Universality of the crossing probability for the Potts model for q=1, 2, 3, 4

Tails of the crossing probability

The scaling of the tails of the probability of a system to percolate only in the horizontal direction pi(hs) was investigated numerically for the correlated site-bond percolation model (q -state Potts model) for q=1 , 2, 3, 4 (where q is the number of spin states). We have to demonstrate that the crossing probability pi(hs) (p) far from the critical point p(c) has the shape pi(hs) (p) similar to D exp [cL (p- p(c) )(nu) ] where nu is the correlation length index, and p=1-exp (-beta) is the probability of a bond to be closed. For the tail region the correlation length is smaller than the lattice size. At criticality the correlation length reaches the sample size and we observe crossover to another scaling pi(hs) (p) similar to A exp {-b [L (p- p(c) )(nu)](x)}. Here x is a scaling index describing the central part of the crossing probability.