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Continuity of the phase transition for planar random-cluster and Potts models with $${1 \leq q \leq 4}$$. (English) Zbl 1357.82011
Summary: 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 $${q \geq 1}$$ (note that $$q$$ is not necessarily an integer) and is based on two ingredients:
$$\bullet$$ The fact that the two-point function for the free state decays sub-exponentially fast for cluster-weights $${1\leq q\leq 4}$$, which is derived studying parafermionic observables on a discrete Riemann surface.
$$\bullet$$ 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 $${q \in [1,4]}$$. 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.

##### MSC:
 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B26 Phase transitions (general) in equilibrium statistical mechanics 82D40 Statistical mechanical studies of magnetic materials 82B27 Critical phenomena in equilibrium statistical mechanics
##### Keywords:
planar Potts model; phase transition; random-cluster model
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