The time of bootstrap percolation with dense initial sets

Abstract

Let $r\in\mathbb{N}$. In $r$-neighbour bootstrap percolation on the vertex set of a graph $G$, vertices are initially infected independently with some probability $p$. At each time step, the infected set expands by infecting all uninfected vertices that have at least $r$ infected neighbours. When $p$ is close to 1, we study the distribution of the time at which all vertices become infected. Given $t=t(n)=o(\log n/\log\log n)$, we prove a sharp threshold result for the probability that percolation occurs by time $t$ in $d$-neighbour bootstrap percolation on the $d$-dimensional discrete torus $\mathbb{T}_{n}^{d}$. Moreover, we show that for certain ranges of $p=p(n)$, the time at which percolation occurs is concentrated either on a single value or on two consecutive values. We also prove corresponding results for the modified $d$-neighbour rule.