1201.6645 (Xavier Bressaud et al.)

Mean-field forest-fire models and pruning of random trees    [PDF]

Xavier Bressaud, Nicolas Fournier

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 $i,j \in \nn$ 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 $n\in \nn$, such a model has a unique equilibrium
state and study in details the asymptotics of this equilibrium as $n\to
\infty$: (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.

View original: http://arxiv.org/abs/1201.6645