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Gibbs distributions for random partitions generated by a fragmentation process

Nathanael Berestycki·Jim Pitman
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Abstract

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.

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Nathanael Berestycki, Jim Pitman. 2006-11-14. Gibbs distributions for random partitions generated by a fragmentation process. https://arxiv.org/abs/math/0512378

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