A Smaller Upper Bound for the $$(4,8^2)$$ Lattice Site Percolation Threshold

Abstract

AbstractThe \((4,8^2)\), or “bathroom tile,” lattice is one of the eleven Archimedean lattices, which are infinite vertex-transitive graphs with edges from the tilings of plane by regular polygons. The site percolation model retains each vertex of an infinite graph independently with probability p, \( 0\le p \le 1\). The site percolation threshold is the critical probability \(p_c^{site}\) above which the subgraph induced by retained vertices contains an infinite connected component almost surely, and below which all components are finite almost surely. Using computational improvements for the substitution method, the upper bound for the site percolation threshold of the \((4,8^2)\) lattice is reduced from 0.785661 to 0.749002.KeywordsSite percolationPercolation thresholdSet partitionsNon-crossing partitionsMSCPrimary 60K35Secondary 05C8005A1882B43