# Renormalization of crossing probabilities in the planar random-cluster model

## TL;DR

This paper develops a renormalization scheme for crossing probabilities in the 2D random-cluster model, classifying their asymptotic behaviors without relying on self-duality, applicable to various graphs and models.

## Contribution

It introduces a new renormalization approach that characterizes crossing probability behaviors in the random-cluster model without self-duality assumptions.

## Key findings

- Identifies four distinct crossing probability behaviors: subcritical, supercritical, critical discontinuous, and critical continuous.
- Provides a framework applicable to arbitrary symmetric graphs and other models like random height models.
- Enables analysis of phase transition properties in a broader class of models.

## Abstract

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.