Cutpoints and resistance of random walk paths

TL;DR

This paper constructs a graph where a simple random walk is transient but its path has finitely many cutpoints, and proves the expected number of cutpoints for any transient Markov chain is infinite, addressing open questions in the field.

Contribution

It provides a novel graph construction demonstrating finite cutpoints in a transient walk and proves the infinite expectation of cutpoints for all transient Markov chains, answering longstanding questions.

Findings

Constructed a bounded degree graph with a transient walk and finitely many cutpoints.

Proved the expected number of cutpoints in any transient Markov chain is infinite.

Derived a lower bound on the expected effective resistance between two vertices in a walk path.

Abstract

We construct a bounded degree graph $G$, such that a simple random walk on it is transient but the random walk path (i.e., the subgraph of all the edges the random walk has crossed) has only finitely many cutpoints, almost surely. We also prove that the expected number of cutpoints of any transient Markov chain is infinite. This answers two questions of James, Lyons and Peres [A Transient Markov Chain With Finitely Many Cutpoints (2007) Festschrift for David Freedman]. Additionally, we consider a simple random walk on a finite connected graph $G$ that starts at some fixed vertex $x$ and is stopped when it first visits some other fixed vertex $y$. We provide a lower bound on the expected effective resistance between $x$ and $y$ in the path of the walk, giving a partial answer to a question raised in [Ann. Probab. 35 (2007) 732--738].