## Electronic Journal of Probability

### CLT for Fluctuations of $\beta$-ensembles with general potential

#### Abstract

We prove a central limit theorem for the linear statistics of one-dimensional log-gases, or $\beta$-ensembles. We use a method based on a change of variables which allows to treat fairly general situations, including multi-cut and, for the first time, critical cases, and generalizes the previously known results of Johansson, Borot-Guionnet and Shcherbina. In the one-cut regular case, our approach also allows to retrieve a rate of convergence as well as previously known expansions of the free energy to arbitrary order.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 115, 31 pp.

Dates
Accepted: 4 August 2018
First available in Project Euclid: 24 November 2018

https://projecteuclid.org/euclid.ejp/1543028703

Digital Object Identifier
doi:10.1214/18-EJP209

Mathematical Reviews number (MathSciNet)
MR3885548

Zentralblatt MATH identifier
07021671

#### Citation

Bekerman, Florent; Leblé, Thomas; Serfaty, Sylvia. CLT for Fluctuations of $\beta$-ensembles with general potential. Electron. J. Probab. 23 (2018), paper no. 115, 31 pp. doi:10.1214/18-EJP209. https://projecteuclid.org/euclid.ejp/1543028703

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