The time of bootstrap percolation with dense initial sets for all thresholds

2012/09/19 by Béla Bollobás, Paul Smith, Bollobás, Béla +3 · 1 citation
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #math.CO #math.PR

paper · pdf · doi:10.48550/arxiv.1209.4339

27 pages, 4 figures

arxiv created 2013/08/14 · arxiv updated 2013/08/15

Abstract

We study the percolation time of the r-neighbour bootstrap percolation model on the discrete torus (\Z/n\Z)^d. For t at most a polylog function of n and initial infection probabilities within certain ranges depending on t, we prove that the percolation time of a random subset of the torus is exactly equal to t with high probability as n tends to infinity. Our proof rests crucially on three new extremal theorems that together establish an almost complete understanding of the geometric behaviour of the r-neighbour bootstrap process in the dense setting. The special case d-r=0 of our result was proved recently by Bollobás, Holmgren, Smith and Uzzell.

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