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 -> infinity. 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 Benaych-Georges and Peche (J Stat Phys 161(3):633-656, 2015) with the explicit form of the intensity measure.