Renormalization of crossing probabilities in the planar random-cluster model

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.