Hostname: page-component-7f64f4797f-8cpv7 Total loading time: 0 Render date: 2025-11-04T15:15:44.235Z Has data issue: false hasContentIssue false
  • English
  • Français

Monotonicity of Avoidance Coupling on KN

Part of: Markov processes

Published online by Cambridge University Press:  03 June 2016

OHAD N. FELDHEIM
OHAD N. FELDHEIM*
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA (e-mail: ohadfeld@stanford.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

Answering a question by Angel, Holroyd, Martin, Wilson and Winkler [1], we show that the maximal number of non-colliding coupled simple random walks on the complete graph KN , which take turns, moving one at a time, is monotone in N. We use this fact to couple [N/4] such walks on KN , improving the previous Ω(N/log N) lower bound of Angel et al. We also introduce a new generalization of simple avoidance coupling which we call partially ordered simple avoidance coupling, and provide a monotonicity result for this extension as well.

MSC classification

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

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 1 , January 2017 , pp. 16 - 23
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Angel, O., Holroyd, A., Martin, J., Wilson, D. and Winkler, P. (2013) Avoidance coupling. Electron. Commun. Probab. 18 1–13.Google Scholar
[2] Benjamini, I., Burdzy, K. and Chen, Z. (2007) Shy couplings. Probab. Theory Rel. Fields 137 345–377.Google Scholar
[3] Bramson, M., Burdzy, K. and Kendall, W. (2010) Shy couplings, CAT(0) spaces, and the lion and man. Ann. Probab. 41 744–784.Google Scholar
[4] Coppersmith, D., Tetali, P. and Winkler, P. (1993) Collisions among random walks on a graph. SIAM J. Discrete Math. 6 363–374.Google Scholar
[5] Gács, P. (2011) Clairvoyant scheduling of random walks. Random Struct. Alg. 39 413–485.Google Scholar
[6] Kendall, W. (2009) Brownian couplings, convexity, and shy-ness. Electron. Commun. Probab. 14 66–80.CrossRefGoogle Scholar
[7] Tetali, P. and Winkler, P. (1993) Simultaneous reversible Markov chains. In Combinatorics: Paul Erdős is Eighty, Vol. 1, János Bolyai Mathematical Society, pp. 433–451.Google Scholar