## 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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