Electronic Journal of Probability

Global fluctuations for 1D log-gas dynamics. Covariance kernel and support

Jeremie Unterberger

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We consider the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations,

\[ d\lambda _t^i=\frac{1} {\sqrt{N} } dW_t^i - V'(\lambda _t^i) dt+ \frac{\beta } {2N} \sum _{j\not =i} \frac{dt} {\lambda ^i_t-\lambda ^j_t}, \qquad i=1,\ldots ,N, \qquad \mbox{(0.1)} \]

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\to \infty $ is known to satisfy a mean-field Mc Kean-Vlasov equation. Fluctuations around this limit have been shown [39] to define a Gaussian process solving some explicit martingale problem written in terms of a generalized transport equation.

We prove a series of results concerning either the Mc Kean-Vlasov equation for the density $\rho _t$, notably regularity results and time-evolution of the support, or the associated hydrodynamic fluctuation process, whose space-time covariance kernel we compute explicitly.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 21, 28 pp.

Received: 6 July 2018
Accepted: 2 March 2019
First available in Project Euclid: 21 March 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60F05: Central limit and other weak theorems 60G20: Generalized stochastic processes 60J60: Diffusion processes [See also 58J65] 60J75: Jump processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

random matrices Dyson’s Brownian motion log-gas beta-ensembles hydrodynamic limit Stieltjes transform fluctuations support

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Unterberger, Jeremie. Global fluctuations for 1D log-gas dynamics. Covariance kernel and support. Electron. J. Probab. 24 (2019), paper no. 21, 28 pp. doi:10.1214/19-EJP288. https://projecteuclid.org/euclid.ejp/1553155301

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