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In r-neighbour bootstrap percolation on the vertex set of a graph G, verticesare initially infected independently with some probability p. At each timestep, the infected set expands by infecting all uninfected vertices that haveat least r infected neighbours. We study the distribution of the time t atwhich all vertices become infected. Given t = t(n) = o(log n/log log n), weprove a sharp threshold result for the probability that percolation occurs bytime t in d-neighbour bootstrap percolation on the d-dimensional discrete torusT_n^d. Moreover, we show that for certain ranges of p = p(n), the time at whichpercolation occurs is concentrated either on a single value or on twoconsecutive values. We also prove corresponding results for the modifiedd-neighbour rule.

The time of bootstrap percolation for dense initial sets