The time of bootstrap percolation with dense initial sets

Let r.N. 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. When p is close to 1, we study the distribution of the time at which all vertices become infected. Given t=t(n)=o(logn/loglogn), 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 Tdn. 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.