Polluted bootstrap percolation with threshold two in all dimensions

In the polluted bootstrap percolation model, the vertices of a graph are independently declared initially occupied with probability p or closed with probability q . At subsequent steps, a vertex becomes occupied if it is not closed and it has at least r occupied neighbors. On the cubic lattice ℤ^d of dimension d≥ 3 with threshold r=2 , we prove that the final density of occupied sites converges to 1 as p and q both approach 0, regardless of their relative scaling. Our result partially resolves a conjecture of Morris, and contrasts with the d=2 case, where Gravner and McDonald proved that the critical parameter is q/p^2 .