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We give an example of a transient reversible Markov chain that almost surelyhas only a finite number of cutpoints. We explain how this is relevant to aconjecture of Diaconis and Freedman and a question of Kaimanovich. We alsoanswer Kaimanovich's question when the Markov chain is a nearest-neighborrandom walk on a tree.

A transient Markov chain with finitely many cutpoints