2019/08/30 by Blanquicett, Daniel
#60C05 #60K35 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1908.11556
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 ai nearest neighbours in the ± ei-directions for each i ∈ \1,2,3\, where a1≤ a2≤ a3. Suppose we infect any healthy vertex x∈ [L]3 already having a3+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 (a1,a2,a3). 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.