Open Access
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

Abstract

Bootstrap percolation on a graph is a deterministic process that iteratively enlarges a set of occupied sites by adjoining points with at least θ 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 Z2×Kn2, where Kn 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 θ exhibits a sharp phase transition, while odd θ yields a gradual percolation transition. We also establish a gradual transition for bootstrap percolation on Z2×Kn. The community structure of the product graphs connects our process to a heterogeneous bootstrap percolation on Z2. This natural relation with a generalization of polluted bootstrap percolation is the leading theme in our analysis.

Citation

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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. https://doi.org/10.1214/19-AAP1497

Information

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

Subjects:
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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