An Improved Upper Bound for the Site Percolation Threshold of the $(4,8^2)$ Archimedean Lattice

Abstract

Percolation models are infinite random graph models which have applications to phase transitions and critical phenomena. In the site percolation model, each vertex in an infinite graph $G$ is retained independently with probability $p$ and deleted otherwise. The percolation threshold is the critical probability $p_c(G)$ such that if $p>p_c(G)$, there is positive probability that the random subgraph induced by the retained vertices has an infinite connected component, while the probability that all of its components are finite is one if $p<p_c(G)$.

There are few lattice graphs for which the site percolation threshold is exactly known, and rigorous bounds for unsolved lattices are very imprecise. The substitution method for computing bounds for the more common class of bond percolation models must be modified to apply to site models. Some modifications will be illustrated with an application to the $(4,8^2)$ Archimedean lattice, which is a vertex-transitive tiling of the plane by squares and regular octagons. An improved upper bound, $p_c^{site}(4,8^2)<0.785661$, is obtained.