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2012 Trickle-down processes and their boundaries
Steven Evans, Rudolf Grübel, Anton Wakolbinger
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Electron. J. Probab. 17: 1-58 (2012). DOI: 10.1214/EJP.v17-1698


It is possible to represent each of a number of Markov chains as an evolving sequence of connected subsets of a directed acyclic graph that grow in the following way: initially, all vertices of the graph are unoccupied, particles are fed in one-by-one at a distinguished source vertex, successive particles proceed along directed edges according to an appropriate stochastic mechanism, and each particle comes to rest once it encounters an unoccupied vertex. Examples include the binary and digital search tree processes, the random recursive tree process and generalizations of it arising from nested instances of Pitman's two-parameter Chinese restaurant process, tree-growth models associated with Mallows' $\phi$ model of random permutations and with Schützenberger's non-commutative $q$-binomial theorem, and a construction due to Luczak and Winkler that grows uniform random binary trees in a Markovian manner. We introduce a framework that encompasses such Markov chains, and we characterize their asymptotic behavior by analyzing in detail their Doob-Martin compactifications, Poisson boundaries and tail $\sigma$-fields.


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Steven Evans. Rudolf Grübel. Anton Wakolbinger. "Trickle-down processes and their boundaries." Electron. J. Probab. 17 1 - 58, 2012.


Accepted: 1 January 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1246.60100
MathSciNet: MR2869248
Digital Object Identifier: 10.1214/EJP.v17-1698

Primary: 60J50
Secondary: 60J10 , 68W40

Keywords: Catalan , Chinese restaurant process , Diffusion limited aggregation , Dirichlet random measure , Ewens sampling formula , GEM distribution , Harmonic function , H-transform , Mallows model , q-binomial , Random recursive tree , search tree , tail sigma-field


Vol.17 • 2012
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