Electronic Communications in Probability

A counterexample to monotonicity of relative mass in random walks

Oded Regev and Igor Shinkar

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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

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

Received: 26 June 2015
Accepted: 21 January 2016
First available in Project Euclid: 5 February 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

continuous-time random walk lamplighter graph

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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

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