Calculation of the constant factor in the six-vertex model

Bleher, Pavel; Bothner, Thomas

Annales de l’Institut Henri Poincaré D, Tome 1 (2014) no. 4, pp. 363-427.

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 α>1, as a=α-1 2, b=α+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 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 α tend simultaneously to ∞ in such a way that N α→t≥0. Then we apply the Toda equation for the tau-function to find a structural form for C, as a function of α, and we combine the structural form of C and the double scaling asymptotic behavior of Z N to calculate C.

Classification : 82-XX, 15-XX
Mots-clés : Six-vertex model, domain wall boundary conditions, critical line between disordered and antiferroelectric phases, asymptotic behavior of the partition function, Riemann– Hilbert problem, Deift–Zhou nonlinear steepest descent method, Toda equation