Polluted Bootstrap Percolation in Three Dimensions

In the polluted bootstrap percolation model, vertices of the cubic lattice Z are independently declared initially occupied with probability p or closed with probability q. 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 p, q → 0. We show that this density converges to 1 if q p(log 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 q > Cp. For the standard model, we establish convergence to 0 under the stronger condition q > Cp.