  We study two closely related processes on the triangular lattice: frozen
percolation, where connected components of occupied vertices freeze (they stop
growing) as soon as they contain at least $N$ vertices, and forest fire
processes, where connected components burn (they become entirely vacant) at
rate $\zeta > 0$. In this paper, we prove that when the density of occupied
sites approaches the critical threshold for Bernoulli percolation, both
processes display a striking phenomenon: the appearance of near-critical
"avalanches".
  More specifically, we analyze the avalanches, all the way up to the natural
characteristic scale of each model, which constitutes an important step toward
understanding the self-organized critical behavior of such processes. For
frozen percolation, we show in particular that the number of frozen clusters
surrounding a given vertex is asymptotically equivalent to $(\log(96/5))^{-1}
\log \log N$ as $N \to \infty$. A similar mechanism underlies forest fires,
enabling us to obtain an analogous result for these processes, but with
substantially more work: the number of burnt clusters is equivalent to
$(\log(96/41))^{-1} \log \log (\zeta^{-1})$ as $\zeta \searrow 0$. Moreover,
almost all of these clusters have a volume $\zeta^{- 91/55 + o(1)}$.
  For forest fires, the percolation process with impurities introduced in
arXiv:1810.08181 plays a crucial role in our proofs, and we extend the results
in that paper, up to a positive density of impurities. In addition, we develop
a novel exploration procedure to couple full-plane forest fires with processes
in finite but large enough (compared to the characteristic scale) domains.
