A Counterexample to Monotonicity of Relative Mass in Random Walks

TL;DR

This paper provides a counterexample in a Cayley graph showing that the relative mass function in a continuous-time random walk can decrease, challenging the assumption of monotonicity suggested by prior conjectures.

Contribution

The authors construct a specific Cayley graph and vertices demonstrating that the ratio of probabilities is not always non-decreasing, answering a question posed by Peres in 2013.

Findings

Counterexample disproves monotonicity of relative mass in certain graphs

Shows that the ratio function can decrease over time

Addresses a long-standing open question in random walk theory

Abstract

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.