Bootstrap percolation on the product of the two-dimensional lattice with a Hamming square

TL;DR

This paper studies bootstrap percolation on a product graph combining a 2D lattice and a complete graph, revealing phase transition behaviors depending on the threshold parity and scaling of initial occupation probability.

Contribution

It characterizes the phase transition types in bootstrap percolation on the product graph rom the scaling of initial occupation probability and threshold parity.

Findings

Sharp phase transition for even rac{ heta}{2}

Gradual transition for odd rac{ heta-1}{2}

Gradual transition on or rac{p}{n} scaling

Abstract

Bootstrap percolation on a graph is a deterministic process that 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 with a low density $p$. Our main focus is on this process on the product graph $\mathbb{Z}^2\times K_n^2$, where $K_n$ is a complete graph. We investigate how $p$ scales with $n$ so that a typical site is eventually occupied. Under critical scaling, the dynamics with even $\theta$ exhibits a sharp phase transition, while odd $\theta$ yields a gradual percolation transition. We also establish a gradual transition for bootstrap percolation on $\mathbb{Z}^2\times K_n$. The main tool is heterogeneous bootstrap percolation on $\mathbb{Z}^2$.