## 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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