In the polluted bootstrap percolation model, vertices of the cubic lattice are independently declared initially occupied with probability p or closed with probability q, where . 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 . We show that this density converges to 1 if 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 . For the standard model, we establish convergence to 0 under the stronger condition .
"Polluted bootstrap percolation in three dimensions." Ann. Appl. Probab. 31 (1) 218 - 246, February 2021. https://doi.org/10.1214/20-AAP1588