Matrix models for circular ensembles

TL;DR

This paper introduces a new matrix model for circular ensembles that captures the Coulomb gas eigenvalue distribution on the unit circle, combining orthogonal polynomial theory with recent matrix models.

Contribution

It provides a tri-diagonal matrix model for the Jacobi ensemble, resolving an open question and linking orthogonal polynomials with Coulomb gas distributions.

Findings

Derived a matrix ensemble matching Coulomb gas eigenvalue distribution

Established a tri-diagonal model for the Jacobi ensemble

Connected orthogonal polynomial methods with random matrix theory

Abstract

We describe an ensemble of (sparse) random matrices whose eigenvalues follow the Gibbs distribution for n particles of the Coulomb gas on the unit circle at inverse temperature beta. Our approach combines elements from the theory of orthogonal polynomials on the unit circle with ideas from recent work of Dumitriu and Edelman. In particular, we resolve a question left open by them: find a tri-diagonal model for the Jacobi ensemble.