The role of orthogonal polynomials in the six-vertex model and its combinatorial applications

TL;DR

This paper explores how orthogonal polynomials can be used to analyze the six-vertex model, simplifying enumeration of Alternating Sign Matrices and deriving new refined enumeration results.

Contribution

It introduces a unified approach using orthogonal polynomials to study the six-vertex model and its combinatorial applications, including new refined enumeration formulas.

Findings

Unified treatment of ASM enumerations via orthogonal polynomials

Derivation of refined 3-enumerations of ASMs

Expression of partition functions in terms of boundary correlation functions

Abstract

The Hankel determinant representations for the partition function and boundary correlation functions of the six-vertex model with domain wall boundary conditions are investigated by the methods of orthogonal polynomial theory. For specific values of the parameters of the model, corresponding to 1-, 2- and 3-enumerations of Alternating Sign Matrices (ASMs), these polynomials specialize to classical ones (Continuous Hahn, Meixner-Pollaczek, and Continuous Dual Hahn, respectively). As a consequence, a unified and simplified treatment of ASMs enumerations turns out to be possible, leading also to some new results such as the refined 3-enumerations of ASMs. Furthermore, the use of orthogonal polynomials allows us to express, for generic values of the parameters of the model, the partition function of the (partially) inhomogeneous model in terms of the one-point boundary correlation functions of the homogeneous one.