Calculation of the constant factor in the six-vertex model

In the present paper we calculate explicitly the constant factor $C$ in the large $N$ asymptotics of the partition function $Z_N$ of the six-vertex model with domain wall boundary conditions on the critical line between the disordered and ferroelectric phases. On the critical line the weights $a,b,c$ of the model are parameterized by a parameter $\al>1$, as $a=\frac{\al-1}{2}$, $b=\frac{\al+1}{2}$, $c=1$. The asymptotics of $Z_N$ on the critical line was obtained earlier in the paper \cite{BL2} of Bleher and Liechty: $Z_N=CF^{N^2}G^{\sqrt{N}}N^{1/4}\big(1+O(N^{-1/2})\big)$, where $F$ and $G$ are given by explicit expressions, but the constant factor $C>0$ was not known. To calculate the constant $C$, we find, by using the Riemann-Hilbert approach, an asymptotic behavior of $Z_N$ in the double scaling limit, as $N$ and $\al$ tend simultaneously to $\infty$ in such a way that $\frac{N}{\al}\to t\ge 0$. Then we apply the Toda equation for the tau-function to find a structural form for $C$, as a function of $\al$, and we combine the structural form of $C$ and the double scaling asymptotic behavior of $Z_N$ to calculate $C$.