Self‐destructive percolation as a limit of forest‐fire models on regular rooted trees1. Issue 1 (7th March 2016)

• Record Type: Journal Article
• Authors: Graf, Robert
• Is Part Of: Random structures & algorithms. Volume 50:Issue 1(2017)
• Page Start: 86
• Page End: 113
• Publication Date: 2016-03-07
• Subjects: forest‐fire model -- self‐destructive percolation -- regular rooted tree; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20646
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract: Let T be a regular rooted tree. For every natural number n, let T n be the finite subtree of vertices with graph distance at most n from the root. Consider the following forest‐fire model on T n : 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/| T n |, where | T n | denotes the number of vertices of T n . We show that then there exist τ, ϵ > 0 such that between time 0 and time τ + ϵ the forest‐fire model on T n 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 τ + ϵ vertices again become occupied at rate 1, independently for all vertices. At time τ + ϵ 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