A 2D forest fire process beyond the critical time

TL;DR

This paper analyzes a 2D forest fire process near and beyond the critical time, revealing that fires are unlikely to occur immediately after the critical point due to the emergence of resilient fire lanes.

Contribution

It provides a detailed asymptotic description of the forest fire process beyond criticality, answering an open problem and extending previous methods to account for recoveries.

Findings

Fires do not occur immediately after the critical time with high probability.

Emergence of fire lanes with negligible density but significant robustness.

Recoveries have negligible macroscopic impact as the threshold N increases.

Abstract

We study forest fire processes in two dimensions. On a given planar lattice, vertices independently switch from vacant to occupied at rate $1$ (initially they are all vacant), and any connected component "is burnt" (its vertices become instantaneously vacant) as soon as its cardinality crosses a (typically large) threshold $N$, the parameter of the model. Our analysis provides a detailed description, as $N \to \infty$, of the process near and beyond the critical time $t_c$ (at which an infinite cluster would arise in the absence of fires). In particular we prove a somewhat counterintuitive result: there exists $\delta > 0$ such that with high probability, the origin does not burn before time $t_c + \delta$. This provides a negative answer to Open Problem 4.1 of van den Berg and Brouwer [Comm. Math. Phys., 2006]. Informally speaking, the result can be explained in terms of the emergence of fire lanes, whose total density is negligible (as $N \to \infty$), but which nevertheless are sufficiently robust with respect to recoveries. We expect that such a behavior also holds for the classical Drossel-Schwabl model. A large part of this paper is devoted to analyzing recoveries during the interval $[t_c, t_c + \delta]$. These recoveries do have a "microscopic" effect, but it turns out that their combined influence on macroscopic scales (and in fact on relevant "mesoscopic" scales) vanishes as $N \to \infty$. In order to prove this, we use key ideas of Kiss, Manolescu and Sidoravicius [Ann. Probab., 2015], introducing a suitable induction argument to extend and strengthen their results. We then use it to prove that a deconcentration result in our earlier joint work with Kiss on volume-frozen percolation also holds for the forest fire process. As we explain, significant additional difficulties arise here, since recoveries destroy the nice spatial Markov property of frozen percolation.