A Counterexample to Monotonicity of Relative Mass in Random Walks

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.