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-equals, slant]l[less-than-or-equals, slant]d, is L~ exp°(l-1)(Cp-1/(d-l+1)), where exp°(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.