The [formula omitted]-dimensional bootstrap percolation models with axial neighbourhoods

Abstract

Fix positive integers d,r and a1≤a2≤⋯≤ad. For large L, each site of {1,…,L}d⊂Zd can be at state 0 or 1 (infected), and its neighbourhood consists of the ak nearest neighbours in the ±ek-directions for each k∈{1,2,…,d}. The state will evolve in discrete time as follows: At time 0, vertices are independently 1 with some probability p. We infect any vertex v∈{1,…,L}d at state 0 already having r infected neighbours, and infected sites remain infected forever.In this paper we study the critical length for percolation, defined by Lc(Nra1,…,ad,p)=min{L∈N:Pp({1,…,L}dis eventually infected)≥1/2}. We determine the (d−1)-times iterated logarithm of Lc(Nra1,…,ad,p) up to a constant factor, for all d-tuples (a1,…,ad) and all r∈{a2+⋯+ad+1,…,a1+a2+⋯+ad}.We conjecture that we can reduce the problem of determining this threshold for all d≥3 and all r∈{ad+1,…,a1+a2+⋯+ad}, to that of determining the threshold for all d≥3 and r∈{ad+1,…,ad−1+ad} only.