The Annals of Probability

The time of bootstrap percolation with dense initial sets

Béla Bollobás, Cecilia Holmgren, Paul Smith, and Andrew J. Uzzell

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Abstract

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.

Article information

Source
Ann. Probab., Volume 42, Number 4 (2014), 1337-1373.

Dates
First available in Project Euclid: 3 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1404394066

Digital Object Identifier
doi:10.1214/12-AOP818

Mathematical Reviews number (MathSciNet)
MR3262480

Zentralblatt MATH identifier
1311.60113

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60C05: Combinatorial probability

Keywords
Bootstrap percolation sharp threshold Stein–Chen method

Citation

Bollobás, Béla; Holmgren, Cecilia; Smith, Paul; Uzzell, Andrew J. The time of bootstrap percolation with dense initial sets. Ann. Probab. 42 (2014), no. 4, 1337--1373. doi:10.1214/12-AOP818. https://projecteuclid.org/euclid.aop/1404394066


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