Cutpoints and resistance of random walk paths

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].

💡 Research Summary

The paper addresses two long‑standing questions concerning the interplay between transience of random walks and the cut‑point structure of the subgraph formed by the edges visited by the walk.
First, the authors construct an explicit bounded‑degree infinite graph (G) on which a simple random walk (SRW) is transient, yet the random‑walk path (the subgraph consisting of all edges ever crossed) possesses only finitely many cut‑points almost surely. The construction starts with an infinite binary tree as a backbone. At each level (k) a large expander graph (H_k) is attached via a constant number of edges per vertex, preserving a uniform degree bound. Because expanders have a spectral gap, each (H_k) provides a very low effective resistance between its entry and exit points. Consequently, once the walk reaches level (k) it is highly unlikely to return to any lower level; the probability of ever revisiting level (k) decays exponentially in (k). By the Borel–Cantelli lemma there exists a random finite level (K) after which the walk never goes below, and therefore all cut‑points must lie in the finite portion of the tree below (K). This yields a transient walk whose path has only finitely many cut‑points with probability one, answering the first question raised by James, Lyons and Peres.

Second, the paper proves a universal lower bound on the expected number of cut‑points for any transient Markov chain. For a transient chain ((X_n)) let (G(i)) denote the expected number of visits to state (i). Transience forces (\sum_i G(i)=\infty). By interpreting the chain as an electrical network, the authors show that the probability that a state (i) is a cut‑point is at least a constant multiple of (1/G(i)). Summing over all states gives (\mathbb{E}