Gibbs distributions for random partitions generated by a fragmentation process
| Autor: | Jim Pitman, Nathanaël Berestycki |
|---|---|
| Přispěvatelé: | University of British Columbia (UBC), Department of Statistics [Berkeley], University of California [Berkeley], University of California-University of California |
| Jazyk: | angličtina |
| Rok vydání: | 2006 |
| Předmět: |
Fragmentation processes
Negative binomial distribution FOS: Physical sciences Poisson distribution MSC: 60J10 60K35 05A15 05A19 01 natural sciences Combinatorics 010104 statistics & probability symbols.namesake [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] Cluster (physics) FOS: Mathematics Partition (number theory) Mathematics - Combinatorics 0101 mathematics [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech] Mathematical Physics Condensed Matter - Statistical Mechanics Mathematics Real number Discrete mathematics Statistical Mechanics (cond-mat.stat-mech) Stochastic process 010102 general mathematics Probability (math.PR) Sampling (statistics) Statistical and Nonlinear Physics [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] symbols Gibbs distributions Gould convolution identities Affine transformation Combinatorics (math.CO) Marcus-Lushnikov processes Mathematics - Probability |
| Popis: | 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. Comment: 38 pages, 2 figures, version considerably modified. To appear in the Journal of Statistical Physics |
| Databáze: | OpenAIRE |
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