SMIRNOV'S FERMIONIC OBSERVABLE AWAY FROM CRITICALITY

• 成果类型: Article
• 署名作者: Beffara, V.; Duminil-Copin, H.
• 署名单位: Ecole Normale Superieure de Lyon (ENS de LYON); University of Geneva
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP689
• 发表日期: 2012
• 页码: 2667-2689
• 关键词: random-cluster model

摘要

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 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 1/2 log(1 + root 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