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

TL;DR

This paper investigates bootstrap percolation thresholds in plane tilings by regular polygons, establishing thresholds for various lattices and showing that most have thresholds below 4, with some exactly 3 or 2.

Contribution

It determines the percolation thresholds for all Archimedean lattices and characterizes a broad class of tilings with thresholds at most 3, revealing new insights into percolation behavior in regular polygon tilings.

Findings

Thresholds for Archimedean lattices are explicitly determined.

No tiling in the considered class has a threshold of 4 or more.

Certain tilings have a threshold exactly equal to 3 or 2.

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.