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

Abstract

The universality of the crossing probability ${\pi }_{hs}$ 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 ${\pi }_{hs}$ has universal form ${\pi }_{hs}=A(q)Q(z)$ for $q=1,2,3,4$ as a function of the scaling variable $z=[b(q)L^{\frac{1}{\nu (q)}}(p-p_c(q,L)){]}^{\zeta (q)}$. Here, $p=1-\exp (-\beta )$ is the probability of a bond to be closed, $A(q)$ is the nonuniversal crossing amplitude, $b(q)$ is the nonuniversal metric factor, $\zeta (q)$ is the nonuniversal scaling index, $\nu (q)$ is the correlation length index. The universal function $Q(x)\simeq \exp (-z)$. Nonuniversal scaling factors were found numerically.

15 pages, 3 figures, revtex4b, (minor errors in text fixed, journal-ref added)

References in corpus (2)

• A fast Monte Carlo algorithm for site or bond percolation
• Convergence of threshold estimates for two-dimensional percolation

Cited by in corpus (2)

• The density of critical percolation clusters touching the boundaries of strips and squares
• The Tails of the Crossing Probability