We introduce a new percolation model on planar lattices. First, impurities (“holes”) are removed independently from the lattice. On the remaining part, we then consider site percolation with some parameter p close to the critical value \(p_c\). The mentioned impurities are not only microscopic, but allowed to be mesoscopic (“heavy-tailed”, in some sense). For technical reasons (the proofs of our results use quite precise bounds on critical exponents in Bernoulli percolation), our study focuses on the triangular lattice. We determine explicitly the range of parameters in the distribution of impurities for which the connectivity properties of percolation remain of the same order as without impurities, for distances below a certain characteristic length. This generalizes a celebrated result by Kesten for classical near-critical percolation (which can be viewed as critical percolation with single-site impurities). New challenges arise from the potentially large impurities. This generalization, which is also of independent interest, turns out to be crucial to study 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 very small rate \(\zeta \), its entire occupied cluster burns immediately, so that all its vertices become vacant. Our results for percolation with impurities are instrumental in 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 \(\zeta \searrow 0\). This surprising behavior, related to the non-monotonicity of these processes, was not predicted in the physics literature.

我们在平面晶格上引入了一个新的渗流模型。首先，独立于晶格去除杂质（“孔”）。在其余部分，我们然后考虑一些参数p接近临界值\（p_c \）的站点渗滤。。提及的杂质不仅是微观的，而且是介观的（在某种意义上为“重尾”）。由于技术原因（我们的结果的证明对伯努利渗滤中的临界指数使用了非常精确的界线），因此我们的研究集中在三角晶格上。我们明确确定了杂质分布中参数的范围，对于这些参数，渗滤的连通性保持与不存在杂质时相同的数量级，且距离小于特定特征长度。这概括了Kesten对于经典近临界渗滤（可以看作是单中心杂质的临界渗滤）的著名结果。潜在的大量杂质带来了新的挑战。这种概括也具有独立意义，事实证明，这对于研究森林火灾（或流行病）模型至关重要。在这些模型中，所有顶点最初都是空的，然后以1的速率被占据。\（\ zeta \），其整个被占用的簇会立即燃烧，因此其所有顶点都将变为空置。我们的杂质渗滤结果有助于分析这些森林火灾模型在关键时刻附近和之后的行为（例如，在此之后的时间里，在没有火灾的森林中，会出现无数的树木丛集）。特别是，我们证明了（到目前为止，对于被烧毁的树木无法恢复的情况）存在一系列“异常尺度”（\（\ zeta \）的函数）。对于边长这样的盒子上的森林，火灾的影响不会消失为\（\ zeta \ searrow 0 \）。与这些过程的非单调性有关的这种令人惊讶的行为，在物理学文献中并未预见到。