Asymptotics of matrix orthogonal polynomials on the real line

TL;DR

This paper derives strong asymptotic results for matrix orthogonal polynomials on the real line with exponential weights, using Riemann-Hilbert techniques and analyzing the matrix Szegő function.

Contribution

It extends asymptotic analysis methods to matrix-valued orthogonal polynomials, highlighting the role of the matrix Szegő function.

Findings

Asymptotics obtained in different regions of the complex plane.

Recurrence coefficients and norms' asymptotic behavior characterized.

Application of Riemann-Hilbert and Deift-Zhou methods to the matrix case.

Abstract

In this paper, we are interested in matrix valued orthogonal polynomials on the real line with respect to exponential weights. We obtain strong asymptotics as the degree tends to infinity in different regions of the complex plane, as well as asymptotic behavior of recurrence coefficients and norms. The main tools are the Riemann-Hilbert formulation and the Deift-Zhou method of steepest descent, adapted to the matrix case. A central role is played by the matrix Szeg\H{o} function, an object that has independent interest.