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Consider critical site percolation on a "nice" planar lattice: each vertex isoccupied 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 $p_c$ isreplaced by $p_c - \varepsilon$, for some small $\varepsilon > 0$. A celebratedresult by Kesten says, informally speaking, that on scales below thecharacteristic length $L(p_c - \varepsilon)$, the connection probabilitiesremain of the same order as before. We prove a substantial and subtlegeneralization to the case where the impurities are not only microscopic, butallowed to be "mesoscopic".  This generalization, which is also interesting in itself, was motivated byour study of models of forest fires (or epidemics). In these models, allvertices are initially vacant, and then become occupied at rate $1$. If anoccupied vertex is hit by lightning, which occurs at a (typically very small)rate $\zeta$, its entire occupied cluster burns immediately, so that all itsvertices become vacant.  Our results for percolation with impurities turn out to be crucial foranalyzing the behavior of these forest fire models near and beyond the criticaltime (i.e. the time after which, in a forest without fires, an infinite clusterof trees emerges). In particular, we prove (so far, for the case when burnttrees do not recover) the existence of a sequence of "exceptional scales"(functions of $\zeta$). For forests on boxes with such side lengths, the impactof fires does not vanish in the limit as $\zeta \searrow 0$.

Near-critical percolation with heavy-tailed impurities, forest fires and frozen percolation