2008/06/18 by Balint, Andras, Camia, Federico, Meester, Ronald
#60K35 #82B20 #82B43 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.0806.3020
The Ising model at inverse temperature β and zero external field can be obtained via the Fortuin-Kasteleyn (FK) random-cluster model with q=2 and density of open edges p=1-e-β by assigning spin +1 or -1 to each vertex in such a way that (1) all the vertices in the same FK cluster get the same spin and (2) +1 and -1 have equal probability. We generalize the above procedure by assigning spin +1 with probability r and -1 with probability 1-r, with r ∈ [0,1], while keeping condition (1). For fixed β, this generates a dependent (spin) percolation model with parameter r. We show that, on the triangular lattice and for β1/2, sharpness of the percolation phase transition (by showing exponential decay of the cluster size distribution for r<1/2), and continuity of the percolation function for all r ∈ [0,1].