We study the distribution of ages in the mean field forest fire model introduced by Ráth and Tóth. This model is an evolving random graph whose dynamics combine Erdős–Rényi edge-addition 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 self-organized 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 measure-valued 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 Bollobás, Janson and Riordan. We then study the evolution of the ages via the multitype Galton–Watson 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.
The first author was supported by the Heilbronn Institute for Mathematical Research. The second author was partially supported by Postdoctoral Fellowship NKFI-PD-121165 and grant NKFI-FK-123962 of NKFI (National Research, Development and Innovation Office), the Bolyai Research Scholarship of the Hungarian Academy of Sciences, the ÚNKP-18-4-BME-124 New National Excellence Program of the Ministry of Human Capacities and the ERC Synergy under Grant No. 810115-DYNASNET. The third author was supported by EPSRC doctoral training grant EP/K503113, the German-Israeli Foundation of Scientific Research and Development Grant I-2494-304.6/2017, by ISF grant 1382/17, and by the Joan and Reginald Coleman–Cohen Fund.
The authors would like to thank both anonymous referees for their numerous constructive comments that improved the quality of this paper.
"Age evolution in the mean field forest fire model via multitype branching processes." Ann. Probab. 49 (4) 2031 - 2075, July 2021. https://doi.org/10.1214/20-AOP1501