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Multiple Meixner–Pollaczek polynomials and the six-vertex model
Authors:Martin Bender  Steven Delvaux  Arno BJ Kuijlaars
Institution:aMSRI, 17 Gauss Way, Berkeley, CA 94720-5070, United States;bDepartment of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium
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 (locally) 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.
Keywords:Multiple orthogonal polynomial  Meixner&ndash  Pollaczek polynomial  Recurrence relation  Block Toeplitz matrix  Potential theory  Six-vertex model
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