2022/01/22 by Daniel Blanquicett, Blanquicett, Daniel
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2201.09029
openalex publication_date 2022/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a p-random subset A of initially infected vertices in the discrete cube [L]d, and assume that the neighbourhood of each vertex consists of the ai nearest neighbours in the ± ei-directions for each i ∈ \1,2,…, d\, where a1≤ a2≤ … ≤ ad. Suppose we infect any healthy vertex v∈ [L]d already having r infected neighbours, and that infected sites remain infected forever. In this paper we determine the (d-1)-times iterated logarithm of the critical length for percolation up to a constant factor, for all d-tuples (a1,… ,ad) and all r∈ \a2+… + ad+1, …, a1+a2+… + ad\. Moreover, we reduce the problem of determining this (coarse) threshold for all d≥ 3 and all r∈ \ad+1, …, a1+a2+… + ad\, to that of determining the threshold for all d≥ 3 and all r∈ \ ad+1, …, ad-1 + ad\.