Square ice, alternating sign matrices, and classical orthogonal polynomials

Abstract

The six-vertex model with domain wall boundary conditions, or square ice, is considered for particular values of its parameters, corresponding to 1-, 2-, and 3-enumerations of alternating sign matrices (ASMs). Using Hankel determinant representations for the partition function and the boundary correlator of homogeneous square ice, it is shown how the ordinary and refined enumerations can be derived in a very simple and straightforward way. The derivation is based on the standard relationship between Hankel determinants and orthogonal polynomials. For the particular sets of parameters corresponding to 1-, 2-, and 3-enumerations of ASMs, the Hankel determinant can be naturally related to Continuous Hahn, Meixner-Pollaczek, and Continuous Dual Hahn polynomials, respectively. This observation allows for a unified and simplified treatment of ASMs enumerations. In particular, along the lines of the proposed approach, we provide a complete solution to the long standing problem of the refined 3-enumeration of AMSs.

The Limit Shape of Large Alternating Sign Matrices

“…In this case we have ∆ = 1/2 and t = 1, and the function h N z; 1 2 , 1 is given by the formula (see, e.g., [25,26])…”

“…The function h N z; − 1 2 ; 1 has been obtained in [25] (see also [26]). The following formulae are valid…”

Emptiness formation probability in the domain-wall six-vertex model

“…Evaluating the integral in RHS of (5.13), one finds that (5.13) is nothing but the relation 16) where E denotes the lower triangular matrix, with entries standing under the main diagonal equal to one and all other entries being zeroes, i.e., E pr = δ p,r+1 . Inverting matrix A in (5.15) with the help of (5.16), one arrives immediately to the expression for H N 's, which follows from formula (5.11) and definition (5.14).…”

On two-point boundary correlations in the six-vertex model with domain wall boundary conditions

“…In this section the results for one-and two-point boundary correlation functions will be analyzed by making use of the orthogonal polynomials theory, along the lines proposed in paper [27]. Here we show that the two-point boundary correlation function, studied in the previous section, is expressible in terms of one-point ones.…”

“…First we note, following paper [27], that the determinant entering the expression for the homogenous model partition function can be related with orthogonal polynomials using the integral representation…”