## The Annals of Probability

- Ann. Probab.
- Volume 39, Number 3 (2011), 1122-1136.

### Cutpoints and resistance of random walk paths

Itai Benjamini, Ori Gurel-Gurevich, and Oded Schramm

#### 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].

#### Article information

**Source**

Ann. Probab., Volume 39, Number 3 (2011), 1122-1136.

**Dates**

First available in Project Euclid: 16 March 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1300281734

**Digital Object Identifier**

doi:10.1214/10-AOP569

**Mathematical Reviews number (MathSciNet)**

MR2789585

**Zentralblatt MATH identifier**

1223.60012

**Subjects**

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G50: Sums of independent random variables; random walks

**Keywords**

Graph random walk path cutpoints

#### Citation

Benjamini, Itai; Gurel-Gurevich, Ori; Schramm, Oded. Cutpoints and resistance of random walk paths. Ann. Probab. 39 (2011), no. 3, 1122--1136. doi:10.1214/10-AOP569. https://projecteuclid.org/euclid.aop/1300281734