TY - JOUR

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

AU - Kawamoto, Yosuke

AU - Osada, Hirofumi

N1 - Funding Information:
H.O. is supported in part by JSPS KAKENHI Grant Nos. 16K13764, 16H02149, 16H06338, and KIBAN-A 24244010. Y.K. is supported by Grant-in-Aid for JSPS Research Fellowships (Grant No. 15J03091).
Funding Information:
H.O. thanks Professor H. Spohn for a useful comment at RIMS in Kyoto University in 2002.
Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2019/6/1

Y1 - 2019/6/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s10959-018-0816-2

DO - 10.1007/s10959-018-0816-2

M3 - Article

AN - SCOPUS:85041905213

VL - 32

SP - 907

EP - 933

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

SN - 0894-9840

IS - 2

ER -