Smirnov’s fermionic observable away from criticality

Abstract

In a recent and celebrated article, Smirnov [Ann. of Math. (2) 172 (2010) 1435-1467] defines an observable for the self-dual random-cluster model with cluster weight q = 2 on the square lattice $\mathbb{Z}^2$, and uses it to obtain conformal invariance in the scaling limit. We study this observable away from the self-dual point. From this, we obtain a new derivation of the fact that the self-dual and critical points coincide, which implies that the critical inverse temperature of the Ising model equals $1/2\log(1+\sqrt{2})$. Moreover, we relate the correlation length of the model to the large deviation behavior of a certain massive random walk (thus confirming an observation by Messikh [The surface tension near criticality of the 2d-Ising model (2006) Preprint]), which allows us to compute it explicitly.