Polluted Modified Bootstrap Percolation

In the polluted modified bootstrap percolation model, sites in the square lattice are independently initially occupied with probability p or closed with probability q. A site becomes occupiedat a subsequent step if it is not closed and has at least one occupied nearest neighbor in each ofthe two coordinates. We study the final density of occupied sites when p and q are both small.We show that this density approaches 0 if q ≥ Cp2/ log p−1 and 1 if q ≤ p2/(log p−1)1+o(1). Thuswe establish a logarithmic correction in the critical scaling, which is known not to be present inthe standard model, settling a conjecture of Gravner and McDonald from 1997.