### 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 |
---|---|

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 |

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### 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.

Research output: Contribution to journal › Article

*Journal of Theoretical Probability*, vol. 32, no. 2, pp. 907-933. https://doi.org/10.1007/s10959-018-0816-2

}

TY - JOUR

T1 - Dynamical Bulk Scaling Limit of Gaussian Unitary Ensembles and Stochastic Differential Equation Gaps

AU - Kawamoto, Yosuke

AU - Osada, Hirofumi

PY - 2019/6/1

Y1 - 2019/6/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85041905213&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85041905213&partnerID=8YFLogxK

U2 - 10.1007/s10959-018-0816-2

DO - 10.1007/s10959-018-0816-2

M3 - Article

VL - 32

SP - 907

EP - 933

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

SN - 0894-9840

IS - 2

ER -