  In r-neighbour 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 neighbours. We study the distribution of the time t at
which all vertices become infected. Given t = t(n) = o(log n/log log n), we
prove a sharp threshold result for the probability that percolation occurs by
time t in d-neighbour bootstrap percolation on the d-dimensional discrete torus
T_n^d. 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-neighbour rule.
