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

TL;DR

This paper derives the large-scale asymptotics of the partition function for the six-vertex model with domain wall boundary conditions in the antiferroelectric phase, confirming a conjecture and employing advanced mathematical techniques.

Contribution

It provides the first rigorous proof of the asymptotic behavior of the partition function in the antiferroelectric phase, confirming Zinn-Justin's conjecture using Riemann-Hilbert and steepest descent methods.

Findings

Asymptotic formula for $Z_n$ involving theta functions and exponential growth.

Confirmation of Zinn-Justin's conjecture on the partition function behavior.

Explicit expressions for parameters $ heta_4(noldsymbol{ extomega})$ and $F$ in the asymptotics.

Abstract

We obtain the large $n$ asymptotics of the partition function $Z_n$ of the six-vertex model with domain wall boundary conditions in the antiferroelectric phase region, with the weights $a=\sinh(\ga-t), b=\sinh(\ga+t), c=\sinh(2\ga), |t|<\ga$. We prove the conjecture of Zinn-Justin, that as $n\to\infty$, $Z_n=C\th_4(n\om) F^{n^2}[1+O(n^{-1})]$, where $\om$ and $F$ are given by explicit expressions in $\ga$ and $t$, and $\th_4(z)$ is the Jacobi theta function. The proof is based on the Riemann-Hilbert approach to the large $n$ asymptotic expansion of the underlying discrete orthogonal polynomials and on the Deift-Zhou nonlinear steepest descent method.