Near-critical Frozen Percolation and Forest Fires

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 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 ratez> 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. Due to this non-monotonicity, new techniques and ideas are required. Indeed, standard tools from statistical mechanics for models on lattices cannot be used.The goal of this research proposal is to improve the mathematical understanding of such models, based on groundbreaking ideas developed recently for near-critical percolation and related processes. In particular, we want to prove the existence of exceptional scales for forest fire processes in finite boxes, as well as a related Poisson frozen percolation model, in boxes whose side length is a suitable function of the ratez. Doing so would require a precise understanding of the cumulative effect of fires up to the critical timetc, when an infinite component of trees would arise if the lightning process was not present. This would emphasize the robustness of the behavior established recently with van den Berg for volume-frozen percolation.