Critical heights of destruction for a forest-fire model on the half-plane

2014/06/08 by Graf, Robert
#60K35 #82C22 (Primary) 82B43 (Secondary) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1406.1992

Abstract

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 upper half-plane, the cluster is instantaneously destroyed, i.e. all of its sites turn vacant. At the critical time t_c := log 2 the process is stopped. Now 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 in the final configuration a.s. only finitely many sites in the cone have been affected by destruction.

Related

• Two-dimensional forest fires with boundary ignitions
• Self-Organized Forest-Fires near the Critical Time
• A 2D forest fire process beyond the critical time
• One-dimensional general forest fire processes
• Self-destructive percolation as a limit of forest-fire models on regular rooted trees