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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 Z2, 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 12 log(1 + √ 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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Vincent Beffara, Hugo Duminil-Copin. Smirnov's fermionic observable away from criticality . Annals of Probability, Institute of Mathematical Statistics, 2012, 40 (6), pp.2667 - 2689. ⟨10.1214/11-AOP689⟩. ⟨ensl-00523497⟩



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