Invariant $β$-ensembles and the Gauss-Wigner crossover

Abstract

We define a new diffusive matrix model converging towards the $β$ -Dyson Brownian motion for all $β\in [0,2]$ that provides an explicit construction of $β$-ensembles of random matrices that is invariant under the orthogonal/unitary group. For small values of $β$, 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.