Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions. Disordered Phase

TL;DR

This paper derives the large N asymptotics of the six-vertex model with domain wall boundary conditions in the disordered phase using Riemann-Hilbert techniques, confirming conjectures about its asymptotic behavior and connecting it to alternating sign matrices.

Contribution

It provides the first large N asymptotic analysis of the six-vertex model with DWBC in the disordered phase, using Riemann-Hilbert methods and confirming Zinn-Justin's conjecture.

Findings

Confirmed the asymptotic form of the partition function as N→∞

Calculated the exact value of the exponent κ in the asymptotics

Compared the six-vertex model results with enumeration of alternating sign matrices

Abstract

The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC) has been introduced and solved for finite $N$ by Korepin and Izergin. The solution is based on the Yang-Baxter equations and it represents the free energy in terms of an $N\times N$ Hankel determinant. Paul Zinn-Justin observed that the Izergin-Korepin formula can be re-expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction. We use this observation to obtain the large $N$ asymptotics of the six-vertex model with DWBC in the disordered phase. The solution is based on the Riemann-Hilbert approach and the Deift-Zhou nonlinear steepest descent method. As was noticed by Kuperberg, the problem of enumeration of alternating sign matrices (the ASM problem) is a special case of the the six-vertex model. We compare the obtained exact solution of the six-vertex model with known exact results for the 1, 2, and 3 enumerations of ASMs, and also with the exact solution on the so-called free fermion line. We prove the conjecture of Zinn-Justin that the partition function of the six-vertex model with DWBC has the asymptotics, $Z_N\sim CN^\kappa e^{N^2f}$ as $N\to\infty$, and we find the exact value of the exponent $\kappa$.