Exploring the phase transition of planar FK-percolation

TL;DR

This paper introduces FK-percolation on 2D lattices, highlighting recent findings on phase transition types and rotational invariance at criticality, with a focus on the impact of cluster weight q.

Contribution

It summarizes recent results on phase transition behavior and rotational invariance in 2D FK-percolation, emphasizing the role of cluster weight q.

Findings

Phase transition is continuous for certain q values.

Phase transition becomes discontinuous for other q values.

Critical phase exhibits asymptotic rotational invariance.

Abstract

The aim of these notes is to give a quick introduction to FK-percolation, focusing on certain recent results about the phase transition of the two dimensional model, namely its continuity or discontinuity depending on the cluster weight $q$, and the asymptotic rotational invariance of the critical phase (when the phase transition is continuous). As such, the main focus is on FK-percolation on $\mathbb Z^2$ with $q \geq 1$, but we do mention some important results valid for general dimension. To favour quick access to recent results, the style is minimal, with certain proofs omitted or left as exercises.