The metastability threshold for modified bootstrap percolation in d dimensions

In the modified bootstrap percolation model, sites in the cube {1,..., L}(d) are initially declared active independently with probability p. At subsequent steps, an inactive site becomes active if it has at least one active nearest neighbour in each of the d dimensions, while an active site remains active forever. We study the probability that the entire cube is eventually active. For all d >= 2 we prove that as L -> infinity and p -> 0 simultaneously, this probability converges to 1 if L >= exp...exp lambda+epsilon/p, and converges to 0 if L <= exp...exp lambda-epsilon/p, for any epsilon > 0. Here the exponential function is iterated d-1 times, and the threshold lambda equals pi(2)/6 for all d.