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Exact solution of the six-vertex model with domain wall boundary conditions: antiferroelectric phase. (English) Zbl 1192.82015

Summary: 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 \(a= \sinh(\gamma-t)\), \(b= \sinh(\gamma+t)\), \(c= \sinh(2\gamma)\), \(|t|<\gamma\). We prove the conjecture of Zinn-Justin, that as \(n\to\infty\), \(Z_n= C\vartheta _4(n\omega)F^{n^2} [1+O(n^{-1})]\), where \(\omega\) and \(F\) are given by explicit expressions in \(\gamma \) and \(t\), and \(\vartheta_4(z)\) 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.

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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