Open Access
2017 Bootstrap percolation on products of cycles and complete graphs
Janko Gravner, David Sivakoff
Electron. J. Probab. 22: 1-20 (2017). DOI: 10.1214/17-EJP43


Bootstrap percolation on a graph 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, and we say that spanning occurs if every point eventually becomes occupied. The main question concerns the critical probability, that is, the minimal initial density that makes spanning likely. The graphs we consider are products of cycles of $m$ points and complete graphs of $n$ points. The major part of the paper focuses on the case when two factors are complete graphs and one factor is a cycle. We identify the asymptotic behavior of the critical probability and show that, when $\theta $ is odd, there are two qualitatively distinct phases: the transition from low to high probability of spanning as the initial density increases is sharp or gradual, depending on the size of $m$.


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Janko Gravner. David Sivakoff. "Bootstrap percolation on products of cycles and complete graphs." Electron. J. Probab. 22 1 - 20, 2017.


Received: 23 May 2016; Accepted: 1 March 2017; Published: 2017
First available in Project Euclid: 24 March 2017

zbMATH: 1361.60092
MathSciNet: MR3629873
Digital Object Identifier: 10.1214/17-EJP43

Primary: 60K35

Keywords: Bootstrap percolation , critical probability , gradual transition , sharp transition

Vol.22 • 2017
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