Percolation on High-dimensional Product Graphs

TL;DR

This paper studies percolation on high-dimensional product graphs, revealing phase transition behaviors similar to binomial random graphs, with detailed results on component sizes in different regimes.

Contribution

It extends known phase transition results to general high-dimensional product graphs with regular base graphs, generalizing previous hypercube findings.

Findings

Largest component is logarithmic in subcritical regime

Main result recovers sharp asymptotics in supercritical regime

Other components are typically logarithmic in size

Abstract

We consider percolation on high-dimensional product graphs, where the base graphs are regular and of bounded order. In the subcritical regime, we show that typically the largest component is of order logarithmic in the number of vertices. In the supercritical regime, our main result recovers the sharp asymptotic of the order of the largest component, and shows that all the other components are typically of order logarithmic in the number of vertices. In particular, we show that this phase transition is quantitatively similar to the one of the binomial random graph. This generalises the results of Ajtai, Koml\'os, and Szemer\'edi and of Bollob\'as, Kohayakawa, and \L{}uczak who showed that the $d$-dimensional hypercube, which is the $d$-fold Cartesian product of an edge, undergoes a phase transition quantitatively similar to the one of the binomial random graph.