A counterexample to monotonicity of relative mass in random walks. (English) Zbl 1343.60054

Regev, Oded; Shinkar, Igor

Electron. Commun. Probab. 21, Paper No. 8, 8 p. (2016).

Summary: 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 \[ r_{u,v}(t) = \frac{p_{u,v}(t)}{p_{u,u}(t)}, \;\;t\geq 0, \] is not monotonically non-decreasing. This answers a question asked by Peres in 2013.

MSC:

60G50	Sums of independent random variables; random walks
60J27	Continuous-time Markov processes on discrete state spaces
05C81	Random walks on graphs

Keywords:

continuous-time random walk; Cayley graph; lamplighter graph; relative mass; monotonicity