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

This is a continuation of the paper [4] of Bleher and Fokin, in which the large $n$ asymptotics is obtained for the partition function $Z_n$ of the six-vertex model with domain wall boundary conditions in the disordered phase. In the present paper we obtain the large $n$ asymptotics of $Z_n$ in the ferroelectric phase. We prove that for any $\text{ep}>0$, as $n\to \mathrm{\infty }$, $Z_n=C{G}^n{F}^{n^2}[1+O(e^{-n^{1-\text{ep}}})]$, and we find the exact value of the constants $C,G$ and $F$. The proof is based on the large $n$ asymptotics for the underlying discrete orthogonal polynomials and on the Toda equation for the tau-function.