Anisotropic Bootstrap Percolation in Three Dimensions

Consider a p-random subset A of initially infected vertices in the discrete cube [L](3), and assume that the neighborhood of each vertex consists of the a(i) nearest neighbors in the +/- e(i)-directions for each i is an element of {1, 2, 3}, where a(1) <= a(2) <= a(3). Suppose we infect any healthy vertex x is an element of [L](3) already having a(3) + 1 infected neighbors, 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.