Polluted bootstrap percolation in three dimensions

Abstract

In the polluted bootstrap percolation model, vertices of the cubic lattice ${\mathbb{Z}}^3$ are independently declared initially occupied with probability p or closed with probability q, where $\mathit{p}+\mathit{q}\le 1$. Under the standard (respectively, modified) bootstrap rule, a vertex becomes occupied at a subsequent step if it is not closed and it has at least 3 occupied neighbors (respectively, an occupied neighbor in each coordinate). We study the final density of occupied vertices as $\mathit{p},\mathit{q}\to 0$. We show that this density converges to 1 if $\mathit{q}\ll {\mathit{p}}^3{(log{\mathit{p}}^{-1})}^{-3}$ for both standard and modified rules. Our principal result is a complementary bound with a matching power for the modified model: there exists C such that the final density converges to 0 if $\mathit{q}>\mathit{C}{\mathit{p}}^3$. For the standard model, we establish convergence to 0 under the stronger condition $\mathit{q}>\mathit{C}{\mathit{p}}^2$.