Dynamical Bulk Scaling Limit of Gaussian Unitary Ensembles and Stochastic Differential Equation Gaps

Yosuke Kawamoto, Hirofumi Osada

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)907-933
Number of pages27
JournalJournal of Theoretical Probability
Volume32
Issue number2
DOIs
Publication statusPublished - Jun 1 2019

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Mathematics(all)
  • Statistics, Probability and Uncertainty

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