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

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 $n\to \infty$, $Z_n=Cþ_4(n\om) F^{n^2}[1+O(n^{-1})]$, where $\om$ and $F$ are given by explicit expressions in $\ga$ and $t$, and ${þ}_4(z)$ 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.

69 pages, 10 figures

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