Graf, Robert
(2017):
SelfDestructive Percolation as a Limit of ForestFire Models on Regular Rooted Trees.
In: Random Structures & Algorithms, Vol. 50, No. 1: pp. 86113

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Abstract
Let T be a regular rooted tree. For every natural number n, let Tn be the finite subtree of vertices with graph distance at most n from the root. Consider the following forestfire model on Tn: Each vertex can be "vacant" or "occupied". At time 0 all vertices are vacant. Then the process is governed by two opposing mechanisms: Vertices become occupied at rate 1, independently for all vertices. Independently thereof and independently for all vertices, "lightning" hits vertices at rate lambda(n) > 0. When a vertex is hit by lightning, its occupied cluster becomes vacant instantaneously. Now suppose that lambda(n) decays exponentially in n but much more slowly than 1/vertical bar Tn vertical bar, where vertical bar Tn vertical bar denotes the number of vertices of Tn. We show that then there exist tau, epsilon > 0 such that between time 0 and time tau + epsilon the forestfire model on Tn tends to the following process on T as n goes to infinity: At time 0 all vertices are vacant. Between time 0 and time t vertices become occupied at rate 1, independently for all vertices. Immediately before time t there are infinitely many infinite occupied clusters. At time tau all these clusters become vacant. Between time tau and time tau + epsilon vertices again become occupied at rate 1, independently for all vertices. At time tau + epsilon all occupied clusters are finite. This process is a dynamic version of selfdestructive percolation. (C) 2016 WileyBlackwell Periodicals, Inc.