Open Access
November 2012 Smirnov’s fermionic observable away from criticality
V. Beffara, H. Duminil-Copin
Ann. Probab. 40(6): 2667-2689 (November 2012). DOI: 10.1214/11-AOP689


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.


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V. Beffara. H. Duminil-Copin. "Smirnov’s fermionic observable away from criticality." Ann. Probab. 40 (6) 2667 - 2689, November 2012.


Published: November 2012
First available in Project Euclid: 26 October 2012

zbMATH: 1339.60136
MathSciNet: MR3050513
Digital Object Identifier: 10.1214/11-AOP689

Primary: 60K35 , 82B20
Secondary: 82B26 , 82B43

Keywords: correlation length , Critical temperature , Ising model , massive harmonic function

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 6 • November 2012
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