Let F (N, m) denote a random forest on a set of N vertices, chosen uniformly from all forests with m edges. Let F (N, p) denote the forest obtained by conditioning the Erdős-Rényi graph G(N, p) to be acyclic. We describe scaling limits for the largest components of F (N, p) and F (N, m), in the critical window p = N −1 + O(N −4/3 ) or m = N/2 + O(N 2/3 ). Aldous [4] described a scaling limit for the largest components of G(N, p) within the critical window in terms of the excursion lengths of a reflected Brownian motion with time-dependent drift. Our scaling limit for critical random forests is of a similar nature, but now based on a reflected diffusion whose drift depends on space as well as on time. , C G(N,p) 2 , . . . be the sequence of component sizes of G(N, p) written in non-increasing order, augmented by zeros. Let B λ (t), t ≥ 0 be a (time-inhomogeneous) reflected Brownian motion, with drift λ − t at time t, and B λ (0) = 0. Let C B λ be the lengths of the excursions of B λ , written in

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.

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