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CLT for fluctuations of $$\beta$$-ensembles with general potential. (English) Zbl 1406.60036
Summary: 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 K. Johansson [Duke Math. J. 91, No. 1, 151–204 (1998; Zbl 1039.82504)], G. Borot and A. Guionnet [Commun. Math. Phys. 317, No. 2, 447–483 (2013; Zbl 1344.60012); “Asymptotic expansion of $$\beta$$ matrix models in the multi-cut regime”, Preprint, arXiv:1303.1045] and M. Shcherbina [J. Stat. Phys. 151, No. 6, 1004–1034 (2013; Zbl 1273.15042)]. 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.

MSC:
 60F05 Central limit and other weak theorems 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60B10 Convergence of probability measures 60B20 Random matrices (probabilistic aspects) 15B52 Random matrices (algebraic aspects) 82B05 Classical equilibrium statistical mechanics (general) 60G15 Gaussian processes
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