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

Yosuke Kawamoto, Hirofumi Osada

    Research output: Contribution to journalArticlepeer-review

    3 Citations (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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