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We study the distribution of ages in the mean field forest fire modelintroduced by R\'ath and T\'oth. This model is an evolving random graph whosedynamics combine Erd\H{o}s-R\'enyi edge-addition with a Poisson rain oflightning strikes. All edges in a connected component are deleted when any ofits vertices is struck by lightning. We consider the asymptotic regime oflightning rates for which the model displays self-organized criticality. Theage of a vertex increases at unit rate, but it is reset to zero at each burningtime. We show that the empirical age distribution converges as a process to adeterministic solution of an autonomous measure-valued differential equation.The main technique is to observe that, conditioned on the vertex ages, thegraph is an inhomogeneous random graph in the sense of Bollob\'as, Janson andRiordan. We then study the evolution of the ages via the multitypeGalton--Watson trees that arise as the limit in law of the component of anidentified vertex at any fixed time. These trees are critical from the gelationtime onwards.

Age evolution in the mean field forest fire model via multitype branching processes