CLT for real beta-ensembles at high temperature

We establish a central limit theorem for the fluctuations of the linear statistics in the $\beta$-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 $\beta>0$ 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 $C^2$ functions. The method that we use is based on a change of variables in the partition function introduced in Johansson [1998] 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 Hardy, Lambert [2019] 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 Schr\"odinger operators theory as well as concentration of measure.