Forest fire processes and near-critical percolation with heavy-tailed impurities
We discuss models of forest fires (or epidemics): on a given planar lattice, all vertices are initially vacant, and then become occupied at rate 1. If an occupied vertex is hit by lightning, which occurs at a (typically very small) rate, all the vertices connected to it "burn" instantaneously, i.e. they become vacant. We want to analyze the behavior of such processes near and beyond the critical time (the time after which, in the absence of fires, infinite connected components would emerge).
We are led to introduce a percolation model where regions ("impurities") are removed from the lattice, in an independent fashion. These impurities are not only microscopic, but also allowed to be mesoscopic. We are interested in whether 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 talk is based on a joint work with Rob van den Berg (CWI and VU, Amsterdam).