Phase Transitions in a Two-Component Site-Bond Percolation Model

TL;DR

This paper introduces a method to analyze a two-component percolation model by reducing it to an effective one-component model, determining percolation thresholds through Monte Carlo simulations, and proposing empirical formulas for these thresholds.

Contribution

The paper presents a novel approach to treat multi-component percolation models as effective single-component models using a scaled control variable, with simulation results and empirical formulas for thresholds.

Findings

Percolation threshold in terms of scaled variable $p_{+}$ is determined for N=2.

Phase transitions are identified in two limits of bond probabilities.

A new site percolation threshold $f_{b}^{c} \,\simeq\, 0.145$ is reported.

Abstract

A method to treat a N-component percolation model as effective one component model is presented by introducing a scaled control variable $p_{+}$. In Monte Carlo simulations on $16^{3}$, $32^{3}$, $64^{3}$ and $128^{3}$ simple cubic lattices the percolation threshold in terms of $p_{+}$ is determined for N=2. Phase transitions are reported in two limits for the bond existence probabilities $p_{=}$ and $p_{\neq}$. In the same limits, empirical formulas for the percolation threshold $p_{+}^{c}$ as function of one component-concentration, $f_{b}$, are proposed. In the limit $p_{=} = 0$ a new site percolation threshold, $f_{b}^{c} \simeq 0.145$, is reported.