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

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 $\theta \ne 0$.

arXiv comment21pages, Appeared in J,Theoretical Probability: https://doi.org/10.1007/s10959-018-0816-2 Key words: the Gaussian Unitary Ensemble, Dyson's model, bulk scaling limit, J. Theoretical Probability 2018

References in corpus (7)

• Determinantal random point fields
• Universality of General $\beta$-Ensembles
• Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials
• Infinite Dimensional Stochastic Differential Equations for Dyson's Model
• Markov property of determinantal processes with extended sine, Airy, and Bessel kernels
• Infinite-dimensional stochastic differential equations arising from Airy random point fields
• Discrete approximations of determinantal point processes on continuous spaces: tree representations and tail triviality

Cited by in corpus (3)

• Ergodicity of unlabeled dynamics of Dyson's model in infinite dimensions
• Infinite-dimensional Stochastic Differential Equations with Symmetry
• Dynamical Universality for Random Matrices