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.
"CLT for Fluctuations of $\beta $-ensembles with general potential." Electron. J. Probab. 23 1 - 31, 2018. https://doi.org/10.1214/18-EJP209