Sum rules and large deviations for spectral matrix measures in the Jacobi ensemble

TL;DR

This paper establishes a large deviation principle and a sum rule for spectral matrix measures in the Jacobi ensemble, extending previous work on classical ensembles and deriving the distribution of canonical moments.

Contribution

It introduces the first large deviation principle and sum rule for spectral matrix measures of the Jacobi ensemble, filling a gap in the random matrix theory literature.

Findings

Large deviation principle for spectral matrix measures in the Jacobi ensemble.

Sum rule relating matrix measures to the Kesten-McKay law.

Distribution of canonical moments for the matrix Jacobi ensemble.

Abstract

We continue to explore the connections between large deviations for objects coming from random matrix theory and sum rules. This connection was established in [17] for spectral measures of classical ensembles (Gauss-Hermite, Laguerre, Jacobi) and it was extended to spectral matrix measures of the Hermite and Laguerre ensemble in [20]. In this paper, we consider the remaining case of spectral matrix measures of the Jacobi ensemble. Our main results are a large deviation principle for such measures and a sum rule for matrix measures with reference measure the Kesten-McKay law. As an important intermediate step, we derive the distribution of canonical moments of the matrix Jacobi ensemble.