Trickle-down processes and their boundaries

TL;DR

This paper introduces a unified framework for representing and analyzing various Markov chains related to growing tree structures, focusing on their asymptotic behavior through advanced probabilistic tools.

Contribution

It develops a comprehensive framework that captures diverse Markov chain models of tree growth and characterizes their asymptotic properties using Doob-Martin compactifications and Poisson boundaries.

Findings

Unified representation of Markov chains for tree growth models

Detailed analysis of asymptotic behavior and boundaries

Characterization of tail sigma-fields in these processes

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 Schuetzenberger'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.