Multiple Meixner-Pollaczek polynomials and the six-vertex model

Abstract

We study multiple orthogonal polynomials of Meixner-Pollaczek type with respect to a symmetric system of two orthogonality measures. Our main result is that the limiting distribution of the zeros of these polynomials is one component of the solution to a constrained vector equilibrium problem. We also provide a Rodrigues formula and closed expressions for the recurrence coefficients. The proof of the main result follows from a connection with the eigenvalues of block Toeplitz matrices, for which we provide some general results of independent interest. The motivation for this paper is the study of a model in statistical mechanics, the so-called six-vertex model with domain wall boundary conditions, in a particular regime known as the free fermion line. We show how the multiple Meixner-Pollaczek polynomials arise in an inhomogeneous version of this model.

32 pages, 4 figures. References added

References in corpus (8)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Another proof of the alternating sign matrix conjecture
• Thermodynamic limit of the Six-Vertex Model with Domain Wall Boundary Conditions
• Six-Vertex Model with Domain Wall Boundary Conditions and One-Matrix Model
• The arctic curve of the domain-wall six-vertex model
• Square ice, alternating sign matrices and classical orthogonal polynomials
• On the partition function of the six-vertex model with domain wall boundary conditions
• Equilibrium problem for the eigenvalues of banded block Toeplitz matrices