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

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

2009-04-20

arxiv.org/abs/0904.3088v2

Physics

Mathematical Physics

69 pages, 10 figures

Scientific paper

10.1007/s00220-008-0709-9

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.

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