JournalsaihpdVol. 1, No. 4pp. 363–427

Calculation of the constant factor in the six-vertex model

  • Pavel Bleher

    Indiana University Purdue University Indianapolis, USA
  • Thomas Bothner

    Université de Montréal, Canada
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We calculate explicitly the constant factor CC in the large NN asymptotics of the partition function ZNZ_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,ca,b,c of the model are parameterized by a parameter α>1\alpha >1, as a=α12a=\frac{\alpha-1}{2}, b=α+12b=\frac{\alpha +1}{2}, c=1c=1. The asymptotics of ZNZ_N on the critical line was obtained earlier in the paper [8] of Bleher and Liechty: ZN=CFN2GNN1/4(1+O(N1/2))Z_N=CF^{N^2}G^{\sqrt{N}}N^{1/4}(1+O(N^{-1/2})), where FF and GG are given by explicit expressions, but the constant factor C>0C>0 was not known. To calculate the constant CC, we find, by using the Riemann–Hilbert approach, an asymptotic behavior of ZNZ_N in the double scaling limit, as NN and α\alpha tend simultaneously to \infty in such a way that Nαt0\frac{N}{\alpha}\to t\ge 0. Then we apply the Toda equation for the tau-function to find a structural form for CC, as a function of α\alpha, and we combine the structural form of CC and the double scaling asymptotic behavior of ZNZ_N to calculate CC.

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. 363–427

DOI 10.4171/AIHPD/11