The Annals of Applied Probability

Sharp thresholds for the random-cluster and Ising models

Benjamin Graham and Geoffrey Grimmett

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A sharp-threshold theorem is proved for box-crossing probabilities on the square lattice. The models in question are the random-cluster model near the self-dual point psd(q)=√q∕(1+√q), the Ising model with external field, and the colored random-cluster model. The principal technique is an extension of the influence theorem for monotonic probability measures applied to increasing events with no assumption of symmetry.

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Ann. Appl. Probab., Volume 21, Number 1 (2011), 240-265.

First available in Project Euclid: 17 December 2010

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 60E15: Inequalities; stochastic orderings

Random-cluster model Potts model Ising model percolation box-crossing influence sharp threshold colored random-cluster model fuzzy Potts model


Graham, Benjamin; Grimmett, Geoffrey. Sharp thresholds for the random-cluster and Ising models. Ann. Appl. Probab. 21 (2011), no. 1, 240--265. doi:10.1214/10-AAP693.

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