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An Improved Upper Bound for Bootstrap Percolation in All Dimensions

Part of: Special processes Combinatorial probability

Published online by Cambridge University Press:  17 June 2019

Andrew J. Uzzell
Andrew J. Uzzell*
Affiliation:
Department of Mathematics and Statistics, Grinnell College, Grinnell, IA 50112, USA
*
Email: uzzellan@grinnell.edu
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Abstract

In r-neighbour bootstrap percolation on the vertex set of a graph G, a set A of initially infected vertices spreads by infecting, at each time step, all uninfected vertices with at least r previously infected neighbours. When the elements of A are chosen independently with some probability p, it is natural to study the critical probability pc(G, r) at which it becomes likely that all of V(G) will eventually become infected. Improving a result of Balogh, Bollobás and Morris, we give a bound on the second term in the expansion of the critical probability when G = [n]d and d ⩾ r ⩾ 2. We show that for all d ⩾ r ⩾ 2 there exists a constant cd,r > 0 such that if n is sufficiently large, then

$$p_c (\left[ n \right]^d ,{\rm{ }}r){\rm{\le }}\left( {\frac{{\lambda (d,r)}}{{\log _{(r - 1)} (n)}} - \frac{{c_{d,r} }}{{(\log _{(r - 1)} (n))^{3/2} }}} \right)^{d - r + 1} ,$$

where λ(d, r) is an exact constant and log(k) (n) denotes the k-times iterated natural logarithm of n.

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60C05: Combinatorial probability

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 6 , November 2019 , pp. 936 - 960
Copyright
© Cambridge University Press 2019 

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Footnotes

†

This work was done while the author was at the University of Memphis.

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