Polluted bootstrap percolation in three dimensions

In the polluted bootstrap percolation model, vertices of cubic lattice Z3 are independently declared initially occupied with probability p or closed q, where p+q≤1. Under standard (respectively, modified) rule, a vertex becomes at subsequent step if it is not and has least 3 neighbors an neighbor in each coordinate). We study final density as p,q→0. show that this converges to 1 q≪p3(logp−1)−3 for both modified rules. Our principal result complementary bound matching power model: there exists C such 0 q>Cp3. For we establish convergence under stronger condition q>Cp2.