Most transient random walks have infinitely many cut times

Abstract

We prove that if $(X_n)_{n\geq 0}$ is a random walk on a transient graph such that the Green's function decays at least polynomially along the random walk, then $(X_n)_{n\geq 0}$ 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. 1980) holds for the same class of Markov chains, and resolve a conjecture of Benjamini, Gurel-Gurevich, and Schramm (Ann. Probab. 2011) on the existence of infinitely many cut times for random walks of positive speed.