Most transient random walks have infinitely many cut times

Abstract

We prove that if ${({\mathit{X}}_{\mathit{n}})}_{\mathit{n}\ge 0}$ is a random walk on a transient graph such that the Green’s function decays at least polynomially along the random walk, then ${({\mathit{X}}_{\mathit{n}})}_{\mathit{n}\ge 0}$ has infinitely many cut times almost surely. This condition applies in particular to any graph of spectral dimension strictly larger than 2. In fact, our proof applies to general (possibly nonreversible) Markov chains satisfying a similar decay condition for the Green’s function that is sharp for birth–death chains. We deduce that a conjecture of Diaconis and Freedman (Ann. Probab. 8 (1980) 115–130) holds for the same class of Markov chains, and resolve a conjecture of Benjamini, Gurel-Gurevich, and Schramm (Ann. Probab. 39 (2011) 1122–1136) on the existence of infinitely many cut times for random walks of positive speed.