Gaussian free fields coupled with multiple SLEs driven by stochastic log-gases
Makoto Katori, Shinji Koshida
备注：
v7: LaTex, 21 pages, no figure. This manuscript was prepared for the proceeding of the workshop of the 12th Mathematical Society of Japan, Seasonal Institute (MSJ-SI), `Stochastic Analysis, Random Fields and Integrable Probability', held at Kyushu University, Fukuoka, Japan, July 31--August 9, 2019. The proceeding will be published in Advanced Studies in Pure Mathematics 87, 2021
发表信息：
Advanced Studies in Pure Mathematics 87 (2021) pp. 315--340
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 $\mathbb{O}$ is locally coupled with a multiple SLE if the multiple SLE is driven by the stochastic log-gas called the Dyson model defined on $\mathbb{R}$ and the Bru--Wishart process defined on $\mathbb{R}_+$, respectively. We obtain pairs of time-evolutionary domains and GFF-valued processes.
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 …
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.
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 …
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…
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.
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 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…
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 …
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…
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.…
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 $…
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 introduce and study stochastic $N$-particle ensembles which are discretizations for general-$β$ log-gases of random matrix theory. The examples include random tilings, families of non-intersecting paths, $(z,w)$-measures, etc. We prove that under technical assumptions on general analytic potential, the global fluctuations for such ensembles are asymptotically Gaussian as $N\to\infty$. The covariance is universal and coincides with its counterpart in random matrix theory. Our main tool is an appropriate discrete version of the Schwinger-Dyson (or loop) equations, which originates in the work…
