The sharp threshold for bootstrap percolation in all dimensions

In r-neighbour bootstrap percolation on a graph G, a (typically random) set A of initially 'infected' vertices spreads by infecting (at each time step) vertices with at least r already-infected neighbours. This process may be viewed as a monotone version of the Glauber dynamics of the Ising model, and has been extensively studied on the d-dimensional grid [n]d. The elements of the set A are usually chosen independently, with some density p, and the main question is to determine, the density at which percolation (infection of the entire vertex set) becomes likely. In this paper we prove, for every pair d, for some constant, and thus prove the existence of a sharp threshold for percolation in any (fixed) number of dimensions. We moreover determine.