Calculation of the constant factor in the six-vertex model

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 alpha > 1, as a = alpha-1/2 b = alpha+1/2 c = 1 The asymptotics of Z(N) on the critical line was obtained earlier in the paper [8] of Bleher and Liechty: Z(N) = C F-N2 G(root N) N-1/4(1 + O(N-1/2)), 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 ZN in the double scaling limit, as N and alpha tend simultaneously to infinity in such a way that N/alpha -> t >= 0. Then we apply the Toda equation for the tau-function to find a structural form a for C, as a function of alpha, and we combine the structural form of C and the double scaling asymptotic behavior of Z(N) to calculate C.