## The Annals of Probability

### 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 $\frac{1}{2}\log(1+\sqrt{2})$. 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.

#### Article information

Source
Ann. Probab., Volume 40, Number 6 (2012), 2667-2689.

Dates
First available in Project Euclid: 26 October 2012

https://projecteuclid.org/euclid.aop/1351258736

Digital Object Identifier
doi:10.1214/11-AOP689

Mathematical Reviews number (MathSciNet)
MR3050513

Zentralblatt MATH identifier
1339.60136

#### Citation

Beffara, V.; Duminil-Copin, H. Smirnov’s fermionic observable away from criticality. Ann. Probab. 40 (2012), no. 6, 2667--2689. doi:10.1214/11-AOP689. https://projecteuclid.org/euclid.aop/1351258736

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