Most transient random walks have infinitely many cut times

TL;DR

This paper proves that under certain decay conditions of the Green's function, transient random walks on graphs have infinitely many cut times, confirming conjectures for broad classes of Markov chains.

Contribution

It establishes that many transient random walks, including those on graphs with spectral dimension greater than 2, have infinitely many cut times, extending previous conjectures.

Findings

Infinite cut times for random walks on graphs with polynomial Green's function decay

Confirmation of Diaconis-Freedman's conjecture for certain Markov chains

Resolution of Benjamini-Gurel-Gurevich-Schramm's conjecture on positive speed walks

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.