May 2023 Monotonicity for continuous-time random walks
Russell Lyons, Graham White
Author Affiliations +
Ann. Probab. 51(3): 1112-1138 (May 2023). DOI: 10.1214/22-AOP1615


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 Z+ is monotone increasing in the rates for arbitrary rate functions, which is not true on all of Z. Various intermediate results are also of interest.

Funding Statement

The first author was supported by the National Science Foundation under grants DMS-1612363 and DMS-1954086 and also by the Simons Foundation.


We thank Kavita Ramanan for early discussions of some of these questions. We also thank Yiping Hu for asking us about convergence to stationarity. Thanks to Yuval Peres, who alerted us to [28] and [13] after seeing an earlier draft of our results. We also thank one of the referees for informing us of [18]. We are grateful to Matthias Weber for permission to include Figure 1.


Download Citation

Russell Lyons. Graham White. "Monotonicity for continuous-time random walks." Ann. Probab. 51 (3) 1112 - 1138, May 2023.


Received: 1 October 2021; Revised: 1 September 2022; Published: May 2023
First available in Project Euclid: 2 May 2023

MathSciNet: MR4583064
zbMATH: 1515.05165
Digital Object Identifier: 10.1214/22-AOP1615

Primary: 05C81 , 60G50
Secondary: 20F55 , 60B15

Keywords: Cayley graphs , Coxeter groups , majorization , Mixing , Speed

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 3 • May 2023
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