Monotonicity for continuous-time random walks

TL;DR

This paper investigates how the speed and convergence properties of continuous-time random walks on Cayley graphs depend on edge rates, revealing monotonicity in certain cases and surprising behaviors in others.

Contribution

It establishes monotonicity results for random walks on Cayley graphs derived from Coxeter systems and abelian groups, and provides counterexamples to existing conjectures.

Findings

Speed is monotone increasing in rates for infinite Coxeter Cayley graphs.

Distance to stationarity decreases monotonically with rates for finite Coxeter and abelian Cayley graphs.

Counterexamples show non-monotonic behavior and challenge previous conjectures.

Abstract

Consider continuous-time random walks on Cayley graphs where the rate assigned to each edge depends only on the corresponding generator. We show that the limiting speed is monotone increasing in the rates for infinite Cayley graphs that arise from Coxeter systems, but not for all Cayley graphs. On finite Cayley graphs, we show that the distance -- in various senses -- to stationarity is monotone decreasing in the rates for Coxeter systems and for abelian groups, but not for all Cayley graphs. We also find several examples of surprising behaviour in the dependence of the distance to stationarity on the rates. This includes a counterexample to a conjecture on entropy of Benjamini, Lyons, and Schramm. We also show that the expected distance at any fixed time for random walks on $\mathbb{Z}^+$ is monotone increasing in the rates for arbitrary rate functions, which is not true on all of $\mathbb{Z}$. Various intermediate results are also of interest.