## The Annals of Applied Probability

### Bootstrap percolation on the Hamming torus

#### 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.

#### Article information

Source
Ann. Appl. Probab., Volume 25, Number 1 (2015), 287-323.

Dates
First available in Project Euclid: 16 December 2014

https://projecteuclid.org/euclid.aoap/1418740187

Digital Object Identifier
doi:10.1214/13-AAP996

Mathematical Reviews number (MathSciNet)
MR3297774

Zentralblatt MATH identifier
1308.60109

#### Citation

Gravner, Janko; Hoffman, Christopher; Pfeiffer, James; Sivakoff, David. Bootstrap percolation on the Hamming torus. Ann. Appl. Probab. 25 (2015), no. 1, 287--323. doi:10.1214/13-AAP996. https://projecteuclid.org/euclid.aoap/1418740187

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