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⩽ℓ⩽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.