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

The universality of the crossing probability π_hs of a system to percolate only in the horizontal direction was investigated numerically by a cluster Monte Carlo algorithm for the q-state Potts model for q=2, 3, 4 and for percolation q=1. We check the percolation through Fortuin-Kasteleyn clusters near the critical point on the square lattice by using representation of the Potts model as the correlated site-bond percolation model. It was shown that probability of a system to percolate only in the horizontal direction π_hs has the universal form π_hs=A(q)Q(z) for q=1,2,3,4 as a function of the scaling variable z={b(q)L^1/ν(q)[p-p_c(q,L)]}^ζ(q). Here, p=1-exp(-β) is the probability of a bond to be closed, A(q) is the nonuniversal crossing amplitude, b(q) is the nonuniversal metric factor, ν(q) is the correlation length index, and ζ(q) is the additional scaling index. The universal function Q(x)≃exp(-|z|). The nonuniversal scaling factors were found numerically.