Zeros and ratio asymptotics for matrix orthogonal polynomials

TL;DR

This paper extends the understanding of ratio asymptotics and zero distributions for matrix orthogonal polynomials, including non-Hermitian cases, with applications to spectral measures and random band matrices.

Contribution

It provides an alternative description of the limiting ratio, generalizes to asymptotically periodic and slowly varying recurrence coefficients, and derives new spectral measure formulas.

Findings

Derived the limiting zero distribution for generalized recurrence coefficients.

Obtained explicit spectral measure formulas for matrix Chebyshev polynomials.

Analyzed asymptotic eigenvalue distribution for new classes of random band matrices.

Abstract

Ratio asymptotics for matrix orthogonal polynomials with recurrence coefficients $A_n$ and $B_n$ having limits $A$ and $B$ respectively (the matrix Nevai class) were obtained by Dur\'an. In the present paper we obtain an alternative description of the limiting ratio. We generalize it to recurrence coefficients which are asymptotically periodic with higher periodicity, and/or which are slowly varying in function of a parameter. Under such assumptions, we also find the limiting zero distribution of the matrix orthogonal polynomials, generalizing results by Dur\'an-L\'opez-Saff and Dette-Reuther to the non-Hermitian case. Our proofs are based on "normal family" arguments and on the solution to a quadratic eigenvalue problem. As an application of our results we obtain new explicit formulas for the spectral measures of the matrix Chebyshev polynomials of the first and second kind, and we derive the asymptotic eigenvalue distribution for a class of random band matrices generalizing the tridiagonal matrices introduced by Dumitriu-Edelman.