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2014 Bootstrap percolation on Galton-Watson trees
Béla Bollobás, Karen Gunderson, Cecilia Holmgren, Svante Janson, Michał Przykucki
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Electron. J. Probab. 19: 1-27 (2014). DOI: 10.1214/EJP.v19-2758

Abstract

Bootstrap percolation is a type of cellular automaton which has been used to model various physical phenomena, such as ferromagnetism. For each natural number $r$, the $r$-neighbour bootstrap process is an update rule for vertices of a graph in one of two states: `infected' or `healthy'. In consecutive rounds, each healthy vertex with at least $r$ infected neighbours becomes itself infected. Percolation is said to occur if every vertex is eventually infected. Usually, the starting set of infected vertices is chosen at random, with all vertices initially infected independently with probability $p$. In that case, given a graph $G$ and infection threshold $r$, a quantity of interest is the critical probability, $p_c(G,r)$, at which percolation becomes likely to occur. In this paper, we look at infinite trees and, answering a problem posed by Balogh, Peres and Pete, we show that for any $b \geq r$ and for any $\epsilon > 0$ there exists a tree $T$ with branching number $\operatorname{br}(T) = b$ and critical probability $p_c(T,r) < \epsilon$. However, this is false if we limit ourselves to the well studied family of Galton--Watson trees. We show that for every $r \geq 2$ there exists a constant $c_r>0$ such that if $T$ is a Galton- Watson tree with branching number $\operatorname{br}(T) = b \geq r$ then $$p_c(T,r) > \frac{c_r}{b} e^{-\frac{b}{r-1}}.$$ We also show that this bound is sharp up to a factor of $O(b)$ by giving an explicit family of Galton--Watson trees with critical probability bounded from above by $C_r e^{-\frac{b}{r-1}}$ for some constant $C_r>0$.

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Béla Bollobás. Karen Gunderson. Cecilia Holmgren. Svante Janson. Michał Przykucki. "Bootstrap percolation on Galton-Watson trees." Electron. J. Probab. 19 1 - 27, 2014. https://doi.org/10.1214/EJP.v19-2758

Information

Accepted: 19 January 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1290.05058
MathSciNet: MR3164766
Digital Object Identifier: 10.1214/EJP.v19-2758

Subjects:
Primary: 05C05
Secondary: 05C80 , 60C05 , 60J80 , 60K35

Keywords: Bootstrap percolation , branching number , Galton--Watson trees , infinite trees

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