We present the convergence of the height function of the six-vertex model to the Gaussian free field (GFF) in the parameter range $-1 \leq \Delta \leq -\frac12$. The convergence holds both for the full-plane measure and in domains with “flat” boundary conditions. The latter result is particularly important for transferring convergence results to other models, such as FK-percolation and the Ashkin–Teller model. As an application, we deduce that the cluster contours of critical FK-percolation with $q = 4$ converge to the conformal loop ensemble $CLE_4$.

We first establish convergence for the full-plane six-vertex measure using a combination of algebraic and percolation techniques. Multi-point correlations in the six-vertex model can be expressed in terms of the spectrum of the transfer matrix, while regularity and mixing estimates are obtained through RSW arguments applied to a suitable percolation representation. Using the BKW correspondence together with a recent result for FK-percolation, we then show that every subsequential scaling limit of the six-vertex height function is rotationally invariant. Combined with an explicit Bethe ansatz computation, these ingredients characterize the multi-point correlations of any scaling limit and establish convergence to the GFF.

We then extend the convergence result to finite domains. The key observation is that the one-point function is harmonic in domains with sufficiently regular boundary conditions. To establish this, we combine a form {\em reflection positivity} and {\em analyticity} to effectively push the boundary away and reduce the problem to the full-plane setting.

Based on joint work with E. Averous, H. Duminil-Copin, T. He, K. Kozlowski and P. Lammers ,