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

We study the percolation time of the r-neighbour bootstrap percolation model on the discrete torus i¾?/ni¾?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 Bollobas, Holmgren, Smith and Uzzell. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 1-29, 2015

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