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February 2015 Bootstrap percolation on the Hamming torus
Janko Gravner, Christopher Hoffman, James Pfeiffer, David Sivakoff
Ann. Appl. Probab. 25(1): 287-323 (February 2015). DOI: 10.1214/13-AAP996

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.

Citation

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Janko Gravner. Christopher Hoffman. James Pfeiffer. David Sivakoff. "Bootstrap percolation on the Hamming torus." Ann. Appl. Probab. 25 (1) 287 - 323, February 2015. https://doi.org/10.1214/13-AAP996

Information

Published: February 2015
First available in Project Euclid: 16 December 2014

zbMATH: 1308.60109
MathSciNet: MR3297774
Digital Object Identifier: 10.1214/13-AAP996

Subjects:
Primary: 60K35

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.25 • No. 1 • February 2015
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