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

Yosuke Kawamoto, Hirofumi Osada

Research output: Contribution to journalArticle

Abstract

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 θ in the support of the semicircle distribution. The limits are always Sine 2 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

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Scaling Limit
Stochastic Equations
Ensemble
Particle System
Point Process
Differential equation
Semicircle
Converge
Dilation
Scaling
kernel
Model
Stochastic differential equations
Point process

All Science Journal Classification (ASJC) codes

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

Cite this

Dynamical Bulk Scaling Limit of Gaussian Unitary Ensembles and Stochastic Differential Equation Gaps. / Kawamoto, Yosuke; Osada, Hirofumi.

In: Journal of Theoretical Probability, Vol. 32, No. 2, 01.06.2019, p. 907-933.

Research output: Contribution to journalArticle

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