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July 2014 The time of bootstrap percolation with dense initial sets
Béla Bollobás, Cecilia Holmgren, Paul Smith, Andrew J. Uzzell
Ann. Probab. 42(4): 1337-1373 (July 2014). DOI: 10.1214/12-AOP818


Let $r\in\mathbb{N}$. In $r$-neighbour bootstrap percolation on the vertex set of a graph $G$, vertices are initially infected independently with some probability $p$. At each time step, the infected set expands by infecting all uninfected vertices that have at least $r$ infected neighbours. When $p$ is close to 1, we study the distribution of the time at which all vertices become infected. Given $t=t(n)=o(\log n/\log\log n)$, we prove a sharp threshold result for the probability that percolation occurs by time $t$ in $d$-neighbour bootstrap percolation on the $d$-dimensional discrete torus $\mathbb{T}_{n}^{d}$. Moreover, we show that for certain ranges of $p=p(n)$, the time at which percolation occurs is concentrated either on a single value or on two consecutive values. We also prove corresponding results for the modified $d$-neighbour rule.


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Béla Bollobás. Cecilia Holmgren. Paul Smith. Andrew J. Uzzell. "The time of bootstrap percolation with dense initial sets." Ann. Probab. 42 (4) 1337 - 1373, July 2014.


Published: July 2014
First available in Project Euclid: 3 July 2014

zbMATH: 1311.60113
MathSciNet: MR3262480
Digital Object Identifier: 10.1214/12-AOP818

Primary: 60K35
Secondary: 60C05

Keywords: Bootstrap percolation , sharp threshold , Stein–Chen method

Rights: Copyright © 2014 Institute of Mathematical Statistics


Vol.42 • No. 4 • July 2014
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