbb{I}_{k=2}\right)- \mu_{k}^2}{4k}\left| \mathbb{P}_{n} \right.\stackrel{p}{\to} 0.
\end{split}$$ So it is enough to prove that $$\beta C_{n,1} + \sum_{k=2}^{m_{n}}\frac{2\mu_{k}\left(C_{n,k}- (n-1)\mathbb{I}_{k=2}\right)- \mu_{k}^2}{4k} \stackrel{d}{\to} N\left( \beta^2+\frac{1}{4}\log(1-4\beta^2), -\beta^2 - \frac{1}{2}\log(1-4\beta^2) \right).$$ On the other hand for any fixed $K$, $$\beta C_{n,1} + \sum_{k=2}^{K}\frac{2\mu_{k}\left(C_{n,k}- (n-1)\mathbb{I}_{k=2}\right)- \mu_{k}^2}{4k}\left| \mathbb{P}_{n} \right.\stackrel{d}{\to} W + \sum_{k=1}^{K-1}\frac{2\mu_{k+1}Z_{k}-\mu_{k+1}^2}{4(k+1)}.$$ Since all the random variables $\beta C_{n,1}$, $\sum_{k=2}^{m_{n}}\frac{2\mu_{k}\left(C_{n,k}- (n-1)\mathbb{I}_{k=2}\right)- \mu_{k}^2}{4k}$ and $\sum_{k=2}^{K}\frac{2\mu_{k}\left(C_{n,k}- (n-1)\mathbb{I}_{k=2}\right)- \mu_{k}^2}{4k}$ have uniformly bounded second moments, they are tight. Hence we have any of their linear combination is also tight. Hence given any subsequence $n_{l} $ there exists a further subsequence $n_{l_{q}}$ such that $$\beta C_{n_{l_{q}},1} + \sum_{k=2}^{m_{n_{l_{q}}}}\frac{2\mu_{k}\left(C_{n_{l_{q}},k}- (n_{l_{q}}-1)\mathbb{I}_{k=2}\right)- \mu_{k}^2}{4k} \left| \mathbb{P}_{n_{l_q}} \right. \stackrel{d}{\to} M\{ n_{l_{q}} \}.$$ Here the notation $M\{ n_{l_{q}} \}$ means that the limiting distribution might possibly depend on the choice of the sub-sequence. On the other hand for every fixed $K$ there is a further subsequence $n_{{l_{q}}_{m}}$ (possibly dependent on $K$) such that $$\begin{split}
&\left(\beta C_{n_{{l_{q}}_{m}},1} + \sum_{k=2}^{m_{n_{{l_{q}}_{m}}}}\frac{2\mu_{k}\left(C_{{n_{l_{q}}}_{m},k}- (n_{{l_{q}}_{m}}-1)\mathbb{I}_{k=2}\right)- \mu_{k}^2}{4k},\beta C_{n_{{l_{q}}_{m}},1} + \sum_{k=2}^{K}\frac{2\mu_{k}\left(C_{{n_{l_{q}}}_{m},k}- (n_{{l_{q}}_{m}}-1)\mathbb{I}_{k=2}\right)- \mu_{k}^2}{4k}\right)\left| \mathbb{P}_{n_{{l_{q}}_{m}}}\right.\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\stackrel{d}{\to}\left( M_{1},M_{2,K} \right).
\end{split}$$ where $M_{1}\stackrel{d}{=}M\{ n_{l_{q}}\}$ and $M_{2,K}\stackrel{d}{=} W+ \sum_{k=1}^{K-1}\frac{2\mu_{k+1}Z_{k}-\mu_{k+1}^2}{4(k+1)}$. Hence $$\sum_{k=K+1}^{m_{n_{{l_{q}}_{m}}}}\frac{2\mu_{k}\left(C_{n_{{l_{q}}_{m}},k}- (n_{{l_{q}}_{m}}-1)\mathbb{I}_{k=2}\right)- \mu_{k}^2}{4k} \left| \mathbb{P}_{n_{{l_{q}}_{m}}} \right. \stackrel{d}{\to} M_{1}-M_{2,K}.$$ On the other hand by Fatou’s lemma, $$\label{balchal}
\liminf\operatorname{E}_{\mathbb{P}_{n_{{l_{q}}_{m}}}}\left[\left( \sum_{k=K+1}^{m_{n}}\frac{2\mu_{k}\left(C_{n_{{l_{q}}_{m}},k}- (n-1)\mathbb{I}_{k=2}\right)- \mu_{k}^2}{4k} \right)^2\right] \ge \operatorname{E}\left[(M_{1}-M_{2,K})^2\right].$$ We know that for large enough value of $n_{{l_{q}}_{m}}$, $$\begin{split}
&\operatorname{E}_{\mathbb{P}_{n_{{l_{q}}_{m}}}}\left[\left( \sum_{k=K+1}^{m_{n_{{l_{q}}_{m}}}}\frac{2\mu_{k}\left(C_{n_{{l_{q}}_{m}},k}- (n-1)\mathbb{I}_{k=2}\right)- \mu_{k}^2}{4k} \right)^2\right] = \operatorname{Var}\left( \sum_{k=K+1}^{m_{n_{{l_{q}}_{m}}}} \frac{2\mu_{k}\left(C_{n_{{l_{q}}_{m}},k}- (n-1)\mathbb{I}_{k=2}\right)}{4k}\right) + \left(\sum_{k=K+1}^{m_{n_{{l_{q}}_{m}}}} \frac{\mu_{k}^2}{4k}\right)^2\\
&= (1+o(1))\sum_{k=K+1}^{m_{n_{{l_{q}}_{m}}}} \frac{\mu_{k}^2}{2k} + \left(\sum_{k=K+1}^{m_{n_{{l_{q}}_{m}}}} \frac{\mu_{k}^2}{4k}\right)^2.
\end{split}$$ Here we have used the identity $\operatorname{E}[X^2]= \operatorname{Var}[X]+ \left(\operatorname{E}[X]\right)^2$. Given any $\epsilon >0$, we now choose $K$ large enough so that $
\sum_{k=K+1}^{\infty} \frac{\mu_{k}^2}{2k} \le \epsilon,
$ implying $\operatorname{E}\left[ (M_{1}-M_{2,K})^2\ \right]\le \epsilon + \epsilon^2/4$. Hence the r.h.s. of converges to $0$ as $K \to \infty$. This implies $W_{2}\left( F^{M_{1}}, F^{M_{2,K}} \right)\to 0$ as $K \to \infty$. Here $F^{M_{1}}$ and $F^{M_{2,K}}$ denote the distribution functions of $M_{1}$ and $M_{2,K}$ respectively. As a consequence, we have $$W+ \sum_{k=1}^{K-1}\frac{2\mu_{k+1}Z_{k}-\mu_{k+1}^2}{4(k+1)} \stackrel{d}{\to} M\{n_{l_{q}}\}.$$ Hence $M\{ n_{l_{q}} \}\stackrel{d}{=} W+ \sum_{k=1}^{\infty}\frac{2\mu_{k+1}Z_{k}-\mu_{k+1}^2}{4(k+1)} $ which does not depend on the specific choice of the subsequence $\{ n_{l_{q}} \}$. This concludes the proof.\
We now give proofs of Theorem \[Thm:approximation\] and Proposition \[prop:signdistr\]
Proof of Theorem \[Thm:approximation\] {#sec:thmapp}
======================================
Although the proof of Theorem \[Thm:approximation\] is similar to the proof of Theorem 3.4 in @Banerjee2017, we sketch the main details here for the shake of completeness. In order to complete the proof we first need some preliminary notations, definitions and some results. All these definitions can be found in @Banerjee2017. One might have a look at Section A.\
The proof of Theorem \[Thm:approximation\] is divided into two parts depending upon $k$ being even or odd. We analyze each case separately. The case $k$ odd is almost similar to the case considered in @Banerjee2017. However the case $k$ even is easier here than the case considered in @Banerjee2017. This due to the fact that when $k$ is even there are words with $l(w)=2k+1$ such that $G_{w}$ is a tree. This creates additional complications. We discuss these things in details now.\
As a general discussion, the fundamental idea is to show that $\operatorname{Tr}\left[ \left( \frac{1}{\sqrt{n}} \right)^{2k+1}\tilde{A}^{2k+1} \right]$ can be written as a linear combination of $C_{n,r}$’s where $r$ is odd. In particular, one shows that the mean of $\operatorname{Tr}\left[ \left( \frac{1}{\sqrt{n}} \right)^{2k+1}\tilde{A}^{2k+1} \right]$ is approximately $0$ and the main contribution to the variance comes from a class of closed words which corresponds to some multiple of the $C_{n,r}$’s. One can show that the contribution by all the other words are negligible. The Chebyshev polynomial approximation is obtained by inverting this relation. On the other hand when $k$ is even, $\operatorname{Tr}\left[\operatorname{Tr}\left[ \left( \frac{1}{\sqrt{n}} \right)^{2k}\tilde{A}^{2k} \right]\right]$ can be written as a linear combination of $C_{n,r}$’s plus an additional term which has two components. The first component is approximately equal to $k\psi_{2k}\left[ \frac{1}{n}\operatorname{Tr}[\tilde{A}_{n}^{2}]- \operatorname{E}\left[ \frac{1}{n}\operatorname{Tr}[\tilde{A}_{n}^{2}] \right] \right]$ and the second component gives a non-trivial contribution to the mean keeping the variance unchanged. These terms correspond to words in $\mathcal{W}_{1,2k}$ and $\mathcal{W}_{3,2k}$ in the proof. Finally the Chebyshev polynomial approximation still holds due to the fact $$\sum_{r=1}^{k} P_{2k}[2r]r\psi_{2r}=0.$$ This cancels out the term corresponding to $k\psi_{2k}\left[ \frac{1}{n}\operatorname{Tr}[\tilde{A}_{n}^{2}]- \operatorname{E}\left[ \frac{1}{n}\operatorname{Tr}[\tilde{A}_{n}^{2}] \right] \right]$.
We now elaborate the above discussion in some details.
(i) **$k$ is odd:**In this part to avoid confusion we shall use the notation $2k+1$ instead of using the terminology $k$ being odd. The proof is completed by first showing that $\operatorname{Tr}\left[ \left( \frac{1}{\sqrt{n}} \right)^{2k+1}\tilde{A}^{2k+1} \right]$ can be approximated as a linear combination of the cycles $C_{n,i}$’s. Then one inverts the relation to get cycles as a linear combination of the traces.\
We at first write down the following identity: $$\begin{split}
\operatorname{Tr}\left[ \left( \frac{1}{\sqrt{n}} \right)^{2k+1}\tilde{A}^{2k+1} \right]& = \left( \frac{1}{n} \right)^{\frac{2k+1}{2}}\sum_{w~|~ \text{closed} ~ l(w)=2k+2} X_{w}
\end{split}$$ where the graph $G_{w}$ has no self loops and for any closed word $w=(i_{0},i_{1},\ldots,i_{2k},i_{0})$ we define $$X_{w}:= A_{i_{0},i_{1}}\ldots A_{i_{2k},i_{0}}.$$ By a direct use of the parity principle (Lemma A.1 in @Banerjee2017) we get that $G_{w}$ can not be a tree. We now divide the class of words with $l(w)=2k+2$ in to two parts. In the first part we consider the cases when $G_{w}$ is an unicyclic graph such that each edge in the bracelet is repeated exactly once in particular $w \in \mathfrak{W}_{2k+2,r,t}$ for some $r$ odd and $2t -r= m$ (to be denoted by $\mathfrak{W}_{2k+2,r}$) and we denote the complement by $\mathcal{W}_{1,2k+1}$. In particular, we write $$\label{eq:deom}
\begin{split}
&\left( \frac{1}{n} \right)^{\frac{2k+1}{2}}\sum_{w~|~ \text{closed} ~ l(w)=2k+2} X_{w}\\
& = \left( \frac{1}{n} \right)^{\frac{2k+1}{2}}\sum_{r=3}^{2k+1}\sum_{w \in \mathfrak{W}_{2k+2,r,t}} X_{w}+ \left( \frac{1}{n} \right)^{\frac{2k+1}{2}} \sum_{w \in \mathcal{W}_{1,2k+1}} X_{w}.
\end{split}$$ Our fundamental goal is to show that $$\operatorname{E}\left[\left(\left( \frac{1}{n} \right)^{\frac{2k+1}{2}} \sum_{w \in \mathcal{W}_{1,2k+1}} X_{w}\right)^2\right] \to 0$$ and $$\label{eq:cyctotr}
\operatorname{E}\left[ \left(\left( \frac{1}{n} \right)^{\frac{2k+1}{2}}\sum_{r=3~|~ r ~ \text{odd}}^{2k+1}\sum_{w \in \mathfrak{W}_{2k+2,r}} X_{w} - \sum_{r=3~|~ r~ \text{odd}}^{2k+1}f(2k+1,r)\frac{2k+1}{r}C_{n,r}\right)^2 \right]\to 0.$$ We at first analyze the second part of . $$\begin{split}
&\operatorname{E}\left[\left(\left( \frac{1}{n} \right)^{\frac{2k+1}{2}} \sum_{w \in \mathcal{W}_{1,2k+1}} X_{w}\right)^2\right]\\
&=\left( \frac{1}{n} \right)^{2k+1} \sum_{w, x \in \mathcal{W}_{1,2k+1}} \operatorname{E}[X_{w}X_{x}].
\end{split}$$ Now $\operatorname{E}[X_{w}X_{x}]=0$ unless all the edges in $a=[w,x]$ are repeated at least twice. Observe that by Holder’s inequality and sub Gaussianity we have $\operatorname{E}[\left| X_{w} X_{x}\right|]\le \left|A_{1,2}\right|^{4k+2}\le \left( C_{1}k \right)^{C_{2}k}$. Here $C_{1}$ and $C_{2}$ are two deterministic constants. We divide this case into two further sub cases. First of all $w$ and $x$ shares an edge in particular $a$ is an weak CLT sentence If $\#V_{a}=t$ by Lemma A.5 in @Banerjee2017, the number of such $a$ is bounded by $2^{4k+4}\left( C_{3}(4k+4) \right)^{C_{4}}\left( 4k+4 \right)^{3(4k+4-2t)} n^{t}$. By Proposition A.6 in @Banerjee2017, $w,x \in \mathcal{W}_{1,2k+1}$, $a$ can not be a CLT word pair. Hence $\#V_{a} < 2k+1$. Now consider the case when $w$ and $x$ don’t share an edge. In this case $\#V_{w}\le \#E_{w}< \frac{2k+1}{2}$ and for any $t_{1}$ such that $\#V_{w}=t_{1}$ we have by Lemma A.4 in @Banerjee2017 we have the cardinality of such $w$ is bounded by $2^{2k+1}(2k+1)^{3(2k-2t_{1}+3)}n^{t_{1}}$. Similarly the number of words $x$ such that for any $t_{2}$ such that $\#V_{x}=t_{2}$ is bounded by $2^{2k+1}(2k+1)^{3(2k-2t_{2}+3)}n^{t_{2}}$. As a consequence, $$\begin{split}
&\operatorname{E}\left[\left(\left( \frac{1}{n} \right)^{\frac{2k+1}{2}} \sum_{w \in \mathcal{W}_{1,2k+1}} X_{w}\right)^2\right]\\
&\le \left( C_{1}k \right)^{C_{2}k}\left[\sum_{t=1}^{2k} 2^{4k+4}\left( C_{3}(4k+4) \right)^{C_{4}}\left( 4k+4 \right)^{3(4k+4-2t)}\left(\frac{1}{n}\right)^{2k+1-t}\right.\\
&~~~~~\left.+ \sum_{t_{1}=1}^{k}\sum_{t_{2}=1}^{k}2^{4k+2}(2k+1)^{3(4k-2t_{2}-2t_{1}+6)}\left(\frac{1}{n}\right)^{2k+1-t_{1}-t_{2}}\right]\\
&~~~~ \to 0
\end{split}$$ whenever $k=o\left( \sqrt{\log n} \right)$. Hence we can neglect the second term in . Now we prove \[eq:cyctotr\].\
First of all for any $w \in \mathfrak{W}_{2k+2,r}$, $\operatorname{E}[X_{w}]=0$ and if $w_{1} \in \mathfrak{W}_{2k+2,r_{1}}$ and $w_{2} \in \mathfrak{W}_{2k+2,r_{2}}$ where $r_{1}\neq r_{2}$ $\operatorname{E}[X_{w_{1}}X_{w_{2}}]=0$. Hence $$\begin{split}
&\operatorname{E}\left[\left(\left( \frac{1}{n} \right)^{\frac{2k+1}{2}}\sum_{r=3~|~ r ~ \text{odd}}^{2k+1}\sum_{w \in \mathfrak{W}_{2k+2,r}} X_{w}\right)^2\right]= \sum_{r=3~|~ r ~ \text{odd}}\operatorname{Var}\left[ \left( \frac{1}{n} \right)^{\frac{2k+1}{2}}\sum_{w \in \mathfrak{W}_{2k+2,r}} X_{w} \right]\\
&= \left(1+O\left( \frac{k^2}{n} \right) \right)\sum_{r=3~|~ r ~ \text{odd}} f^{2}(2k+1,r)\frac{2(2k+1)^{2}}{r}.
\end{split}$$ Here we have used Proposition A.6 in @Banerjee2017 and Lemma A.10 in @Banerjee2017. On the other hand by proof of Proposition \[prop:signdistr\] we have $$\begin{split}
&\operatorname{E}\left[ \left(\sum_{r=3~|~ r~ \text{odd}}^{2k+1}f(2k+1,r)\frac{2k+1}{r}C_{n,r}\right)^2 \right]\\
&= \left(1+O\left( \frac{k^2}{n} \right) \right)\sum_{r=3~|~ r ~ \text{odd}} f^{2}(2k+1,r)\frac{2(2k+1)^2}{r}.
\end{split}$$ Now $$\begin{split}
&\operatorname{E}\left[ \left(\left( \frac{1}{n} \right)^{\frac{2k+1}{2}}\sum_{r=3~|~ r ~ \text{odd}}^{2k+1}\sum_{w \in \mathfrak{W}_{2k+2,r}} X_{w}\right) \left( \sum_{r=3~|~ r~ \text{odd}}^{2k+1}f(2k+1,r)\frac{2k+1}{r}C_{n,r} \right) \right]\\
&= \sum_{r=3~|~ r ~ \text{odd}}f(2k+1,r)\frac{2k+1}{r}{\text{Cov}}\left[ \left( \frac{1}{n} \right)^{\frac{2k+1}{2}}\sum_{w \in \mathfrak{W}_{2k+2,r}} X_{w},C_{n,r} \right]\\
&= \sum_{r=3~|~ r ~ \text{odd}}\left( 1+ O\left( \frac{k^2}{n} \right) \right) f(2k+1,r)\frac{2k+1}{r} 2f(2k+1,r) \frac{2k+1}{r}\\
&= \sum_{r=3~|~ r ~ \text{odd}}\left( 1+ O\left( \frac{k^2}{n} \right) \right) f^{2}(2k+1,r)\frac{2(2k+1)^2}{r}
\end{split}$$ by Lemma A.10 in @Banerjee2017. As a consequence, $$\begin{split}
&\operatorname{E}\left[ \left(\left( \frac{1}{n} \right)^{\frac{2k+1}{2}}\sum_{r=3~|~ r ~ \text{odd}}^{2k+1}\sum_{w \in \mathfrak{W}_{2k+2,r}} X_{w} - \sum_{r=3~|~ r~ \text{odd}}^{2k+1}f(2k+1,r)\frac{2k+1}{r}C_{n,r}\right)^2 \right]\\
&= \sum_{r=3~|~ r \text{odd}} O\left( \frac{k^2}{n} \right)f^{2}(2k+1,r)\frac{2(2k+1)^2}{r}\\
&= O\left(\frac{\text{Poly}(k)2^{4k}}{n}\right)\to 0.
\end{split}$$ Here $\text{Poly}(k)$ is a deterministic polynomial in $k$. It is known that for example see @Lang that $f(m,r)\frac{m}{r}=\binom{m}{\frac{m+r}{2}}$ when $m$ and $r$ have same parity. By comparing coefficient it can be checked that $$\left(
\begin{array}{lllll}
1&0 &\ldots & 0 & 0\\
\binom{3}{2}&1&\ldots&0& 0\\
\vdots & \ddots & \ldots & 1& 0\\
\binom{2k+1}{k+1} & \binom{2k+1}{k+2} & \ldots & \binom{2k+1}{2k} & 1
\end{array}
\right)^{-1}= D.$$ Here $D$ is a lower triangular matrix and $D_{i,j}= P_{2i+1}[2j+1]$ where $P_{j}[i]$ is the coefficient of $z^{i}$ in $\left( z + \frac{1}{z} \right)^{j}$. Now $$\label{eq:quadbound}
\begin{split}
&\operatorname{E}\left[\left(C_{n,2k+1}- \operatorname{Tr}\left[ P_{2k+1}\left[\frac{1}{\sqrt{n}}\tilde{A}\right] \right]\right)^2\right]\\
& \le \sum_{r=3~|~ r ~ \text{odd}}^{2k+1}\operatorname{E}\left[ \left(C_{n,2r+1}- \operatorname{Tr}\left[ P_{2r+1}\left[\frac{1}{\sqrt{n}}\tilde{A}\right] \right]\right)^2 \right]\\
& = \operatorname{E}\left[ \zeta'D' D \zeta \right]\\
&\le \sup_{i,j}| D'D_{i,j}| \operatorname{E}\left[ \sum_{i=1}^{k}\zeta_{i}^{2} + 2\sum_{i<j}\left|\zeta_{i}\zeta_{j}\right| \right]\\
&\le \sup_{i,j}| D'D_{i,j}|k \operatorname{E}\left[ \sum_{i=1}^{k} \zeta_{i}^{2} \right]\to 0.
\end{split}$$ Here $$\zeta=\left(
\begin{array}{l}
0 \\
\operatorname{Tr}\left[\left(\frac{1}{\sqrt{n}}\right)^{3}\tilde{A}^{3}\right]- \binom{3}{2}C_{n,3}\\
\vdots\\
\operatorname{Tr}\left[ \left(\frac{1}{\sqrt{n}}\right)^{2k+1}\tilde{A}^{2k+1} \right]- \sum_{r=3~|~ r ~\text{odd}}^{2k+1} \binom{2k+1}{\frac{2k+1+r}{2}} C_{n,r}
\end{array}
\right)$$ and the last step of follows from the fact that the coefficients of the Chebyshev polynomial grows at most exponentially. This completes the proof of the odd case.
(ii) **$k$ is even:** Here again to avoid confusion we use the notation $2k$ instead of using the term $k$ even. Like the $2k+1$ case we at first write $$\label{eq:sumeven}
\begin{split}
& \operatorname{Tr}\left[ \left(\frac{1}{\sqrt{n}}\right)^{2k}\tilde{A}^{2k} \right] = \left( \frac{1}{n} \right)^{k}\sum_{w~|~ \text{closed} ~ l(w)=2k+1} X_{w}.
\end{split}$$ However unlike the $2k+1$ case, in this case there are words $w$ such that $G_{w}$ is a tree. Keeping this in mind, we decompose $\operatorname{Tr}\left[ \left(\frac{1}{\sqrt{n}}\right)^{2k}\tilde{A}^{2k} \right]$ in the following way: $$\begin{split}
\operatorname{Tr}\left[ \left(\frac{1}{\sqrt{n}}\right)^{2k}\tilde{A}^{2k} \right]= \left( \frac{1}{n} \right)^{k} \left[ \sum_{w \in \mathcal{W}_{1,2k}} X_{w} + \sum_{w \in \mathcal{W}_{2,2k}} X_{w}+ \sum_{w \in \mathcal{W}_{3,2k}}X_{w}+ \sum_{w \in \mathcal{W}_{4,2k}}X_{w}\right]
\end{split}$$ where
1. $\mathcal{W}_{1,2k}$ is the collection of all Wigner words(See Definition A.3 in @Banerjee2017). In this case $G_{w}$ is a tree and each is traversed exactly twice.
2. $\mathcal{W}_{2,2k}:= \cup_{r=4~:~ r ~ \text{even}}^{2k}\mathfrak{W}_{2k+1,r}$.
3. $\mathcal{W}_{3,2k}$ is the collection of words such that $G_{w}$ is either an unicyclic graph with all edges repeated exactly twice or $G_{w}$ is a tree with exactly one edge repeated exactly four times and all other edges are repeated exactly twice.
4. $\mathcal{W}_{4,2k}$ is collection of all other words.
We at first prove that $$\label{eq:decomsecondeven}
\begin{split}
\left( \frac{1}{n} \right)^{2k}\operatorname{E}\left[\left(\sum_{w \in \mathcal{W}_{4,2k}}X_{w}\right)^2 \right]\to 0.
\end{split}$$ Note that $$\operatorname{E}\left[\left(\sum_{w \in \mathcal{W}_{4,2k}}X_{w}\right)^2 \right] = \sum_{w,x \in \mathcal{W}_{4,2k}} \operatorname{E}[X_{w}X_{x}].$$ Alike the odd case $\operatorname{E}[X_{w}X_{x}]=0$ unless all the edges in the sentence $a=[w,x]$ are repeated at least twice. Now we consider two cases firstly when the sentence $a$ is a weak CLT sentence i.e. the graphs $G_w$ and $G_x$ share an edge. In this case we prove that $\#V_{a}<2k$. We do a case by case analysis here. Since all the edges in $a$ are repeated at least twice $\#E_{a}\le 2k$. Whenever $\#E_{a}=2k$, there are two cases when $\#V_{a}\ge 2k$. Firstly the graph $G_{a}$ is a tree and each edge is repeated exactly twice. This is impossible since otherwise in both the words $w$ and $x$ there will be an edge which are repeated exactly once however both $G_{w}$ and $G_{x}$ are tree. Now consider the other case when $\#E_{a}=\#V_{a}=2k$ in this case $G_{w}$ and $G_{x}$ are both unicyclic graphs with common bracelet. This is impossible by definition of $\mathcal{W}_{4,2k}$. Finally the when $G_{w}$ is a tree and $\#E_{a}=2k-1$ is also impossible since in this case the only possibility is both $w, x \in \mathcal{W}_{1,2k}$. As a consequence $\#V_{a}<2k$. Now we consider the other case when $G_{w}$ and $G_{x}$ don’t share an edge. In this case we have all the edges in both $G_{w}$ and $G_{x}$ are repeated at least twice. However all the words of such type having $\#V_{w}\ge k$ or $\#V_{x}\ge k$ are covered in $W_{1,2k}$, $W_{2,2k}$ and $W_{3,2k}$. Hence both $\#V_{w}$ and $\#V_{x}$ are strictly less than $k$. Now arguments exactly similar to the analysis of the second term of proves . By arguments similar to the odd case, it can be proved that $$\begin{split}
\operatorname{E}\left[\left(\left( \frac{1}{n} \right)^{k}\sum_{w \in \mathcal{W}_{2,2k}}X_{w} - \sum_{r=4 ~ r ~ even}^{2k} f(2k,r)\frac{2k}{r}C_{n,r}\right)^{2}\right]\le O\left(\frac{\text{Poly}(k)2^{4k}}{n}\right).
\end{split}$$ Next we prove that $\left( \frac{1}{n} \right)^{k} \sum_{w \in \mathcal{W}_{1,2k}}X_{w}- \operatorname{E}\left[ \left( \frac{1}{n} \right)^{k} \sum_{w \in \mathcal{W}_{1,2k}}X_{w} \right]$ can be approximated by $k\psi_{2k}\left[ \frac{1}{n}\operatorname{Tr}[\tilde{A}_{n}^{2}]- \operatorname{E}\left[ \frac{1}{n}\operatorname{Tr}[\tilde{A}_{n}^{2}] \right] \right]$. This is done by second moment calculation. Observe that $$\label{eq:secontr}
\begin{split}
&\operatorname{E}\left[ \left( \left( \frac{1}{n} \right)^{k} \sum_{w \in \mathcal{W}_{1,2k}}X_{w}- \operatorname{E}\left[ \left( \frac{1}{n} \right)^{k} \sum_{w \in \mathcal{W}_{1,2k}}X_{w} \right]- k\psi_{2k}\left[ \frac{1}{n}\operatorname{Tr}[\tilde{A}_{n}^{2}]- \operatorname{E}\left[ \frac{1}{n}\operatorname{Tr}[\tilde{A}_{n}^{2}] \right] \right] \right)^2 \right]\\
&= \operatorname{Var}\left[ \left( \frac{1}{n} \right)^{k} \sum_{w \in \mathcal{W}_{1,2k}}X_{w} \right]-2 k\psi_{2k}{\text{Cov}}\left[ \left( \frac{1}{n} \right)^{k} \sum_{w \in \mathcal{W}_{1,2k}}X_{w},\frac{1}{n}\operatorname{Tr}[\tilde{A}_{n}^{2}] \right]+ k^{2}\psi_{2k}^{2}\operatorname{Var}\left[ \frac{1}{n}\operatorname{Tr}[\tilde{A}_{n}^{2}] \right]
\end{split}$$ Now we analyze each term separately. First $$\label{eq:varw1}
\begin{split}
&\operatorname{Var}\left[ \left( \frac{1}{n} \right)^{k} \sum_{w \in \mathcal{W}_{1,2k}}X_{w} \right]\\
&= \left( \frac{1}{n} \right)^{2k} \sum_{w,x \in \mathcal{W}_{1,2k}}\operatorname{E}\left[ \left(X_{w}-\operatorname{E}[X_{w}]\right)\left( X_{x}- \operatorname{E}[X_{x}] \right) \right]\\
&= \left( \frac{1}{n} \right)^{2k} \sum_{w,x\in \mathcal{W}_{1,2k};a=[w,x]~\text{weak CLT sentence}}\operatorname{E}\left[ \left(X_{w}-\operatorname{E}[X_{w}]\right)\left( X_{x}- \operatorname{E}[X_{x}] \right) \right]
\end{split}$$ Since $\# \left( E_{w} \cap E_{x} \right)\ge 1$, we have $\#E_{a}\le 2k-1$. Hence $\#V_{a}\le 2k$. The equality $\#V_{a}=2k$ occurs when $w$ and $x$ share exactly one edge. We at first fix a word $w$, then the number of $x$’s such that $\#V_{a}=2k$ holds can be enumerated as follows. In the graph $G_{w}$ there are $k$ distinct edges we chose one of them which shares the edge with the word $x$. After fixing this edge we choose the equivalence class of the word $x$. There are $\psi_{2k}$ many of them and once an equivalence class is fixed, there are $k$ choices for the edge in $G_{x}$ to be shared. Now once this edge is also fixed there are two choices such that this edge is same as the chosen edge in $G_{w}$, one in the same order as the edge in $G_{w}$ and other in the reverse order. Now fixing all these choices there are $n^{k-1}\left( 1+ O \left( \frac{k^2}{n} \right) \right)$ choices for other vertices in $G_{x}$. As a consequence, given $w$ there are $2k^{2}\psi_{2k}n^{k-1}\left( 1+ O \left( \frac{k^2}{n} \right) \right)$ choices of $x$ such that $a$ is a weak CLT sentence and $\#V_{a}=2k$. Finally there are $\psi_{2k}n^{k+1}\left( 1+ O \left( \frac{k^2}{n} \right) \right)$. Combining all these, we have the number of $a$ such that $\#V_{a}=2k$ is given by $2k^{2}\psi_{2k}^{2}n^{2k}\left( 1+ O \left( \frac{k^2}{n} \right) \right)$. In this case $\operatorname{E}\left[ \left(X_{w}-\operatorname{E}[X_{w}]\right)\left( X_{x}- \operatorname{E}[X_{x}] \right) \right]=2$ (by Gaussianity). Finally for all the cases the number of $a$’s such that $\#V_{a}=t$ is bounded by $n^{t}2^{4k+2}\left( C_{1}(4k+2) \right)^{C_{2}}\left( 4k+2 \right)^{3\left( 4k+2 -2t\right)}$. Finally for any $a$, $\operatorname{E}\left[ \left|\left(X_{w}-\operatorname{E}[X_{w}]\right)\left( X_{x} - \operatorname{E}[X_{x}]\right)\right| \right]\le \left( C_{3}k \right)^{C_{4}k}$ for some deterministic constant $C_{3}$ and $C_{4}$. Plugging all these estimates in we have, $$\begin{split}
& Var\left[ \left( \frac{1}{n} \right)^{k} \sum_{w \in \mathcal{W}_{1,2k}}X_{w} \right]\\
&= 4k^{2}\psi_{2k}^{2}\left( 1+ O\left( \frac{k^2}{n} \right) \right) + \mathbf{E}
\end{split}$$ where $$\mathbf{E}\le 2^{4k+2} \left( C_{3}k \right)^{C_{4}k}\sum_{t=1}^{2k-1} \left( C_{1}(4k+2) \right)^{C_{2}}\left( \frac{\left( 4k+2 \right)^{6}}{n} \right)^{2k-t}\to 0$$ whenever $k= o\left( \sqrt{\log n}\right)$. Similar arguments can be used to prove that $$\begin{split}
&{\text{Cov}}\left[ \left( \frac{1}{n} \right)^{k} \sum_{w \in \mathcal{W}_{1,2k}}X_{w},\frac{1}{n}\operatorname{Tr}[\tilde{A}_{n}^{2}] \right]= 4k \psi_{2k}\left( 1+ O\left( \frac{k^2}{n}\right) \right)\\
&\operatorname{Var}\left[ \frac{1}{n}\operatorname{Tr}[\tilde{A}_{n}^{2}] \right]= 4\left( 1+ O\left( \frac{1}{n} \right) \right).
\end{split}$$ Plugging these estimates in we have $$\begin{split}
&\operatorname{E}\left[ \left( \left( \frac{1}{n} \right)^{k} \sum_{w \in \mathcal{W}_{1,2k}}X_{w}- \operatorname{E}\left[ \left( \frac{1}{n} \right)^{k} \sum_{w \in \mathcal{W}_{1,2k}}X_{w} \right]- k\psi_{2k}\left[ \frac{1}{n}\operatorname{Tr}[\tilde{A}_{n}^{2}]- \operatorname{E}\left[ \frac{1}{n}\operatorname{Tr}[\tilde{A}_{n}^{2}] \right] \right] \right)^2 \right]\\
&\le 4k^2\psi_{2k}^{2} O \left( \frac{k^2}{n} \right) + \mathbf{E}\to 0.
\end{split}$$ Finally for each word $w\in \mathcal{W}_{3,2k}$, we have $\#V_{w}=k$. These words give nontrivial contribution to the mean. However it can be checked that $$\operatorname{Var}\left[ \left( \frac{1}{n} \right)^{k}\sum_{w \in \mathcal{W}_{3,2k}} X_{w}\right]\le 2^{4k}\frac{\text{Poly}(k)}{n} \to 0.$$ In particular, one also has an explicit expression for the mean $$\begin{split}
\operatorname{E}\left[ \left( \frac{1}{n} \right)^{k}\sum_{w \in \mathcal{W}_{3,2k}} X_{w} \right]= \left( 1+ O\left( \frac{k^2}{n}\right) \right)\left[\sum_{r=3}^{k} f(2k,2r)\frac{k(r+1)}{r} + 3 f(2k,4)\frac{k}{2}\right].
\end{split}$$ Finally the proof for the even case can be completed by following the arguments similar to and using the fact that $$\label{eq:cheidentity}
\sum_{r=1}^{k} P_{2k}[2r]r\psi_{2r}=0.$$ One might check @Banerjee2017 for a proof of .
Proof of Proposition \[prop:signdistr\] {#sec:propsign}
=======================================
We at first give the proofs of part $(1)$ and $(3)$. The proof of part $(2)$ will be given separately.\
**Proof of part $(1)$ and $(3)$:** We start with a very basic but fundamental observation. Note that for any $k\ge 3$ $$\begin{split}
C_{n,k}& = \left(\frac{1}{n}\right)^{\frac{k}{2}}\sum_{w \in \mathfrak{W}_{k+1,k}} X_{w}.
\end{split}$$ It is easy to see that for $k\ge 2$, $\operatorname{Var}\left[C_{n,k}\right]\to 2k$. The proof is completed by method of moments and Wick’s formula. We at first give formal statements of these results: At first we state the method of moments.
\[lem:mom\] Let $(Y_{n,1},\ldots, Y_{n,l})$ be a sequence of random vectors of $l$ dimension. Then $(Y_{n,1},\ldots, Y_{n,l}) \stackrel{d}{\to} (Z_1,\ldots,Z_{l})$ if the following conditions are satisfied:
i) $$\label{eqn:momcond}
\lim_{n \to \infty}\operatorname{E}[X_{n,1}\ldots X_{n,m}]$$
exists for any fixed $m$ and $X_{n,i} \in \{ Y_{n,1},\ldots,Y_{n,l} \}$ for $1\le i \le m$.
ii) (Carleman’s Condition)[@Carl26] $$\sum_{h=1}^{\infty} \left(\lim_{n \to \infty}\operatorname{E}[X_{n,i}^{2h}]\right)^{-\frac{1}{2h}} =\infty ~~ \forall ~ 1\le i \le l.$$
Further, $$\lim_{n \to \infty}\operatorname{E}[X_{n,1}\ldots X_{n,m}]= \operatorname{E}[X_{1}\ldots X_{m}].$$ Here $X_{n,i} \in \{ Y_{n,1},\ldots,Y_{n,l} \}$ for $1\le i \le m$ and $X_{i}$ is the in distribution limit of $X_{n,i}$. In particular, if $X_{n,i}= Y_{n,j}$ for some $j \in \{1,\ldots,l \}$ then $X_{i}= Z_{j}$.
The method of moments is very well known and much useful in probability theory. We omit its proof.
Now we state the Wick’s formula for Gaussian random variables which was first proved by Isserlis(1918)[@I18] and later on introduced by @W50 in the physics literature in 1950.
(Wick’s formula)\[lem:wick\]@W50 Let $(Y_1,\ldots, Y_{l})$ be a multivariate mean $0$ random vector of dimension $l$ with covariance matrix $\Sigma$(possibly singular). Then $((Y_1,\ldots, Y_{l}))$ is jointly Gaussian if and only if for any integer $m$ and $X_{i} \in \{ Y_1,\ldots,Y_{l} \}$ for $1\le i \le m$ $$\label{eqn:wick}
\operatorname{E}[X_1\ldots X_{m}]=\left\{
\begin{array}{ll}
\sum_{\eta} \prod_{i=1}^{\frac{m}{2}} \operatorname{E}[X_{\eta(i,1)}X_{\eta(i,2)}] & ~ \text{for $m$ even}\\
0 & \text{for $m$ odd.}
\end{array}
\right.$$ Here $\eta$ is a partition of $\{1,\ldots,m \}$ into $\frac{m}{2}$ blocks such that each block contains exactly $2$ elements and $\eta(i,j)$ denotes the $j$ th element of the $i$ th block of $\eta$ for $j=1,2$.
The proof of aforesaid lemma is also omitted.\
We at first verify the asymptotic CLT, then variance calculation will be given. Given $1\le k_{1} < k_{2}< \ldots < k_{l}= o\left( \sqrt{\log n} \right)$, we consider the random variables a $(Y_{n,1},\ldots, Y_{n,l})$ as follows: $$Y_{n,k_{j}}=\left\{
\begin{array}{ll}
\frac{C_{n,k_{j}}}{\sqrt{2k_{j}}} & \text{if $k_{j}\neq 2$}\\
\frac{C_{n,2}- (n-1)}{2} & \text{otherwise.}
\end{array}
\right.$$ Now fix $m$ and take $X_{n,1},\ldots, X_{n,m}\in \{ Y_{n,1},\ldots, Y_{n,l} \}$. Let $l_{1},\ldots,l_{m}$ be the corresponding lengths of the cycles. Now observe that $$\label{eq:expectm}
\begin{split}
&\operatorname{E}_{\mathbb{P}_{n}}\left[ X_{n,1},\ldots, X_{n,m} \right]\\
&= \left(\frac{1}{n}\right)^{\frac{\sum_{i=1}^{m}l_{i}}{2}}\prod_{i=1}^{m}\left( \frac{1}{\mathbb{I}_{l_{i}\neq 1}\sqrt{2l_{i}}+ \mathbb{I}_{l_{i}=1}} \right) \sum_{a=[w_{1}\ldots w_{m}]~|~ a ~ \text{weak CLT sentence}} \operatorname{E}\left[ \left( X_{w_{1}}-\operatorname{E}[X_{w_{1}}] \right)\ldots \left( X_{w_{l}} - \operatorname{E}[X_{w_{l}}]\right) \right].
\end{split}$$ By arguments similar to the arguments given right after , we have $$\operatorname{E}\left[ \left| \left( X_{w_{1}}-\operatorname{E}[X_{w_{1}}] \right)\ldots \left( X_{w_{l}} - \operatorname{E}[X_{w_{l}}]\right) \right| \right]\le \left( C_{1}\left( \sum_{i=1}^{m} l_{i} \right) \right)^{C_{2}\left( \sum_{i=1}^{m} l_{i} \right)}.$$ By Proposition A.2 in @Banerjee2017, we have $\#V_{a}< \sum_{i=1}^{m}\frac{l_{i}}{2}$ unless $a$ is a CLT sentence. Further the CLT sentences only exists if $m$ is even. Finally, we prove we can neglect the sum corresponding all the sentences which are not CLT sentences. $$\label{eq:weakcltbal}
\begin{split}
&\left(\frac{1}{n}\right)^{\frac{\sum_{i=1}^{m}l_{i}}{2}}\prod_{i=1}^{m}\left( \frac{1}{\mathbb{I}_{l_{i}\neq 1}\sqrt{2l_{i}}+ \mathbb{I}_{l_{i}=1}} \right) \sum_{a=[w_{1}\ldots w_{m}]~|~ ~ \#V_{a}< \sum_{i=1}^{m}\frac{l_{i}}{2}} \operatorname{E}\left[ \left|\left( X_{w_{1}}-\operatorname{E}[X_{w_{1}}] \right)\ldots \left( X_{w_{l}} - \operatorname{E}[X_{w_{l}}]\right)\right| \right]\\
&\le \left( C_{1}\left( \sum_{i=1}^{m} l_{i} \right) \right)^{C_{2}\left( \sum_{i=1}^{m} l_{i} \right)} 2^{\sum_{i=1}^{m}(l_{i}+1)} \left( C_{3} \sum_{i=1}^{m} (l_{i}+1) \right)^{C_{4}m} \sum_{t < \sum_{i=1}^{m}\frac{l_{i}}{2}}\left( \sum_{i=1}^{m} (\l_{i}+1) \right)^{3 (\sum_{i=1}^{m}(l_{i}+1)-2t)}\left( \frac{1}{n}\right)^{\sum_{i=1}^{m} \frac{l_{i}}{2}-t} \to 0.
\end{split}$$ With in hand, we are only left with CLT sentences. In particular for every $w_{i}$, there exists an unique $w_{j}$ such that $G_{w_{i}}$ shares an edge with $G_{w_{j}}$. On the other hand in these cases in order to get $$\operatorname{E}\left[ \left(X_{w_{i}}-\operatorname{E}\left[ X_{w_{j}} \right]\right)\left( X_{w_{j}}-\operatorname{E}\left[ X_{w_{j}} \right] \right) \right]\neq 0,$$ we need $G_{w_{i}}=G_{w_{j}}$. Further the random variables $X_{w_{i}}$ and $X_{w_{j}}$ are mutually independent if $G_{w_{i}}$ and $G_{w_{j}}$ are disjoint. Hence asymptotically satisfies Wick’s formula with appropriate variance. This concludes the proof of part $(1)$ and $(3)$.\
**Proof of part $(2)$:** We at first give a proof for $k=2$ case. Observe that under $\mathbb{Q}_{n,\sigma}$, $$\begin{split}
&\left( \frac{1}{n} \right)\sum_{i,j} A_{i,j}^{2}\\
&= \left( \frac{1}{n} \right) \sum_{i,j}\left( B_{i,j} + \frac{(2\beta)}{\sqrt{n}} \sigma_{i}\sigma_{j} \right)^2\\
&= \left( \frac{1}{n} \right) \sum_{i,j} \left( B_{i,j}^2+ \frac{4\beta \sigma_{i}\sigma_{j}}{\sqrt{n}} B_{i,j} + \frac{4\beta^2}{n} \right)\\
&= \left( \frac{1}{n} \right) \sum_{i,j} B_{i,j}^2 + \frac{4\beta \sigma_{i}\sigma_{j}}{n^{\frac{3}{2}}}\sum_{i,j} B_{i,j} + (1+ o(1)) 4\beta^2.
\end{split}$$ Here $B_{i,j}\sim_{i.i.d.} N(0,1)$. By CLT we have for any $\sigma$, $$\left( \frac{4\beta \sigma_{i}\sigma_{j}}{n^{\frac{3}{2}}} \right) \sum_{i,j} B_{i,j} \stackrel{p}{\to} 0.$$ This completes the proof for $k=2$ case.
Now we move on to the other cases. Observe that under $\mathbb{Q}_{n,\sigma}$ $$\label{eq:expandcyclealt}
\begin{split}
C_{n,k}&= \left( \frac{1}{\sqrt{n}} \right)^{k}\sum_{w \in \mathfrak{W}_{k+1,k}} \prod_{j=0}^{k-1}A_{i_{j},i_{j+1}}\\
&= \left( \frac{1}{\sqrt{n}} \right)^{k}\sum_{w \in \mathfrak{W}_{k+1,k}} \prod_{j=0}^{k-1} \left( B_{i_{j},i_{j+1}} + \frac{2\beta\sigma_{i_{j}}\sigma_{i_{j+1}}}{\sqrt{n}} \right)\\
&= \left( \frac{1}{\sqrt{n}} \right)^{k}\sum_{w \in \mathfrak{W}_{k+1,k}}\prod_{j=0}^{k-1}B_{i_{j},i_{j+1}} + \sum_{w \in \mathfrak{W}_{k+1,k}} V_{n,k,w} + \left( 1+ O \left( \frac{k^2}{n} \right) \right)(2\beta)^{k}.
\end{split}$$ Here $V_{n,k,w}$ is obtained by expanding the product in and taking all the residual terms in the product. Observe that $w=(i_{0},i_{1},\ldots,i_{k})$, $$\begin{split}
V_{n,k,w}&= \left( \frac{1}{n} \right)^{\frac{k}{2}} \sum_{ \emptyset \subsetneq E_{f}\subsetneq E_{w}} \prod_{e \in E_{f}} \sigma_{e} (\frac{2\beta}{\sqrt{n}}) \prod_{e \in E_{w} \backslash E_{f}} B_{e}
\end{split}$$ Here for any edge $\{i,j\}$, $B_{e}=B_{i,j}$ and $\sigma_{e}={\sigma_{i}\sigma_{j}}$. Observe that for any $\sigma$, $\operatorname{E}[V_{n,k,w}]=0$. We now prove that $$\operatorname{E}\left[\left( \sum_{w \in \mathfrak{W}_{k+1,k}}V_{n,k,w} \right)^{2}\right] \to 0.$$ We have that $$\begin{split}
& \operatorname{E}\left[\left( \sum_{w \in \mathfrak{W}_{k+1,k}}V_{n,k,w} \right)^{2}\right]\\
&= \left( \frac{1}{n}\right)^{k} \sum_{w,x \in \mathfrak{W}_{k+1,k}} \operatorname{E}\left[ V_{n,k,w} V_{n,k,x}\right].
\end{split}$$ We now find an upper bound to $\operatorname{E}\left[ V_{n,k,w} V_{n,k,x} \right]$. At first fix any word $w$ and the set $\emptyset \subsetneq E_{f} \subsetneq E_{w}$ and consider all the words $x$ such that $E_{{w}}\cap E_{x}= E_{w} \backslash E_{f}$. As every edge in $G_{w}$ and $G_{x}$ appear exactly once, $$\begin{split}
& \operatorname{E}\left[ V_{n,k,w} V_{n,k,x} \right]\\
&= \left( \frac{1}{n} \right)^{k} \sum_{E_w\backslash E' \subset E_{w} \backslash E_{f}} \prod_{e \in E'}\left(\pm \frac{4\beta^2}{n}\right)\operatorname{E}\left[\prod_{e \in E_{w}\backslash E'} B_{e}^2\right]\\
&\le \left( \frac{1}{n} \right)^{k} \sum_{E_w\backslash E' \subset E_{w} \backslash E_{f}}\left( \frac{4\beta^2}{n} \right)^{\#E'}\\
&\le \left( \frac{1}{n} \right)^{k+ \#E_{f}}2^{k}.
\end{split}$$ The last inequality holds since $\#E'\ge \#E_{f}$ and $\#(E_w\backslash E' \subset E_{w} \backslash E_{f})\le 2^{k}$.
Observe that the graph corresponding to the edges $E_{w} \backslash E_{f}$ is a disjoint collection of straight lines. Let the number of such straight lines be $\zeta$. Obviously $\zeta \le \#(E_{w} \backslash E_{f})$. The number of ways these $\zeta$ components can be placed in $x$ is bounded by $k^{\zeta}\le k^{\#(E_{w} \backslash E_{f})}$ and all other nodes in $x$ can be chosen freely. So there are at most $n^{k-\#V_{E_{w}\backslash E_f}}k^{\#(E_{w} \backslash E_{f})}$ choices of such $x$. Here $V_{E_{w}\backslash E_{f}}$ is the set of vertices of the graph corresponding to $(E_{w} \backslash E_{f})$. Observe that, whenever $k>\#E_{f}>0$, $E_{w}\backslash E_{f}$ is a forest so $$\#V_{E_{w}\backslash E_{f}}\ge \#(E_{w} \backslash E_{f})+1 \Leftrightarrow k-\#V_{E_{w}\backslash E_f} \le \#E_{f}-1.$$ As a consequence, $$\label{eqn:booboo}
\sum_{x~|~ E_{{w}}\cap E_{x}= E_{w} \backslash E_{f}}\operatorname{Cov}(V_{n,k,w}, V_{n,k,x})\le (2)^{k}\frac{1}{n^{k+\#E_{f}}}n^{\#E_{f}-1}k^{\#(E_{w} \backslash E_{f})}\le (2)^{k} \frac{1}{n^{k+1}}k^{k} .$$ The right hand side of does not depend on $E_{f}$ and there are at most $2^{k}$ nonempty subsets $E_{f}$ of $E^{w}$. So $$\sum_{x}\operatorname{Cov}(V_{n,k,w}, V_{n,k,x}) \le (4)^{k} k^{k}\frac{1}{n^{k+1}}.$$ Finally there are at most $n^{k}$ many $w$. So $$\label{eqn:final}
\sum_{w}\sum_{x}\operatorname{Cov}(V_{n,k,w}, V_{n,k,x})\le (4)^{k}k^{k}\frac{1}{n}.$$ Now we use the fact $k=o(\sqrt{\log(n)})$. In this case $$k\log(4)+k\log(k)\le \sqrt{log(n)}\log(\sqrt{\log n}) = o(log(n)) \Leftrightarrow (4)^{k}k^{k}=o(n).$$ This concludes the proof.
**Acknowledgment:** The author acknowledges Prof. Jinho Baik for suggesting this problem to him while he was visiting U Michigan for a summer school. The author also thanks Prof. Wei Kuo Chen and the referees for comments.
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abstract: 'Multifrequency media access control has been well understood in general wireless ad hoc networks, while in wireless sensor networks, researchers still focus on single frequency solutions. In wireless sensor networks, each device is typically equipped with a single radio transceiver and applications adopt much smaller packet sizes compared to those in general wireless ad hoc networks. Hence, the multifrequency MAC protocols proposed for general wireless ad hoc networks are not suitable for wireless sensor network applications, which we further demonstrate through our simulation experiments. In this article, we propose MMSN, which takes advantage of multifrequency availability while, at the same time, takes into consideration the restrictions of wireless sensor networks. Through extensive experiments, MMSN exhibits the prominent ability to utilize parallel transmissions among neighboring nodes. When multiple physical frequencies are available, it also achieves increased energy efficiency, demonstrating the ability to work against radio interference and the tolerance to a wide range of measured time synchronization errors[^1].'
author:
- Gang Zhou
- Yafeng Wu
- 'John A. Stankovic'
- Ting Yan
- Tian He
- Chengdu Huang
- 'Tarek F. Abdelzaher'
subtitle: This is a subtitle
title: A Multifrequency MAC Specially Designed for Wireless Sensor Network Applications
---
<ccs2012> <concept> <concept\_id>10010520.10010553.10010562</concept\_id> <concept\_desc>Computer systems organization Embedded systems</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010520.10010575.10010755</concept\_id> <concept\_desc>Computer systems organization Redundancy</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10010520.10010553.10010554</concept\_id> <concept\_desc>Computer systems organization Robotics</concept\_desc> <concept\_significance>100</concept\_significance> </concept> <concept> <concept\_id>10003033.10003083.10003095</concept\_id> <concept\_desc>Networks Network reliability</concept\_desc> <concept\_significance>100</concept\_significance> </concept> </ccs2012>
[^2]
[^1]: This is an abstract footnote
[^2]: This work is supported by the National Science Foundation, under grant CNS-0435060, grant CCR-0325197 and grant EN-CS-0329609.
Author’s addresses: G. Zhou, Computer Science Department, College of William and Mary; Y. Wu [and]{} J. A. Stankovic, Computer Science Department, University of Virginia; T. Yan, Eaton Innovation Center; T. He, Computer Science Department, University of Minnesota; C. Huang, Google; T. F. Abdelzaher, (Current address) NASA Ames Research Center, Moffett Field, California 94035.
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---
abstract: 'This paper presents a robotic pick-and-place system that is capable of grasping and recognizing both known and novel objects in cluttered environments. The key new feature of the system is that it handles a wide range of object categories without needing any task-specific training data for novel objects. To achieve this, it first uses an object-agnostic grasping framework to map from visual observations to actions: inferring dense pixel-wise probability maps of the affordances for four different grasping primitive actions. It then executes the action with the highest affordance and recognizes picked objects with a cross-domain image classification framework that matches observed images to product images. Since product images are readily available for a wide range of objects (e.g., from the web), the system works out-of-the-box for novel objects without requiring any additional data collection or re-training. Exhaustive experimental results demonstrate that our multi-affordance grasping achieves high success rates for a wide variety of objects in clutter, and our recognition algorithm achieves high accuracy for both known and novel grasped objects. The approach was part of the MIT-Princeton Team system that took 1st place in the stowing task at the 2017 Amazon Robotics Challenge. All code, datasets, and pre-trained models are available online at [http://arc.cs.princeton.edu](http://arc.cs.princeton.edu/)'
author:
- |
Andy Zeng$^{1}$, Shuran Song$^{1}$, Kuan-Ting Yu$^{2}$, Elliott Donlon$^{2}$, Francois R. Hogan$^{2}$, Maria Bauza$^{2}$, Daolin Ma$^{2}$,\
Orion Taylor$^{2}$, Melody Liu$^{2}$, Eudald Romo$^{2}$, Nima Fazeli$^{2}$, Ferran Alet$^{2}$, Nikhil Chavan Dafle$^{2}$, Rachel Holladay$^{2}$,\
Isabella Morona$^{2}$, Prem Qu Nair$^{1}$, Druck Green$^{2}$, Ian Taylor$^{2}$, Weber Liu$^{1}$, Thomas Funkhouser$^{1}$, Alberto Rodriguez$^{2}$\
$^{1}$Princeton University$^{2}$Massachusetts Institute of Technology\
[http://arc.cs.princeton.edu](http://arc.cs.princeton.edu/)\
<https://youtu.be/6fG7zwGfIkI> [^1]
bibliography:
- 'main.bib'
title: |
**Robotic Pick-and-Place of Novel Objects in Clutter\
with Multi-Affordance Grasping and Cross-Domain Image Matching**
---
Introduction
============
A human’s remarkable ability to grasp and recognize unfamiliar objects with little prior knowledge of them is a constant inspiration for robotics research. This ability to grasp the unknown is central to many applications: from picking packages in a logistic center to bin-picking in a manufacturing plant; from unloading groceries at home to clearing debris after a disaster. The main goal of this work is to demonstrate that it is possible – and practical – for a robotic system to pick and recognize novel objects with very limited prior information about them (e.g. with only a few representative images scraped from the web).
Despite the interest of the research community, and despite its practical value, robust manipulation and recognition of novel objects in cluttered environments still remains a largely unsolved problem. Classical solutions for robotic picking require recognition and pose estimation prior to model-based grasp planning, or require object segmentation to associate grasp detections with object identities. These solutions tend to fall short when dealing with novel objects in cluttered environments, since they rely on 3D object models and/or large amounts of training data to achieve robust performance. Although there has been inspiring recent work on detecting grasps directly from RGB-D pointclouds as well as learning-based recognition systems to handle the constraints of novel objects and limited data, these methods have yet to be proven in the constraints and accuracy required by a real task with heavy clutter, severe occlusions, and object variability.
![[**Our picking system**]{} computing pixel-wise affordances for grasping over visual observations of bins full of objects (a), grasping a towel and holding it up away from clutter, and recognizing it by matching observed images of the towel (b) to an available representative product image. The entire system works out-of-the-box for novel objects (unseen in training) without the need for any additional data collection or re-training. ](figures/v3-teaser.pdf){width="\linewidth"}
\[fig:teaser\]
In this paper, we propose a system that picks and recognizes objects in cluttered environments. We have designed the system specifically to handle a wide range of objects novel to the system without gathering any task-specific training data for them. To make this possible, our system consists of two components. The first is a multi-affordance grasping framework which uses fully convolutional networks (FCNs) to take in visual observations of the scene and output a dense grid of values (arranged with the same size and resolution as the input data) measuring the affordance (or probability of picking success) for four different grasping primitive actions over a pixel-wise sampling of end effector orientations and locations. The primitive action with the highest inferred affordance value determines the grasping action executed by the robot. This grasping framework operates without a priori object segmentation and classification and hence is agnostic to object identity. The second component of the system is a cross-domain image matching framework for recognizing grasped objects by matching them to product images useing a two-stream convolutional network (ConvNet) architecture. This framework adapts to novel objects without additional re-training. Both components work hand-in-hand to achieve robust picking performance of novel objects in heavy clutter. We provide exhaustive experiments and ablation studies to evaluate both components. We demonstrate that our affordance-based algorithm for grasp planning achieves high success rates for a wide variety of objects in clutter, and the recognition algorithm achieves high accuracy for known and novel grasped objects. These algorithms were developed as part of the MIT-Princeton Team system that took 1st place in the stowing task of the Amazon Robotics Challenge (ARC), being the only system to have successfully stowed all known and novel objects from an unstructured tote into a storage system within the allotted time frame. [Fig. \[fig:teaser\]]{} shows our robot in action during the competition.
In summary, our main contributions are:
- [An object-agnostic grasping framework using four primitive grasping actions for fast and robust picking, utilizing fully convolutional networks for inferring the pixel-wise affordances of each primitive (Section \[sec:manipulation\]).]{}
- [A perception framework for recognizing both known and novel objects using only product images without extra data collection or re-training (Section \[sec:recognition\]).]{}
- [A system combining these two frameworks for picking novel objects in heavy clutter.]{}
All code, datasets, and pre-trained models are available online at [http://arc.cs.princeton.edu](http://arc.cs.princeton.edu/) [@projectwebpage]. We also provide a video summarizing our approach at <https://youtu.be/6fG7zwGfIkI>.
Related Work
============
In this section, we review works related to robotic picking systems. Works specific to grasping (Section \[sec:manipulation\]) and recognition (Section \[sec:recognition\]) are in their respective sections.
Recognition followed by Model-based Grasping
--------------------------------------------
A large number of autonomous pick-and-place solutions follow a standard two-step approach: object recognition and pose estimation followed by model-based grasp planning. For example, Jonschkowski et al. [@jonschkowski2016probabilistic] designed object segmentation methods over handcrafted image features to compute suction proposals for picking objects with a vacuum. More recent data-driven approaches [@hernandez2016team; @zeng2016multi; @schwarz2017nimbro; @wong2017segicp] use ConvNets to provide bounding box proposals or segmentations, followed by geometric registration to estimate object poses, which ultimately guide handcrafted picking heuristics [@Bicchi2000RoboticContact; @Miller2003]. Nieuwenhuisen et al. [@nieuwenhuisen2013mobile] improve many aspects of this pipeline by leveraging robot mobility, while Liu et al. [@liu2012fast] adds a pose correction stage when the object is in the gripper. These works typically require 3D models of the objects during test time, and/or training data with the physical objects themselves. This is practical for tightly constrained pick-and-place scenarios, but is not easily scalable to applications that consistently encounter novel objects, for which only limited data (i.e. product images from the web) is available.
Recognition in parallel with Object-Agnostic Grasping
-----------------------------------------------------
It is also possible to exploit local features of objects without object identity to efficiently detect grasps [@morales2004; @lenz2015rss; @redmon2015; @ten2015using; @pinto2016; @pinto2017; @mahler2017; @gualtieri2017; @levine2016learning]. Since these methods are agnostic to object identity, they better adapt to novel objects and experience higher picking success rates by eliminating error propagation from a prior recognition step. Matsumoto et al. [@matsumotoend] apply this idea in a full picking system by using a ConvNet to compute grasp proposals, while in parallel inferring semantic segmentations for a fixed set of known objects. Although these pick-and-place systems use object-agnostic grasping methods, they still require some form of in-place object recognition in order to associate grasp proposals with object identities, which is particularly challenging when dealing with novel objects in clutter.
![[**The bin and camera setup.**]{} Our system consists of 4 units (top), where each unit has a bin with 4 stationary cameras: two overlooking the bin (bottom-left) are used for inferring grasp affordances while the other two (bottom-right) are used for recognizing grasped objects.[]{data-label="fig:setup-wide"}](figures/v3-setup.pdf){width="\linewidth"}
Active Perception
-----------------
Active perception – exploiting control strategies for acquiring data to improve perception [@bajcsy1992active; @chen2011active] – can facilitate the recognition of novel objects in clutter. For example, Jiang et al. [@jiang2016novel] describe a robotic system that actively rearranges objects in the scene (by pushing) in order to improve recognition accuracy. Other works [@wu2015active; @jayaraman2016look] explore next-best-view based approaches to improve recognition, segmentation and pose estimation results. Inspired by these works, our system applies active perception by using a grasp-first-then-recognize paradigm where we leverage object-agnostic grasping to isolate each object from clutter in order to significantly improve recognition accuracy for novel objects.
System Overview
===============
We present a robotic pick-and-place system that grasps and recognizes both known and novel objects in cluttered environments. The “known” objects are provided to the system at training time, both as physical objects and as representative product images (images of objects available on the web); while the “novel” objects are provided only at test time in the form of representative product images.
[**Overall approach.**]{} The system follows a *grasp-first-then-recognize* work-flow. For each pick-and-place operation, it first uses FCNs to infer the pixel-wise affordances of four different grasping primitive actions: from suction to parallel-jaw grasps (Section \[sec:manipulation\]). It then selects the grasping primitive action with the highest affordance, picks up one object, isolates it from the clutter, holds it up in front of cameras, recognizes its category, and places it in the appropriate bin. Although the object recognition algorithm is trained only on known objects, it is able to recognize novel objects through a learned cross-domain image matching embedding between observed images of held objects and product images (Section \[sec:recognition\]).
[**Advantages.**]{} This system design has several advantages. First, the affordance-based grasping algorithm is model-free and agnostic to object identities and generalizes to novel objects without re-training. Second, the category recognition algorithm works without task-specific data collection or re-training for novel objects, which makes it scalable for applications in warehouse automation and service robots where the range of observed object categories is large and dynamic. Third, our grasping framework supports multiple grasping modes with a multi-functional gripper and thus handles a wide variety of objects. Finally, the entire processing pipeline requires only a few forward passes through deep networks and thus executes quickly (Table \[table:speed\]).
![[**Multi-functional gripper**]{} with a retractable mechanism that enables quick and automatic switching between suction (pink) and grasping (blue).[]{data-label="fig:gripper"}](figures/v3-gripper.pdf){width="\linewidth"}
[**System setup.**]{} Our system features a 6DOF ABB IRB 1600id robot arm next to four picking work-cells. The robot arm’s end-effector is a multi-functional gripper with two fingers for parallel-jaw grasps and a retractable suction cup ([Fig. \[fig:gripper\]]{}). This gripper was designed to function in cluttered environments: finger and suction cup length are specifically chosen such that the bulk of the gripper body does not need to enter the cluttered space. Each work-cell has a storage bin and four statically-mounted RealSense SR300 RGB-D cameras ([Fig. \[fig:setup-wide\]]{}): two cameras overlooking the storage bins are used to infer grasp affordances, while the other two pointing towards the robot gripper are used to recognize objects in the gripper. Although our experiments were performed with this setup, the system was designed to be flexible for picking and placing between any number of reachable work-cells and camera locations. Furthermore, all manipulation and recognition algorithms in this paper were designed to be easily adapted to other system setups.
![[**Multiple motion primitives**]{} for suction and grasping to ensure successful picking for a wide variety of objects in any orientation.[]{data-label="fig:primitives"}](figures/primitives.pdf){width="\linewidth"}
Multi-Affordance Grasping {#sec:manipulation}
=========================
The goal of the first step in our system is to robustly grasp objects from a cluttered scene without relying on their object identities or poses. To this end, we define a set of four grasping primitive actions that are complementary to each other in terms of utility across different object types and scenarios – empirically maximizing the variety of objects and orientations that can be picked with at least one primitive. Given RGB-D images of the cluttered scene at test time, we infer the dense pixel-wise affordances for all four primitives. A task planner then selects and executes the primitive with the highest affordance (more details of this planner can be found in the Appendix).
{width="\linewidth"}
Grasping Primitives
-------------------
We define four grasping primitives to achieve robust picking for typical household objects. [Fig. \[fig:primitives\]]{} shows example motions for each primitive. Each of them are implemented as a set of guarded moves, with collision avoidance and quick success or failure feedback mechanisms: for suction, this comes from flow sensors; for grasping, this comes from contact detection via force feedback from sensors below the work-cell. Robot arm motion planning is automatically executed within each primitive with stable IK solves [@diankov_thesis]. These primitives are as follows:
[**Suction down**]{} grasps objects with a vacuum gripper vertically. This primitive is particularly robust for objects with large and flat suctionable surfaces (e.g. boxes, books, wrapped objects), and performs well in heavy clutter.
[**Suction side**]{} grasps objects from the side by approaching with a vacuum gripper tilted an an angle. This primitive is robust to thin and flat objects resting against walls, which may not have suctionable surfaces from the top.
[**Grasp down**]{} grasps objects vertically using the two-finger parallel-jaw gripper. This primitive is complementary to the suction primitives in that it is able to pick up objects with smaller, irregular surfaces (e.g. small tools, deformable objects), or made of semi-porous materials that prevent a good suction seal (e.g. cloth).
[**Flush grasp**]{} retrieves unsuctionable objects that are flushed against a wall. The primitive is similar to grasp down, but with the additional behavior of using a flexible spatula to slide one finger in between the target object and the wall.
Learning Affordances with Fully Convolutional Networks
------------------------------------------------------
Given the set of pre-defined grasping primitives and RGB-D images of the scene, we train FCNs [@long_shelhamer_fcn] to infer the affordances for each primitive across a dense pixel-wise sampling of end-effector orientations and locations (*i.e.* each pixel correlates to a different position on which to execute the primitive). Our approach relies on the assumption that graspable regions can be deduced from the local geometry and material properties, as reflected in visual information. This is inspired by recent data-driven methods for grasp planning [@morales2004; @lenz2015rss; @redmon2015; @pinto2016; @pinto2017; @mahler2017; @gualtieri2017; @levine2016learning], which do not rely on object identities or state estimation. [**Inferring Suction Affordances.**]{} We define suction points as 3D positions where the vacuum gripper’s suction cup should come in contact with the object’s surface in order to successfully grasp it. Good suction points should be located on suctionable (*e.g.* nonporous) surfaces, and nearby the target object’s center of mass to avoid an unstable suction seal (e.g. particularly for heavy objects). Each suction proposal is defined as a suction point, its local surface normal (computed from the projected 3D point cloud), and its affordance value. Each pixel of an RGB-D image (with a valid depth value) maps surjectively to a suction point.
We train a fully convolutional residual network (ResNet-101 [@he2016deep]), that takes a $640\times480$ RGB-D image as input, and outputs a densely labeled pixel-wise map (with the same image size and resolution as the input) of affordance values between 0 and 1. Values closer to one imply a more preferable suction location. Visualizations of these densely labeled affordance maps are shown as heat maps in the first row of [Fig. \[fig:passive-network\]]{}. Our network architecture is multi-modal, where the color data (RGB) is fed into one ResNet-101 tower, and 3-channel depth (DDD, cloned across channels, normalized by subtracting mean and dividing by standard deviation) is fed into another ResNet-101 tower. Features from the ends of both towers are concatenated across channels, followed by 3 additional spatial convolution layers to merge the features; then spatially bilinearly upsampled and softmaxed to output a binary probability map representing the inferred affordances.
Our FCN is trained over a manually annotated dataset of RGB-D images of cluttered scenes with diverse objects, where pixels are densely labeled either positive, negative, or neither. Pixel regions labeled as neither are trained with 0 loss backpropagation. We train our FCNs by stochastic gradient descent with momentum, using fixed learning rates of $10^{-3}$ and momentum of 0.99. Our models are trained in Torch/Lua with an NVIDIA Titan X on an Intel Core i7-3770K clocked at 3.5 GHz.
During testing, we feed each captured RGB-D image through our trained network to generate dense suction affordances for each view of the scene. As a post-processing step, we use calibrated camera intrinsics and poses to project the RGB-D data and aggregate the affordances onto a combined 3D point cloud. We then compute surface normals for each 3D point (using a local region around it), which are used to classify which suction primitive (down or side) to use for the point. To handle objects without depth, we use a simple hole filling algorithm [@NYUdataset] on the depth images, and project inferred affordance values onto the hallucinated depth. We filter out suction points from the background by performing background subtraction [@zeng2016multi] between the captured RGB-D image of the scene with objects and an RGB-D image of the scene without objects (captured automatically before any objects are placed into the picking work-cells).
{width="\linewidth"}
[**Inferring Grasp Affordances.**]{} Grasp proposals are represented by 1) a 3D position which defines the middle point between the two fingers during top-down parallel-jaw grasping, 2) an angle which defines the orientation of the gripper around the vertical axis along the direction of gravity, 3) the width between the gripper fingers during the grasp, and 4) its affordance value.
Two RGB-D views of the scene are aggregated into a registered 3D point cloud, which is then orthographically back-projected upwards in the gravity direction to obtain a “heightmap" image representation of the scene with both color (RGB) and height-from-bottom (D) channels. Each pixel of the heightmap represents a $2^2$mm vertical column of 3D space in the scene. Each pixel also correlates bijectively to a grasp proposal whose 3D position is naturally computed from the spatial 2D position of the pixel relative to the heightmap image and the height value at that pixel. The gripper orientation of the grasp proposal is horizontal with respect to the frame of the heightmap. Analogous to our deep network inferring suction affordances, we feed this RGB-D heightmap as input to a fully convolutional ResNet-101 [@he2016deep], which densely infers affordance values (between 0 and 1) for each pixel – thereby for all top-down parallel-jaw grasping primitives executed with a horizontally orientated gripper across all 3D locations in heightmap of the scene sampled at pixel resolution. Visualizations of these densely labeled affordance maps are also shown as heat maps in the second row of [Fig. \[fig:passive-network\]]{}. By rotating the heightmap of the scene with $n$ different angles prior to feeding as input to the FCN, we can account for $n$ different gripper orientations around the vertical axis. For our system $n=16$; hence we compute affordances for all top-down parallel-jaw grasping primitives with $16$ forward passes of our FCN to generate $16$ output affordance maps.
We train our FCN over a manually annotated dataset of RGB-D heightmaps, where each positive and negative grasp label is represented by a pixel on the heightmap as well as an angle correlated to the orientation of gripper. We trained this FCN with the same optimization parameters as that of the FCN used for inferring suction affordances.
During post-processing, the width between the gripper fingers for each grasp proposal is determined by using the local geometry of the 3D point cloud. We also use the location of each proposal relative to the bin to classify which grasping primitive (down or flush) should be used: flush grasp is executed for pixels located near the sides of the bins; grasp down is executed for all other pixels. To handle objects without depth, we triangulate no-depth regions in the heightmap using both RGB-D camera views of the scene, and fill in these regions with synthetic height values of 3cm prior to feeding into the FCN. We filter out inferred grasp proposals in the background by using background subtraction with the RGB-D heightmap of an empty work-cell.
Recognizing Novel Objects {#sec:recognition}
=========================
After successfully grasping an object and isolating it from clutter, the goal of the second step in our system is to recognize the identity of the grasped object.
Since we encounter both known and novel objects, and we have only product images for the novel objects, we address this recognition problem by retrieving the best match among a set of product images. Of course, observed images and product images can be captured in significantly different environments in terms of lighting, object pose, background color, post-process editing, etc. Therefore, we require an algorithm that is able to find the semantic correspondences between images from these two different domains. While this is a task that appears repeatedly in a variety of research topics (*e.g.* domain adaptation, one-shot learning, meta-learning, visual search, etc.), in this paper we simply refer to it as a *cross-domain image matching* problem [@saenko2010adapting; @shrivastava2011data; @bell2015learning].
Metric Learning for Cross-Domain Image Matching
-----------------------------------------------
To perform the cross-domain image matching between observed images and product images, we learn a metric function that takes in an observed image and a candidate product image and outputs a distance value that models how likely the images are of the same object. The goal of the metric function is to map both the observed image and product image onto a meaningful feature embedding space so that smaller $\ell_2$ feature distances indicate higher similarities. The product image with the smallest metric distance to the observed image is the final matching result.
We model this metric function with a two-stream convolutional neural network (ConvNet) architecture where one stream computes features for the observed images, and a different stream computes features for the product images. We train the network by feeding it a balanced 1:1 ratio of matching and non-matching image pairs (one observed image and one product image) from the set of known objects, and backpropagate gradients from the distance ratio loss (Triplet loss [@hoffer2016deep]). This effectively optimizes the network in a way that minimizes the $\ell_2$ distances between features of matching pairs while pulling apart the $\ell_2$ distances between features of non-matching pairs. By training over enough examples of these image pairs across known objects, the network learns a feature embedding that encapsulates object shape, color, and other visual discriminative properties, which can generalize and be used to match observed images of novel objects to their respective product images (Fig. \[fig:active-vision\]). [**Avoiding metric collapse by guided feature embeddings.**]{} One issue commonly encountered in metric learning occurs when the number of training object categories is small – the network can easily overfit its feature space to capture only the small set of training categories, making generalization to novel object categories difficult. We refer to this problem as metric collapse. To avoid this issue, we use a model pre-trained on ImageNet [@deng2009imagenet] for the product image stream and train only the stream that computes features for observed images. ImageNet contains a large collection of images from many categories, and models pre-trained on it have been shown to produce relatively comprehensive and homogenous feature embeddings for transfer tasks [@huh2016makes] – i.e. providing discriminating features for images of a wide range of objects. Our training procedure trains the observed image stream to produce features similar to the ImageNet features of product images – i.e., it learns a mapping from observed images to ImageNet features. Those features are then suitable for direct comparison to features of product images, even for novel objects not encountered during training.
[**Using multiple product images.**]{} For many applications, there can be multiple product images per object. However, with multiple product images, supervision of the two-stream network can become confusing - on which pair of matching observed and product images should the backpropagated gradients be based? To solve this problem, we add a module we call a “multi-anchor switch” in the network. During training, this module automatically chooses which “anchor” product image to compare against based on nearest neighbor $\ell_2$ distance. We find that allowing the network to select its own criterion for choosing “anchor” product images provides a significant boost in performance in comparison to alternative methods like random sampling.
Two Stage Framework for a Mixture of Known and Novel Objects
------------------------------------------------------------
In settings where both types of objects are present, we find that training two different network models to handle known and novel objects separately can yield higher overall matching accuracies. One is trained to be good at “over-fitting" to the known objects (K-net) and the other is trained to be better at “generalizing" to novel objects (N-net).
Yet, how do we know which network to use for a given image? To address this issue, we execute our recognition pipeline in two stages: a “recollection” stage that determines whether the observed object is known or novel, and a “hypothesis” stage that uses the appropriate network model based on the first stage’s output to perform image matching.
First, the recollection stage infers whether the input observed image from test time is that of a known object that has appeared during training. Intuitively, an observed image is of a novel object if and only if its deep features cannot match to that of any images of known objects. We explicitly model this conditional by thresholding on the nearest neighbor distance to product image features of known objects. In other words, if the $\ell_2$ distance between the K-net features of an observed image and the nearest neighbor product image of a known object is greater than some threshold k, then the observed images is a novel object.
In the hypothesis stage, we perform object recognition based on one of two network models: K-net for known objects and N-net for novel objects. The K-net and N-net share the same network architecture. However, the K-net has an additional auxiliary classification loss during training for the known objects. This classification loss increases the accuracy of known objects at test time to near perfect performance, and also boosts up the accuracy of the recollection stage, but fails to maintain the accuracy of novel objects. On the other hand, without the restriction of the classification loss, N-net has a lower accuracy for known objects, but maintains a better accuracy for novel objects.
By adding the recollection stage, we can exploit both the high accuracy of known objects with K-net and good accuracy of novel objects with N-net, though incurring a cost in accuracy from erroneous known vs novel classification. We find that this two stage system overall provides higher total matching accuracy for recognizing both known and novel objects (mixed) than all other baselines (Table \[table:recognition\]).
Experiments
===========
In this section, we evaluate our affordance-based grasping framework, our recognition algorithm over both known and novel objects, as well as our full system in the context of the Amazon Robotics Challenge 2017.
Evaluating Multi-affordance Grasping
------------------------------------
[**Datasets.**]{} To generate datasets for learning affordance-based grasping, we designed a simple labeling interface that prompts users to manually annotate suction and grasp proposals over RGB-D images collected from the real system. For suction, users who have had experience working with our suction gripper are asked to annotate pixels of suctionable and non-suctionable areas on raw RGB-D images overlooking cluttered bins full of various objects. Similarly, users with experience using our parallel-jaw gripper are asked to sparsely annotate positive and negative grasps over re-projected heightmaps of cluttered bins, where each grasp is represented by a pixel on the heightmap and an angle corresponding to the orientation (parallel-jaw motion) of the gripper. On the interface, users directly paint labels on the images with wide-area circular (suction) or rectangular (grasping) brushstrokes. The diameter and angle of the strokes can be adjusted with hotkeys. The color of the strokes are green for positive labels and red for negative labels. Examples of images and labels from this dataset can be found in [Fig. \[fig:dataset\]]{} of the Appendix. During training, we further augment each grasp label by adding additional labels via small jittering (less than 1.6cm). In total, the dataset contains 1837 RGB-D images with suction and grasp labels. We use a 4:1 training/testing split across this dataset to train and evaluate different models.
[**Evaluation.**]{} In the context of our grasping framework, a method is robust if it is able to consistently find at least one suction or grasp proposal that works. To reflect this, our evaluation metric is the precision of inferred proposals versus manual annotations. For suction, a proposal is considered a true positive if its pixel center is manually labeled as a suctionable area (false positive if manually labeled as an non-suctionable area). For grasping, a proposal is considered a true positive if its pixel center is nearby within 4 pixels and 11.25 degrees from a positive grasp label (false positive if nearby a negative grasp label). We report the precision of our inferred proposals for different confidence percentiles in Table \[table:affordance-prediction\]. The precision of the top-1 proposal is reliably above 90% for both suction and grasping. We further compare our methods to heuristic-based baseline algorithms that compute suction affordances by estimating surface normal variance over the observed 3D point cloud (lower variance = higher affordance), and computes anti-podal grasps by detecting hill-like geometric structures in the 3D point cloud. Baselines details and code are available on our project webpage [@projectwebpage].
------------------------------------------------------------------ -- -- -- -- --
**Primitive & **Method &**Top-1 & **Top 1% &**Top 5% &**Top 10%\
& Baseline & 35.2 & 55.4 & 46.7 & 38.5\
& ConvNet & **92.4 & 83.4 & 66.0 & 52.0\
& Baseline & 92.5 & 90.7 & 87.2 & 73.8\
& ConvNet & **96.7 & 91.9 & 87.6 & 84.1\
****************
------------------------------------------------------------------ -- -- -- -- --
: Multi-affordance Grasping Performance
\[table:affordance-prediction\]\
% precision of grasp proposals across different confidence percentiles.
[**Speed.**]{} Our suction and grasp affordance algorithms were designed to achieve fast run-time speeds during test time by densely inferring affordances over images of the entire scene. In Table \[table:speed\], we compare our run-time speeds to several state-of-the-art alternatives for grasp planning. Our own numbers measure the time of each FCN forward pass, reported with an NVIDIA Titan X on an Intel Core i7-3770K clocked at 3.5 GHz, excluding time for image capture and other system-related overhead. Our FCNs run at a fraction of the time required by most other methods, while also being significantly deeper (with 101 layers) than all other deep learning methods.
Evaluating Novel Object Recognition
-----------------------------------
We evaluate our recognition algorithms using a 1 vs 20 classification benchmark. Each test sample in the benchmark contains 20 possible object classes, where 10 are known and 10 are novel, chosen at random. During each test sample, we feed the recognition algorithm the product images for all 20 objects as well as an observed image of a grasped object. In Table \[table:recognition\], we measure performance in terms of average % accuracy of the top-1 nearest neighbor product image match of the grasped object. We evaluate our method against a baseline algorithm, a state-of-the-art network architecture for both visual search [@bell2015learning] and one-shot learning without retraining [@koch2015siamese], and several variations of our method. The latter provides an ablation study to show the improvements in performance with every added component:
----------------------------------------------------- --
**Method &**Time\
Lenz et al. [@lenz2015rss] & 13.5\
Zeng et al. [@zeng2016multi] & 10 - 15\
Hernandez et al. [@hernandez2016team] & 5 - 40 ^a^\
Schwarz et al. [@schwarz2017nimbro] & 0.9 - 3.3\
Dex-Net 2.0 [@mahler2017] & 0.8\
Matsumoto et al. [@matsumotoend] & 0.2\
Redmon et al. [@redmon2015] & 0.07\
Ours (suction) & 0.06\
Ours (grasping) & 0.05$\times{n}$ ^b^\
****
----------------------------------------------------- --
: Grasp Planning Run-Times (sec.)
\
^a^ times reported from [@matsumotoend] derived from [@hernandez2016team].\
^b^ $n$ = number of possible grasp angles. \[table:speed\]
[**Nearest neighbor**]{} is a baseline algorithm where we compute features of product images and observed images using a ResNet-50 pre-trained on ImageNet, and use nearest neighbor matching with $\ell_2$ distance. [**Siamese network with weight sharing**]{} is a re-implementation of Bell et al. [@bell2015learning] for visual search and Koch et al. [@koch2015siamese] for one shot recognition without retraining. We use a Siamese ResNet-50 pre-trained on ImageNet and optimized over training pairs in a Siamese fashion. The main difference between this method and ours is that the weights between the networks computing deep features for product images and observed images are shared.
[**Two-stream network without weight sharing**]{} is a two-stream network, where the networks’ weights for product images and observed images are not shared. Without weight sharing the network has more flexibility to learn the mapping function and thus achieves higher matching accuracy. All the later models describe later in this section use this two stream network without weight sharing.
[**Two-stream + guided-embedding (GE)**]{} includes a guided feature embedding with ImageNet features for the product image stream. We find this model has better performance for novel objects than for known objects.
[**Two-stream + guided-embedding (GE) + multi-product-images (MP)**]{} By adding a multi-anchor switch, we see more improvements to accuracy for novel objects. This is the final network architecture for N-net.
[**Two-stream + guided-embedding (GE) + multi-product-images (MP) + auxiliary classification (AC)**]{} By adding an auxiliary classification, we achieve near perfect accuracy of known objects for later models, however, at the cost of lower accuracy for novel objects. This also improves known vs novel (K vs N) classification accuracy for the recollection stage. This is the final network architecture for K-net.
[**Two-stage system**]{} As described in Section \[sec:recognition\], we combine the two different models - one that is good at known objects (K-net) and the other that is good at novel objects (N-net) - in the two stage system. This is our final recognition algorithm, and it achieves better performance than any single model for test cases with a mixture of known and novel objects.
------------------------------------------------------------------------------ -- -- -- --
**Method &**K vs N &**Known &**Novel &**Mixed\
Nearest Neighbor & 69.2 & 27.2 & 52.6 & 35.0\
Siamese ([@bell2015learning; @koch2015siamese]) & 70.3 & 76.9 & 68.2 & 74.2\
Two-stream & 70.8 & 85.3 & 75.1 & 82.2\
Two-stream + GE & 69.2 & 64.3 & 79.8 & 69.0\
Two-stream + GE + MP (N-net) & 69.2 & 56.8 & **82.1 & 64.6\
N-net + AC (K-net) & **93.2 & **99.7 & 29.5 & 78.1\
Two-stage K-net + N-net & 93.2 & 93.6 & 77.5 & **88.6\
******************
------------------------------------------------------------------------------ -- -- -- --
: Recognition Evaluation (% Accuracy of Top-1 Match)
\[table:recognition\]
Full System Evaluation in Amazon Robotics Challenge
---------------------------------------------------
To evaluate the performance of our system as a whole, we used it as part of our MIT-Princeton entry for the 2017 Amazon Robotics Challenge (ARC), where state-of-the-art pick-and-place solutions competed in the context of a warehouse automation task. Participants were tasked with designing a robot system to grasp and recognize a large variety of different objects in unstructured storage systems. The objects were characterized by a number of difficult-to-handle properties. Unlike earlier versions of the competition [@Correll2016], half of the objects were novel in the 2017 edition of the competition. The physical objects as well as related item data (i.e. product images, weight, 3D scans), were given to teams just 30 minutes before the competition. While other teams used the 30 minutes to collect training data for the new objects and re-train models, our unique system did not require any of that during those 30 minutes. [**Setup.**]{} Our system setup for the competition features several differences. We incorporated weight sensors to our system, using them as a guard to signal stop for grasping primitive behaviors during execution. We also used the measured weights of objects provided by Amazon to boost recognition accuracy to near perfect performance. Green screens made the background more uniform to further boost accuracy of the system in the recognition phase. For inferring affordances, Table \[table:affordance-prediction\] shows that our data-driven methods with ConvNets provide more precise affordances for both suction and grasping than the baseline algorithms. For the case of parallel-jaw grasping, however, we did not have time to develop a fully stable network architecture before the day of the competition, so we decided to avoid risks and use the baseline grasping algorithm. The ConvNet-based approach became stable with the reduction to inferring only horizontal grasps and rotating the input heightmaps. This is discussed more in depth in the Appendix, along with a state tracking/estimation algorithm used for the picking task of the ARC.
[**Results.**]{} During the ARC 2017 final stowing task, we had a 58.3% pick success with suction, 75% pick success with grasping, and 100% recognition accuracy during the stow task of the ARC, stowing all 20 objects within 24 suction attempts and 8 grasp attempts. Our system took 1st place in the stowing task, being the only system to have successfully stowed all known and novel objects and to have finished the task well within the allotted time frame.
Discussion and Future Work
==========================
We present a system to pick and recognize novel objects with very limited prior information about them (a handful of product images). The system first uses an object-agnostic visuomotor affordance-based algorithm to select among four different grasping primitive actions, and then recognizes grasped objects by matching them to their product images. We evaluate both components and demonstrate their combination in a robot system that picks and recognizes novel objects in heavy clutter, and that took 1st place in the stowing task of the Amazon Robotics Challenge 2017. Here are some of the most salient features/limitations of the system:
[**Object-Agnostic Manipulation.**]{} The system finds grasp affordances directly in the RGB-D image. This proved faster and more reliable than doing object segmentation and state estimation prior to grasp planning [@zeng2016multi]. The ConvNet learns the visual features that make a region of an image graspable or suctionable. It also seems to learn more complex rules, e.g., that tags are often easier to suction that the object itself, or that the center of a long object is preferable than its ends. It would be interesting to explore the limits of the approach. For example learning affordances for more complex behaviors, e.g., scooping an object against a wall, which require a more global understanding of the geometry of the environment.
[**Pick First, Ask Questions Later.**]{} The standard grasping pipeline is to first recognize and then plan a grasp. In this paper we demonstrate that it is possible and sometimes beneficial to reverse the order. Our system leverages object-agnostic picking to remove the need for state estimation in clutter. Isolating the picked object drastically increases object recognition reliability, especially for novel objects. We conjecture that “pick first, ask questions later” is a good approach for applications such as bin-picking, emptying a bag of groceries, or clearing debris. It is, however, not suited for all applications – nominally when we need to pick a particular object. In that case, the described system needs to be augmented with state tracking/estimation algorithms.
[**Towards Scalable Solutions.**]{} Our system is designed to pick and recognize novel objects without extra data collection or re-training. This is a step forward towards robotic solutions that scale to the challenges of service robots and warehouse automation, where the daily number of novel objects ranges from the tens to the thousands, making data-collection and re-training cumbersome in one case and impossible in the other. It is interesting to consider what data, besides product images, is available that could be used for recognition using out-of-the-box algorithms like ours.
[**Limited to Accessible Grasps.**]{} The system we present in this work is limited to picking objects that can be directly perceived and grasped by one of the primitive picking motions. Real scenarios, especially when targeting the grasp of a particular object, often require plans that deliberately sequence different primitive motions. For example, when removing an object to pick the one below, or when separating two objects before grasping one. This points to a more complex picking policy with a planning horizon that includes preparatory primitive motions like pushing whose value is difficult to reward/label in a supervised fashion. Reinforcement learning of policies that sequence primitive picking motions is a promising alternative approach worth exploring.
[**Open-loop vs. Closed-loop Grasping**]{} Most existing grasping approaches, whether model-based or data-driven are for the most part, based on open-loop executions of planned grasps. Our system is no different. The robot decides what to do and executes it almost blindly, except for simple feedback to enable guarded moves like move until contact. Indeed, the most common failure modes are when small errors in the estimated affordances lead to fingers landing on top of an object rather than on the sides, or lead to a deficient suction latch, or lead to a grasp that is only marginally stable and likely to fail when the robot lifts the object. It is unlikely that the picking error rate can be trimmed to industrial grade without the use of explicit feedback for closed-loop grasping during the approach-grasp-retrieve operation.
Appendix {#sec:appendix}
========
Task Planner Details
--------------------
Our full system for the ARC also includes a task planner that selects and executes the suction or grasp proposal with the highest affordance value. Prior to this, affordance values are scaled by a factor $\gamma_\psi$ that is specific to the proposals’ primitive action types $\psi\in\{\mathrm{sd,ss,gd,fg}\}$: suction down (sd), suction side (ss), grasp down (gd), or flush grasp (fg). The value of $\gamma_\psi$ is determined by several task-specific heuristics that induce more efficient picking under competition settings. Here we briefly describe these heuristics:
[**Suction first, grasp later.**]{} We empirically find suction to be more reliable than parallel-jaw grasping when picking in scenarios with heavy clutter (10+ objects). Hence, to reflect a greedy picking strategy that initially favors suction over grasping, $\gamma_\mathrm{gd} = 0.5$ and $\gamma_\mathrm{fg} = 0.5$ for the first 3 minutes of either ARC task (stowing or picking).
[**Avoid repeating unsuccessful attempts.**]{} It is possible for the system to get stuck repeatedly executing the same (or similar) suction or grasp proposal as no change is made to the scene (and hence affordance estimates remain the same). Therefore, after each unsuccessful suction or parallel-jaw grasping attempt, the affordances of the proposals (for the same primitive action) nearby within radius 2cm of the unsuccessful attempt are set to 0.
[**Encouraging exploration upon repeat failures.**]{} The planner re-weights grasping primitive actions $\gamma_\psi$ depending on how often they fail. For primitives that have been unsuccessful for two times in the last 3 minutes, $\gamma_\psi = 0.5$; if unsuccessful for more than three times, $\gamma_\psi = 0.25$. This not only helps the system avoid repeating unsuccessful actions, but also prevents it from excessively relying on any one primitive that doesn’t work as expected (*e.g.* in the case of an unexpected hardware failure preventing suction air flow).
[**Leveraging dense affordances for speed picking.**]{} Our FCNs densely infer affordances for all visible surfaces in the scene, which enables the robot to attempt multiple different suction or grasping proposals (at least 3cm apart from each other) in quick succession until at least one of them is successful (given by immediate feedback from flow sensors or gripper finger width). This improves picking efficiency.
State Tracking/Estimation
-------------------------
While the system described in the main paper works well out-of-the-box for the stowing task of the ARC, it requires an additional state tracking/estimation algorithm in order to perform competitively during the picking task of the ARC, where the goal is to pick *target* objects out of a storage system (*e.g.* shelves, separate work-cells) and place them into specific boxes for order fulfillment. Our state tracking algorithm is built around the assumption that each object in the storage system has been placed by another automated system – hence the identities of the objects and their positions in the storage system can be tracked over time as the storage system is stocked (*e.g.* from being completely empty to being full of objects).
The goal of our state tracking algorithm is to track the objects (their identities, 6D poses, amodal bounding boxes, and support relationships) as they are individually placed into the storage system one after the other. This information can then later be used by the task planner during the picking task to prioritize certain grasp proposals (close to, or above target objects) over others. After executing a grasp, our system continues to perform the recognition algorithm described in the main paper as a final verification step before placing it into a box. Objects that are not the intended target objects for order fulfillment are placed into another (relatively empty) bin in the storage system.
When autonomously adding an object into the storage system (*e.g.* during the stowing task), our state tracking algorithm captures RGB-D images of the storage system at time $t$ (before the object is placed) and at time $t+1$ (after the object is placed). The difference between the RGB-D images captured at $t+1$ and $t$ provides an estimate for the visible surfaces of the newly placed object (*i.e.* near the pixel regions with the largest change). 3D models of the objects (either constructed from the same RGB-D data captured during recognition or given by another system) are aligned to these visible surfaces via ICP-based pose estimation [@zeng2016multi]. To reduce the uncertainty and noise of these pose estimates, the placing primitive actions are gently executed – *i.e.* the robot arm holding the object moves down slowly until contact between the object and storage system is detected with weight sensors, upon which then the gripper releases the object.
To handle placing into boxes with different sizes, our recognition framework simultaneously estimates a 3D bounding box of the grasped object (using the same RGB-D data captured for the recognition framework). The bounding box enables the placing primitives to re-orient grasped objects such that they can fit into the target boxes.
![[**Images and annotations from the grasping dataset**]{} with labels for suction (top row) and parallel-jaw grasping (bottom row). Positive labels appear in green while negative labels appear in red.[]{data-label="fig:dataset"}](figures/dataset.pdf){width="\linewidth"}
Other Network Architectures for Parallel-Jaw Grasping
-----------------------------------------------------
A significant challenge during the development of our system was designing a deep network architecture for inferring dense affordances for parallel-jaw grasping that 1) supports various gripper orientations and 2) could quickly converge during training with less than 2000 manually labeled images. It took several iterations of network architecture designs before discovering one that worked (described in the main paper). Here, we briefly review the deprecated architectures and their primary drawbacks:
**Parallel trunks and branches ($n$ copies).** This design consists of $n$ separate FCNs, each responsible for inferring the output affordances for one of $n$ grasping angles. Each FCN shares the same architecture: a multi-modal trunk (with color (RGB) and depth (DDD) data fed into two ResNet-101 towers pre-trained on ImageNet, where features at the ends of both towers are concatenated across channels), followed by 3 additional spatial convolution layers to merge the features; then spatially bilinearly upsampled and softmaxed to output an affordance map. This design is similar to our final network design, but with two key differences: 1) there are multiple FCNs, one for each grasping angle, and 2) the input data is not rotated prior to feeding as input to the FCNs. This design is sample inefficient, since each network during training is optimized to learn a different set of visual features to support a specific grasping angle, thus requiring a substantial amount of training samples with that specific grasping angle to converge. Our small manually annotated dataset is characterized by an unequal distribution of training samples across different grasping angles, some of which have as little as less than 100 training samples. Hence, only a few of the FCNs (for grasping angles of which have more than 1,000 training samples) are able to converge during training. Furthermore, attaining the capacity to pre-load all $n$ FCNs into GPU memory for test time requires multiple GPUs.
**One trunk, split to $n$ parallel branches.** This design consists of a single FCN architecture, which contains a multi-modal ResNet-101 trunk followed by a split into $n$ parallel, individual branches, one for each grasping angle. Each branch contains 3 spatial convolution layers followed by spatial bilinearly upsampling and softmax to output affordance maps. While more lightweight in terms of GPU memory consumption (*i.e.* the trunk is shared and only the 3-layer branches have multiple copies), this FCN still runs into similar training convergence issues as the previous architecture, where each branch during training is optimized to learn a different set of visual features to support a specific grasping angle. The uneven distribution of limited training samples in our dataset made it so that only a few branches are able to converge during training.
**One trunk, rotate, one branch.** This design consists of a single FCN architecture, which contains a multi-modal ResNet-101 trunk, followed by a spatial transform layer [@jaderberg2015spatial] to rotate the intermediate feature map from the trunk with respect to an input grasp angle (such that the gripper orientation is aligned horizontally to the feature map), followed by a branch with 3 spatial convolution layers, spatially bilinearly upsampled, and softmaxed to output a single affordance map for the input grasp angle. This design is even more lightweight than the previous architecture in terms of GPU memory consumption, performs well with grasping angles for which there is a sufficient amount of training samples, but continues to performs poorly for grasping angles with very few training samples (less than 100).
**One trunk and branch (rotate $n$ times).** This is the final network architecture design as proposed in the main paper, which differs from the previous design in that the rotation occurs directly on the input image representation prior to feeding through the FCN (rather than in the middle of the architecture). This enables the entire network to share visual features across different grasping orientations, enabling it to generalize for grasping angles of which there are very few training samples.
### Perception Sensor Mounting
Goal: make affordance prediction and recognition easy and fast.
Stationary multi-camera setup (as opposed to camera mounted on gripper) for more speed and viewing angles. Trade-off: frequent disconnections for camera driver, buffer issues.
Each work-cell is comprised of a bin, a set of four statically-mounted cameras, and two weight sensors. Two cameras look inside the bin for generating picking proposals and the other two look above the bin for recognition of a picked object. We refer to the former set as passive cameras and the latter as active cameras [Fig. \[fig:setup-wide\]]{}. Weight sensors are used for object identification and contact sensing during primitive execution. Relative to other camera mounting strategies there are two advantages to static mounting.
- [$\Cdot$]{}Save time: no extra robot motion is required to take pictures and the recognition can happen when robot is executing other motions;
- [$\Cdot$]{}Save space: no camera on the gripper yields a slimmer end-effector.
There are two disadvantages
- [$\Cdot$]{}Interference: the laser projectors used for depth measurement interfere between camera individuals, which forces us to use one camera at a time;
- [$\Cdot$]{}Unstable connections: one camera failure can cause the driver to crash. This happened in our final competition run. Also, we spent much engineering effort to connect all USB cameras to our computers reliably.
In the recognition phase, green screens make the background more uniform to boost accuracy of the system.
### Gripper
Our strategy was to enable multiple grasp behaviors – ensuring that all objects can be picked at least one way in all orientations. We facilitate this with a hybrid grasp-suction strategy featuring a high-fidelity Gelsight sensor\cite{}, a thin, spatula-like finger to slide into tight spaces, and an actuated mechanism for switching between high-powered suction and fingertip grasping [Fig. \[fig:gripper\]]{}.
![Our gripper switches between grasping and suction, allowing us to have a large, high-flow suction system while also being able to grasp with unimpeded fingertips.[]{data-label="fig:gripper"}](figures/Placeholdergripper.jpg){width="\linewidth"}
While many works choose to simplify manipulation systems by removing or minimizing clutter, this gripper was designed to function in cluttered environments. Even though the design is comparatively large overall, finger and suction cup length are specifically chosen such that the bulk of the gripper will never need to enter cluttered spaces. The thin spatula finger is actuated and can be flushed outward against bin walls, or inward to make stable, four-point grasps.
### Suction
Suction primitive picks or places object with the suction cup with suction proposals provided by perception and high-level planner. Suction proposal is the position where suction cup should be sealed to the object, which is the most important decision for successful suction, with the fact that only certain areas of an object might be suctionable. Suction primitive takes care of all the necessary actions needed, such as motion planning as well as interactive motion design with feedback from flow sensor and weight sensors, after taking inputs from high-level planner. For objects near boundary of the totes, option of side suction with tilted suction cup is also provided to make a good sealing.
### Grasping
The grasping primitive is used to retrieve objects with two fingers from the storage system. Due to the uncertainty regarding the position of the target objects, the grasping primitive heavily relies on system sensors (gelsight in fingers and weight sensors under storage system) to detect when contact has been made between the robot and the storage system. This reactive behavior is used to locate the position of the object about the vertical direction and to ensure that no damage is done to the objects, gripper, and storage system.
The input to the grasping primitive is a target position and orientation of the gripper as well as the desired gripper opening. In order to get the robot from the initial configuration to this desired pose, we follow a set of safe motions in order to minimize collisions with other objects and the storage system:
- move the gripper to a distance $D$ above the target grasp in the desired orientation;
- open spatula and gripper with desired opening;
- move downwards towards the object while monitoring for unexpected contact forces (weight sensors);
- once contact is detected, close spatula and gripper with force control;
- move the gripper at the height of the cameras to identify the object retrieved;
- return success or failure of grasping primitive.
The steps described above are useful for objects that are located at a safe distance from the walls of the storage system. When objects are flushed against the side of the wall, we can reorient the gripper, so that the flexible spatula can be pressed against and sneak in between the object and the wall. This variant of grasping is referred to as “flush grasping.”
[^1]: The authors would like to thank the MIT-Princeton ARC team members for their contributions to this project, and ABB Robotics, Mathworks, Intel, Google, NSF (IIS-1251217 and VEC 1539014/1539099), NVIDIA, and Facebook for hardware, technical, and financial support.
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---
abstract: 'In social networks, it is conventionally thought that two individuals with more overlapped friends tend to establish a new friendship, which could be stated as homophily breeding new connections. While the recent hypothesis of maximum information entropy is presented as the possible origin of effective navigation in small-world networks. We find there exists a competition between information entropy maximization and homophily in local structure through both theoretical and experimental analysis. This competition means that a newly built relationship between two individuals with more common friends would lead to less information entropy gain for them. We conjecture that in the evolution of the social network, both of the two assumptions coexist. The rule of maximum information entropy produces weak ties in the network, while the law of homophily makes the network highly clustered locally and the individuals would obtain strong and trust ties. Our findings shed light on the social network modeling from a new perspective.'
author:
- 'Jichang Zhao$^{1,\star}$, Xiao Liang$^{2}$ and Ke Xu$^{3}$'
title: Competition Between Homophily and Information Entropy Maximization in Social Networks
---
Introduction {#sec:motivation}
============
The last decade has witnessed tremendous research interests in complex networks [@newman_survey; @evolution-zhou; @xu-finacial-network], including the evolution of social networks [@newman_cluster; @jin_structure; @homophily; @yang_growing; @clustering_citation]. It has been found that in many social networks from different circumstances, the probability of having a friend at a distance $r$ is $p(r)\propto r^{-1}$, which is stated as the spacial scaling law [@distance_notdead]. Recent work [@max_entropy] presents a possible origin that explains the emergence of this scaling law with the hypothesis of maximum information entropy with energy constrains. The authors assume that human social behavior is based on gathering maximum information through various activities and making friends is one of them. However, it is also found conventionally that homophily leads to connections in social networks [@homophily; @transitive; @jin_structure; @davidsen_emergence; @cnn; @cnnr; @toivonen_model; @holme_growing; @link_prediction]. Homophily is the principle that a contact between similar individuals occurs at a higher rate than among dissimilar ones [@homophily]. For instance, in social networks, two individuals with more common friends are easier to get connected, where the number of overlapped friends could represent the strength of homophily. Both of the above rules might drive the growth of the network in local structure simultaneously, however, to our best knowledge, little has been done to unveil the relationship between them. In this paper, we try to fill this gap from the perspective of network evolution in local structure.
Theoretical Analysis {#sec:ta}
====================
A social network can be modeled as a simple undirected graph $G(V,E)$, where $V$ is the set of individuals (nodes) and $E$ is the set of friendships (ties) among them. As shown in FIG. \[fig:example\_1\], node 1 may obtain information from nodes 2, 3, 4 and their friends 5, 7. Therefore, as defined in [@max_entropy], the information sequence for node 1 is {2, 3, 4, 5, 7} and the frequency of each node appears in the sequence is $q_2=q_3=q_4=q_5=q_7=1/5$ for nodes 2, 3, 4, 5 and 7 respectively, while $q_6=0$ for node 6. Then the information entropy for node 1 can be obtained as $$\epsilon
(1)=-\sum_{i=1}^{7}{q_i\log{q_i}}=1.61.$$ Next, we assume the social network evolves to the one as shown in FIG. \[fig:example\_2\] under the rule of homophily. For example, node 1 and node 5 may establish a new friendship because they share the common friend node 2. Therefore, the updated information sequence for node 1 is {2, 3, 4, 5, 5, 7, 2} currently. Then the new frequency of each node appears in the sequence is $q_2{'}=q_5{'}=2/7$, $q_3{'}=q_4{'}=q_7{'}=1/7$, and $q_6{'}=0$. We recompute the information entropy of node 1 as depicted above and obtain $$\epsilon^{'}(1)=-\sum_{1}^{7}{q_{i}{'}\log{q_{i}{'}}}=1.55.$$ It can be easily observed that $\Delta
\epsilon(1)=\epsilon{'}(1)-\epsilon(1)<0$ after node 1 built a new tie with node 5, which means in the evolution dominated by homophily, the information entropy for node 1 decreases. It is an intuitive observation that the rule of homophily is incompatible with the law of maximum information entropy, and a general explanation is introduced as follows. Note that here we mainly discuss the network evolution in local structure, in which ties are newly built only with nodes two hops away. Because of this, with the aim of simplification, conditions of limited energy and nodes’ distances are not considered in the following analytical framework. Besides, the magnificent development of the online social network has facilitated our daily social activity greatly[@trustties; @ahn-cyworld-wwww], so here the cost of establishing a new tie is assumed to be a constant and it is independent to the distance in social networks.
We define $n(i)$ as the set of individual $i$’s initial friends and $k_i$ is $i$’s degree, i.e., the number of its friends. Then the set of overlapped friends between $i$ and $j$ is $c(i,j)=n(i)\cap n(j)$ and $c_{ij}=|c(i,j)|$ is the number of their common friends. We define $U=\cup_{q\in n(i)}{n(q)} \cup n(i).$ We also define $\Psi=\{j\} \cup c(i,j),$ where $j$ is a random individual appearing in $i$’s information sequence $s(i)$ and $j\notin n(i)$. Based on the definition of information entropy in [@max_entropy], we can obtain the information entropy for node $i$ is $$\begin{aligned}
\epsilon(i)_j=&-\sum_{q\in U/\Psi}{\frac{n_q}{s_i}\log{\frac{n_q}{s_i}}} -
\sum_{l\in c(i,j)}{\frac{n_l}{s_i}\log{\frac{n_l}{s_i}}} \notag \\
&-\frac{c_{ij}}{s_i}\log{\frac{c_{ij}}{s_i}},\end{aligned}$$ where $n_q$ is the count that $q$ appears in $s(i)$ and $s_i$ is the length of $s(i)$. Since we mainly investigate the evolution in local structure, here only friends of $i$ and friends of its friends are considered during the computation of the entropy. Then we assume that a new friendship is established between $i$ and $j$ and the current entropy for $i$ is $$\begin{aligned}
\epsilon{'}(i)_j=&-\sum_{q\in
U/\Psi}{\frac{n_q}{s_i'}\log{\frac{n_q}{s_i'}}} - \sum_{l\in
c(i,j)}{\frac{n_l+1}{s_i'}\log{\frac{n_l+1}{s_i'}}} \notag \\
&-\frac{c_{ij}+1}{s_i'}\log{\frac{c_{ij}+1}{s_i'}}-
(k_j-c_{ij})\frac{1}{s_i'}\log{\frac{1}{s_i'}},\end{aligned}$$ where $s_i{'}=s_i+k_j-c_{ij}+1+c_{ij}=s_i+k_j+1$, which is the length of the updated information sequence, where $k_j$ is the initial degree of $j.$ Therefore, the change of entropy for $i$ caused by the new tie with $j,$ i.e., $\Delta \epsilon(i)_j=\epsilon{'}(i)_j-\epsilon(i)_j$ could be rewritten as $$\begin{aligned}
\Delta \epsilon(i)_j = &\sum_{q\in
U/\Psi}{(\frac{n_q}{s_i}\log{\frac{n_q}{s_i}} -
\frac{n_q}{s_i'}\log{\frac{n_q}{s_i'}})} \notag \\
&+ \sum_{l\in c(i,j)}{(\frac{n_l}{s_i}\log{\frac{n_l}{s_i}}-\frac{n_l+1}{s_i'}\log{\frac{n_l+1}{s_i'}})}\notag \\
&+ (\frac{c_{ij}}{s_i}\log{\frac{c_{ij}}{s_i}} -
\frac{c_{ij}+1}{s_i'}\log{\frac{c_{ij}+1}{s_i'}}) \notag \\
&- (k_j-c_{ij})\frac{1}{s_i'}\log{\frac{1}{s_i'}}.\end{aligned}$$ Assume $f(x) = x\log x$, $$f(x+\Delta x) = f(x) + f'(x)\Delta x +
o({(\Delta x)^2}),$$ therefore, $$\frac{n_l+1}{s_i'}\log{\frac{n_l+1}{s_i'}} =
\frac{n_l}{s_i'}\log{\frac{n_l}{s_i'}} + (\log{\frac{n_l}{s_i'}} +
1)\frac{1}{s_i'} + o(\frac{1}{{s_i'}^2})$$ and $$\frac{c_{ij}+1}{s_i'}\log{\frac{c_{ij}+1}{s_i'}} =
\frac{c_{ij}}{s_i'}\log{\frac{c_{ij}}{s_i'}} +
(\log{\frac{c_{ij}}{s_i'}}+1)\frac{1}{s_i'} + o(\frac{1}{{s_i'}^2}).$$ Then for Equation (3) we have (for details, see [*Appendix*]{}), $$\begin{aligned}
\Delta \epsilon(i)_j =& -\frac{k_j+1}{s_i'}\epsilon(i)_j - \sum_{l\in
\Psi}{\frac{1}{s_i'}\log{n_l}} - \frac{c_{ij}+1}{s_i'} \notag \\
&+\frac{k_j+1}{s_i'}\log{s_i'} - (c_{ij} +
1)o(\frac{1}{{s_i'}^2}).\end{aligned}$$ Suppose that $k_j$ is fixed, it can be easily obtained that as $c_{ij}$ grows, $\Delta \epsilon(i)_j$ decreases. Given the network is undirected, so this conclusion is also proper for $j$. Then we can conclude that if we build a new tie between $i$ and $j$, the information entropy gain $\Delta \epsilon(i,j)=\Delta
\epsilon(i)_j+\Delta \epsilon(j)_i$ produced by this new friendship for the two nodes decreases as $c_{ij}$ increases. It tells us that for the nodes with more common friends, establishing a new tie between them produces less information entropy gain for them. Be brief, there is a competition between homophily and information entropy in breeding a new connection. Note that $\Delta \epsilon(i,j)$ declining with $c_{ij}$ might be very slow, because generally $s_i'$ is much greater than $c_{ij}.$
In fact, the information entropy for $i$ represents the diversity of its information sources. If we create ties between $i$ and other nodes who have overlapped friends with it, these nodes will appear more frequently in its information sequence and even become the dominating sources of the information. Then the diversity of the information source is weaken and the gain of the information entropy decays accordingly.
Empirical Analysis {#sec:ea}
==================
In order to validate the above analysis, we employ several data sets, including both synthetic and real-world networks, for further empirical study. The synthetic data sets are generated by BA [@ba_model], Small World [@sw_model] and CNNR [@cnnr] models. BA is a classic model to generate scale-free networks with the mechanism of preferential attachment. We denote the data set it generates as BA$(N,m)$, where $N$ is the size of the network and $m$ is the number of initial ties that would be connected when a new node is added. Small World model is a random model with probability $p$ to rewire and produce long range ties, it can be denoted as SW$(N,K,p)$, where $2K$ is the averaged degree. CNNR model is modified from CNN [@cnn] for generating social networks, especially online social networks. We denote it as CNNR$(N,u,r)$, where $u(1-r)$ is the probability to covert the potential edges into real ties. The averaged degree of the network it generates is approximately $2/(1-u)$. The real-world data sets come from different fields. For example, `CA-HepPh` is a collaboration network from the e-print arXiv[^1] and covers scientific collaborations between authors of papers submitted to High Energy Physics [@ca_hepPh_dataset]. `NewOrleans` is the Facebook network in New Orleans [@neworleans_dataset]. `Email-Enron` is an email communication network that covers all the email communication within a data set of around half million emails [@email_dataset]. The basic properties of theses data sets we utilize in following experiments are listed in Tab. \[tab:dataset\].
[lll]{} Data set & $N$ & $|E|$\
`BA(20000,10)` & 20000 & 199352\
`SW(20000,10,0.1)` & 20000 & 200000\
`CNNR(20000,0.9,0.04)` & 20000 & 187215\
`CA-HepPh` & 12006 & 118489\
`NewOrleans` & 63392 & 816886\
`Email-Enron` & 36692 & 183831\
\
\
As discussed before, establishing a new friendship may affect the entropy of the both ends. In the above networks, we characterize the relation between $c_{ij}$ and $\Delta \epsilon(i,j)$ in the following steps: For each tie between $i$ and $j$, we first obtain $\epsilon{'}(i)_j+\epsilon{'}(j)_i$ in the origin network; Secondly, we delete this tie and get $\epsilon(i)_j+\epsilon(j)_i$; Thirdly, the tie is restored. For different $\Delta \epsilon(i,j)$ for the same $c_{ij}$, we get the maximum, mean and minimum values, respectively. The change of entropy for other nodes in the network is not considered here for the reason that we assume the establishment of a tie between $i$ and $j$ is a personal activity with local information solely. As shown in FIG. \[fig:cnn\_delta\_e\], in all networks, $\Delta
\epsilon(i,j)$ decreases as $c_{ij}$ grows, which is consistent with our above analysis, especially for the small world network in FIG. \[fig:sw\_cnn\_delta\_e\]. At the start stage, the diverge between the maximum and mean of $\Delta \epsilon(i,j)$ is large, then it decays quickly as $c_{ij}$ increases. It is also observed that for the nodes with tremendous common friends, building a new friendship between them may even lead to entropy loss. To sum up, the empirical results testify our statement further that increment of homophily would reduce the information entropy gain, which indicates a competition between the two evolving rules.
Discussion {#sec:discuss}
==========
[lll]{} Data set & $\tau$ & $c$\
`NewOrleans` & 0.70 & 0.22\
`Email-Enron` & 0.56 & 0.50\
`CA-HepPh` & 0.50 & 0.61\
The growing of a social network could be simply regarded as establishing new ties among individuals. From the perspective of information entropy maximization, a tie should be established to gain more entropy for both ends. Therefore, we could distinguish the tie that makes the entropy of its ends gain as the positive tie, while the one that leads to entropy loss as the negative tie. Then we define the positiveness of the social network as the fraction of positive ties, which is denoted as $\tau$. Larger $\tau$ means more ties in the network are established to increase their ends’ entropy gain. As shown in Tab. \[tab:real\_c\_e\], we list $\tau$ of the real-world network, where $c$ is the clustering of the network. It is interesting that for the network with higher $c$, its $\tau$ is lower generally. We also investigate this finding on the network with various clusterings generated by BA and Small World models. For the BA model, we employ the method of tuning clustering while keeping its degree distribution stable [@kim_clustering; @ma_clustering]. We only perform experiments of tuning the clustering on BA(1000,4), because it is too much time consuming for BA(20000,10). For the model of Small World, we just vary $p$. As shown in FIG. \[fig:c\_e\], for both of models, the positiveness of network decreases as $c$ grows. In fact, the clustering of the network could be rewritten [@rewrite_c] as $$c=\frac{1}{|V|}\sum_{\forall (i,j) \in E}{\frac{c_{ij}}{{k_i \choose 2}}}.$$ For this reason, with respect to the rule of homophily, a new tie added preferentially between nodes with overlapped friends would also lead to new triangles constructed in local structure. That is to say, the clustering of the network, i.e., $c,$ would be increased when its evolution is driven by the homophily. Because of this, homophily dominated evolution leads to the decrement of $\tau$. However, with respect to the information entropy maximization, the new tie is established to increase the diversity of the information source and gain more entropy, which would improve $\tau$ by importing more positive ties.
![$\tau$ of the network varies as $c$ increases. \[fig:c\_e\]](c_map_e.eps){width="3in"}
The strength of a social tie can be defined as the number of overlapped friends between its ends. For example, the strength of a tie between $i$ and $j$ could be defined as $w_{ij}=c_{ij}/(k_i-1+k_j-1-c_{ij})$ [@mobile_network; @bridgeness; @tie-role], where lower $w_{ij}$ stands for a weak tie. It is obvious that if $i$ and $j$ share a lot of common friends, the strength of the tie between them is strong. Conventionally, it is thought that the weak tie is helpful in getting the new information [@weakties], while the strong tie means the relationship is trustful [@trustties]. Therefore, based on the above discussion, it seems that the evolution supervised by homophily could lead to generations of strong ties in the network, because it renders the network highly clustered. In order to validate this, we observe the cumulative distribution function(CDF) of $w_{ij}$ for each tie in the network. As shown in FIG. \[fig:c\_cdf\], as $c$ of the network decreases, the CDF curve moves to the left, which indicates the increment of the fraction of weak ties [@cdf_wij]. It validates our conjecture that in both synthetic and real-world data sets, highly clustered networks caused by homophily contain more strong ties, while the ones with lower clusterings contain more weak ties, which are produced by the law of maximum information entropy.
Conclusion and Future Work {#sec:conclusion}
==========================
In summary, both theoretical analysis and experimental results show that the rule of homophily is competing with the law of information entropy maximization in social networks. Moreover, the rule of homophily driven evolution makes the network highly clustered and increases the certainty of the information source for a node. Contrarily, the rule of maximum entropy leads to the diversity of information sources. Based on the definition of weak ties, we can conclude that the rule of maximum information entropy leads to the generation of weak ties in the network, while the homophily produces strong ties between nodes with overlapped friends. Corresponding to the fact that both the weak and strong ties coexist in the network, we conjecture that both of the evolving rules might coexist in growth of the social networks. Therefore, in the view of maximum information entropy, the social network is not efficient, however, it owns many strong ties which may deliver trust information. Our findings could provide insights for modeling social network evolution as a competition of different rules.
Given the tremendous development of the online social network, the cost of social activity in the epoch of the Internet continues to decrease [@trustties; @ahn-cyworld-wwww]. Because of this, we neglect the cost of establishing ties of different strengths for simplifying the analytical framework in this paper. While in the real world, the social activity is constrained by the personal cognition limit and social cost [@offlinenetworksize] and the Dunbar’s number [@dunbarnumber] still exists in the online social network [@dunbar-zhao; @golder:rhythms; @ahn-cyworld-wwww]. Hence in the future work, we would take the cost of establish different ties into consideration and build an evolution model of social networks based on the competition of strong and weak ties.
Acknowledgment {#sec:ack .unnumbered}
==============
Jichang Zhao was partially supported by the Fundamental Research Funds for the Central Universities (Grant Nos. YWF-14-RSC-109 and YWF-14-JGXY-001).
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Appendix {#sec:append .unnumbered}
========
$$\begin{aligned}
\Delta \epsilon(i)_j &=& \sum_{q\in
U/\Psi}{(\frac{n_q}{s_i}\log{\frac{n_q}{s_i}} -
\frac{n_q}{s_i'}\log{\frac{n_q}{s_i'}})} \\ &+& \sum_{l\in
c(i,j)}(\frac{n_l}{s_i}\log{\frac{n_l}{s_i}} -
\frac{n_l}{s_i'}\log{\frac{n_l}{s_i'}} - (\log{\frac{n_l}{s_i'}} +
1)\frac{1}{s_i'}\\&-&o(\frac{1}{{s_i'}^2})) \\
&+& (\frac{c_{ij}}{s_i}\log{\frac{c_{ij}}{s_i}} -
\frac{c_{ij}}{s_i'}\log{\frac{c_{ij}}{s_i'}} - (\log{\frac{c_{ij}}{s_i'}} +
1)\frac{1}{s_i'}-o(\frac{1}{{s_i'}^2})) \\
&-& (k_j - c_{ij})\frac{1}{s_i'}\log{\frac{1}{s_i'}} \\
&=& \sum_{q\in U}{(\frac{n_q}{s_i}\log{\frac{n_q}{s_i}} -
\frac{n_q}{s_i'}\log{\frac{n_q}{s_i'}})} - \sum_{l\in
\Psi}{\frac{1}{s_i'}(\log{\frac{n_l}{s_i'}} + 1)}\\
&-& (k_j-c_{ij})\frac{1}{s_i'}\log{\frac{1}{s_i'}} - (c_{ij} +
1)o(\frac{1}{{s_i'}^2}) \\
&=& \sum_{q\in U}{(\frac{n_q}{s_i}\log{\frac{n_q}{s_i}} -
\frac{n_q}{s_i'}(\log{\frac{n_q}{s_i}} + \log{\frac{s_i}{s_i'}}))}\\ &-&
\sum_{l\in \Psi}{\frac{1}{s_i'}(\log{\frac{n_l}{s_i'}} + 1)} -
(k_j-c_{ij})\frac{1}{s_i'}\log{\frac{1}{s_i'}}\\ &-& (c_{ij} +
1)o(\frac{1}{{s_i'}^2}) \\
&=& (1-\frac{s_i}{s_i'})\sum_{q\in U}{\frac{n_q}{s_i}\log{\frac{n_q}{s_i}}} - \frac{s_i}{s_i'}\log{\frac{s_i}{s_i'}}\\
&-& \sum_{l\in \Psi}{\frac{1}{s_i'}(\log{\frac{n_l}{s_i'}} + 1)} -
(k_j-c_{ij})\frac{1}{s_i'}\log{\frac{1}{s_i'}} \\&-& (c_{ij} +
1)o(\frac{1}{{s_i'}^2})\\
&=& -(1-\frac{s_i}{s_i'})\epsilon(i)_j - \sum_{l\in
\Psi}{\frac{1}{s_i'}(\log{\frac{n_l}{s_i'}} + 1)}\\& -&
(k_j-c_{ij})\frac{1}{s_i'}\log{\frac{1}{s_i'}} - (c_{ij} +
1)o(\frac{1}{{s_i'}^2}) \\
&=& -(1-\frac{s_i}{s_i'})\epsilon(i)_j - \sum_{l\in
\Psi}{\frac{1}{s_i'}\log{\frac{n_l}{s_i'}}} - \frac{c_{ij}+1}{s_i'} \\&-&
(k_j-c_{ij})\frac{1}{s_i'}\log{\frac{1}{s_i'}} - (c_{ij} +
1)o(\frac{1}{{s_i'}^2})\\
&=& -(1-\frac{s_i}{s_i'})\epsilon(i)_j - \sum_{l\in
\Psi}{\frac{1}{s_i'}\log{n_l}} - \frac{c_{ij}+1}{s_i'} \\&+&
(k_j+1)\frac{1}{s_i'}\log{s_i'} - (c_{ij} +
1)o(\frac{1}{{s_i'}^2})\\
&=& -\frac{k_j+1}{s_i'}\epsilon(i)_j - \sum_{l\in
\Psi}{\frac{1}{s_i'}\log{n_l}} - \frac{c_{ij}+1}{s_i'}\\ &+&
\frac{k_j+1}{s_i'}\log{s_i'} - (c_{ij} +
1)o(\frac{1}{{s_i'}^2})\\\end{aligned}$$
[^1]: http://www.arxiv.org
|
---
abstract: 'We present the construction of the $\lambda$-deformation of $AdS_5\times S^5$ superstring from the four dimensional Chern-Simons-type gauge theory. The procedure is applicable to all the semi-symmetric coset models and generalizes the previous construction of $\lambda$-deformation of the symmetric coset models.'
author:
- 'Jia Tian$^{1,3}$, Yi-Jun He$^{1}$,and Bin Chen$^{1,2,3}$[^1]'
title: '**$\lambda$-deformed $AdS_5\times S^5$ superstring from 4D Chern-Simons theory**'
---
[ *$^{1}$School of Physics and State Key Laboratory of Nuclear Physics and Technology,\
Peking University, No.5 Yiheyuan Rd, Beijing 100871, P. R. China\
$^{2}$Collaborative Innovation Center of Quantum Matter, No.5 Yiheyuan Rd, Beijing 100871, P. R. China\
$^{3}$Center for High Energy Physics, Peking University, No.5 Yiheyuan Rd, Beijing 100871, P. R. China* ]{}
Introduction
============
The classical integrable systems can be described in the Lax formalism. Therefore in some sense classifying the classical integrable systems is equivalent to classifying Lax connections. For classical Hamiltonian systems there does exist a general procedure, due to Zakharov and Shabat, to construct integrable systems from the Lax pairs [@ZS]. In this so-called Zakharov-Shabat construction, the Lax pairs are characterized by their analytic properties and the underlying semi-simple Lie algebras[^2]. There is also a field theory analogue of Zakharov-Shabat construction [@ZSfield] by extending the semi-simple Lie algebra to an infinite dimensional algebra such as the central-extended loop algebra or affine Kac-Moody algebra. But this construction has not been carefully considered or fully explored. The recently developed affine Gaudin model approach is closest to such a construction [@Gaudin]. The affine Gaudin model is in the Hamiltonian formulation while sometimes it is more convenient to describe classical field theories in the Lagrangian formulation.
In the past few years, a new approach to study integrable models from the point of view of four dimensional Chern-Simons-type (4D CS) gauge theory [@Costello:2013zra] has been developed in a series of papers [@Costello:2017dso; @Costello:2018gyb; @CY3]. In particular, Costello and Yamazaki [@CY3] proposed a Lagrangian approach to construct two dimensional integrable field theories (2D IFTs) from a 4D CS gauge theory with a meromorphic one-form $\omega$. It turns out that the 4D approach is closely related to the affine Gaudin model approach [@Relation]. From the 4D CS point of view, the resulted 2D IFTs are completely characterized by the choice of the one-form $\omega$ and the boundary conditions of the gauge fields. The construction has shown its power by covering a wide variety of 2D IFTs, including principle chiral models (PCM), Wess-Zumino-Witten models, sigma models whose target space are symmetric cosets, the superstring on AdS$_5 \times $ S$^5$, etc.[@CY3]. It was conjectured that the all known 2D IFTs could arise from such a construction. Following the work in [@CY3], many interesting 2D IFTs with deformations, including the Yang-Baxter deformation [@BosonicYangBaxter; @DelducEta; @Yoshida] and $\lambda$-deformation [@Lambda; @LambdaCoset], have been realized in this approach [@JT; @Uni; @COUPLED; @HOLOLAMBDA; @YB1; @YB2; @STRING].
In this paper our goal is to realize the $\lambda$-deformation of the $AdS_5\times S^5$ superstring or other semi-symmetric coset models[^3]. The strategy comes from the observations that in the 4D approach the $\lambda$-deformation can be performed by splitting the double poles of the one-form $\omega$ into a pair of simple poles, and the symmetric coset models can be constructed by introducing the covering space [@CY3; @JT].
The paper has the following organization. In section \[Review\] we briefly describe the 4D CS approach. In section \[lambda\] and section \[Superstring\] we show the constructions of $\lambda$-deformation of Principle Chiral Model (PCM) and $AdS_5\times S_5$ superstring which are two main ingredients of our following construction. The section \[LambdaSuperstring\] is devoted to the derivation of the $\lambda$-deformation of $AdS_5\times S_5$ superstring in the pure spinor formulation and in the Appendix, we give the construction in Green-Schwarz formulation.
2D IFTs from 4D CS {#Review}
==================
In this section we briefly describe how to derive 2D integrable sigma models from 4D Chern-Simons-type gauge theory. We begin with the action of the 4D Chern-Simons-type gauge theory [^4] \[4Daction\] S\[A\]= \_[\^[2]{} \^1]{} CS(A), where &&A=A\_d+A\_d+A\_[|[z]{}]{}d|[z]{},=(z)dz,\
&&CS(A)=A,dA+AA. Here $\omega(z)$ is a meromorphic function on $\cp$ and it can be identified with the twist function which also plays the crucial role in the construction of integrable field theories from the affine Gaudin model approach [@Gaudin]. Varying the action with respect to the gauge field $A$ gives the bulk equation of motion \[BulkEOM\] F(A)=0,F(A)=dA+AA=0, and the boundary equation of motion \[BEOM\] dA,A=0. The bulk equation of motion is just the flatness condition of the gauge field. To study the boundary equation of motion it is useful to rewrite in terms of the coordinates: \[Boundary\] \_[x]{}\_[p0]{}\^(\_x \^p\_x )\_[ij]{} \^p\_[\_x]{}A\_i, A\_j |\_x=0, where $\mathfrak{p}$ is the set of poles of the one-form $\omega$, $\xi_x$ is a local holomorphic coordinate around $x \in \mathfrak{p}$ and $i,j= \sigma,\tau$ are the coordinates of $\mathbb{R}^{2}$. In our construction we will only encounter $\omega$ with simple poles and double poles. For the double poles we always choose the Dirichlet boundary condition, i.e. \[Dirichlet\] A=A=0. Considering only the simple poles, the boundary equation of motion simplifies to \[BSimple\] \_[x]{}(\_x )\_[ij]{}A\_i ,A\_j|\_x=0. The zeros of $\omega$ constrain the gauge fields as well. To have a non-degenerate propagator the gauge fields must have the poles at the positions of the zeros of $\omega$. This could be induced by introducing some defect operators[^5] inserted at those positions. Then the 2D IFT is totally determined by the boundary conditions and the defect operators.
The Lax connection of the 2D IFT is directly related to the 4D gauge field through a gauge transformation \[ConnectionLax\] A=-d \^[-1]{} +L\^[-1]{}, for some regular $\hat{g}:\mathbb{R}^{2}\times \cp \rightarrow G$ such that $L_{\bar{z}}=0$. In terms of the Lax connection the bulk equation of motion reads [^6] \[\_++L\_+,\_-+L\_-\]=0,\_[|[z]{}]{}L(z,,)=0. The second identity implies that the positions of the poles of $L$ coincide with the positions of the zeros of $\omega$. When the 4D field $\hat{g}$ satisfies the archipelago conditions introduced in [@Uni] we can localize the four dimensional field to a two dimensional field $g_x$ such that the 4D action is reduced to a 2D action [@Uni] \[Action\] S\[{g\_x}\_[x]{}\]= \_[x]{} \_[\^[2]{}]{} \_x L,g\_x\^[-1]{}dg\_x-\_[x]{}(\_x ) I\_[WZ]{}\[g\_x\]. This resulted 2D theory has two kinds of gauge redundancies. Firstly there is always an overall gauge transformation $g_x \rightarrow g_x h, h \in G$ which reflects the redundancy in the definition of $g_x$ in terms of $A$. Secondly if the gauge field $A$ at $x\in \mathfrak{p}$ does not vanish but takes values in some algebra $\mathfrak{h}$ whose corresponding group is $H$ then we can perform a gauge transformation $g_x \rightarrow u_x g_x, u_x\in H$.
To conclude this section, let us summarize the general procedures to construct a 2D IFT:
1. Choose a meromorphic one-form $\omega=\omega(z)dz$.
2. Specify the boundary conditions at the positions of the poles of $\omega$.
3. Remove the gauge redundancies [^7] in $g_x,x\in \mathfrak{p}$.
4. Make an ansatz of the Lax connection with the poles at the positions of the zeros of $\omega$.
5. Solve the Lax connection by substituting into the boundary conditions.
6. Substitute the Lax connection into to get the action of the 2D IFT.
$\lambda$-deformation of PCM {#lambda}
============================
The prototypical example is the construction of the principle chiral model (PCM) [@Uni]. The corresponding one-form has two double poles and two simple zeros. To construct the $\lambda$-deformed PCM, one can split one double pole into two simple poles. As a result the one-form can be chosen to be [@Uni] \[lambdaOneform\] =dz. The boundary condition which needs to be satisfied is \_[ij]{}(A\_i ,A\_j|\_-A\_i ,A\_j|\_[-]{})=0. The solution which leads to the $\lambda$-deformation is requiring $(A|_\eta,A|_{-\eta})\in \mathfrak{g}^\delta$ to take values in a Lagrangian subalgebra of $\mathfrak{g}\oplus \mathfrak{g}\equiv \mathfrak{d}$ as[^8] \[BoundaryLambda\] \^={(x,x)|x}(\^, ) In other words, we will identify the gauge fields at the two boundaries $z=\pm \eta$. Since $z=\infty$ is a double pole, we will use the Dirichlet boundary condition. To remove the gauge redundancy we first use the overall gauge symmetry to set $g_\infty=I$ then use the local gauge symmetry $H=\{(h,h)|h\in G\}$ to fix $(g_\eta,g_{-\eta})=(g,1)$. Because the one-form has zeros at $\pm 1$ and the gauge field vanishes at $z=\infty$ so the ansatz of the Lax connection can be \[ansatzLambda\] \_+=,\_-=, where $V_\pm$ are regular functions. Rewriting the gauge fields in terms of the Lax connection through and substituting into the boundary condition give a set of equations of $V_\pm$. The solution is &&V\_+=(-1)(\_g+)\^[-1]{}\_g j\_+,\
&&V\_-=(+1)(\_g+)\^[-1]{}\_g j\_-, where $\operatorname{Ad}_g x=g x g^{-1}$ is the adjoint conjugation. Introducing the parameters $\lambda$ and $k$ as \[Para\] =, k=-1/(4), and substituting the Lax connection into leads to the action of $\lambda$–deformed PCM [@Lambda] &&\[g\]=k d\_+d\_- +k I\_[WZ]{}(g).
To conclude this section, let us make some comments about the construction.
- In it seems that $\lambda$ and $k$ are related to each other, however we have freedom to put a prefactor in the one-form such that these parameters become independent.
- In the ansatz of the Lax connection , we intentionally distribute the two poles into the two components of the Lax connection to avoid the appearance of the double poles in the flatness condition. In principle the most general ansatz should be =+.
- Because we have assumed the algebra $\mathfrak{g}$ to be real in the beginning so the resulted 2D IFT is real. Otherwise one has to consider additional reality conditions [@Uni].
- To get a non-trivial 2D IFT, the one-form must at least have two poles, otherwise one can always use the overall gauge symmetry to trivialize the field $g=I$.
- To construct the symmetric coset model, one can start with the PCM and modulo a $\mathbb{Z}_2$ discrete symmetry. This quotient will not identify the fields at the two boundaries instead the quotient relates the two fields through a $\mathbb{Z}_2$ involution as $g_2=\rho (g_1),\rho^2=1$. To construct the $\lambda$ deformation of the coset model, one can split both of the double poles into two pairs of simple poles and apply the boundary condition [@JT].
- One 2D IFT can be constructed from different set-ups of the 4D CS theory. For example, the (Yang-Baxter deformation or $\lambda$-deformation of) symmetric coset model has been obtained in [@YB1] and in [@JT] with different settings.
The $AdS_5\times S^5$ superstring {#Superstring}
=================================
Let us start with the undeformed theory. In the original paper [@CY3], the $AdS_5\times S^5$ superstring was constructed as a special example of the generalized Riemannian symmetric spaces. Recently it was constructed from a different setting of the 4D CS theory [@YB2]. Interestingly the authors of [@YB2] found that by choosing different boundary conditions the same one-form can lead to either the $AdS_5\times S^5$ superstring or homogeneous Yang-Baxter deformation of the superstring. To construct the $\lambda$-deformation of the superstring, we find that starting with original setting in [@CY3] is more convenient.
The semi-symmetric coset space $AdS_5\times S^5$ is associated with a $\mathbb{Z}_4$ graded algebra $\mathfrak{psu}(2,2|4)=\mathfrak{g}=\mathfrak{g}^{(0)}\oplus \mathfrak{g}^{(1)}\oplus \mathfrak{g}^{(2)}\oplus \mathfrak{g}^{(3)}$. The construction is also suitable for other semi-symmetric coset spaces with $\mathbb{Z}_4$ grading, for example $AdS_3\times S^3$. There exists an automorphism $\rho$ of order 4 for the lie algebra $\mathfrak{g}$. Starting with a Riemann surface equipped with a holomorphic 1-form $dz$, we introduce a cut of degree $4$ at $[0,1]$ and when the gauge fields cross this cut we apply the automorphism $\rho$. Then we go to the 4-fold covering space of the $z$–plane by introducing the new coordinate $u$ through z=-. In the 4-fold covering space, the one-form $dz$ pulls back to[^9] \[SuperOneForm\] =, it has four double poles at ={1,i} and two triple zeros at ={0,}. At the double poles we again take the Dirichlet boundary conditions . It was already pointed out in [@CY3] that there is a subtlety about the formulation of superstring coset model. In the Green-Schwarz formulation [@GSF], the kinetic term of the coset Lagrangian is degenerate on the fermionic fields but the degeneracy can be resolved by introducing the $\kappa$-symmetry. So in order to construct the integrable superstring theory in this formulation from the 4D perspective, the 4D CS theory has to be modified to incorporate the $\kappa$–symmetry. This modified 4D CS theory was recently proposed in [@STRING]. To avoid this subtlety we will use the pure spinor formulation [@PureF] following the convention in [@CY3]. In other words, we want to construct an integrable supercoset model whose kinetic term is non-degenerate. It turns out that the corresponding Lax connection coincides with pure-spinor Lax connection [@SpinConnetion]. Since $\omega$ has zeros of order $3$, in our ansatz of the Lax connection there should be the poles at the positions of the zeros with orders up to three: \[LaxAnsatz\] &&\_+=V\^0\_++V\^1\_+ u\^[-3]{}+V\^2\_+ u\^[-2]{}+V\^3\_+ u\^[-1]{},&&\_-=V\^0\_-+V\^1\_- u\^[1]{}+V\^2\_- u\^[2]{}+V\^3\_- u\^[3]{}. Let us first take the field contents at the four boundaries to be \[Field\] |\_[z=1]{}=g\_1,|\_[z=i]{}=g\_2,|\_[z=-1]{}=g\_3,|\_[z=-i]{}=g\_4. Considering that $\pm i$ and $\pm 1$ have the same preimage, we can set them to be connected by the $Z_4$ automorphsim $\rho$ as \[FieldZ4\] g\_2=(g\_1),g\_3=\^2(g\_1),g\_4=\^3(g\_1),\^4=1, which remove the overall gauge redundancy [@JT]. Because of the Dirichlet boundary conditions, all gauge field $A$ vanish at the boundaries so there is no local gauge redundancy to remove. Substituting the and into boundary equations gives &&j\_[1,]{}=V\^0\_+V\^1\_+V\^2\_+V\^3\_,j\_[2,]{}=V\^0\_+iV\^1\_-V\^2\_-iV\^3\_,&&j\_[3,]{}=V\^0\_-V\^1\_+V\^2\_-V\^3\_,j\_[4,]{}=V\^0\_-iV\^1\_-V\^2\_+iV\^3\_, where we have defined the left-invariant currents j\_[i,]{}=g\_i\^[-1]{}\_g\_i. These equations can be solved by &&V\^0\_=,V\^1\_=,&&V\^2\_=,V\^3\_=. Evaluating the residues at the positions of the poles gives \[ResSuper\] &&\_1((u)\_)=(3 j\_[1,]{}+(i1)j\_[2,]{}j\_[3,]{}-(i1)j\_[4,]{}),\
&&\_i((u)\_)=(3 j\_[2,]{}+(i1)j\_[3,]{}j\_[4,]{}-(i1)j\_[1,]{}),&&\_[-1]{}((u)\_)=(3 j\_[3,]{}+(i1)j\_[4,]{}j\_[1,]{}-(i1)j\_[2,]{}),&&\_[-i]{}((u)\_)=(3 j\_[4,]{}+(i1)j\_[1,]{}j\_[2,]{}-(i1)j\_[3,]{}). To proceed, let us consider the $\mathbb{Z}_4$ automorphism on the $\mathbb{Z}_4$ graded algebra \[Z4\] \^s(\^[(l)]{})=i\^[sl]{}\^[l]{}, l{0,1,2,3}. The $\mathbb{Z}_4$ transformation induces a transformation on the left-invariant currents as \[currentZ4\] j\_k=\^[k-1]{}(j\_1),k{1,2,3,4}. If we decompose the left-invariant currents into four components according to the $Z_4$-grading as \[DecomCurrent\] j\_k=j\_k\^[(0)]{}+j\_[k]{}\^[(1)]{}+j\_k\^[(2)]{}+j\_k\^[(3)]{}, then the $\mathbb{Z}_4$ transformation implies $$\begin{aligned}
\label{GradedCurrent}
&j_{1}\equiv j^{(0)}+j^{(1)}+j^{(2)}+j^{(3)},\\
&j_{2}= j^{(0)}+ij^{(1)}-j^{(2)}-ij^{(3)},\nn
&j_{3}= j^{(0)}-j^{(1)}+j^{(2)}-j^{(3)},\nn
&j_{4}= j^{(0)}-ij^{(1)}-j^{(2)}+ij^{(3)}\nonumber.\end{aligned}$$ Substituting and into the expression of the 2D action, we end up with S\[g\]=(3 j\_+\^[(3)]{} j\_-\^[(1)]{}+2 j\_+\^[(2)]{} j\_-\^[(2)]{}+j\_+\^[(1)]{} j\_-\^[(3)]{}) d\_+d\_-, which coincides with the action of $AdS_5\times S^5$ superstring in the pure spinor formulation [@PureF].
$\lambda$-deformation of $AdS_5\times S^5$ superstring {#LambdaSuperstring}
======================================================
The $\lambda$-deformation of $AdS_5\times S^5$ superstring has been proposed in the Green-Schwarz formulation [@SuperLambda] and in the pure spinor formulation [@Spin], respectively. In this section we will use the 4D CS theory to derive it in the pure spinor formulation and put the derivation in Green-Schwarz formulation in the Appendix. We start by splitting the four double poles in into four pairs of simple poles so the one-form can be $$\begin{aligned}
\label{equ:1.8}
\omega=\frac{u^3}{[(u-1)^2-\alpha^2][(u-i)^2+\alpha^2][(u+i)^2+\alpha^2][(u+1)^2-\alpha^2]}du ,\end{aligned}$$ with eight simple poles $$\begin{aligned}
\label{equ:1.9}
\mathfrak{p}=\{1\pm\alpha,i(1\pm\alpha),-(1\pm\alpha),-i(1\pm\alpha)\}\end{aligned}$$ and two triple zeros $$\begin{aligned}
\label{equ:1.10}
\mathfrak{z}=\{0,\infty\}. \end{aligned}$$ It is straightforward to get $$\begin{aligned}
\label{equ:1.11}
&\operatorname{Res}_{1+\alpha}\omega=\operatorname{Res}_{i(1+\alpha)}\omega=\operatorname{Res}_{-(1+\alpha)}\omega=\operatorname{Res}_{-i(1+\alpha)}\omega\equiv K, \notag\\
&\operatorname{Res}_{1-\alpha}\omega=\operatorname{Res}_{i(1-\alpha)}\omega=\operatorname{Res}_{-(1-\alpha)}\omega=\operatorname{Res}_{-i(1-\alpha)}\omega\equiv -K.\end{aligned}$$ As we discussed before, the boundary conditions should be taken to be \[equ:1.12\] &&A|\_[1+]{}=A|\_[1-]{}, A|\_[i(1+)]{}=A|\_[i(1-)]{},&&A|\_[-(1+)]{}=A|\_[-(1-)]{},A|\_[-i(1+)]{}=A|\_[-i(1-)]{}. In other words we require at each pair of the simple poles the gauge fields to take values in the Lagrangian subalgebra . Following the argument after we can using the local gauge symmetry to set the fields at the boundaries to be $$\begin{gathered}
\hat{g}|_{(1+\alpha)}=g_1,\quad
\hat{g}|_{i(1+\alpha)}=g_2,\quad
\hat{g}|_{-(1+\alpha)}=g_3,\qquad
\hat{g}|_{-i(1+\alpha)}=g_4\\
\hat{g}|_{(1-\alpha)}=\hat{g}|_{i(1-\alpha)}=\hat{g}|_{-(1-\alpha)}=\hat{g}|_{-i(1-\alpha)}=1,\end{gathered}$$ which through lead to $$\begin{gathered}
\label{equ:1.13}
A|_{(1+\alpha)}=-\mathrm{d}g_1 g_1^{-1}+\operatorname{Ad}_{g_1}\mathcal{L}|_{(1+\alpha)},\qquad A|_{i(1+\alpha)}=-\mathrm{d}g_2 g_2^{-1}+\operatorname{Ad}_{g_2}\mathcal{L}|_{i(1+\alpha)},\\
A|_{-(1+\alpha)}=-\mathrm{d}g_3 g_3^{-1}+\operatorname{Ad}_{g_3}\mathcal{L}|_{-(1+\alpha)},\qquad A|_{-i(1+\alpha)}=-\mathrm{d}g_4 g_4^{-1}+\operatorname{Ad}_{g_4}\mathcal{L}|_{-i(1+\alpha)},\\
A|_{(1-\alpha)}=\mathcal{L}|_{(1-\alpha)},\qquad
A|_{i(1-\alpha)}=\mathcal{L}|_{i(1-\alpha)},\\
A|_{-(1-\alpha)}=\mathcal{L}|_{-(1-\alpha)},\qquad
A|_{-i(1-\alpha)}=\mathcal{L}|_{-i(1-\alpha)}. \end{gathered}$$
Though we can use the same ansatz for the Lax connection, it is more convenient to use the following ansatz[^10] $$\begin{aligned}
\label{equ:1.14}
\mathcal{L}_+(u)&=U_{+0}+\frac{u}{1+\alpha}U_{+1}+\frac{u^2}{(1+\alpha)^2}U_{+2}+\frac{u^3}{(1+\alpha)^3}U_{+3},\\
\label{equ:1.14.1}
\mathcal{L}_-(u)&=\frac{(1+\alpha)^3}{u^3}U_{-3}+\frac{(1+\alpha)^2}{u^2}U_{-2}+\frac{1+\alpha}{u}U_{-1}+U_{-0},\end{aligned}$$ where $U_{\pm0,\pm1,\pm2,\pm3}$ are regular and take value in $\mathfrak{g}=\mathfrak{p s u}(2,2 | 4)$. As we discussed before, the boundary conditions are $$\begin{gathered}
\label{equ:1.15}
-\mathrm{d}g_1 g_1^{-1}+\operatorname{Ad}_{g_1}\mathcal{L}|_{(1+\alpha)}=\mathcal{L}|_{(1-\alpha)}\\\label{equ:1.15.1}
-\mathrm{d}g_2 g_2^{-1}+\operatorname{Ad}_{g_2}\mathcal{L}|_{i(1+\alpha)}=\mathcal{L}|_{i(1-\alpha)}\\\label{equ:1.15.2}
-\mathrm{d}g_3 g_3^{-1}+\operatorname{Ad}_{g_3}\mathcal{L}|_{-(1+\alpha)}=\mathcal{L}|_{-(1-\alpha)}\\\label{equ:1.15.3}
-\mathrm{d}g_4 g_4^{-1}+\operatorname{Ad}_{g_4}\mathcal{L}|_{-i(1+\alpha)}=\mathcal{L}|_{-i(1-\alpha)}. \end{gathered}$$ Put ansatz for $\mathcal{L}$ in these conditions and define $\lambda\equiv\frac{1-\alpha}{1+\alpha}$, we have the following equations Rewriting the gauge fields in terms of the Lax connection through and substituting into the boundary condition give a set of equations &&-\_g\_1 g\_1\^[-1]{}+\_[g\_1]{}\_[k=0]{}\^3 U\_[k]{}=\_[k=0]{}\^3\^[k]{}U\_[k]{},\
&& -\_g\_2 g\_2\^[-1]{}+\_[g\_2]{}\_[k=0]{}\^3 i\^[k]{}U\_[i]{}=\_[k=0]{}\^3(i)\^[k]{}U\_[k]{},\
&& -\_g\_3 g\_3\^[-1]{}+\_[g\_3]{}\_[k=0]{}\^3 (-1)\^[k]{}U\_[i]{}=\_[k=0]{}\^3(-)\^[k]{}U\_[k]{},\
&& -\_g\_4 g\_4\^[-1]{}+\_[g\_4]{}\_[k=0]{}\^3 (-i)\^[k]{}U\_[i]{}=\_[k=0]{}\^3(-i)\^[k]{}U\_[k]{}, which are equivalent to \[LambdaEqn\] &&j\_[1,]{}=\_[k=0]{}\^3(1-\^[k]{}\_[g\_1]{}\^[-1]{}) U\_[k]{},j\_[2,]{}=\_[k=0]{}\^3i\^[k]{}(1-\^[k]{}\_[g\_2]{}\^[-1]{}) U\_[k]{},\
&&j\_[3,]{}=\_[k=0]{}\^3(-1)\^[k]{}(1-\^[k]{}\_[g\_3]{}\^[-1]{}) U\_[k]{},j\_[4,]{}=\_[k=0]{}\^3(-i)\^[k]{}(1-\^[k]{}\_[g\_4]{}\^[-1]{}) U\_[k]{}. Here we have defined $j_{k}\equiv g^{-1}_k\mathrm{d}g_k$, and the parameter $\lambda=(1-\alpha)/(1+\alpha)$.
To proceed, we need to impose $\mathbb{Z}_4$ symmetry and , and then we have the decomposition of the currents and . Let us parametrize a group field $g$ as \[Groupfield\] g=(\_[k=0]{}\^3\_[i\_k=1]{}\^[(\^[(k)]{})]{}\_[i\_k]{}\^[(k)]{}T\^[(k)]{}\_[i\_k]{}), then the $\mathbb{Z}_4$ action on the field $g$ is explicitly given by $$\begin{aligned}
\label{Z4group}
\rho(g)=\operatorname{exp}\left(\sum_{k=0}^3\sum_{i_k=1}^{\operatorname{dim}(\mathfrak{g}^{(k)})}\theta_{i_k}^{(k)}\rho(T^{(k)}_{i_k})\right)=\operatorname{exp}\left(\sum_{k=0}^3\sum_{i_k=1}^{\operatorname{dim}(\mathfrak{g}^{(k)})}i^k \theta_{i_k}^{(k)}T^{(k)}_{i_k}\right),\end{aligned}$$ where $T^{(k)}_{i_k}$ are the generators of the subalgebra $\mathfrak{g}^{(k)}$. Using and and following [@YB2; @JT] one can derive the following identity $$\begin{aligned}
\label{Ad}
P^{(m)} \circ \mathrm{Ad}_{g_{k}}^{-1}=\sum_{r=0}^{3} i^{(m-r)(k-1)} P^{(m)} \circ \mathrm{Ad}_{g_1}^{-1} \circ P^{(r)},\end{aligned}$$ where $P^{(m)}$ denote projection operator onto the subalgebra $\mathfrak{g}^{(m)}$. For convenience, we define $$\begin{gathered}
\label{equ:1.26}
\mathbf{Ad}_g^{-1(p)}:=\frac{1}{4}\left(\operatorname{Ad}_{g_1}^{-1}+i^p\operatorname{Ad}_{g_2}^{-1}+i^{2p}\operatorname{Ad}_{g_3}^{-1}+i^{3p}\operatorname{Ad}_{g_3}^{-1}\right),\end{gathered}$$ which implies $$\begin{aligned}
\label{equ:1.29}
P^{(m)} \circ \mathbf{Ad}_g^{-1(p)}=P^{(m)} \circ \operatorname{Ad}_{g_1}^{-1} \circ P^{(r)},\end{aligned}$$ where $r=p+m\, \text{mod}\, 4$ and $r\in\{0,1,2,3\}$. Substituting into the we find the following rewriting of \[Rewriting\] (1-\_g\^[-1(0)]{})U\_[+0]{}-\_g\^[-1(1)]{}U\_[+1]{}-\^2 \_g\^[-1(2)]{}U\_[+2]{}-\^3\_g\^[-1(3)]{}U\_[+3]{}=&j\_+\^[(0)]{},\
-\_g\^[-1(3)]{}U\_[+0]{}+(1-\_g\^[-1(0)]{})U\_[+1]{}-\^2 \_g\^[-1(1)]{}U\_[+2]{}-\^3 \_g\^[-1(2)]{}U\_[+3]{}=&j\_+\^[(1)]{},\
-\_g\^[-1(2)]{}U\_[+0]{}-\_g\^[-1(3)]{}U\_[+1]{}+(1-\^2\_g\^[-1(0)]{})U\_[+2]{}-\^3\_g\^[-1(1)]{}U\_[+3]{}=&j\_+\^[(2)]{},\
-\_g\^[-1(1)]{}U\_[+0]{}-\_g\^[-1(2)]{}U\_[+1]{}-\^2\_g\^[-1(3)]{}U\_[+2]{}+(1-\^3\_g\^[-1(0)]{})U\_[+1]{}=&j\_+\^[(3)]{},\
-\^[-3]{}\_g\^[-1(1)]{}U\_[-3]{}-\^[-2]{}\_g\^[-1(2)]{}U\_[-2]{}-\^[-1]{}\_g\^[-1(3)]{}U\_[-1]{}+(1-\_g\^[-1(0)]{})U\_[-0]{}=&j\_-\^[(0)]{},\
(1-\^[-3]{}\_g\^[-1(0)]{})U\_[-3]{}-\^[-2]{}\_g\^[-1(1)]{}U\_[-2]{}-\^[-1]{}\_g\^[-1(2)]{}U\_[-1]{}-\_g\^[-1(3)]{}U\_[-0]{}=&j\_-\^[(1)]{},\
-\^[-3]{}\_g\^[-1(3)]{}U\_[-3]{}+(1-\^[-2]{}\_g\^[-1(0)]{})U\_[-2]{}-\^[-1]{}\_g\^[-1(1)]{}U\_[-1]{}-\_g\^[-1(2)]{}U\_[-0]{}=&j\_-\^[(2)]{},\
-\^[-3]{}\_g\^[-1(2)]{}U\_[-3]{}-\^[-2]{}\_g\^[-1(3)]{}U\_[-2]{}+(1-\^[-1]{}\_g\^[-1(0)]{})U\_[-1]{}-\_g\^[-1(1)]{}U\_[-0]{}=&j\_-\^[(3)]{}.After this rewriting it is easy to see that the equations can be solved by $$\begin{aligned}
\label{Solution}
U_+=\frac{1}{1-\operatorname{Ad}_{g_1}^{-1}\circ\Omega_+}j_+,\qquad &\Omega_+=P^{(0)}+\lambda P^{(1)}+\lambda^2 P^{(2)}+\lambda^3 P^{(3)},\\
U_-=\frac{1}{1-\operatorname{Ad}_{g_1}^{-1}\circ\Omega_-}j_-, \qquad &\Omega_-=P^{(0)}+\lambda^{-3} P^{(1)}+\lambda^{-2}P^{(2)}+\lambda^{-1}P^{(3)},\end{aligned}$$ with $$\begin{aligned}
\label{}
&U_{+0}=P^{(0)}U_{+}\qquad U_{+1}=P^{(1)}U_{+}\qquad U_{+2}=P^{(2)}U_{+}\qquad U_{+3}=P^{(3)}U_{+},\\
&U_{-0}=P^{(0)}U_{-}\qquad U_{-3}=P^{(1)}U_-\qquad U_{-2}=P^{(2)}U_-\qquad U_{-1}=P^{(3)}U_-.\end{aligned}$$ In the end substituting the Lax connection into the we obtain the 2D action[^11] S&=&\_[k=1]{}\^4d\_+d\_-&=&d\_+d\_-&=&d\_+d\_-, where we have used $(\rho^k)^\dagger=\rho^{4-k}$ and $\Omega^{T}_+\Omega_-=\mathbf{1}$ with $$\begin{aligned}
\label{equ:1.31}
\Omega^{T}_+=P^{(0)}+\lambda^3 P^{(1)}+\lambda^2 P^{(2)}+\lambda P^{(3)}.\end{aligned}$$ This action coincides with the one in [@Spin] up to a prefactor, if we ignore the terms involved with the auxiliary fields.
Discussion
==========
In this paper, we have successfully constructed the $\lambda$-deformation of the $AdS_5\times S^5$ superstring from the 4D CS theory. The same analysis is applicable for other superstring theories with $Z_4$-grading superalgebra [@Chen:2005uj]. It is known that the $\lambda$-deformation is Poisson-Lie-T-dual to the $\eta$-deformation [@PT1; @PT2]. With the same one-form but choosing the other Mannin pair $(\mathfrak{g}_R,\mathfrak{d})$ as boundary conditions one should be able to derive $\eta$-deformation of the $AdS_5\times S^5$ superstring.
From the construction we realize that the discrete symmetry plays a crucial role. By orbifolding the Riemann surface we can add several copies of 2D IFTs which are related by discrete symmetry transformations. This supplies the complement of the gluing process [@CY3] for constructing new IFTs. One example is the Yang-Baxter model in the trigonometric description [@YB1]. We expect that the asymmetric $\lambda$-deformation [@ASY] , the anisotropic $\lambda$-deformation [@SU2] and the generalized $\lambda$-deformation [@GenLambda1] can be constructed in this fashion. It may also be helpful to understand how to realize the sine-Gordon model. The sine-Gordon model can be reproduced from the affine Gaudin model by considering the Coxeter automorphism [@Gaudin]. Since the Gaudin model approach and the 4D CS theory approach are closely related [@Relation], hence it would be interesting to figure out how the Coxeter automorphism is implemented in the 4D CS theory.
Acknowledgments {#acknowledgments .unnumbered}
===============
JT and YJH would like to thank the Tohoko University for the hospitality during the 14th Kavli Asian Winter School and to thank Jue Hou, Han Liu and Jun-Bao Wu for useful discussion. The work was in part supported by NSFC Grant No. 11335012, No. 11325522 and No. 11735001.
Appendix: the Green Schwarz formulation {#GS .unnumbered}
=======================================
In our discussion in the pure spinor formulation, we chose the ansatz so that the highest order of the poles of the ansatz is equal to the order of the zeros of the one-form $\omega$. If we disregard the degeneracy of the 4D action we can construct the $AdS_5\times S^5$ superstring in the Green Schwarz formulation. To do that we simply replace the ansatz with $$\begin{aligned}
\label{equ:2.1}
&\mathcal{L}_{+}(u)=\frac{(1+\alpha)^2}{u^2}V_{-2,+}+\frac{1+\alpha}{u}V_{-1,+}+V_{0,+}+\frac{u}{1+\alpha}V_{1,+},\\
&\mathcal{L}_{-}(u)=\frac{1+\alpha}{u}V_{-1,-}+V_{0,-}+\frac{u}{1+\alpha}V_{1,-}+\frac{u^2}{(1+\alpha)^2}V_{2,-}. \end{aligned}$$ Then the counterpart of is
$$\begin{aligned}
\label{equ:2.3}
-\lambda^2 \mathbf{Ad}_g^{-1(2)}V_{-2,+}-\lambda\mathbf{Ad}_g^{-1(3)}V_{-1,+}+(1-\mathbf{Ad}_g^{-1(0)})V_{0,+}-\frac{1}{\lambda}\mathbf{Ad}_g^{-1(1)}V_{1,+}=&j_+^{(0)},\notag\\
-\lambda^2 \mathbf{Ad}_g^{-1(1)}V_{-2,+}-\lambda\mathbf{Ad}_g^{-1(2)}V_{-1,+}-\mathbf{Ad}_g^{-1(3)}V_{0,+}+(1-\frac{1}{\lambda}\mathbf{Ad}_g^{-1(0)})V_{1,+}=&j_+^{(1)},\notag\\
(1-\lambda^2 \mathbf{Ad}_g^{-1(0)})V_{-2,+}-\lambda\mathbf{Ad}_g^{-1(1)}V_{-1,+}-\mathbf{Ad}_g^{-1(2)}V_{0,+}-\frac{1}{\lambda}\mathbf{Ad}_g^{-1(3)}V_{1,+}=&j_+^{(2)},\notag\\
-\lambda^2 \mathbf{Ad}_g^{-1(3)}V_{-2,+}+(1-\lambda\mathbf{Ad}_g^{-1(0)})V_{-1,+}-\mathbf{Ad}_g^{-1(1)}V_{0,+}-\frac{1}{\lambda}\mathbf{Ad}_g^{-1(2)}V_{1,+}=&j_+^{(3)},\notag\\
-\lambda \mathbf{Ad}_g^{-1(3)}V_{-1,-}+(1-\mathbf{Ad}_g^{-1(0)})V_{0,-}-\frac{1}{\lambda}\mathbf{Ad}_g^{-1(1)}V_{1,-}-\frac{1}{\lambda^2}\mathbf{Ad}_g^{-1(2)}V_{2,-}=&j_-^{(0)},\notag\\
-\lambda \mathbf{Ad}_g^{-1(2)}V_{-1,-}-\mathbf{Ad}_g^{-1(3)}V_{0,-}+(1-\frac{1}{\lambda}\mathbf{Ad}_g^{-1(0)})V_{1,-}-\frac{1}{\lambda^2}\mathbf{Ad}_g^{-1(1)}V_{2,-}=&j_-^{(1)},\notag\\
-\lambda \mathbf{Ad}_g^{-1(1)}V_{-1,-}-\mathbf{Ad}_g^{-1(2)}V_{0,-}-\frac{1}{\lambda}\mathbf{Ad}_g^{-1(3)}V_{1,-}+(1-\frac{1}{\lambda^2}\mathbf{Ad}_g^{-1(0)})V_{2,-}=&j_-^{(2)},\notag\\
(1-\lambda \mathbf{Ad}_g^{-1(0)})V_{-1,-}-\mathbf{Ad}_g^{-1(1)}V_{0,-}-\frac{1}{\lambda}\mathbf{Ad}_g^{-1(2)}V_{1,-}-\frac{1}{\lambda^2}\mathbf{Ad}_g^{-1(3)}V_{2,-}=&j_-^{(3)}.\end{aligned}$$
Using , these equations can be solved to be $$\begin{aligned}
\label{equ:2.4}
V_+=\frac{1}{1-\operatorname{Ad}_{g_1}^{-1}\circ\tilde{\Omega}_+}j_+,\qquad &\tilde{\Omega}_+=P^{(0)}+\lambda^{-1} P^{(1)}+\lambda^2 P^{(2)}+\lambda P^{(3)},\\
V_-=\frac{1}{1-\operatorname{Ad}_{g_1}^{-1}\circ\tilde{\Omega}_-}j_-, \qquad &\tilde{\Omega}_-=P^{(0)}+\lambda^{-1} P^{(1)}+\lambda^{-2}P^{(2)}+\lambda P^{(3)}, \end{aligned}$$ which also satisfy $\tilde{\Omega}^{T}_+\tilde{\Omega}_-=\tilde{\Omega}^{T}_-\tilde{\Omega}_+=\mathbf{1}$. To be more explicit, we have $$\begin{aligned}
\label{equ:2.5}
&V_{0,+}=P^{(0)}V_{+}\qquad V_{1,+}=P^{(1)}V_{+}\qquad V_{-2,+}=P^{(2)}V_{+}\qquad V_{-1,+}=P^{(3)}V_{+},\\
&V_{0,-}=P^{(0)}V_{-}\qquad V_{1,-}=P^{(1)}V_-\qquad V_{2,-}=P^{(2)}V_-\qquad V_{-1,-}=P^{(3)}V_-. \end{aligned}$$ Substituting into , the kinetic term of effective action in the Green Schwarz formulation is given by $$\begin{aligned}
\label{equ:2.6}
S_{\text{kin}}&\propto\int\left\{\left\langle V_+,j_-\right\rangle-\left\langle V_-,j_+\right\rangle\right\}\notag\\
&\propto\int\left\{\left\langle \frac{1}{1-\operatorname{Ad}_{g_1}^{-1}\circ\tilde{\Omega}_+}j_+,j_-\right\rangle-\left\langle j_+,\frac{1}{1-\operatorname{Ad}_{g_1}^{-1}\circ\tilde{\Omega}_-}j_-\right\rangle\right\}\notag\\
&\propto\int\left\{\left\langle j_+,j_-\right\rangle-2\left\langle j_+,\frac{1}{1-\operatorname{Ad}_{g_1}^{-1}\circ\tilde{\Omega}_-}j_-\right\rangle\right\}.\end{aligned}$$ It is same as the results in previous literature [@SuperLambda] up to a prefactor.
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[^1]: wukongjiaozi, yjhe96, bchen01@pku.edu.cn
[^2]: For details, see the classic book [@YellowBook].
[^3]: The homogeneous Yang-Baxter deformation of $AdS_5\times S^5$ superstring was considered in [@YB2].
[^4]: Here we follow the convention in [@Uni].
[^5]: These are called the disorder defects in [@CY3].
[^6]: The light-cone coordinates are defined as $\sigma_\pm=\frac{1}{2}(\tau\pm i\sigma)$.
[^7]: One can also choose to remove the gauge redundancies in the end.
[^8]: Actually we have an Manin triple $\left(\mathfrak{d}, \mathfrak{g}_{R}, \mathfrak{g}^{\delta}\right)$ for some solutions $R$ of modified classical Yang-Baxter equation and $\mathfrak{g}_{R}:=\{((R-1){x},(R+1){x}|{x}\in\mathfrak{g}\}$. The other Manin pair $(\mathfrak{g}_R,\mathfrak{d})$ will lead to Yang-Baxter model [@Uni].
[^9]: This is also the one-form considered in [@YB2].
[^10]: Note that if we take the orders of the poles in the ansatz to be at most two, we will get the $AdS_5\times S^5$ model in the Green-Schwarz formulation instead of the pure spinor formulation, so as its $\lambda$-deformation, whose kinematic term is non-degenerate.
[^11]: Here we have ignored the topological term.
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