Matrix Models for Beta Ensembles

TL;DR

This paper develops new tridiagonal matrix models for general beta-ensembles, extending classical cases and removing quantization constraints, with applications and open problems discussed.

Contribution

It introduces generalized tridiagonal matrix models for beta-ensembles that extend classical models and eliminate quantization in the Laguerre case.

Findings

New matrix models for all beta > 0

Elimination of exponent quantization in Laguerre models

Applications and open problems discussed

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.