Near-critical 2D percolation with heavy-tailed impurities, forest fires and frozen percolation

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 $$p_c$$ 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 $$\zeta $$ ζ , 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 $$\zeta $$ ζ ). For forests on boxes with such side lengths, the impact of fires does not vanish in the limit as $$\zeta \searrow 0$$ ζ ↘ 0 . This surprising behavior, related to the non-monotonicity of these processes, was not predicted in the physics literature.

Development of radar glacier zones on the King George Island ice cap, Antarctica, during austral summer 1996/97 as observed in ERS-2 SAR data

Based on a time series of European remote-sensing satellite (ERS-2) synthetic-aperture radar (SAR) images from 1996/97, ablation on the King George Island (Antarctica) ice cap is documented. Snowmelt patterns were monitored by mapping the dynamic evolution of radar glacier zones and their boundaries. On the ice cap, all major radar glacier zones except the dry-snow radar zone were identified during the observed period While winter was characterized by a frozen-percolation radar zone, the ablation season was characterized by wet-snow and bare-ice radar zones. A striking bright backscatter signature indicated the presence of a highly reflective zone in the lower parts of the wet-snow zone. It was attributed to a phase 2 melt (P2) radar zone, which is characterized by a metamorphosed and roughened surface of a melting snow cover. Due to the absence of simultaneously acquired ground-truth information, concurrent meteorological data proved to be essential for interpreting the SAR images. Although the maximum elevation of the ice cap does not exceed 680 ma.s.L, ablation patterns obviously reflect altitudinal control. Melt onset up to 530 m a.s.l. was initiated by an advection event at the end of October 1996. A wet snowpack on the entire ice cap corresponds with a prolonged period of high temperatures in January 1997. However, the highest parts of the ice cap were affected by occasional melt-freeze cycles. The transient snowline at the end of February was determined as being at 250 m a i l. This late-summer snowline was regarded as an approximation of the equilibrium-line altitude for the 1996/97 ablation season.