Abstract
We establish a central limit theorem for the fluctuations of the linear statistics in the β-ensemble of dimension N at a temperature proportional to N and with confining smooth potential. In this regime, the particles do not accumulate in a compact set as in the fixed case which results in an equilibrium measure supported on the whole real line. The space of test functions for which the CLT holds includes bounded functions. The method that we use is based on a change of variables in the partition function introduced in [30] and allows to deduce the convergence of the Laplace transform of the recentred linear statistics towards the Laplace transform of the normal distribution. It is obtained by the inversion of the master operator, which is the main contribution of the present paper, by following the scheme developed in [29] in the compact case. In the high-temperature regime, the master operator contains an additional differential term due to entropic effects which makes it an unbounded operator. The techniques used in this article involve Schrödinger operators theory as well as concentration of measure.
Funding Statement
This project has received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (grant agreement No. 884584).
Acknowledgments
The authors wish to thank Alice Guionnet, Karol Kozlowski and an anonymous referee for their helpful suggestions. We also thank Arnaud Debussche for pointing out the link with Schrödinger operators theory and Gautier Lambert for pointing out [33]. We would also like to thank Jeanne Boursier, Corentin Le Bihan and Jules Pitcho for their intuition about the regularity of the inverse operator. We would like to thank Jean-Christophe Mourrat for telling us about a more general framework for Poincaré inequalities.
Citation
Charlie Dworaczek Guera. Ronan Memin. "CLT for real β-ensembles at high temperature." Electron. J. Probab. 29 1 - 45, 2024. https://doi.org/10.1214/24-EJP1233
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