From Random Matrices to Quasiperiodic Jacobi Matrices via Orthogonal Polynomials

TL;DR

This paper reviews asymptotic behaviors of orthogonal polynomials on multiple intervals, highlighting their spectral properties and connections to quasiperiodic Jacobi matrices, with implications for random matrix eigenvalue distributions.

Contribution

It introduces a unified perspective on orthogonal polynomials and quasiperiodic Jacobi matrices, revealing their spectral relationships and isospectral deformations.

Findings

Construction of double infinite quasiperiodic Jacobi matrices

Identification of isospectral deformations between matrices

Discussion of asymptotic eigenvalue distributions in random matrices

Abstract

We present an informal review of results on asymptotics of orthogonal polynomials, stressing their spectral aspects and similarity in two cases considered. They are polynomials orthonormal on a finite union of disjoint intervals with respect to the Szego weight and polynomials orthonormal on R with respect to varying weights and having the same union of intervals as the set of oscillations of asymptotics. In both cases we construct double infinite Jacobi matrices with generically quasiperiodic coefficients and show that each of them is an isospectral deformation of another. Related results on asymptotic eigenvalue distribution of a class of random matrices of large size are also shortly discussed.