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

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.

Talk presented by F.C. at the Short Program of the Centre de Recherches Mathematiques: Random Matrices, Random Processes and Integrable Systems, Montreal, June 20 - July 8, 2005

Cited by in corpus (11)

• The arctic curve of the domain-wall six-vertex model
• On the weighted enumeration of alternating sign matrices and descending plane partitions
• Emptiness formation probability in the domain-wall six-vertex model
• Functional relations for the six vertex model with domain wall boundary conditions
• Diagonally and antidiagonally symmetric alternating sign matrices of odd order
• Multiply-refined enumeration of alternating sign matrices
• A new representation for the partition function of the six vertex model with domain wall boundaries
• New enumeration formulas for alternating sign matrices and square ice partition functions
• A doubly-refined enumeration of alternating sign matrices and descending plane partitions
• Six-vertex model on a finite lattice: integral representations for nonlocal correlation functions
• Exact results for the six-vertex model with domain wall boundary conditions and a partially reflecting end