an:06680501
Zbl 1357.82011
Duminil-Copin, Hugo; Sidoravicius, Vladas; Tassion, Vincent
Continuity of the phase transition for planar random-cluster and Potts models with \({1 \leq q \leq 4}\)
EN
Commun. Math. Phys. 349, No. 1, 47-107 (2017).
00362310
2017
j
82B20 60K35 82B26 82D40 82B27
planar Potts model; phase transition; random-cluster model
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.