Universality of the Crossing Probability for the Potts Model for q=1,2,3,4

Abstract

The universality of the crossing probability $π_{hs}$ of a system to percolate only in the horizontal direction, was investigated numerically by using 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 universal form $π_{hs}=A(q) Q(z)$ for $q=1,2,3,4$ as a function of the scaling variable $z= [ b(q)L^{\frac{1}{ν(q)}}(p-p_{c}(q,L)) ]^{ζ(q)}$. 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, $ζ(q)$ is the nonuniversal scaling index, $ν(q)$ is the correlation length index. The universal function $Q(x) \simeq \exp(-z)$. Nonuniversal scaling factors were found numerically.