A Mean-field forest-fire model

We consider a family of discrete coagulation-fragmentation equations closely related to the one-dimensional forest-fire model of statistical mechanics: each pair of particles with masses i, j is an element of N merge together at rate 2 to produce a single particle with mass i + j, and each particle with mass i breaks into i particles with mass 1 at rate (i - 1)/n. The (large) parameter n controls the rate of ignition and there is also an acceleration factor (depending on the total number of particles) in front of the coagulation term. We prove that for each n is an element of N, such a model has a unique equilibrium state and study in details the asymptotics of this equilibrium as n -> infinity: (I) the distribution of the mass of a typical particle goes to the law of the number of leaves of a critical binary Galton-Watson tree, (II) the distribution of the mass of a typical size-biased particle converges, after rescaling, to a limit profile, which we write explicitly in terms of the zeroes of the Airy function and its derivative. We also indicate how to simulate perfectly a typical particle and a size-biased typical particle by pruning some random trees.