Universality for eigenvalue correlations from the modified Jacobi unitary ensemble

TL;DR

This paper proves that the universal eigenvalue correlation behaviors known for the Jacobi Unitary Ensemble also hold for a generalized version with a modified weight function, extending the universality principle in random matrix theory.

Contribution

The authors demonstrate that the sine and Bessel kernel behaviors persist for the Modified Jacobi Unitary Ensemble, generalizing known asymptotics to a broader class of weights.

Findings

Eigenvalue correlations follow sine and Bessel kernel asymptotics.

Universality extends to ensembles with analytic, strictly positive modifications.

Results rely on orthogonal polynomial asymptotics and recent analytical techniques.

Abstract

The eigenvalue correlations of random matrices from the Jacobi Unitary Ensemble have a known asymptotic behavior as their size tends to infinity. In the bulk of the spectrum the behavior is described in terms of the sine kernel, and at the edge in terms of the Bessel kernel. We will prove that this behavior persists for the Modified Jacobi Unitary Ensemble. This generalization of the Jacobi Unitary Ensemble is associated with the modified Jacobi weight w(x)=(1-x)^\alpha (1+x)^\beta h(x) where the extra factor h is assumed to be real analytic and strictly positive on [-1,1]. We use the connection with the orthogonal polynomials with respect to the modified Jacobi weight, and recent results on strong asymptotics derived by K.T-R McLaughlin, W. Van Assche and the authors.