Gibbs Distributions for Random Partitions Generated by a Fragmentation Process |
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Authors: | Nathanaël Berestycki Jim Pitman |
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Institution: | (1) University of British Columbia, Vancouver, B.C., Canada, V6T 1Z2;(2) Department of Statistics, University of California, Berkeley, CA, USA |
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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.
Research supported in part by N.S.F. Grant DMS-0405779. |
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Keywords: | Fragmentation processes Gibbs distributions Marcus– Lushnikov processes Gould convolution identities |
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