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.
Categories: math.PR