Bootstrap percolation on products of cycles and complete graphs

TL;DR

This paper investigates bootstrap percolation on product graphs of cycles and complete graphs, analyzing the critical initial density needed for full occupation and revealing phase transition behaviors depending on graph sizes and parameters.

Contribution

It characterizes the asymptotic critical probability and identifies phase transition types for bootstrap percolation on specific product graphs, especially when two factors are complete graphs and one is a cycle.

Findings

Critical probability asymptotics for product graphs

Existence of sharp or gradual phase transitions depending on parameters

Distinct behaviors for odd threshold values

Abstract

Bootstrap percolation on a graph iteratively enlarges a set of occupied sites by adjoining points with at least $\theta$ occupied neighbors. The initially occupied set is random, given by a uniform product measure, and we say that spanning occurs if every point eventually becomes occupied. The main question concerns the critical probability, that is, the minimal initial density that makes spanning likely. The graphs we consider are products of cycles of $m$ points and complete graphs of $n$ points. The major part of the paper focuses on the case when two factors are complete graphs and one factor is a cycle. We identify the asymptotic behavior of the critical probability and show that, when $\theta$ is odd, there are two qualitatively distinct phases: the transition from low to high probability of spanning as the initial density increases is sharp or gradual, depending on the size of $m$.