Matrix Models for Beta Ensembles

Abstract

This paper constructs tridiagonal random matrix models for general (\(\beta>0\)) \(\beta\)-Hermite (Gaussian) and \(\beta\)-Laguerre (Wishart) ensembles. These generalize the well-known Gaussian and Wishart models for \(\beta = 1,2,4\). Furthermore, in the cases of the \(\beta\)-Laguerre ensembles, we eliminate the exponent quantization present in the previously known models. We further discuss applications for the new matrix models, and present some open problems.