Repository landing page
Gibbs distributions for random partitions generated by a fragmentation process
Abstract
38 pages, 2 figures, version considerably modified. To appear in the Journal of Statistical Physics.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- info:eu-repo/semantics/preprint
- Preprints, Working Papers, ...
- Fragmentation processes
- Gibbs distributions
- Marcus-Lushnikov processes
- Gould convolution identities
- MSC: 60J10, 60K35, 05A15, 05A19
- [MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
- [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech]
- [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]