%0 Journal Article %1 BPGibbs %A Berestycki, Nathanaël %A Pitman, Jim %D 2007 %J J. Stat. Phys. %K Dept_Mathematics_Berkeley Dept_Statistics_Berkeley Gibbs_distribution fragmentation random_partitions myown %N 2 %P 381--418 %R 10.1007/s10955-006-9261-1 %T Gibbs distributions for random partitions generated by a fragmentation process %V 127 %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.

@article{BPGibbs, 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 = {2008-01-20T23:58:29.000+0100}, arxiv = {math.PR/0512378}, author = {Berestycki, Nathana{\"e}l and Pitman, Jim}, biburl = {https://www.bibsonomy.org/bibtex/2d3f4513fec11fb06b86053a92abfeac9/pitman}, doi = {10.1007/s10955-006-9261-1}, fjournal = {Journal of Statistical Physics}, interhash = {11ba891a1f67b2b77cabc91c6b2868a4}, intrahash = {d3f4513fec11fb06b86053a92abfeac9}, issn = {0022-4715}, journal = {J. Stat. Phys.}, keywords = {Dept_Mathematics_Berkeley Dept_Statistics_Berkeley Gibbs_distribution fragmentation random_partitions myown}, mrclass = {60Kxx, 60J10, 60K35, 05A15, 05A19 (82B44)}, mrnumber = {MR2314353}, number = 2, pages = {381--418}, timestamp = {2010-10-30T22:51:58.000+0200}, title = {Gibbs distributions for random partitions generated by a fragmentation process}, volume = 127, year = 2007 }