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

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 $\theta ≠0$.