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 $$ _2$$ 2 point processes under bulk scaling limits. These scalings are parameterized by a macro-position $$ \theta $$ θ in the support of the semicircle distribution. The limits are always Sine $$ _{2}$$ 2 point processes and independent of the macro-position $$ \theta $$ θ 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 $$ \theta $$ θ , whereas the N-particle SDEs depend on $$ \theta $$ θ and are different from the ISDE in the limit whenever $$ \theta \not = 0 $$ θ ≠ 0 .