Abstract
We study the distribution of ages in the mean field forest fire model introduced by Rath and Toth. This model is an evolving random graph whose dynamics combine ErdosRenyi edgeaddition with a Poisson rain of lightning strikes. All edges in a connected component are deleted when any of its vertices is struck by lightning. We consider the asymptotic regime of lightning rates for which the model displays selforganized criticality. The age of a vertex increases at unit rate, but it is reset to zero at each burning time. We show that the empirical age distribution converges as a process to a deterministic solution of an autonomous measurevalued differential equation. The main technique is to observe that conditioned on the vertex ages the graph is an inhomogeneous random graph in the sense of Bollobas, Janson and Riordan. We then study the evolution of the ages via the multitype GaltonWatson trees that arise as the limit in law of the component of an identified vertex at any fixed time. These trees are critical from the gelation time onwards.
Original language  English 

Number of pages  46 
Journal  Annals of Probability 
Publication status  Accepted/In press  10 Dec 2020 
Keywords
 inhomogeneous random graph
 multitype branching process
 selforganized criticality
 Perron–Frobenius theory
 differential equations
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Profiles

Dr Edward T Crane
 School of Mathematics  Heilbronn Senior Research Fellow
 Probability, Analysis and Dynamics
 Heilbronn Institute for Mathematical Research
 Pure Mathematics
Person: Academic , Member