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Majority Bootstrap Percolation on the Hypercube

Published online by Cambridge University Press:  01 March 2009

JÓZSEF BALOGH ,
BÉLA BOLLOBÁS  and
ROBERT MORRIS
JÓZSEF BALOGH
Affiliation:
Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, USA (e-mail: jobal@math.uiuc.edu)
BÉLA BOLLOBÁS
Affiliation:
Trinity College, Cambridge CB2 1TQ, UK and Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA (e-mail: B.Bollobas@dpmms.cam.ac.uk)
ROBERT MORRIS
Affiliation:
Murray Edwards College, University of Cambridge, Cambridge CB3 0DF, UK (e-mail: rdm30@cam.ac.uk)
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Abstract

In majority bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: if at least half of the neighbours of a vertex v are already infected, then v is also infected, and infected vertices remain infected forever. We say that percolation occurs if eventually every vertex is infected.

The elements of the set of initially infected vertices, A ⊂ V(G), are normally chosen independently at random, each with probability p, say. This process has been extensively studied on the sequence of torus graphs [n]d, for n = 1,2, . . ., where d = d(n) is either fixed or a very slowly growing function of n. For example, Cerf and Manzo [17] showed that the critical probability is o(1) if d(n) ≤ log*n, i.e., if p = p(n) is bounded away from zero then the probability of percolation on [n]d tends to one as n → ∞.

In this paper we study the case when the growth of d to ∞ is not excessively slow; in particular, we show that the critical probability is 1/2 + o(1) if d ≥ (log log n)2 log log log n, and give much stronger bounds in the case that G is the hypercube, [2]d.

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