Cutpoints and Exchangeable Events for Random Walks

For a Markov chain {Sn}, call Sk a cutpoint, and k a cut-epoch, if there is no possible transition from Si to Sj whenever i < k < j. We show that a transient random walk of bounded stepsize on an integer lattice has infinitely many cutpoints almost surely. For simple random walk on Zd , $d \g 4$, this is due to Lawler. Furthermore, let G be a finitely generated group of growth at least polynomial of degree 5; then for any symmetric random walk on G such that the steps have a bounded support that generates G, the cut-epochs have positive density. We also show that for any Markov chain which has infinitely many cutpoints almost surely, the eventual occupation numbers generate the exchangeable $\sigma$-field. Combining these results answers a question posed by Kaimanovich, and partially resolves a conjecture of Diaconis and Freedman.