Bootstrap percolation on the Hamming torus

TL;DR

This paper analyzes bootstrap percolation on the Hamming torus, establishing critical thresholds for the probability of the entire graph becoming open based on dimension, threshold, and scaling parameters.

Contribution

It provides new asymptotic thresholds for bootstrap percolation on high-dimensional Hamming tori, especially for large thresholds and specific dimensions.

Findings

Critical exponent $oldsymbol{eta}$ is about $oldsymbol{1+d/ heta}$ for $i=1$.

Critical exponent $oldsymbol{eta}$ is about $oldsymbol{1+2/ heta+ ext{lower order terms}}$ for $i extgreater 1$.

Exact critical probability computed for the case $ heta=3$ in three dimensions.

Abstract

The Hamming torus of dimension $d$ is the graph with vertices $\{1,\dots,n\}^d$ and an edge between any two vertices that differ in a single coordinate. Bootstrap percolation with threshold $\theta$ starts with a random set of open vertices, to which every vertex belongs independently with probability $p$, and at each time step the open set grows by adjoining every vertex with at least $\theta$ open neighbors. We assume that $n$ is large and that $p$ scales as $n^{-\alpha}$ for some $\alpha>1$, and study the probability that an $i$-dimensional subgraph ever becomes open. For large $\theta$, we prove that the critical exponent $\alpha$ is about $1+d/\theta$ for $i=1$, and about $1+2/\theta+\Theta(\theta^{-3/2})$ for $i\ge2$. Our small $\theta$ results are mostly limited to $d=3$, where we identify the critical $\alpha$ in many cases and, when $\theta=3$, compute exactly the critical probability that the entire graph is eventually open.