Invariant Beta Ensembles and the Gauss-Wigner Crossover

We define a new diffusive matrix model converging toward the β-Dyson Brownian motion for all β is an element of [0,2] that provides an explicit construction of beta ensembles of random matrices that is invariant under the orthogonal or unitary group. For small values of β, our process allows one to interpolate smoothly between the Gaussian distribution and the Wigner semicircle. 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 semicircle.