Divergence of the correlation length for critical planar FK percolation with $1\le q\le4$ via parafermionic observables

TL;DR

This paper uses parafermionic observables to prove that the correlation length diverges at the critical point for planar FK percolation with cluster-weight between 1 and 4, on the universal cover of the punctured plane.

Contribution

It demonstrates the divergence of the correlation length at criticality for FK percolation with 1≤q≤4 using parafermionic observables and universal cover techniques.

Findings

Correlation length diverges at critical point for 1≤q≤4

Properties of parafermionic observables are crucial for the proof

FK percolation on the universal cover is key to the analysis

Abstract

Parafermionic observables were introduced by Smirnov for planar FK percolation in order to study the critical phase $(p,q)=(p_c(q),q)$. This article gathers several known properties of these observables. Some of these properties are used to prove the divergence of the correlation length when approaching the critical point for FK percolation when $1\le q\le 4$. A crucial step is to consider FK percolation on the universal cover of the punctured plane. We also mention several conjectures on FK percolation with arbitrary cluster-weight $q>0$.