Gibbs distributions for random partitions generated by a fragmentation process

Nathanael Berestycki (1) , Jim Pitman (2)

1 UBC - University of British Columbia
2 Department of Statistics [Berkeley]

In this paper we study random partitions of 1,...n, where every cluster of size j can be in any of w_j possible internal states. The Gibbs (n,k,w) distribution is obtained by sampling uniformly among such partitions with k clusters. We provide conditions on the weight sequence w allowing construction of a partition valued random process where at step k the state has the Gibbs (n,k,w) distribution, so the partition is subject to irreversible fragmentation as time evolves. For a particular one-parameter family of weight sequences w_j, the time-reversed process is the discrete Marcus-Lushnikov coalescent process with affine collision rate K_{i,j}=a+b(i+j) for some real numbers a and b. Under further restrictions on a and b, the fragmentation process can be realized by conditioning a Galton-Watson tree with suitable offspring distribution to have n nodes, and cutting the edges of this tree by random sampling of edges without replacement, to partition the tree into a collection of subtrees. Suitable offspring distributions include the binomial, negative binomial and Poisson distributions.

Mots clés: Fragmentation processes; Gibbs distributions; Marcus-Lushnikov processes; Gould convolution identities

Domaines: Probabilités [math.PR]; Mécanique statistique [cond-mat.stat-mech]; Combinatoire [math.CO]

Nathanael Berestycki, Jim Pitman. Gibbs distributions for random partitions generated by a fragmentation process. 2006. ⟨hal-00015991v2⟩