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

TL;DR

This paper analyzes a forest-fire model on the half-plane lattice, showing that only finitely many sites within any non-horizontal cone are affected by destruction at the critical time.

Contribution

It establishes that at the critical time, the destruction impact remains localized within any such cone in the half-plane.

Findings

Finitely many sites affected in any cone at critical time

Destruction does not percolate infinitely in the cone

Model behavior at criticality is spatially localized

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.