Two-dimensional forest fires with boundary ignitions

• 成果类型: Article
• 署名作者: Van Den Berg, Jacob; Nolin, Pierre
• 署名单位: Centrum Wiskunde & Informatica (CWI); City University of Hong Kong
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-025-01407-3
• 发表日期: 2026-06
• 页码: 521-546
• 关键词: near-critical percolation; Forest fires; self-organized criticality; critical percolation

摘要

In the classical Drossel-Schwabl forest fire process, vertices of a lattice become occupied at rate 1, and they are hit by lightning at some tiny rate zeta > 0, which causes entire connected components to burn. In this paper, we study a variant where fires are coming from the boundary of the forest instead. In particular we prove for every positive zeta (including zeta = infinity) that, for the forest fire process without recoveries on an N x N box in the triangular lattice, where each point on the boundary of the box has ignition rate zeta, the probability that the center of the box gets burnt tends to 0 as N -> infinity (but substantially slower than the one-arm probability of critical Bernoulli percolation). And, for the case where the forest is the upper-half plane, we show (still for the version without recoveries) that no infinite occupied cluster emerges. We also discuss analogs of some of these results for the corresponding models with recoveries, and explain how our results and proofs give valuable insight on a process considered earlier by Graf (Electron J Probab 19:8, 2014), (Electron Commun Probab 21:39, 2016).