Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with $${1 \le q \le 4}$$1≤q≤4

AbstractThis 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 $${\mathbb{Z}^2}$$Z2 is continuous for $${q \in \{2,3,4\}}$$q∈{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 $${q \ge 1}$$q≥1 (note that q is not necessarily an integer) and is based on two ingredients: The fact that the two-point function for the free state decays sub-exponentially fast for cluster-weights $${1\le q\le 4}$$1≤q≤4, which is derived studying parafermionic observables on a discrete Riemann surface.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 has important consequences toward the study of the scaling limit of the random-cluster model with $${q \in [1,4]}$$q∈[1,4]. 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.