Rotational invariance in critical planar lattice models

Abstract

We prove that the large-scale properties of a number of two-dimensional lattice models are rotationally invariant. More precisely, we prove that the random-cluster model on the square lattice with cluster-weight $1\le q\le 4$ exhibits rotational invariance at large scales. This covers the case of Bernoulli percolation on the square lattice as an important example. We deduce that the correlations of the critical Potts models with $q\in \{2,3,4\}$ colours are rotationally invariant at large scales. Our result is instrumental in proving the convergence of the six-vertex model to the Gaussian Free Field in a separate paper.

This new version of the paper substantially simplifies and streamlines the proof. In particular, it does not make use of Bethe ansatz computations to deduce that the drift of the transformations is null. It also contains improvements of the results: quantitative bounds and uniformity in angles for the universality theorem