Continuity of the phase transition for planar random-cluster and Potts models with $1\le q\le4$

TL;DR

This paper proves the continuity of phase transitions in planar q-state Potts models for q=2,3,4, using random-cluster representations and parafermionic observables, with implications for scaling limits and critical exponents.

Contribution

It establishes the continuous nature of phase transitions for q=2,3,4 in planar Potts models and links this to properties of the random-cluster model, including decay of correlations and interface tightness.

Findings

Phase transition is continuous for q=2,3,4.

Two-point function decays sub-exponentially for these q values.

Interfaces are tight in the scaling limit, parametrized by Loewner chains.

Abstract

This article studies the planar Potts model and its random-cluster representation. We show that the phase transition of the nearest-neighbor ferromagnetic $q$-state Potts model on $\mathbb Z^2$ is continuous for $q\in\{2,3,4\}$, in the sense that there exists a unique Gibbs state, or equivalently that there is no ordering for the critical Gibbs states with monochromatic boundary conditions. The proof uses the random-cluster model with cluster-weight $q\ge1$ (note that $q$ is not necessarily an integer) and is based on two ingredients: 1. The fact that the two-point function for the free state decays sub-exponentially fast for cluster-weights $1\le q\le 4$, which is derived studying parafermionic observables on a discrete Riemann surface. 2. A new result proving the equivalence of several properties of critical random-cluster models: - the absence of infinite-cluster for wired boundary conditions, - the uniqueness of infinite-volume measures, - the sub-exponential decay of the two-point function for free boundary conditions, - a Russo-Seymour-Welsh type result on crossing probabilities in rectangles with arbitrary boundary conditions. The result leads to a number of consequences concerning the scaling limit of the random-cluster model with $1\le q \le 4$. It shows that the family of interfaces (for instance for Dobrushin boundary conditions) are tight when taking the scaling limit and that any sub-sequential limit can be parametrized by a Loewner chain. We also study the effect of boundary conditions on these sub-sequential limits. Let us mention that the result should be instrumental in the study of critical exponents as well.