Multiple Meixner-Pollaczek polynomials and the six-vertex model

TL;DR

This paper investigates multiple Meixner-Pollaczek polynomials, revealing their zeros' distribution as a solution to a vector equilibrium problem, and connects these polynomials to the six-vertex model in statistical mechanics.

Contribution

It introduces a new class of multiple orthogonal polynomials, derives their properties, and links them to a key model in statistical mechanics, expanding understanding of their applications.

Findings

Zeros' distribution solves a constrained vector equilibrium problem

Provides Rodrigues formula and recurrence coefficients

Connects polynomials to the six-vertex model in statistical mechanics

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.