The Tails of the Crossing Probability

Abstract

The scaling of the tails of the probability of a system to percolate only in the horizontal direction $π_{hs}$ was investigated numerically for correlated site-bond percolation model for $q=1,2,3,4$.We have to demonstrate that the tails of the crossing probability far from the critical point have shape $π_{hs}(p) \simeq D \exp(c L[p-p_{c}]^ν)$ where $ν$ is the correlation length index, $p=1-\exp(-β)$ is the probability of a bond to be closed. At criticality we observe crossover to another scaling $π_{hs}(p) \simeq A \exp (-b {L [p-p_{c}]^ν}^{z})$. Here $z$ is a scaling index describing the central part of the crossing probability.