Monte-Carlo simulation study of the two-stage percolation transition in enhanced binary trees

TL;DR

This study uses Monte-Carlo simulations to analyze bond percolation on enhanced binary trees, revealing a two-stage transition with an intermediate phase featuring criticality and multiple infinite clusters.

Contribution

It demonstrates the existence of two distinct percolation thresholds and characterizes the intermediate critical phase in enhanced binary trees.

Findings

Two percolation thresholds $p_{c1}$ and $p_{c2}$ identified

Intermediate phase exhibits critical behavior with infinite clusters

Fractal exponent varies continuously from 0 to 1 in the intermediate phase

Abstract

We perform Monte-Carlo simulations to study the Bernoulli ($p$) bond percolation on the enhanced binary tree which belongs to the class of nonamenable graphs with one end. Our numerical results show that the system has two different percolation thresholds $p_{c1}$ and $p_{c2}$. All the points in the intermediate phase $(p_{c1} < p < p_{c2})$ are critical and there exist infinitely many infinite clusters in the intermediate phase. In this phase the corresponding fractal exponent continuously increases with $p$ from zero to unity.