The time of bootstrap percolation with dense initial sets for all   thresholds
  We study the percolation time of the $r$-neighbour bootstrap percolation model on the discrete torus $(\Z/n\Z)^d$. For $t$ at most a polylog function of $n$ and initial infection probabilities within certain ranges depending on $t$, we prove that the percolation time of a random subset of the torus is exactly equal to $t$ with high probability as $n$ tends to infinity. Our proof rests crucially on three new extremal theorems that together establish an almost complete understanding of the geometric behaviour of the $r$-neighbour bootstrap process in the dense setting. The special case $d-r=0$ of our result was proved recently by Bollob\'as, Holmgren, Smith and Uzzell. 
B\'ela Bollob\'as, Paul Smith, and Andrew J. Uzzell
math.PR math.CO