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February 2020 Bootstrap percolation on the product of the two-dimensional lattice with a Hamming square
Janko Gravner, David Sivakoff
Ann. Appl. Probab. 30(1): 145-174 (February 2020). DOI: 10.1214/19-AAP1497


Bootstrap percolation on a graph is a deterministic process that iteratively enlarges a set of occupied sites by adjoining points with at least $\theta $ occupied neighbors. The initially occupied set is random, given by a uniform product measure with a low density $p$. Our main focus is on this process on the product graph $\mathbb{Z}^{2}\times K_{n}^{2}$, where $K_{n}$ is a complete graph. We investigate how $p$ scales with $n$ so that a typical site is eventually occupied. Under critical scaling, the dynamics with even $\theta $ exhibits a sharp phase transition, while odd $\theta $ yields a gradual percolation transition. We also establish a gradual transition for bootstrap percolation on $\mathbb{Z}^{2}\times K_{n}$. The community structure of the product graphs connects our process to a heterogeneous bootstrap percolation on $\mathbb{Z}^{2}$. This natural relation with a generalization of polluted bootstrap percolation is the leading theme in our analysis.


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Janko Gravner. David Sivakoff. "Bootstrap percolation on the product of the two-dimensional lattice with a Hamming square." Ann. Appl. Probab. 30 (1) 145 - 174, February 2020.


Received: 1 November 2018; Revised: 1 April 2019; Published: February 2020
First available in Project Euclid: 25 February 2020

zbMATH: 07200525
MathSciNet: MR4068308
Digital Object Identifier: 10.1214/19-AAP1497

Primary: 60K35 , 82B43

Keywords: Bootstrap percolation , cellular automaton , critical scaling , final density , heterogeneous bootstrap percolation

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.30 • No. 1 • February 2020
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