A new computation of the critical point for the planar random-cluster model with $q\ge1$

TL;DR

This paper introduces a new method to compute the critical point of the planar random-cluster model with q ≥ 1, offering advantages over previous approaches by extending to other graphs and avoiding certain assumptions.

Contribution

The paper provides an alternative computation of the critical point for the planar random-cluster model that extends to other graphs and uses RSW-type arguments without relying on self-duality.

Findings

New computation of the critical point for q ≥ 1

Method extends to other planar graphs with symmetries

Avoids self-duality and symmetric event assumptions

Abstract

We present a new computation of the critical value of the random-cluster model with cluster weight $q\ge 1$ on $\mathbb{Z}^2$. This provides an alternative approach to the result of Beffara and Duminil-Copin. We believe that this approach has several advantages. First, most of the proof can easily be extended to other planar graphs with sufficient symmetries. Furthermore, it invokes RSW-type arguments which are not based on self-duality. And finally, it contains a new way of applying sharp threshold results which avoid the use of symmetric events and periodic boundary conditions.