The Tails of the Crossing Probability

TL;DR

This paper investigates the scaling behavior of the tails of the crossing probability in correlated percolation models, revealing different exponential scaling regimes far from and near the critical point.

Contribution

It demonstrates the specific exponential scaling forms of the crossing probability tails in correlated percolation, including crossover behavior at criticality.

Findings

Tails follow an exponential form far from criticality

Crossover to a different exponential scaling at criticality

Identification of scaling indices $ u$ and $z$ in the models

Abstract

The scaling of the tails of the probability of a system to percolate only in the horizontal direction $\pi_{hs}$ 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 $\pi_{hs}(p) \simeq D \exp(c L[p-p_{c}]^{\nu})$ where $\nu$ is the correlation length index, $p=1-\exp(-\beta)$ is the probability of a bond to be closed. At criticality we observe crossover to another scaling $\pi_{hs}(p) \simeq A \exp (-b {L [p-p_{c}]^{\nu}}^{z})$. Here $z$ is a scaling index describing the central part of the crossing probability.