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In ongoing work with Erin Russell, we are studying a directed version of the mean field forest fire model that was introduced by Balazs Rath and Balint Toth in 2009. This is not a realistic model of real-world forest fires! Instead, it is a mathematically tractable dynamic random graph model that displays self-organized criticality.
In the first half of the talk, we will explain a coupling between the directed and undirected versions of this forest fire model, and some consequences. For example, for suitable initial conditions, we expect that the out-graph of a tagged vertex in the directed model is approximated by a critical multitype Bienayme-Galton-Watson (BGW) tree. This tree evolves in time by growth events and by pruning events whose rate at each vertex is type-dependent.
In the second half of the talk, we will look at some simpler dynamic random tree processes of a similar flavour, which have a single-type BGW tree as their stationary law. For example, consider a critical binary BGW tree whose vertices carry independent exponential alarm clocks of rate 1. When the clock at any vertex v rings at time t, the entire subtree above v is instantaneously resampled. This means that it is replaced with a new critical binary BGW tree rooted at v, which is independent of the history before time t. We show that a.s. there are no exceptional times at which this dynamic critical tree is infinite.
