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

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.