Electronic Communications in Probability

Coalescing random walk on unimodular graphs

Eric Foxall, Tom Hutchcroft, and Matthew Junge

Full-text: Open access


Coalescing random walk on a unimodular random rooted graph for which the root has finite expected degree visits each site infinitely often almost surely. A corollary is that an opinion in the voter model on such graphs has infinite expected lifetime. Additionally, we deduce an adaptation of our main theorem that holds uniformly for coalescing random walk on finite random unimodular graphs with degree distribution stochastically dominated by a probability measure with finite mean.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 62, 10 pp.

Received: 6 April 2018
Accepted: 2 May 2018
First available in Project Euclid: 13 September 2018

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

coalescing random walk unimodular random graph voter model

Creative Commons Attribution 4.0 International License.


Foxall, Eric; Hutchcroft, Tom; Junge, Matthew. Coalescing random walk on unimodular graphs. Electron. Commun. Probab. 23 (2018), paper no. 62, 10 pp. doi:10.1214/18-ECP136. https://projecteuclid.org/euclid.ecp/1536804170

Export citation


  • [AL07] David Aldous and Russell Lyons. Processes on unimodular random networks. Electron. J. Probab., 12:1454–1508, 2007.
  • [BC12] Itai Benjamini and Nicolas Curien. Ergodic theory on stationary random graphs. Electron. J. Probab., 17(93):1–20, 2012.
  • [BFGG$^{+}$16] Itai Benjamini, Eric Foxall, Ori Gurel-Gurevich, Matthew Junge, and Harry Kesten. Site recurrence for coalescing random walk. Electron. Commun. Probab., 21:12 pp., 2016.
  • [BPP14] Itai Benjamini, Elliot Paquette, and Joshua Pfeffer. Anchored expansion, speed, and the hyperbolic poisson voronoi tessellation. arXiv preprint arXiv:1409.4312, 2014.
  • [BPS12] Martin T. Barlow, Yuval Peres, and Perla Sousi. Collisions of random walks. Ann. Inst. Henri Poincaré Probab. Stat., 48(4):922–946, 2012.
  • [BS01] Itai Benjamini and Oded Schramm. Recurrence of distributional limits of finite planar graphs. Electron. J. Probab., 6(23):13, 2001.
  • [Cur] Nicolas Curien. Random graphs: The local convergence point of view. Unpublished lecture notes. Available at https://www.math.u-psud.fr/~curien/cours/cours-RG-V3.pdf.
  • [Cur16] Nicolas Curien. Planar stochastic hyperbolic triangulations. Probab. Theory Related Fields, 165(3–4):509–540, 2016.
  • [Gri78] David Griffeath. Annihilating and coalescing random walks on $\mathbb{Z} ^d$. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 46(1):55–65, 1978.
  • [HP15] Tom Hutchcroft and Yuval Peres. Collisions of random walks in reversible random graphs. Electron. Commun. Probab., 20:6 pp., 2015.
  • [Pen03] Mathew Penrose. Random geometric graphs, volume 5 of Oxford Studies in Probability. Oxford University Press, Oxford, 2003.
  • [Rou15] Arnaud Rousselle. Recurrence or transience of random walks on random graphs generated by point processes in $\mathbb{R} ^d$. Stochastic Process. Appl., 125(12):4351–4374, 2015.