## Electronic Journal of Probability

### Trickle-down processes and their boundaries

#### Abstract

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

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

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

https://projecteuclid.org/euclid.ejp/1465062323

Digital Object Identifier
doi:10.1214/EJP.v17-1698

Mathematical Reviews number (MathSciNet)
MR2869248

Zentralblatt MATH identifier
1246.60100

Rights

#### Citation

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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