The distributions of N-particle systems of Gaussian unitary ensembles converge to Sine2 point processes under bulk scaling limits. These scalings are parameterized by a macro-position θ in the support of the semicircle distribution. The limits are always Sine2 point processes and independent of the macro-position 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 θ, whereas the N-particle SDEs depend on θ and are different from the ISDE in the limit whenever θ≠ 0.
CITATION STYLE
Kawamoto, Y., & Osada, H. (2019). Dynamical Bulk Scaling Limit of Gaussian Unitary Ensembles and Stochastic Differential Equation Gaps. Journal of Theoretical Probability, 32(2), 907–933. https://doi.org/10.1007/s10959-018-0816-2
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