2014/04/01 by Graf, Robert · 1 citation
#60K35 #82C22 (Primary) 82B43 (Secondary) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1404.0325
Let T be a regular rooted tree. For every natural number n, let Bn be the finite subtree of vertices with graph distance at most n from the root. Consider the following forest-fire model on Bn: 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 instantaneously becomes vacant. Now suppose that λ(n) decays exponentially in n but much more slowly than 1/|Bn|. We show that then there exist a supercritical time τ and ε> 0 such that the forest-fire model on Bn between time 0 and time τ+ ε 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. At time τ all infinite occupied 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.