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

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.