A Note on Bootstrap Percolation Thresholds in Plane Tilings using Regular Polygons

Abstract

In \emph{$k$-bootstrap percolation}, we fix $p\in (0,1)$, an integer $k$, and a plane graph $G$. Initially, we infect each face of $G$ independently with probability $p$. Infected faces remain infected forever, and if a healthy (uninfected) face has at least $k$ infected neighbors, then it becomes infected. For fixed $G$ and $p$, the \emph{percolation threshold} is the largest $k$ such that eventually all faces become infected, with probability at least $1/2$. For a large class of infinite graphs, we show that this threshold is independent of $p$. We consider bootstrap percolation in tilings of the plane by regular polygons. A \emph{vertex type} in such a tiling is the cyclic order of the faces that meet a common vertex. First, we determine the percolation threshold for each of the Archimedean lattices. More generally, let $\mathcal{T}$ denote the set of plane tilings $T$ by regular polygons such that if $T$ contains one instance of a vertex type, then $T$ contains infinitely many instances of that type. We show that no tiling in $\mathcal{T}$ has threshold 4 or more. Further, the only tilings in $\mathcal{T}$ with threshold 3 are four of the Archimedean lattices. Finally, we describe a large subclass of $\mathcal{T}$ with threshold 2.