Hostname: page-component-cb9f654ff-lqqdg Total loading time: 0 Render date: 2025-09-01T05:28:58.001Z Has data issue: false hasContentIssue false
  • English
  • Français

An Upper Bound on the Size of Avoidance Couplings

Part of: Graph theory Markov processes

Published online by Cambridge University Press:  11 December 2018

ERIK BATES  and
LISA SAUERMANN
ERIK BATES
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA (e-mail: ewbates@stanford.edu, lsauerma@stanford.edu)
LISA SAUERMANN
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA (e-mail: ewbates@stanford.edu, lsauerma@stanford.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that a coupling of non-colliding simple random walkers on the complete graph on n vertices can include at most n - log n walkers. This improves the only previously known upper bound of n - 2 due to Angel, Holroyd, Martin, Wilson and Winkler (Electron. Commun. Probab.18 (2013)). The proof considers couplings of i.i.d. sequences of Bernoulli random variables satisfying a similar avoidance property, for which there is separate interest.

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 05C81: Random walks on graphs

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 3 , May 2019 , pp. 325 - 334
Copyright
Copyright © Cambridge University Press 2018 

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

Footnotes

†

Research was partially supported by NSF grant DGE-114747.

References

[1] Angel, O., Holroyd, A. E., Martin, J., Wilson, D. B. and Winkler, P. (2013) Avoidance coupling. Electron. Commun. Probab. 18 58.Google Scholar
[2] Balister, P. N., Bollobás, B. and Stacey, A. M. (2000) Dependent percolation in two dimensions. Probab. Theory Rel. Fields 117 495–513.Google Scholar
[3] Basu, R., Sidoravicius, V. and Sly, A. (2014) Scheduling of non-colliding random walks. arXiv:1411.4041Google Scholar
[4] Benjamini, I., Burdzy, K. and Chen, Z.-Q. (2007) Shy couplings. Probab. Theory Rel. Fields 137 345–377.Google Scholar
[5] Bramson, M., Burdzy, K. and Kendall, W. (2013) Shy couplings, CAT(0) spaces, and the Lion and Man. Ann. Probab. 41 744–784.Google Scholar
[6] Bramson, M., Burdzy, K. and Kendall, W. S. (2014) Rubber bands, pursuit games and shy couplings. Proc. Lond. Math. Soc. (3) 109 121–160.Google Scholar
[7] Coppersmith, D., Tetali, P. and Winkler, P. (1993) Collisions among random walks on a graph. SIAM J. Discrete Math. 6 363–374.Google Scholar
[8] Feldheim, O. N. (2017) Monotonicity of avoidance coupling on KN. Combin. Probab. Comput. 26 16–23.Google Scholar
[9] Gács, P. (2011) Clairvoyant scheduling of random walks. Random Struct. Alg. 39 413–485.Google Scholar
[10] Infeld, E. J. (2016) Uniform avoidance coupling, design of anonymity systems and matching theory. PhD thesis, Dartmouth College.Google Scholar
[11] Kendall, W. S. (2009) Brownian couplings, convexity, and shy-ness. Electron. Commun. Probab. 14 66–80.Google Scholar
[12] 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
[13] Tsianos, K. I. (2013) The role of the network in distributed optimization algorithms: Convergence rates, scalability, communication/computation tradeoffs and communication delays. PhD thesis, McGill University.Google Scholar
[14] Winkler, P. (2000) Dependent percolation and colliding random walks. Random Struct. Alg. 16 58–84.Google Scholar