Asymptotics of matrix orthogonal polynomials on the real line

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.