We calculate explicitly the constant factor in the large asymptotics of the partition function 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 of the model are parameterized by a parameter , as , , . The asymptotics of on the critical line was obtained earlier in the paper  of Bleher and Liechty: , where and are given by explicit expressions, but the constant factor was not known. To calculate the constant , we find, by using the Riemann–Hilbert approach, an asymptotic behavior of in the double scaling limit, as and tend simultaneously to in such a way that . Then we apply the Toda equation for the tau-function to find a structural form for , as a function of , and we combine the structural form of and the double scaling asymptotic behavior of to calculate .
Cite this article
Pavel Bleher, Thomas Bothner, Calculation of the constant factor in the six-vertex model. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 1 (2014), no. 4, pp. 363–427DOI 10.4171/AIHPD/11