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The distributions of $ N $-particle systems of Gaussian unitary ensemblesconverge to Sine$_2$ point processes under bulk-scaling limits. These scalingsare parameterized by a macro-position $ \theta $ in the support of thesemicircle distribution. The limits are always Sine$_{2}$ point processes andindependent of the macro-position $ \theta $ up to the dilations ofdeterminantal kernels. We prove a dynamical counter part of this fact. We provethat the solution of the $ N $-particle systems given by stochasticdifferential equations (SDEs) converges to the solution of theinfinite-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 aredifferent from the ISDE in the limit whenever $ \theta \not= 0 $.

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