# Annales de l’Institut Henri Poincaré D

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**Volume 1, Issue 4, 2014, pp. 363–427**

**DOI: 10.4171/AIHPD/11**

Published online: 2014-12-31

Calculation of the constant factor in the six-vertex model

Pavel Bleher^{[1]}and Thomas Bothner

^{[2]}(1) Indiana University Purdue University Indianapolis, USA

(2) Université de Montréal, Canada

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 $C$.

*Keywords: *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

Bleher Pavel, Bothner Thomas: Calculation of the constant factor in the six-vertex model. *Ann. Inst. Henri Poincaré Comb. Phys. Interact.* 1 (2014), 363-427. doi: 10.4171/AIHPD/11