The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 21, Number 1 (2011), 240-265.
Sharp thresholds for the random-cluster and Ising models
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.
Ann. Appl. Probab., Volume 21, Number 1 (2011), 240-265.
First available in Project Euclid: 17 December 2010
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Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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. https://projecteuclid.org/euclid.aoap/1292598033