A Probabilistic Model for Forest Fires

TL;DR

The paper develops a rigorous probabilistic framework for a two-dimensional, discrete forest-fire process where each site's ignition depends on its left and upper neighbors. By defining the binary field X and the anti-diagonal sums Y_n, it derives conditional expectations, martingale structures, and moment bounds, and analyzes limiting behavior across regimes determined by alpha+beta and gamma, including a borderline case. Key results show extinction of the fire (Y_n -> 0 a.s.) in certain parameter ranges, while the case alpha+beta>1 with gamma=1 yields substantial fire activity with quantified bounds on zero-fire probability and expected burn counts; an open question remains on the distributional convergence of Y_n/n. The work also connects the 2D process to a simpler one-dimensional appendix, and outlines natural extensions to more ignition sources, more states, higher dimensions, or a continuous analogue, highlighting the model's rich phase-transition-like behavior and potential practical insights for wildfire dynamics.

Abstract

We propose a discrete two-dimensional mathematical model for forest fires and we derive certain results describing its limiting behavior. We also pose a relevant open question.