Gaussian free field convergence of the six-vertex model with $-1\le \Delta \le -\frac{1}{2}$

Abstract

We study the isotropic six-vertex model on ${\mathbb{Z}}^2$ with spectral parameter $\Delta \in [-1,-1/2]$, that is, with weights $\mathbf{a}=\mathbf{b}=1$ and $\mathbf{c}\in [\sqrt{3},2]$. We show that the associated height function converges, in the scaling limit, to a properly scaled full-plane Gaussian free field. The result extends to anisotropic weights $\mathbf{a}\ne \mathbf{b}$ upon using a suitable embedding of the lattice.

132 pages, 31 figures