Cluster size in bond percolation on the Platonic solids

Abstract

The main objective of this paper is to study the size of a typical cluster of bond percolation on each of the five Platonic solids: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. Looking at the clusters from a dynamical point of view, i.e., comparing the clusters with birth processes, we first prove that the first and second moments of the cluster size are bounded by their counterparts in a certain branching process, which results in explicit upper bounds that are accurate when the density of open edges is small. Using that vertices surrounded by closed edges cannot be reached by an open path, we also derive upper bounds that, on the contrary, are accurate when the density of open edges is large. These upper bounds hold in fact for all regular graphs. Specializing in the five~Platonic solids, the exact value of (or lower bounds for) the first and second moments are obtained from the inclusion-exclusion principle and a computer program. The goal of our program is not to simulate the stochastic process but to compute exactly sums of integers that are too large to be computed by hand so these results are analytical, not numerical.