The $d$-dimensional bootstrap percolation models with threshold at least double exponential

Abstract

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 $a_i$ nearest neighbours in the $\pm e_i$-directions for each $i \in \{1,2,\dots, d\}$, where $a_1\le a_2\le \dots \le a_d$. Suppose we infect any healthy vertex $v\in [L]^d$ 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 $(a_1,\dots ,a_d)$ and all $r\in \{a_2+\dots + a_d+1, \dots, a_1+a_2+\dots + a_d\}$. Moreover, we reduce the problem of determining this (coarse) threshold for all $d\ge 3$ and all $r\in \{a_d+1, \dots, a_1+a_2+\dots + a_d\}$, to that of determining the threshold for all $d\ge 3$ and all $r\in \{ a_d+1, \dots, a_{d-1} + a_d\}$.