Forest Fire Processes After the Critical Time
DescriptionIn statistical physics, self-organized criticality is a fascinating phenomenon that can be used to explain the emergence of complexity in nature. Forest fire processes, a model of excitable media introduced by Drossel and Schwabl in 1992, provide a paradigmatic example where this phenomenon arises. In such a process, new trees arrive on a lattice with rate 1, while lightning hits the lattice with some rate ζ > 0 (typically very small): when a tree is hit, it burns and fire spreads instantaneously to its neighbors, so that the whole connected component of trees disappears immediately.Even if forest fire processes attracted a lot of attention, little is known about their long-time behavior. They are notoriously difficult to study, due to the existence of competing effects on the connectivity of the forest: as soon as large components of trees arise, they create lasting “scars” on the lattice when they burn. Because of this non-monotonicity, standard tools from lattice models cannot be used, so that specific techniques and ideas are required. The goal of this research proposal is to improve the mathematical understanding of such processes, based on groundbreaking ideas developed for near-critical percolation and related processes. We recently proved new results about forest fire processes run in finite boxes, whose side length is a suitable function of the rate ζ. For that, we were able to understand precisely the cumulative effect of fires up to the critical timetc(when an infinite component of trees would arise in the absence of ignitions).In this proposal, we want in particular to analyze near-critical “avalanches” around a given vertex, i.e. the successive fires occurring in a near-critical interval just aftertc, with length tending to 0 as a small power of ζ (we expect of order log log(1/ζ ) such fires). We also want to understand the effect of “boundary rules” for forest fire processes, as well as for the related model known as frozen percolation. In some cases, we already know that such rules can significantly perturb the macroscopic behavior of the process, even if they do not look so consequential at first sight. Understanding their effect, which is also interesting in itself, should be particularly important when trying to follow more closely the near-critical dynamics of forest fires.
|Effective start/end date||1/01/20 → …|