A 2D forest fire process beyond the critical time
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→∞, of the process near and beyond the critical time tc (at which an infinite cluster would arise in the absence of fires). In particular we prove a somewhat counterintuitive result: there exists δ>0 such that with high probability, the origin does not burn before time tc+δ. 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→∞), 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 [tc,tc+δ]. 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→∞. 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.