Asymptotics of Certain Coagulation-Fragmentation Processes and Invariant Poisson-Dirichlet Measures
Eddy Mayer-Wolf, Technion
Ofer Zeitouni, Technion
Martin P.W. Zerner, Stanford University
Abstract
We consider Markov chains on the space of (countable) partitions of the
interval [0,1], obtained first by size biased sampling twice (allowing repetitions)
and then merging the parts with probability beta_m (if the sampled parts are
distinct) or splitting the part with probability beta_s, according to a
law sigma (if the same part was sampled twice). We characterize invariant
probability measures for such chains. In particular, if sigma is the
uniform measure, then the Poisson-Dirichlet law is an invariant probability
measure, and it is unique within a suitably defined class of "analytic"
invariant measures. We also derive transience and recurrence criteria for
these chains.
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