Electronic Journal of Probability

Trickle-down processes and their boundaries

Steven Evans, Rudolf Grübel, and Anton Wakolbinger

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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.

Article information

Electron. J. Probab. Volume 17 (2012), paper no. 1, 58 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J50: Boundary theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 68W40: Analysis of algorithms [See also 68Q25]

harmonic function h-transform tail sigma-field diffusion limited aggregation search tree Dirichlet random measure random recursive tree Chinese restaurant process Ewens sampling formula GEM distribution Mallows model q-binomial Catalan

This work is licensed under a Creative Commons Attribution 3.0 License.


Evans, Steven; Grübel, Rudolf; Wakolbinger, Anton. Trickle-down processes and their boundaries. Electron. J. Probab. 17 (2012), paper no. 1, 58 pp. doi:10.1214/EJP.v17-1698. https://projecteuclid.org/euclid.ejp/1465062323

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