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

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.