Hostname: page-component-84b7d79bbc-x5cpj Total loading time: 0 Render date: 2024-07-28T07:17:13.694Z Has data issue: false hasContentIssue false
  • English
  • Français

Mixing Times and Moving Targets

Published online by Cambridge University Press:  14 November 2013

PERLA SOUSI  and
PETER WINKLER
PERLA SOUSI
Affiliation:
DPMMS, University of Cambridge, Cambridge, UK (e-mail: p.sousi@statslab.cam.ac.uk)
PETER WINKLER
Affiliation:
Department of Mathematics, Dartmouth College, Hanover, NH 03755-3551, USA (e-mail: peter.winkler@dartmouth.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider irreducible Markov chains on a finite state space. We show that the mixing time of any such chain is equivalent to the maximum, over initial states x and moving large sets (As)s, of the hitting time of (As)s starting from x. We prove that in the case of the d-dimensional torus the maximum hitting time of moving targets is equal to the maximum hitting time of stationary targets. Nevertheless, we construct a transitive graph where these two quantities are not equal, resolving an open question of Aldous and Fill on a ‘cat and mouse’ game.

Keywords

Primary 60J10Secondary 60J2760G40
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 3 , May 2014 , pp. 460 - 476
Copyright
Copyright © Cambridge University Press 2013 

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.)

References

[1]Aldous, D. and Fill, J. Reversible Markov chains and random walks on graphs. In preparation. http://www.stat.berkeley.edu/~aldous/RWG/book.htmlGoogle Scholar
[2]Burchard, A. and Schmuckenschläger, M. (2001) Comparison theorems for exit times. Geom. Funct. Anal. 11 651–692.Google Scholar
[3]Griffiths, S., Kang, R. J., Oliveira, R. I. and Patel, V. (2012) Tight inequalities among set hitting times in Markov chains. ArXiv e-prints.Google Scholar
[4]Oliveira, R. I. (2011) Mixing and hitting times for finite Markov chains. Electron. J. Probab. 17, no. 70, 1–12.Google Scholar
[5]Levin, D. A., Peres, Y. and Wilmer, E. L. (2009) Markov Chains and Mixing Times, AMS.Google Scholar
[6]Lyons, R. and OveisGharan, S. Gharan, S. (2012) Sharp bounds on random walk eigenvalues via spectral embedding. ArXiv e-prints.Google Scholar
[7]Oliveira, R. I. (2012) On the coalescence time of reversible random walks. Trans. Amer. Math. Soc. 364 2109–2128.Google Scholar
[8]Peres, Y. and Sousi, P. Mixing times are hitting times of large sets. J. Theoret. Probab., to appear.Google Scholar
[9]Peres, Y. and Sousi, P. (2012) An isoperimetric inequality for the Wiener sausage. Geom. Funct. Anal. 22 1000–1014.Google Scholar