Near-critical percolation with heavy-tailed impurities, forest fires and frozen percolation

Abstract

Consider critical site percolation on a "nice" planar lattice: each vertex is occupied with probability $p = p_c$, and vacant with probability $1 - p_c$. Now, suppose that additional vacancies ("holes", or "impurities") are created, independently, with some small probability, i.e. the parameter $p_c$ is replaced by $p_c - \varepsilon$, for some small $\varepsilon > 0$. A celebrated result by Kesten says, informally speaking, that on scales below the characteristic length $L(p_c - \varepsilon)$, the connection probabilities remain of the same order as before. We prove a substantial and subtle generalization to the case where the impurities are not only microscopic, but allowed to be "mesoscopic". This generalization, which is also interesting in itself, was motivated by our study of models of forest fires (or epidemics). In these models, 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 $\zeta$, its entire occupied cluster burns immediately, so that all its vertices become vacant. Our results for percolation with impurities turn out to be crucial for analyzing the behavior of these forest fire models near and beyond the critical time (i.e. the time after which, in a forest without fires, an infinite cluster of trees emerges). In particular, we prove (so far, for the case when burnt trees do not recover) the existence of a sequence of "exceptional scales" (functions of $\zeta$). For forests on boxes with such side lengths, the impact of fires does not vanish in the limit as $\zeta \searrow 0$.