Sharp metastability threshold for two-dimensional bootstrap percolation

In the bootstrap percolation model, sites in an L by L square are initially independently declared active with probability p. At each time step, an inactive site becomes active if at least two of its four neighbours are active. We study the behaviour as p→0 and L→∞ simultaneously of the probability I(L,p) that the entire square is eventually active. We prove that I(L,p)→1 if , and I(L,p)→0 if , where λ=π2/18. We prove the same behaviour, with the same threshold λ, for the probability J(L,p) that a site is active by time L in the process on the infinite lattice. The same results hold for the so-called modified bootstrap percolation model, but with threshold λ′=π2/6. The existence of the thresholds λ,λ′ settles a conjecture of Aizenman and Lebowitz [3], while the determination of their values corrects numerical predictions of Adler, Stauffer and Aharony [2].