Bootstrap percolation in three dimensions

By bootstrap percolation we mean the following deterministic process on a graph $G$. Given a set $A$ of vertices “infected” at time 0, new vertices are subsequently infected, at each time step, if they have at least $r\in \mathbb{N}$ previously infected neighbors. When the set $A$ is chosen at random, the main aim is to determine the critical probability $p_c(G,r)$ at which percolation (infection of the entire graph) becomes likely to occur. This bootstrap process has been extensively studied on the $d$-dimensional grid $[n{]}^d$: with $2\le r\le d$ fixed, it was proved by Cerf and Cirillo (for $d=r=3$), and by Cerf and Manzo (in general), that $$p_c([n{]}^d,r)=\mathrm{\Theta }(\frac{1}{{\log }_{(r-1)}n}{)}^{d-r+1},$$ where ${\log }_{(r)}$ is an $r$-times iterated logarithm. However, the exact threshold function is only known in the case $d=r=2$, where it was shown by Holroyd to be $(1+o(1))\frac{{\pi }^2}{18\log n}$. In this paper we shall determine the exact threshold in the crucial case $d=r=3$, and lay the groundwork for solving the problem for all fixed $d$ and $r$.