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

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 community structure of the product graphs connects our process to a heterogeneous bootstrap percolation on $\mathbb{Z}^{2}$. This natural relation with a generalization of polluted bootstrap percolation is the leading theme in our analysis.