The d-dimensional bootstrap percolation models with axial neighbourhoods

Fix positive integers $d,r$ and $a_1\le a_2\le \cdots \le a_d$. For large $L$, each site of ${\{1,\dots ,L\}}^d\subset Z^d$ can be at state 0 or 1 (infected), and its neighbourhood consists of the $a_k$ nearest neighbours in the $\pm e_k$-directions for each $k\in \{1,2,\dots ,d\}$. The state will evolve in discrete time as follows: At time 0, vertices are independently 1 with some probability $p$. We infect any vertex $v\in {\{1,\dots ,L\}}^d$ 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 $L_c(N_r^{a_1,\dots ,a_d},p)=min\{L\in N:P_p\left({\{1,\dots ,L\}}^d\text{is eventually infected}\right)\ge 1/2\}$. We determine the $(d-1)$-times iterated logarithm of $L_c(N_r^{a_1,\dots ,a_d},p)$ up to a constant factor, for all $d$-tuples $(a_1,\dots ,a_d)$ and all $r\in \{a_2+\cdots +a_d+1,\dots ,a_1+a_2+\cdots +a_d\}$.

We conjecture that we can reduce the problem of determining this threshold for all $d\ge 3$ and all $r\in \{a_d+1,\dots ,a_1+a_2+\cdots +a_d\}$, to that of determining the threshold for all $d\ge 3$ and $r\in \{a_d+1,\dots ,a_{d-1}+a_d\}$ only.