Electronic Communications in Probability

Avoidance Coupling

Omer Angel, Alexander Holroyd, James Martin, Peter Winkler, and David Wilson

Full-text: Open access


We examine the question of whether a collection of random walks on a graph can be coupled so that they never collide.  In particular, we show that on the complete graph on n vertices, with or without loops, there is a Markovian coupling keeping apart Omega(n/log n) random walks, taking turns to move in discrete time.

Article information

Electron. Commun. Probab. Volume 18 (2013), paper no. 58, 13 pp.

Accepted: 9 July 2013
First available in Project Euclid: 7 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

coupling coloring

This work is licensed under a Creative Commons Attribution 3.0 License.


Angel, Omer; Holroyd, Alexander; Martin, James; Winkler, Peter; Wilson, David. Avoidance Coupling. Electron. Commun. Probab. 18 (2013), paper no. 58, 13 pp. doi:10.1214/ECP.v18-2275. https://projecteuclid.org/euclid.ecp/1465315597.

Export citation


  • David Aldous and James Fill. hrefhttp://www.stat.berkeley.edu/~aldous/RWG/book.htmlReversible Markov Chains and Random Walks on Graphs. 2002. Draft, http://www.stat.berkeley.edu/~aldous/RWG/book.html.
  • Benjamini, Itai; Burdzy, Krzysztof; Chen, Zhen-Qing. Shy couplings. Probab. Theory Related Fields 137 (2007), no. 3-4, 345–377.
  • Maury Bramson, Krzysztof Burdzy, and Wilfrid S. Kendall. Shy couplings, CAT(0) spaces, and the lion and man. 2010. arXiv:1007.3199.
  • Coppersmith, Don; Tetali, Prasad; Winkler, Peter. Collisions among random walks on a graph. SIAM J. Discrete Math. 6 (1993), no. 3, 363–374.
  • Gács, Peter. Clairvoyant scheduling of random walks. Random Structures Algorithms 39 (2011), no. 4, 413–485.
  • Kendall, Wilfrid S. Brownian couplings, convexity, and shy-ness. Electron. Commun. Probab. 14 (2009), 66–80.
  • Levin, David A.; Peres, Yuval; Wilmer, Elizabeth L. Markov chains and mixing times. With a chapter by James G. Propp and David B. Wilson. American Mathematical Society, Providence, RI, 2009. xviii+371 pp. ISBN: 978-0-8218-4739-8 http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf
  • Tetali, P.; Winkler, P. Simultaneous reversible Markov chains. Combinatorics, Paul ErdÅ‘s is eighty, Vol. 1, 433–451, Bolyai Soc. Math. Stud., János Bolyai Math. Soc., Budapest, 1993.