  Consider critical site percolation on a "nice" planar lattice: each vertex is
occupied with probability $p = p_c$, and vacant with probability $1 - p_c$.
Now, suppose that additional vacancies ("holes", or "impurities") are created,
independently, with some small probability, i.e. the parameter $p_c$ is
replaced by $p_c - \varepsilon$, for some small $\varepsilon > 0$. A celebrated
result by Kesten says, informally speaking, that on scales below the
characteristic length $L(p_c - \varepsilon)$, the connection probabilities
remain of the same order as before. We prove a substantial and subtle
generalization to the case where the impurities are not only microscopic, but
allowed to be "mesoscopic".
  This generalization, which is also interesting in itself, was motivated by
our study of 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 (typically very small)
rate $\zeta$, its entire occupied cluster burns immediately, so that all its
vertices become vacant.
  Our results for percolation with impurities turn out to be crucial for
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 $\zeta \searrow 0$.
