Matrix models for beta ensembles

Abstract

This paper constructs tridiagonal random matrix models for general (β>0) β-Hermite (Gaussian) and β-Laguerre (Wishart) ensembles. These generalize the well-known Gaussian and Wishart models for β=1,2,4. Furthermore, in the cases of the β-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.