Anisotropic bootstrap percolation in three dimensions

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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 $\pm e_{i}$-directions for each $i\in \{1,2,3\}$, where $a_{1}\le a_{2}\le a_{3}$. Suppose we infect any healthy vertex $x\in [L]^{3}$ 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.