Cutpoints for Random Walks on Quasi-Transitive Graphs

TL;DR

This paper proves that simple random walks on certain quasi-transitive graphs with super-polynomial volume growth almost surely have infinitely many cutpoints, confirming a conjecture for vertex-transitive graphs.

Contribution

It establishes the existence of infinitely many cutpoints for random walks on quasi-transitive graphs with super-polynomial volume growth, extending previous conjectures.

Findings

Random walks on these graphs have infinitely many cutpoints almost surely.

The result applies to graphs with volume growth faster than degree 4 polynomial.

Confirms a conjecture for vertex-transitive graphs regarding cutpoints.

Abstract

We prove that a simple random walk on quasi-transitive graphs with the volume growth being faster than any polynomial of degree 4 has a.s. infinitely many cut times, and hence infinitely many cutpoints. This confirms a conjecture raised by I. Benjamini, O. Gurel-Gurevich and O. Schramm [2011, Cutpoints and resistance of random walk paths, {\it Ann. Probab.} {\bf 39(3)}, 1122-1136] that PATH of simple random walk on any transient vertex-transitive graph has a.s. infinitely many cutpoints in the corresponding case.