Conformal invariance of the six-vertex model's height function

Abstract

In this paper, we show that the height function of the six-vertex model in simply connected domains with piecewise constant boundary conditions converges to the Gaussian Free Field (GFF) with Dirichlet boundary conditions. This implies the conformal invariance of the scaling limit of the model. The argument relies on the full plane GFF convergence combined with Reflexion Positivity and qualitative estimates. The proof works for a wide range of spectral parameters $Δ$ (among which $Δ$ close to $0$, and between $-1$ and $-1/2$) making this paper the first instance of a conformal invariance result for a continuum of models. The paper has consequences for other models, including the FK percolation with cluster-weight $q=4$ and the Ashkin-Teller model, which we will pursue in future work.