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

In r-neighbor bootstrap percolation on the vertex set of a graph G, vertices are initially infected independently with some probability p. At each time step, the infected set expands by infecting all uninfected vertices that have at least r infected neighbors. We study the distribution of the time at which all vertices become infected. Given d ≥ r ≥ 2 and t = t(n) = o ( (log n/ log logn)1/(d−r+1) ) , we prove a sharp threshold result for the probability that percolation occurs by time t in r-neighbor bootstrap percolation on the d-dimensional discrete torus Tn. Moreover, we show that for certain ranges of p = p(n), the time at which percolation occurs is concentrated either on a single value or on two consecutive values. We also prove corresponding results for the modified d-neighbor rule and for the subcritical case d + 1 ≤ r ≤ 2d. This is joint work with B. Bollobás, C. Holmgren, and P. J. Smith.