  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 $.
