Smirnov's fermionic observable away from criticality

Details Smirnov's fermionic observable away from criticality Smirnov's fermionic observable away from criticality

2010-10-04

arxiv.org/abs/1010.0526v2

Mathematics

Probability

19 pages, 6 figures

Scientific paper

In a recent and celebrated article, Smirnov 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 $\frac12\log(1+\sqrt2)$. 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), which allows us to compute it explicitly.

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