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 \({\mathbb {Z}}^d\) of dimension \(d\ge 3\) wi...More