A transient Markov chain with finitely many cutpoints

TL;DR

This paper presents an example of a transient reversible Markov chain with finitely many cutpoints, addressing conjectures and questions in the theory of Markov processes and random walks.

Contribution

It provides a specific example of such a chain and clarifies the behavior of nearest-neighbor random walks on trees regarding cutpoints.

Findings

Existence of a transient reversible Markov chain with finitely many cutpoints

Resolution of Kaimanovich's question for nearest-neighbor walks on trees

Implications for conjectures by Diaconis and Freedman

Abstract

We give an example of a transient reversible Markov chain that almost surely has only a finite number of cutpoints. We explain how this is relevant to a conjecture of Diaconis and Freedman and a question of Kaimanovich. We also answer Kaimanovich's question when the Markov chain is a nearest-neighbor random walk on a tree.