Gaussian beta ensembles at high temperature: eigenvalue fluctuations and bulk statistics

We study the limiting behavior of Gaussian beta ensembles in the regime where $\beta n=const$ as $n\to \mathrm{\infty }$. The results are (1) Gaussian fluctuations for linear statistics of the eigenvalues, and (2) Poisson convergence of the bulk statistics. (2) is an alternative proof of the result by F.~Benaych-Georges and S.~P'ech'e (2015) with the explicit form of the intensity measure.