2012/01/31 by Xavier Bressaud, Bressaud, Xavier, Nicolas Fournier +1
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #82B05 #FOS: Mathematics #FOS: Physical sciences #Financial Risk and Volatility Modeling #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR #msc:82B05
paper · pdf · doi:10.48550/arxiv.1201.6645
arxiv created 2012/01/31 · openalex publication_date 2012/01/31 · arxiv updated 2012/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ∈ \nn 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∈ \nn, such a model has a unique equilibrium state and study in details the asymptotics of this equilibrium as n→ ∞: (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, which allows us to give some probabilistic interpretations of the above results in terms of pruned Galton-Watson trees and pruned continuum random trees.