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

  • Yosuke Kawamoto
  • Hirofumi Osada


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 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 \( \theta \), whereas the N-particle SDEs depend on \( \theta \) and are different from the ISDE in the limit whenever \( \theta \not = 0 \).


Gaussian unitary ensembles Dyson’s model Bulk scaling limit Interacting Brownian motion Infinite-dimensional stochastic differential equation 

Mathematics Subject Classification (2010)

60J60 60J70 15A52 60F17 60J65 



H.O. thanks Professor H. Spohn for a useful comment at RIMS in Kyoto University in 2002.


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Authors and Affiliations

  1. 1.Faculty of MathematicsKyushu UniversityFukuokaJapan

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