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Smirnov's fermionic observable away from criticality

Published in Annals of Probability. 2012, vol. 40, no. 6, p. 2667-2689
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 $1/2log(1+sqrt{2})$. 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.
arXiv: 1010.0526
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BEFFARA, Vincent, DUMINIL-COPIN, Hugo. Smirnov's fermionic observable away from criticality. In: Annals of Probability, 2012, vol. 40, n° 6, p. 2667-2689. doi: 10.1214/11-AOP689 https://archive-ouverte.unige.ch/unige:30545

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Deposited on : 2013-10-21

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