Abstract
We introduce a new percolation model on planar lattices. First, impurities (“holes”) are removed independently from the lattice. On the remaining part, we then consider site percolation with some parameter p close to the critical value p_c. The mentioned impurities are not only microscopic, but allowed to be mesoscopic (“heavy-tailed”, in some sense). For technical reasons (the proofs of our results use quite precise bounds on critical exponents in Bernoulli percolation), our study focuses on the triangular lattice. We determine explicitly the range of parameters in the distribution of impurities for which the connectivity properties of percolation remain of the same order as without impurities, for distances below a certain characteristic length. This generalizes a celebrated result by Kesten for classical near-critical percolation (which can be viewed as critical percolation with single-site impurities). New challenges arise from the potentially large impurities. This generalization, which is also of independent interest, turns out to be crucial to study 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 very small rate zeta , its entire occupied cluster burns immediately, so that all its vertices become vacant. Our results for percolation with impurities are instrumental in 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. This surprising behavior, related to the non-monotonicity of these processes, was not predicted in the physics literature.
Highlights
Even though forest fire processes attracted a lot of attention, very little is known about their long-time behavior
We describe in more detail the processes studied in, or relevant for, this paper: Bernoulli percolation, frozen percolation, and, one of the two main topics in this work, forest fire processes
Our main motivation in this paper comes from the forest fire without recovery (FFWoR) model, but we expect that the techniques on near-critical percolation with impurities that we develop are instrumental for obtaining new results about other related models
Summary
We describe in more detail the processes studied in, or relevant for, this paper: Bernoulli percolation, frozen percolation, and, one of the two main topics in this work, forest fire processes (the other one being percolation with “heavy-tailed” impurities as will be explained later). We studied versions of the following model in [10] and, together with Kiss, in [9], motivated by work by Aldous [2], who in turn was inspired by phenomena concerning sol-gel transitions [45] We will call it the N -volume-frozen percolation model, or, sometimes, parameter-N model. The role of the parameter N , i.e. that a cluster with size ≤ N cannot burn, Fig. 1 Final configuration (i.e. at time t = +∞) for the forest fires without recovery (FFWoR) process on Z2 with rate ζ = 0.01, in a box with side length 200. Apart from the fact that this replacement is too naive, the arguments become considerably more complicated, due to quite delicate problems concerning what we call “near-critical percolation with impurities”, as we heuristically indicate
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