Electronic Journal of Probability
- Electron. J. Probab.
- Volume 23 (2018), paper no. 115, 31 pp.
CLT for Fluctuations of $\beta $-ensembles with general potential
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.
Electron. J. Probab., Volume 23 (2018), paper no. 115, 31 pp.
Received: 6 February 2018
Accepted: 4 August 2018
First available in Project Euclid: 24 November 2018
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Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60F05: Central limit and other weak theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60B10: Convergence of probability measures 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 82B05: Classical equilibrium statistical mechanics (general) 60G15: Gaussian processes
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