Calculation of the constant factor in the six-vertex model

Source abstract

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=\frac{\alpha-1}{2} , b=\frac{\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=CF^{N^2}G^{\sqrt{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 Z_N in the double scaling limit, as N and \alpha tend simultaneously to \infty in such a way that \frac{N}{\alpha}\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 \alpha , and we combine the structural form of C and the double scaling asymptotic behavior of Z_N to calculate C .