Ed Crane

Age evolution in the Rath-Toth mean field forest fire model

The mean field forest fire model was introduced in a 2009 paper of Balazs Rath and Balint Toth. It combines the dynamical Erdos-Renyi random graph process on the complete graph K_n with an extra ingredient, a low intensity Poisson rain of lightning strikes. A lightning strike causes all the edges of the connected component that it hits to burn instantaneously: this is a forest fire. The vertices survive the fire and continue to form new edges. Rath and Toth studied the evolution of the distribution of component sizes in a wide range of asymptotic regimes of the lightning rate as n tends to infinity. When the global lightning rate diverges but the lightning rate per vertex tends to zero, they showed that the competition between the arrival of new edges and the forest fires leads the system to organize itself into a critical state, and remain critical thereafter. This was notable for being a mathematically rigorous result in the literature on self-organized criticality, most of which is experimental.

In recent joint work with Balazs Rath and Dominic Yeo (arXiv: 1811.07981) we study the mean field forest fire model through a different lens, by looking at the empirical distribution of the ages of the vertices. The age of a vertex increases at rate 1 but is reset to 0 each time the vertex participates in a forest fire. Conditioned on the ages of all the vertices, the model is an inhomogeneous random graph of the type studied by Bollobas, Janson and Riordan. It is locally approximated by a multitype Galton-Watson tree, whose offspring distributions depend on the asymptotic age distribution. Once the system is critical, this random tree is critical. Assuming convergence of the initial age distributions, we derive an autonomous ODE describing the asymptotic evolution of the empirical age distribution. This ODE involves the principal eigenfunction of the Perron-Frobenius operator associated to the multitype Galton-Watson tree, and the overall burning rate is identified as a certain nonlinear functional of this eigenfunction.