Invariant $\beta$-ensembles and the Gauss-Wigner crossover

TL;DR

This paper introduces a new matrix model that interpolates between Gaussian and Wigner semi-circle distributions for all eta in [0,2], providing explicit constructions and finite-size corrections for eta-ensembles.

Contribution

The authors construct a diffusive matrix model invariant under orthogonal/unitary groups that smoothly interpolates eta-ensembles for all eta in [0,2], including explicit limit distributions.

Findings

Explicit eta-ensemble constructions for all eta in [0,2]

Smooth interpolation between Gaussian and Wigner semi-circle distributions

Finite-size corrections to the semi-circle law

Abstract

We define a new diffusive matrix model converging towards the $\beta$ -Dyson Brownian motion for all $\beta\in [0,2]$ that provides an explicit construction of $\beta$-ensembles of random matrices that is invariant under the orthogonal/unitary group. For small values of $\beta$, our process allows one to interpolate smoothly between the Gaussian distribution and the Wigner semi-circle. The interpolating limit distributions form a one parameter family that can be explicitly computed. This also allows us to compute the finite-size corrections to the semi-circle.