期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2014
卷号：XI
页码：589-614
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：We consider a family of discrete coagulation-fragmentation equationsclosely related to the one-dimensional forest-fire model of statistical mechanics:each pair of particles with masses i, j ∈ N merge together at rate 2 to produce asingle particle with mass i+j, and each particle with mass i breaks into i particleswith mass 1 at rate (i−1)/n. The (large) parameter n controls the rate of ignitionand 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 ∈ N, such a model hasa unique equilibrium state and study in details the asymptotics of this equilibriumas n → ∞: (I) the distribution of the mass of a typical particle goes to the law ofthe number of leaves of a critical binary Galton-Watson tree, (II) the distributionof the mass of a typical size-biased particle converges, after rescaling, to a limitprofile, which we write explicitly in terms of the zeroes of the Airy function andits derivative. We also indicate how to simulate perfectly a typical particle and asize-biased typical particle by pruning some random trees.
