Dynamical Bulk Scaling limit of Gaussian Unitary Ensembles and Stochastic Differential Equation gaps

TL;DR

This paper proves that the stochastic dynamics of Gaussian Unitary Ensembles converge to a universal infinite-dimensional Dyson model, independent of macro-position, revealing a dynamical bulk scaling limit in random matrix theory.

Contribution

It establishes the convergence of N-particle SDE solutions to an infinite-dimensional Dyson model, showing independence from macro-position in the bulk scaling limit.

Findings

N-particle SDEs depend on macro-position $\theta$

Infinite-dimensional Dyson model is independent of $\theta$

Convergence holds under bulk-scaling limits

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 counter part of this fact. We prove that the solution of the $ N $-particle systems given by stochastic differential equations (SDEs) converges to the solution of the infinite-dimensional Dyson model. We prove 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 $.