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Consider a p-random subset A of initially infected vertices in the discrete cube [L] d , and assume that the neighbourhood of each vertex consists of the a i nearest neighbours in the ±e i -directions for each i ∈ {1, 2, . . . , d}, where a 1 a 2 . . . a d . Suppose we infect any healthy vertex v ∈ [L] d already having r infected neighbours, and that infected sites remain infected forever. In this paper we determine the (d − 1)-times iterated logarithm of the critical length for percolation up to a constant factor, for all d-tuples (a 1 , . . . , a d ) and all r ∈ {a 2Moreover, we reduce the problem of determining this (coarse) threshold for all d 3 and all r ∈ {a d + 1, . . . , a 1 + a 2 + • • • + a d }, to that of determining the threshold for all d 3 and all r ∈ {a d + 1, . . . , a d−1 + a d }. L c (N a 1 ,...,a d r , p) := min{L ∈ N : P p ([A] = [L] d ) 1/2}.

Consider a p-random subset A of initially infected vertices in the discrete cube [L] 3 , and assume that the neighbourhood of each vertex consists of the a i nearest neighbours in the ±e i -directions for each i ∈ {1, 2, 3}, where a 1 a 2 a 3 . Suppose we infect any healthy vertex x ∈ [L] 3 already having a 3 + 1 infected neighbours, and that infected sites remain infected forever. In this paper we determine the critical length for percolation up to a constant factor in the exponent, for all triples (a 1 , a 2 , a 3 ). To do so, we introduce a new algorithm called the beams process and prove an exponential decay property for a family of subcritical two-dimensional bootstrap processes.