Electronic Communications in Probability

A counterexample to monotonicity of relative mass in random walks

Oded Regev and Igor Shinkar

Full-text: Open access

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.

Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 8, 8 pp.

Dates
Received: 26 June 2015
Accepted: 21 January 2016
First available in Project Euclid: 5 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1454682824

Digital Object Identifier
doi:10.1214/16-ECP4392

Mathematical Reviews number (MathSciNet)
MR3485377

Zentralblatt MATH identifier
1343.60054

Subjects
Primary: 60J27: Continuous-time Markov processes on discrete state spaces

Keywords
continuous-time random walk lamplighter graph

Rights
Creative Commons Attribution 4.0 International License.

Citation

Regev, Oded; Shinkar, Igor. A counterexample to monotonicity of relative mass in random walks. Electron. Commun. Probab. 21 (2016), paper no. 8, 8 pp. doi:10.1214/16-ECP4392. https://projecteuclid.org/euclid.ecp/1454682824


Export citation

References

  • [1] Jeff Cheeger, 2015, Private communication.
  • [2] Amir Dembo, Jian Ding, Jason Miller, and Yuval Peres, Cut-off for lamplighter chains on tori: dimension interpolation and phase transition, (2015), arXiv:1312.4522.
  • [3] Júlia Komjáthy, Jason Miller, and Yuval Peres, Uniform mixing time for random walk on lamplighter graphs, Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), no. 4, 1140–1160.
  • [4] David A. Levin, Yuval Peres, and Elizabeth L. Wilmer, Markov chains and mixing times, American Mathematical Society, Providence, RI, 2009.
  • [5] Yuval Peres, 2013, Private communication.
  • [6] Yuval Peres and David Revelle, Mixing times for random walks on finite lamplighter groups, Electron. J. Probab. 9 (2004), no. 26, 825–845.
  • [7] Tom Price, Is the heat kernel more spread out with a smaller metric? MathOverflow, http://mathoverflow.net/questions/186428/, 2014.
  • [8] Oded Regev and Noah Stephens-Davidowitz, An inequality for Gaussians on lattices, (2015), arXiv:1502.04796.