Consider a “nice” planar lattice, such as the square or the triangular lattice. We introduce the following percolation model. First, regions (“impurities”) are removed from the lattice, in some independent fashion, and we then consider site percolation on the remaining vertices. The mentioned impurities are not only microscopic, but also allowed to be mesoscopic (“heavy-tailed”, in some sense).
We are typically interested in whether, on the randomly “perforated” lattice, the connectivity properties of percolation remain of the same order as without impurities, for values of the percolation parameter close to the critical value. This generalizes a celebrated result by Kesten for near-critical percolation (that can be viewed as critical percolation with single-site impurities).
This generalization arises naturally when studying models of forest fires (or epidemics). Our results for percolation with impurities are instrumental in analyzing the behavior of such processes near and beyond the critical time (i.e. the time after which, in the absence of fires, infinite connected components would emerge).
This talk is based on a joint work with Rob van den Berg (CWI and VU, Amsterdam).