Self-Destructive Percolation as a Limit of Forest-Fire Models on Regular Rooted Trees

Graf, Robert (2017): Self-Destructive Percolation as a Limit of Forest-Fire Models on Regular Rooted Trees. In: Random Structures & Algorithms, Bd. 50, Nr. 1: S. 86-113

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 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 T-n vertical bar, where vertical bar T-n vertical bar denotes the number of vertices of T-n. We show that then there exist tau, epsilon > 0 such that between time 0 and time tau + epsilon 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 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 self-destructive percolation. (C) 2016 Wiley-Blackwell Periodicals, Inc.

• Dokumententyp: Zeitschriftenartikel
• Fakultät: Mathematik, Informatik und Statistik > Mathematik
• Themengebiete: 500 Naturwissenschaften und Mathematik > 510 Mathematik
• ISSN: 1042-9832
• Sprache: Englisch