Exact Solution of the Six‐Vertex Model with Domain Wall Boundary Conditions: Antiferroelectric Phase

Abstract

AbstractWe obtain the large‐n asymptotics of the partition function Zn of the six‐vertex model with domain wall boundary conditions in the antiferroelectric phase region, with the weights a = sinh(γ − t), b = sinh(γ + t), c = sinh(2γ), |t| < γ. We prove the conjecture of Zinn‐Justin, that as n → ∞, Zn = Cϑ4(nω)F [1 + O(n−1)], where ω and F are given by explicit expressions in γ and t, and ϑ4(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. © 2009 Wiley Periodicals, Inc.