Self‐destructive percolation as a limit of forest‐fire models on regular rooted trees

Abstract

AbstractLet 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 forest‐fire 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 λ(n) > 0. When a vertex is hit by lightning, its occupied cluster becomes vacant instantaneously. Now suppose that λ(n) decays exponentially in n but much more slowly than 1/|Tn|, where |Tn| denotes the number of vertices of Tn. We show that then there exist such that between time 0 and time the forest‐fire 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 τ vertices become occupied at rate 1, independently for all vertices. Immediately before time τ there are infinitely many infinite occupied clusters. At time τ all these clusters become vacant. Between time τ and time vertices again become occupied at rate 1, independently for all vertices. At time all occupied clusters are finite. This process is a dynamic version of self‐destructive percolation. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 86–113, 2017