Site Percolation on Planar $\Phi^{3}$ Random Graphs

TL;DR

This study investigates site percolation on random planar $$ graphs using Monte Carlo simulations, identifying a percolation threshold and critical exponents consistent with bond percolation theory.

Contribution

It introduces a numerical approach to analyze site percolation on $$ planar graphs and determines the percolation threshold and critical exponents.

Findings

Percolation threshold at p_c=0.7360(5)

Critical exponents match bond percolation

Percolation occurs above p_c

Abstract

In this paper, site percolation on random $\Phi^{3}$ planar graphs is studied by Monte-Carlo numerical techniques. The method consists in randomly removing a fraction $q=1-p$ of vertices from graphs generated by Monte-Carlo simulations, where $p$ is the occupation probability. The resulting graphs are made of clusters of occupied sites. By measuring several properties of their distribution, it is shown that percolation occurs for an occupation probability above a percolation threshold $p_{c}$=0.7360(5). Moreover, critical exponents are compatible with those analytically known for bond percolation.