Smirnov’s fermionic observable away from criticality

• Autores: V. Beffara, H. Duminil-Copin
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 40, Nº. 6, 2012, págs. 2667-2689
• Idioma: inglés
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• Resumen

  • 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 12log(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.