Zeros and ratio asymptotics for matrix orthogonal polynomials

Ratio asymptotics for matrix orthogonal polynomials with recurrence coefficients An and Bn having limits A and B, respectively, (the matrix Nevai class) were obtained by Durán. 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 as a function of a parameter. Under such assumptions, we also find the limiting zero distribution of the matrix orthogonal polynomials, thus generalizing results by Durán-López-Saff and Dette-Reuther to the non-Hermitian case. Our proofs are based on “normal family” arguments and on the solution of 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 derive the asymptotic eigenvalue distribution for a class of random band matrices which generalize the tridiagonal matrices introduced by Dumitriu-Edelman.