Global fluctuations for 1D log-gas dynamics

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:128

Article

Unterberger, Jeremie¹ 

[1] Univ Lorraine, Lab Associe CNRS UMR 7502, Inst Elie Cartan, BP 239, F-54506 Vandoeuvre Les Nancy, France

关键词: Random matrices;    Dyson's Brownian motion;    Log-gas;    Beta-ensembles;    Hydrodynamic limit;    Stieltjes transform;    Entropy;   

DOI  :  10.1016/j.spa.2018.01.008

来源: Elsevier

【 摘 要 】

We study in this article the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations, d lambda(i)(t)( )= 1/root NdWti - V'(lambda(i)(t))dt + beta/2N Sigma(j not equal i )dt/lambda(i)(t )- lambda(j)(t), i = 1,..., N, with beta > 1, sometimes called generalized Dyson's Brownian motion, describing the dissipative dynamics of a log-gas of N equal charges with equilibrium measure corresponding to a beta-ensemble, with sufficiently regular convex potential V. The limit N -> infinity is known to satisfy a mean-field Mc-Kean-Vlasov equation. We prove that, for suitable initial conditions, fluctuations around the limit are Gaussian and satisfy an explicit PDE. The proof is very much indebted to the harmonic potential case treated in Israelsson (2001). Our key argument consists in showing that the time-evolution generator may be written in the form of a transport operator on the upper half-plane, plus a bounded non-local operator interpreted in terms of a signed jump process. (C) 2018 Elsevier B.V. All rights reserved.

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