Generalizations of forest fires with ignition at origin

TL;DR

This paper explores various generalizations of the forest fire model, including spatially dependent growth rates and long-range burning, establishing bounds on the time to reach distant sites.

Contribution

It introduces new variants of the forest fire model with location-dependent growth and long-range effects, providing bounds on propagation times.

Findings

Time to reach site x is at most logarithmic in x with a specific exponent.

Models exhibit similar propagation bounds despite different generalizations.

Results extend understanding of critical behavior in forest fire models.

Abstract

We study generalizations of the Forest Fire model introduced in [van den Berg, J., and J\'arai, A. A. "On the asymptotic density in a one-dimensional self-organized critical forest-fire model". Comm. Math. Phys. 253 (2005)] and [Volkov, Stanislav. "Forest fires on $\mathbb{Z}_+$ with ignition only at 0". ALEA 6 (2009)] by allowing the rates at which the tree grow to depend on their location, introducing long-range burning, as well as continuous-space generalization of the model. We establish that in all the models in consideration the time required to reach site at distance $x$ from the origin is of order at most $(\log x)^{(\log 2)^{-1}+\delta}$ for any $\delta>0$.