The Annals of Probability

Smirnov’s fermionic observable away from criticality

V. Beffara and H. Duminil-Copin

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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

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

First available in Project Euclid: 26 October 2012

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
Secondary: 82B26: Phase transitions (general) 82B43: Percolation [See also 60K35]

Ising model correlation length critical temperature massive harmonic function


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.

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