Recoveries and Avalanches in Forest Fires and Related Models

Project: Research

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In 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 paradigmaticexample 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 > 0 (typically very small): when a tree is hit, it burns and fire spreads instantaneously to its neighbors, so that thewhole connected component of trees disappears immediately. Even though 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. Indeed, as the density of trees increases, large connected components arise, helping fires to spread. When such large connected components burn, they create “fire lines”, preventing (for some time) new largescale connections from appearing. Because of this “non-monotonicity”, standard tools formodels on lattices cannot be used, and specific techniques and ideas must be developed. 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 power law in the ignition rate . For that, we were able to understand precisely the cumulative effect of fires up to the critical time tc (when an infinite component of trees would arise in the absence of ignitions). We want to use this understanding to control the effect of recovering trees after time tc, that is, when the forest has been strongly disconnected by fires: how much time does it take for new large connected components to arise? In a first step, we will study a related process called self-destructive percolation. For this process, some important properties are already known, and we plan to improve them in several directions. Forest fires are a paradigmatic example, and we expect our work to be applicable to other models with a similar flavor, in particular epidemics models. We also plan to analyze the near-critical avalanches uncovered in a recent work: sequences of burnt clusters surrounding any given vertex. We want to improve our understanding of the initial stages of such avalanches, to describe how randomness spreadsdown to smaller scales. This should provide valuable information on the region where a given vertex lies slightly after the critical time.  


Project number9043413
Grant typeGRF
Effective start/end date1/01/23 → …