Global spectrum fluctuations for Gaussian beta ensembles: a martingale approach

TL;DR

This paper investigates the asymptotic spectral distribution of Gaussian beta ensembles with varying parameters, establishing convergence to the semicircle law and deriving fluctuations using a martingale method.

Contribution

It introduces a martingale approach to analyze the global fluctuations of Gaussian beta ensembles with a varying parameter, extending previous fixed-parameter results.

Findings

Empirical distribution converges to the semicircle law as n→∞ with nβ→∞.

Gaussian fluctuations around the limit are characterized.

The approach applies to ensembles with a parameter β that varies with matrix size.

Abstract

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 $n \to \infty$ with $n\beta \to \infty$, the empirical distribution converges weakly to the semicircle distribution, almost surely. The Gaussian fluctuation around the limit is then derived by a martingale approach.