The threshold regime of finite volume bootstrap percolation

We prove that the threshold regime for bootstrap percolation in a d-dimensional box of diameter L with parameters p and l, where 3 less than or equal to l less than or equal to d, is L similar to exp(o(l-1))(Cp-1/(d-l+1)) where exp(o(l-1)) is the exponential iterated l-1 times and C is bounded from above and from below by two positive constants depending on d, l only. (C) 2002 Elsevier Science B.V. All rights reserved.