Global spectrum fluctuations for Gaussian beta ensembles: a martingale approach

The paper describes the global limiting behavior of Gaussian beta ensembles where the parameter $\beta$ is allowed to vary with the matrix size $n$. In particular, we show that as $n\to \mathrm{\infty }$ with $n\beta \to \mathrm{\infty }$, the empirical distribution converges weakly to the semicircle distribution, almost surely. The Gaussian fluctuation around the limit is then derived by a martingale approach.