### 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 language | English |
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Pages (from-to) | 907-933 |

Number of pages | 27 |

Journal | Journal of Theoretical Probability |

Volume | 32 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 1 2019 |

### All Science Journal Classification (ASJC) codes

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