  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.
