Moscow Mathematical Journal

Volume 20, Issue 4, October–December 2020 pp. 711–740.

Renormalization of Crossing Probabilities in the Planar Random-Cluster Model

Authors: Hugo Duminil-Copin (1) and Vincent Tassion (2)
Author institution:(1) Université de Genève, 2-4 rue du Lièvre, 1211 Genève, Switzerland;
Institut des Hautes Études Scientifiques, 35 route de Chartres, 91440 Bures sur Yvette, France
(2) ETH Zurich, Department of Mathematics Group 3 HG G 66.5 Rämistrasse 101 8092, Zurich, Switzerland

Summary:

The study of crossing probabilities (i.e., probabilities of existence of paths crossing rectangles) has been at the heart of the theory of two-dimensional percolation since its beginning. They may be used to prove a number of results on the model, including speed of mixing, tails of decay of the connectivity probabilities, scaling relations, etc. In this article, we develop a renormalization scheme for crossing probabilities in the two-dimensional random-cluster model. The outcome of the process is a precise description of an alternative between four behaviors:

Subcritical: Crossing probabilities, even with favorable boundary conditions, converge exponentially fast to 0.
Supercritical: Crossing probabilities, even with unfavorable boundary conditions, converge exponentially fast to 1.
Critical discontinuous: Crossing probabilities converge to 0 exponentially fast with unfavorable boundary conditions and to 1 with favorable boundary conditions.
Critical continuous: Crossing probabilities remain bounded away from 0 and 1 uniformly in the boundary conditions.

The approach does not rely on self-duality, enabling it to apply in a much larger generality, including the random-cluster model on arbitrary graphs with sufficient symmetry, but also other models like certain random height models.

2010 Math. Subj. Class. 82B43.

Keywords: Crossing probabilities, percolation, random-cluster model.

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