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

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.

3 figures

References in corpus (2)

• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• On invariant 2x2 β-ensembles of random matrices

Cited by in corpus (4)

• Invariant $\beta$-Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law
• Large deviations of the top eigenvalue of large Cauchy random matrices
• Random matrix ensembles: Wang-Landau algorithm for spectral densities
• Hard edge for beta-ensembles and Painlevé III