Limit theorems and soft edge of freezing random matrix models via dual orthogonal polynomials

TL;DR

This paper derives simplified formulas for covariance matrices in freezing regimes of certain random matrix models using dual orthogonal polynomials, and analyzes their asymptotic behavior at the soft edge as the matrix size grows.

Contribution

It introduces new, simpler formulas for covariance matrices in $eta$-Hermite and $eta$-Laguerre ensembles using dual orthogonal polynomials, and studies their asymptotics at the soft edge.

Findings

New formulas for covariance matrices derived from inverse matrices.

Asymptotic analysis of soft edge behavior using Airy functions.

Different limit expressions for $eta$-Hermite ensembles compared to previous work.

Abstract

$N$-dimensional Bessel and Jacobi processes describe interacting particle systems with $N$ particles and are related to $\beta$-Hermite, $\beta$-Laguerre, and $\beta$-Jacobi ensembles. For fixed $N$ there exist associated weak limit theorems (WLTs) in the freezing regime $\beta\to\infty$ in the $\beta$-Hermite and $\beta$-Laguerre case by Dumitriu and Edelman (2005) with explicit formulas for the covariance matrices $\Sigma_N$ in terms of the zeros of associated orthogonal polynomials. Recently, the authors derived these WLTs in a different way and computed $\Sigma_N^{-1}$ with formulas for the eigenvalues and eigenvectors of $\Sigma_N^{-1}$ and thus of $\Sigma_N$. In the present paper we use these data and the theory of finite dual orthogonal polynomials of de Boor and Saff to derive formulas for $\Sigma_N$ from $\Sigma_N^{-1}$ where, for $\beta$-Hermite and $\beta$-Laguerre ensembles, our formulas are simpler than those of Dumitriu and Edelman. We use these polynomials to derive asymptotic results for the soft edge in the freezing regime for $N\to\infty$ in terms of the Airy function. For $\beta$-Hermite ensembles, our limit expressions are different from those of Dumitriu and Edelman.