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

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
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