Dynamical Bulk Scaling Limit of Gaussian Unitary Ensembles and Stochastic Differential Equation Gaps

Yosuke Kawamoto,Hirofumi Osada

doi:10.1007/s10959-018-0816-2

Abstract

The distributions of N-particle systems of Gaussian unitary ensembles converge to Sine _2 point processes under bulk scaling limits. These scalings are parameterized by a macro-position theta in the support of the semicircle distribution. The limits are always Sine _{2} point processes and independent of the macro-position theta up to the dilations of determinantal kernels. We prove a dynamical counterpart of this fact. We prove that the solution to the N-particle system given by a stochastic differential equation (SDE) converges to the solution of the infinite-dimensional Dyson model. We prove that the limit infinite-dimensional SDE (ISDE), referred to as Dyson’s model, is independent of the macro-position theta , whereas the N-particle SDEs depend on theta and are different from the ISDE in the limit whenever theta not = 0 .

Highlights

Gaussian unitary ensembles (GUE) are Gaussian ensembles defined on the space of random matrices M N (N ∈ N) with independent random variables, the matrices of which are Hermitian
Because the logarithmic potential is by its nature long-ranged, the effect of initial distributions μθN still remains in the limit infinite-dimensional SDE (ISDE), and the rigidity of the Sine2 point process makes the residual effect a non-random drift term θ dt
The proof in [16] is algebraic and valid only for dimension d = 1 and inverse temperature β = 2 with the logarithmic potential. It relies on an explicit calculation of the spacetime correlation functions, the strong Markov property of the stochastic dynamics given by the algebraic construction, the identity of the associated Dirichlet forms constructed by two completely different methods, and the uniqueness of solutions of ISDE (1.7)

Summary

Introduction

Gaussian unitary ensembles (GUE) are Gaussian ensembles defined on the space of random matrices M N (N ∈ N) with independent random variables, the matrices of which are Hermitian. Because the logarithmic potential is by its nature long-ranged, the effect of initial distributions μθN still remains in the limit ISDE, and the rigidity of the Sine point process makes the residual effect a non-random drift term θ dt. The proof in [16] is algebraic and valid only for dimension d = 1 and inverse temperature β = 2 with the logarithmic potential It relies on an explicit calculation of the spacetime correlation functions, the strong Markov property of the stochastic dynamics given by the algebraic construction, the identity of the associated Dirichlet forms constructed by two completely different methods, and the uniqueness of solutions of ISDE (1.7). As a corollary and an application, Theorem 1.1 proves the weak convergence of finite-dimensional distributions explicitly given by the space-time correlation functions.

Preliminaries from General Theory

Infinite-Dimensional SDEs

Finite-Particle Approximations

Oscillator Wave Functions

Determinantal Kernels of N-Particle Systems