CLT for real \b{eta}-Ensembles at High Temperature

Charlie Dworaczek Guera, Ronan Memin

Published 2023-01-13Version 1

We establish a central limit theorem for the fluctuations of the empirical measure in the $\beta$- ensemble of dimension $N$ at a temperature proportional to $N$ and with convex, smooth potential. The space of test functions for which the CLT holds includes $\mathcal{C}^1$, vanishing functions at infinity. It is obtained by the inversion of an operator which is a pertubation of a Sturm-Liouville operator. The method that we use is based on a change of variables introduced by Bekerman, Guionnet and Figalli in arXiv:1311.2315 and by Shcherbina in arXiv:1310.7835.