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.
"Sharp thresholds for the random-cluster and Ising models." Ann. Appl. Probab. 21 (1) 240 - 265, February 2011. https://doi.org/10.1214/10-AAP693