2022/01/27 by Daniel Blanquicett, Blanquicett, Daniel
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2201.11365
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 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∈ \a3+1, …, a3+a2\ with a3≥ a1+a2. We moreover give upper bounds for all remaining cases a3 < a1+a2 and believe that they are tight up to a constant factor.