Deconcentration Property for Near-Critical Forest Fire Processes

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 ζ > 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: 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 time tc(when an infinite component of trees would arise in the absence of ignitions).In this proposal, we want to further analyze the successive fires around a given vertex, that occur in a near-critical interval just after tc (with length tending to 0 as a small power of ζ). More precisely, we obtained recently a surprising deconcentration result for a related model, known as volume-frozen percolation. We want to adapt this result to forest fire processes, which would provide a very accurate description of their qualitative behavior right after tc. We hope to achieve this by considering successively several versions of forest fire processes, in order to isolate the potential difficulties.