Majority bootstrap percolation on the hypercube

  • Authors:

  • JÓzsef Balogh;BÉla BollobÁs;Robert Morris

  • Affiliations:

  • Department of mathematics, university of illinois, 1409 w. green street, urbana, il 61801, usa (e-mail: jobal@math.uiuc.edu);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);Murray edwards college, university of cambridge, cambridge cb3 0df, uk (e-mail: rdm30@cam.ac.uk)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2009

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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.