Determinantal Point Processes, Stochastic Log-Gases, and Beyond
Makoto Katori
备注：
v2: LaTeX 79 pages, no figure. This manuscript was prepared for the mini course given at `Workshop on Probability and Stochastic Processes' held at Orange County, Coorg, India, from 23rd to 26th February, 2020, which was organized by the Indian Academy of Sciences, Bangalore. arXiv admin note: substantial text overlap with arXiv:1903.04945
A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures, whose correlation functions are all given by determinants specified by an integral kernel called the correlation kernel. First we show our new scheme of DPPs in which a notion of partial isometies between a pair of Hilbert spaces plays an important role. Many examples of DPPs in one-, two-, and higher-dimensional spaces are demonstrated, where several types of weak convergence from finite DPPs to infinite DPPs are given. Dynamical extensions of DPP are realized in one-dimensional systems of diffusive particles conditioned never to collide with each other. They are regarded as one-dimensional stochastic log-gases, or the two-dimensional Coulomb gases confined in one-dimensional spaces. In the second section, we consider such interacting particle systems in one dimension. We introduce a notion of determinantal martingale and prove that, if the system has determinantal martingale representation (DMR), then it is a determinantal stochastic process (DSP) in the sense that all spatio-temporal correlation function are expressed by a determinant. In the last section, we construct processes of Gaussian free fields (GFFs) on simply connected proper subdomains of $\mathbb{C}$ coupled with interacting particle systems defined on boundaries of the domains. There we use multiple Schramm--Loewner evolutions (SLEs) driven by the interacting particle systems. We prove that, if the driving processes are time-changes of the log-gases studied in the second section, then the obtained GFF with multiple SLEs are stationary. The stationarity defines an equivalence relation of GFFs, which will be regarded as a generalization of the imaginary surface studied by Miller and Sheffield.
We consider random normal matrix and planar symplectic ensembles, which can be interpreted as two-dimensional Coulomb gases having determinantal and Pfaffian structures, respectively. For general radially symmetric potentials, we derive the asymptotic expansions of the log-partition functions up to and including the $O(1)$-terms as the number $N$ of particles increases. Notably, our findings stress that the formulas of the $O(\log N)$- and $O(1)$-terms in these expansions depend on the connectivity of the droplet. For random normal matrix ensembles, our formulas agree with the predictions prop…
We study the outliers for two models which have an interesting connection. On the one hand, we study a specific class of planar Coulomb gases which are determinantal. It corresponds to the case where the confining potential is the logarithmic potential of a radial probability measure. On the other hand, we study the zeros of random polynomials that appear to be closely related to the first model. Their behavior far from the origin is shown to depend only on the decaying properties of the probability measure generating the potential. A similar feature is observed for their behavior near the ori…
We use techniques in the shuffle algebra to present a formula for the partition function of a one-dimensional log-gas comprised of particles of (possibly) different integer charges at certain inverse temperature $β$ in terms of the Berezin integral of an associated non-homogeneous alternating tensor. This generalizes previously known results by removing the restriction on the number of species of odd charge. Our methods provide a unified framework extending the de Bruijn integral identities from classical $β$-ensembles ($β$ = 1, 2, 4) to multicomponent ensembles, as well as to iterated integra…
We prove a Central Limit Theorem for the linear statistics of two-dimensional Coulomb gases, with arbitrary inverse temperature and general confining potential, at the macroscopic and mesoscopic scales and possibly near the boundary of the support of the equilibrium measure. This can be stated in terms of convergence of the random electrostatic potential to a Gaussian Free Field. Our result is the first to be valid at arbitrary temperature and at the mesoscopic scales, and we recover previous results of Ameur-Hendenmalm-Makarov and Rider-Virág concerning the determinantal case, with weaker ass…
We study Fredholm determinants of a class of integral operators, whose kernels can be expressed as double contour integrals of a special type. Such Fredholm determinants appear in various random matrix and statistical physics models. We show that the logarithmic derivatives of the Fredholm determinants are directly related to solutions of the Painlevé II hierarchy. This confirms and generalizes a recent conjecture by Le Doussal, Majumdar, and Schehr. In addition, we obtain asymptotics at $\pm\infty$ for the Painlevé transcendents and large gap asymptotics for the corresponding point processes.
We study linear statistics of a class of determinantal processes which interpolate between Poisson and GUE/Ginibre statistics in dimension 1 or 2. These processes are obtained by performing an independent Bernoulli percolation on the particle configuration of of a log-gas confined in a general potential. We show that, depending on the expected number of deleted particles, there is a universal transition for mesoscopic statistics. Namely, at small scales, the point process still behaves according to random matrix theory, while, at large scales, it needs to be renormalized because the variance o…
An exact map was established by Lacroix-A-Chez-Toine, Majumdar, and Schehr in [44] between the $N$ complex eigenvalues of complex non-Hermitian random matrices from the Ginibre ensemble, and the positions of $N$ non-interacting Fermions in a rotating trap in the ground state. An important quantity is the statistics of the number of Fermions $\mathcal{N}_a$ in a disc of radius $a$. Extending the work [44] covering Gaussian and rotationally invariant potentials $Q$, we present a rigorous analysis in planar complex and symplectic ensembles, which both represent 2D Coulomb gases. We show that the …
We deduce a structurally inductive description of the determinantal probability measure associated with Kalai's celebrated enumeration result for higher--dimensional spanning trees of the $n-1$--simplex. As a consequence, we derive the marginal distributions of the simplex links in such random trees. Along the way, we also characterize the higher--dimensional spanning trees of every other simplicial cone in terms of the higher--dimensional rooted forests of the underlying simplicial complex. We also apply these new results to random topology, the spectral analysis of random graphs, and the the…
We give an expression for the partition function of a one-dimensional log-gas comprised of particles of (possibly) different integer charge at inverse temperature β = 1 (restricted to the line in the presence of a neutralizing field) in terms of the Berezin integral of an associated non- homogeneous alternating tensor. This is the analog of the de Bruijn integral identities [3] (for β = 1 and β = 4) ensembles extended to multicomponent ensembles.
Product approximations for uniform spanning trees and determinantal processes
HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利科学院阿尔弗雷德·雷尼数学研究所）；Einstein Institute of Mathematics, Hebrew University of Jerusalem（耶路撒冷希伯来大学爱因斯坦数学研究所）
Determinantal point processes (DPP) serve as a practicable modeling for many applications of repulsive point processes. A known approach for simulation was proposed in \cite{Hough(2006)}, which generate the desired distribution point wise through rejection sampling. Unfortunately, the size of rejection could be very large. In this paper, we investigate the application of perfect simulation via coupling from the past (CFTP) on DPP. We give a general framework for perfect simulation on DPP model. It is shown that the limiting sequence of the time-to-coalescence of the coupling is bounded by $K|Λ…
We study the hydrodynamic limits of three kinds of one-dimensional stochastic log-gases known as Dyson's Brownian motion model, its chiral version, and the Bru-Wishart process studied in dynamical random matrix theory. We define the measure-valued processes so that their Cauchy transforms solve the complex Burgers-type equations. We show that applications of the method of characteristic curves to these partial differential equations provide the functional equations relating the Cauchy transforms of measures at an arbitrary time with those at the initial time. We transform the functional equati…
Coulomb and log-gases are exchangeable singular Boltzmann-Gibbs measures appearing in mathematical physics at many places, in particular in random matrix theory. We explore experimentally an efficient numerical method for simulating such gases. It is an instance of the Hybrid or Hamiltonian Monte Carlo algorithm, in other words a Metropolis-Hastings algorithm with proposals produced by a kinetic or underdamped Langevin dynamics. This algorithm has excellent numerical behavior despite the singular interaction, in particular when the number of particles gets large. It is more efficient than the …
These are the lecture notes of a course taught at the Park City Mathematics Institute in June 2017. They are intended to review some recent results, obtained in large part with Thomas Leblé, on the statistical mechanics of systems of points with logarithmic or Coulomb interactions. After listing some motivations, we describe the "electric approach" which allows to get concentration results, Central Limit Theorems for fluctuations, and a Large Deviations Principle expressed in terms of the microscopic state of the system.
Dyson's model on interacting Brownian particles is a stochastic dynamics consisting of an infinite amount of particles moving in $ \R $ with a logarithmic pair interaction potential. For this model we will prove that each pair of particles never collide. The equilibrium state of this dynamics is a determinantal random point field with the sine kernel. We prove for stochastic dynamics given by Dirichlet forms with determinantal random point fields as equilibrium states the particles never collide if the kernel of determining random point fields are locally Lipschitz continuous, and give example…
We study expected Riesz s-energies and linear statistics of some determinantal processes on the sphere. In particular, we compute the expected Riesz and logarithmic energies of the determinantal processes given by the reproducing kernel of the space of spherical harmonics. This kernel defines the so called harmonic ensemble on the sphere. With these computations we improve previous estimates for the discrete minimal energy of configurations of points in the sphere. We prove a comparison result for Riesz 2-energies of points defined through determinantal point processes associated to isotropic …
We study the hole probabilities for ${\mathcal X}_{\infty}^{(α)}$ ($α>0$), a determinantal point process in the complex plane with the kernel $\mathbb K_{\infty}^{(α)}(z,w)=\fracα{2π}E_{\frac{2}α,\frac{2}α}(z\bar w)e^{-\frac{|z|^α}{2}-\frac{|w|^α}{2}}$ with respect to Lebesgue measure on the complex plane, where $E_{a,b}(z)$ denotes the Mittag-Leffler function. Let $U$ be an open subset of $D(0,(\frac{2}α)^{\frac{1}α})$ and ${\mathcal X}_{\infty}^{(α)}(rU)$ denote the number of points of ${\mathcal X}_{\infty}^{(α)}$ that fall in $rU$. Then, under some conditions on $U$, we show that $$ \lim_{…
We study a system of N particles with logarithmic, Coulomb or Riesz pairwise interactions, confined by an external potential. We examine a microscopic quantity, the tagged empirical field, for which we prove a large deviation principle at speed N. The rate function is the sum of an entropy term, the specific relative entropy, and an energy term, the renormalized energy introduced in previous works, coupled by the temperature. We deduce a variational property of the sine-beta processes which arise in random matrix theory. We also give a next-to-leading order expansion of the free energy of the …
We consider the correlations of invariant observables for the $O(N)$ and $\mathbb{C}\mathbb{P}^{N-1}$ models at zero coupling, namely, with respect to the natural group-invariant measure. In the limit where one takes a large power of the integrand, we show that these correlations become inverse powers of the Kirchhoff polynomial. The latter therefore provide a simplified toy model for the investigation of inequalities between products of correlations. Properties such as ferromagnetic behavior for spin model correlations correspond, in this asymptotic limit, to log-ultramodularity which is a co…
We study the statistical mechanics of a one-dimensional log gas with general potential and arbitrary beta, the inverse of temperature, according to the method we introduced for two-dimensional Coulomb gases in [SS2]. Such ensembles correspond to random matrix models in some particular cases. The formal limit beta infinite corresponds to "weighted Fekete sets" and is also treated. We introduce a one-dimensional version of the "renormalized energy" of [SS1], measuring the total logarithmic interaction of an infinite set of points on the real line in a uniform neutralizing background. We show tha…
The logarithmic derivative of a point process plays a key role in the general approach, due to the third author, to constructing diffusions preserving a given point process. In this paper we explicitly compute the logarithmic derivative for determinantal processes on $\mathbb{R}$ with integrable kernels, a large class that includes all the classical processes of random matrix theory as well as processes associated with de Branges spaces. The argument uses the quasi-invariance of our processes established by the first author.
We explore probabilistic consequences of correspondences between $q$-Whittaker measures and periodic and free boundary Schur measures established by the authors in the recent paper [arXiv:2106.11922]. The result is a comprehensive theory of solvability of stochastic models in the KPZ class where exact formulas descend from mapping to explicit determinantal and pfaffian point processes. We discover new variants of known results as determinantal formulas for the current distribution of the ASEP on the line and new results such as Fredholm pfaffian formulas for the distribution of the point-to-po…
Miller and Sheffield introduced the notion of an imaginary surface as an equivalence class of pairs of simply connected proper subdomains of $\mathbb{C}$ and Gaussian free fields (GFFs) on them under the conformal equivalence. They considered the situation in which the conformal maps are given by a chordal Schramm--Loewner evolution (SLE). In the present paper, we construct GFF-valued processes on $\mathbb{H}$ (the upper half-plane) and $\mathbb{O}$ (the first orthant of $\mathbb{C}$) by coupling a GFF with a multiple SLE evolving in time on each domain. We prove that a GFF on $\mathbb{H}$ and…
We derive new HWI inequalities for the granular media equation, which external potential $V$ and interaction potential $W$ are only strictly convex on complementary parts of the space. Particularly, potentials are not assumed convex. After solving technicalities related to the singularity of a logarithmic $W$, we apply our result to obtain stability rates of log gases under non-strictly convex or quartic external potentials. We prove that the distribution of a log gas converges towards an equilibrium with respect to the Wasserstein distance at a square root rate. Finally, we establish exponent…
In Nonequilibrium Thermodynamics and Information Theory, the relative entropy (or, KL divergence) plays a very important role. Consider a Hölder Jacobian $J$ and the Ruelle (transfer) operator $\mathcal{L}_{\log J}.$ Two equilibrium probabilities $μ_1$ and $μ_2$, can interact via a discrete-time {\it Thermodynamic Operation} described by the action {\it of the dual of the Ruelle operator} $ \mathcal{L}_{\log J}^*$. We argue that the law $μ\to \mathcal{L}_{\log J}^*(μ)$, producing nonequilibrium, can be seen as a Thermodynamic Operation after showing that it's a manifestation of the Second Law …
The Bessel process with parameter $D>1$ and the Dyson model of interacting Brownian motions with coupling constant $β>0$ are extended to the processes in which the drift term and the interaction terms are given by the logarithmic derivatives of Jacobi's theta functions. They are called the elliptic Bessel process, eBES$^{(D)}$, and the elliptic Dyson model, eDYS$^{(β)}$, respectively. Both are realized on the circumference of a circle $[0, 2 πr)$ with radius $r >0$ as temporally inhomogeneous processes defined in a finite time interval $[0, t_*), t_* < \infty$. Transformations of them to Schrö…
Precise universal edge asymptotics for planar $β=2$ Coulomb gases with radial external fields
School of Mathematical Sciences \& Laboratory of Mathematics；Complex Systems of Ministry of Education, Beijing Normal University, 100875 Beijing, China.
The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue statistics of large real and complex Hermitian matrices with independent, identically distributed entries are universal in a sense that they depend only on the symmetry class of the matrix and otherwise are independent of the details of the distribution. We present the recent solution to this half-century old conjecture. We explain how stochastic tools, such as the Dyson Brownian motion, and PDE ideas, such as De Giorgi-Nash-Moser regularity theory, were combined in the solution. We also show related results for log-gas…
We review what is known, unknown and expected about the mathematical properties of Coulomb and Riesz gases. Those describe infinite configurations of points in $\mathbb{R}^d$ interacting with the Riesz potential $\pm |x|^{-s}$ (resp. $-\log|x|$ for $s=0$). Our presentation follows the standard point of view of statistical mechanics, but we also mention how these systems arise in other important situations (e.g. in random matrix theory). The main question addressed in the article is how to properly define the associated infinite point process and characterize it using some (renormalized) equili…
We study a general class of log-gas ensembles on (shifted) quadratic lattices. We prove that the corresponding empirical measures satisfy a law of large numbers and that their global fluctuations are Gaussian with a universal covariance. We apply our general results to analyze the asymptotic behavior of a $q$-boxed plane partition model introduced by Borodin, Gorin and Rains. In particular, we show that the global fluctuations of the height function on a fixed slice are described by a one-dimensional section of a pullback of the two-dimensional Gaussian free field. Our approach is based on a $…
For general dimension $d$ we prove the equidistribution of energy at the micro-scale in $\mathbb R^d$, for the optimal point configurations appearing in Coulomb gases at zero temperature. At the microscopic scale, i.e. after blow-up at the scale corresponding to the interparticle distance, in the case of Coulomb gases we show that the energy concentration is precisely determined by the macroscopic density of points, independently of the scale. This uses the "jellium energy" which was previously shown to control the next-order term in the large particle number asymptotics of the minimum energy.…
Eugene Wigner's revolutionary vision predicted that the energy levels of large complex quantum systems exhibit a universal behavior: the statistics of energy gaps depend only on the basic symmetry type of the model. Simplified models of Wigner's thesis have recently become mathematically accessible. For mean field models represented by large random matrices with independent entries, the celebrated Wigner-Dyson-Gaudin-Mehta (WDGM) conjecture asserts that the local eigenvalue statistics are universal. For invariant matrix models, the eigenvalue distributions are given by a log-gas with potential…
We consider systems of multiple Brownian particles in one dimension that repel mutually via a logarithmic potential on the real line, more specifically the Dyson model. These systems are characterized by a parameter that controls the strength of the interaction, $k>0$. In spite of being a one-dimensional system, this system is interesting due to the properties that arise from the long-range interaction between particles. It is a well-known fact that when $k$ is small enough, particle collisions occur almost surely, while when $k$ is large, collisions never occur. However, aside from this fact …
We define a notion of logarithmic, Coulomb and Riesz interactions in any dimension for random systems of infinite charged point configurations with a uniform background of opposite sign. We connect this interaction energy with the "renormalized energy" studied by Serfaty et al., which appears in the free energy functional governing the microscopic behavior of logarithmic, Coulomb and Riesz gases. Minimizers of this functional include the Sine-beta processes in the one-dimensional Log-gas case. Using our explicit expression (inspired by the work of Borodin-Serfaty) we prove their convergence to…
