Short proof of the sharpness of the phase transition for the random-cluster model with $q=2$

TL;DR

This paper offers a concise proof demonstrating the sharpness of the phase transition in the random-cluster model with q=2, extending previous methods and confirming exponential decay in the subcritical Ising model.

Contribution

It introduces a short proof of phase transition sharpness for q=2, adapting techniques from the q=1 case to the Ising model.

Findings

Proves phase transition sharpness for q=2 in the random-cluster model.

Establishes exponential decay of two-point correlation in the subcritical Ising model.

Extends existing approach from q=1 to q=2.

Abstract

The purpose of this modest note is to provide a short proof of the sharpness of the phase transition for the Random-cluster model with $q=2$ by extending the approach developed by Duminil-Copin and Tassion for $q=1$. This in particular implies the exponential decay of the two point-correlation function in the subcritical Ising model.