Three-dimensional 2-critical bootstrap percolation: The stable sets approach

Abstract

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 v ∈ [L] 3 already having r infected neighbours, and that infected sites remain infected forever. In this paper we determine log of the critical length for percolation up to a constant factor, for all r ∈ {a 3 + 1, . . . , a 3 + a 2 } with a 3 a 1 + a 2 . We moreover give upper bounds for all remaining cases a 3 < a 1 + a 2 and believe that they are tight up to a constant factor.