Open Access
2016 Critical heights of destruction for a forest-fire model on the half-plane
Robert Graf
Electron. Commun. Probab. 21: 1-10 (2016). DOI: 10.1214/16-ECP4786


Consider the following forest-fire model on the upper half-plane of the triangular lattice: Each site can be “vacant” or “occupied by a tree”. At time $0$ all sites are vacant. Then the process is governed by the following random dynamics: Trees grow at rate $1$, independently for all sites. If an occupied cluster reaches the boundary of the half-plane or if it is about to become infinite, the cluster is instantaneously destroyed, i.e. all of its sites turn vacant.

Let $t_c = \log 2$ denote the critical time after which an infinite cluster first appears in the corresponding pure growth process, where there is only the growth of trees but no destruction mechanism. Choose an arbitrary infinite cone in the half-plane whose apex lies on the boundary of the half-plane and whose boundary lines are non-horizontal. We prove that at time $t_c$ almost surely only finitely many sites inside the cone have been affected by destruction in the forest-fire process.


Download Citation

Robert Graf. "Critical heights of destruction for a forest-fire model on the half-plane." Electron. Commun. Probab. 21 1 - 10, 2016.


Received: 29 December 2015; Accepted: 31 March 2016; Published: 2016
First available in Project Euclid: 10 May 2016

zbMATH: 1336.60188
MathSciNet: MR3510247
Digital Object Identifier: 10.1214/16-ECP4786

Primary: 60K35 , 82C22
Secondary: 82B43

Keywords: forest-fire model , half-plane , Self-organized criticality

Back to Top