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.
%0 Generic
%1 arXiv:math.PR/0512378
%A Berestycki, Nathanael
%A Pitman, Jim
%D 2005
%K author_pitman_from_arxiv
%T Gibbs distributions for random partitions generated by a fragmentation process.
%U http://arxiv.org/abs/math.PR/0512378
%X 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.
@misc{arXiv:math.PR/0512378,
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.},
added-at = {2008-01-25T05:29:59.000+0100},
arxiv = {arXiv:math.PR/0512378},
author = {Berestycki, Nathanael and Pitman, Jim},
biburl = {https://www.bibsonomy.org/bibtex/2eed595ab42d2cb3573242c8def337a28/pitman},
interhash = {c3546b968ea6c28dac2eac09f269e9a1},
intrahash = {eed595ab42d2cb3573242c8def337a28},
keywords = {author_pitman_from_arxiv},
timestamp = {2008-01-25T05:33:08.000+0100},
title = {{Gibbs distributions for random partitions generated by a fragmentation process.}},
url = {http://arxiv.org/abs/math.PR/0512378},
year = 2005
}