Statistics Seminars: Forest fires and the continuum random tree
10 June 2013 14:00 in CM221In joint work with Nic Freeman, Christina Goldschmidt, James Martin and Bálint Tóth, we are studying the mean field forest fire model introduced by Ráth and Tóth in 2009. This model resembles the Erdős-Rényi graph process on a set of n vertices, but with the additional feature of lightning strikes which occur independently of the edge additions. Lightning strikes each vertex independently at times of a Poisson process with constant rate lambda. When lightning strikes a vertex, all edges in the connected component of that vertex are instantaneously deleted. We study the regime in which 1/n << lambda << 1, for which Ráth and Tóth showed that the model displays self-organized criticality in the limit as n tends to infinity. We propose a candidate for the Benjamini-Schramm limit of the stationary state of the finite model. That is, we ask what the forest fire process looks like from the point of view of a fixed vertex, and let n tend to infinity. The proof of convergence to our candidate limit is work in progress, so in this talk we will treat the candidate limit process as an interesting process in its own right. It is a stationary process supported on the set of finite rooted trees, and we can describe many aspects of its distribution precisely. In particular if we condition on the size of the tree being k, then let k tend to infinity, the resulting sequence of random trees has the Brownian continuum random tree as a scaling limit.
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