## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 21 (2016), paper no. 8, 8 pp.

### A counterexample to monotonicity of relative mass in random walks

#### Abstract

For a finite undirected graph $G = (V,E)$, let $p_{u,v}(t)$ denote the probability that a continuous-time random walk starting at vertex $u$ is in $v$ at time $t$. In this note we give an example of a Cayley graph $G$ and two vertices $u,v \in G$ for which the function \[ r_{u,v}(t) = \frac{p_{u,v}(t)} {p_{u,u}(t)} \qquad t \geq 0 \] is not monotonically non-decreasing. This answers a question asked by Peres in 2013.

#### Article information

**Source**

Electron. Commun. Probab., Volume 21 (2016), paper no. 8, 8 pp.

**Dates**

Received: 26 June 2015

Accepted: 21 January 2016

First available in Project Euclid: 5 February 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1454682824

**Digital Object Identifier**

doi:10.1214/16-ECP4392

**Mathematical Reviews number (MathSciNet)**

MR3485377

**Zentralblatt MATH identifier**

1343.60054

**Subjects**

Primary: 60J27: Continuous-time Markov processes on discrete state spaces

**Keywords**

continuous-time random walk lamplighter graph

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Regev, Oded; Shinkar, Igor. A counterexample to monotonicity of relative mass in random walks. Electron. Commun. Probab. 21 (2016), paper no. 8, 8 pp. doi:10.1214/16-ECP4392. https://projecteuclid.org/euclid.ecp/1454682824