Exact solution of the six-vertex model with domain wall boundary conditions. Ferroelectric phase

TL;DR

This paper derives the exact large $n$ asymptotics of the partition function for the six-vertex model with domain wall boundary conditions in the ferroelectric phase, extending previous work in the disordered phase.

Contribution

It provides the first exact asymptotic formula for the partition function in the ferroelectric phase, including explicit constants, using discrete orthogonal polynomials and Toda equations.

Findings

Asymptotic formula for $Z_n$ in the ferroelectric phase

Explicit values for constants $C$, $G$, and $F$

Method based on orthogonal polynomials and Toda equations

Abstract

This is a continuation of the paper [4] of Bleher and Fokin, in which the large $n$ asymptotics is obtained for the partition function $Z_n$ of the six-vertex model with domain wall boundary conditions in the disordered phase. In the present paper we obtain the large $n$ asymptotics of $Z_n$ in the ferroelectric phase. We prove that for any $\ep>0$, as $n\to\infty$, $Z_n=CG^nF^{n^2}[1+O(e^{-n^{1-\ep}})]$, and we find the exact value of the constants $C,G$ and $F$. The proof is based on the large $n$ asymptotics for the underlying discrete orthogonal polynomials and on the Toda equation for the tau-function.