Electronic Communications in Probability

Sensitivity of mixing times

Jian Ding and Yuval Peres

Full-text: Open access

Abstract

In this note, we demonstrate an instance of bounded-degree graphs of size $n$, for which the total variation mixing time for the random walk is decreased by a factor of $\log n/ \log\log n$ if we multiply the edge-conductances by bounded factors in a certain way.

Article information

Source
Electron. Commun. Probab., Volume 18 (2013), paper no. 88, 6 pp.

Dates
Accepted: 11 November 2013
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465315627

Digital Object Identifier
doi:10.1214/ECP.v18-2765

Mathematical Reviews number (MathSciNet)
MR3141797

Zentralblatt MATH identifier
1307.60051

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

Keywords
mixing time sensitivity geometric bounds

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

Citation

Ding, Jian; Peres, Yuval. Sensitivity of mixing times. Electron. Commun. Probab. 18 (2013), paper no. 88, 6 pp. doi:10.1214/ECP.v18-2765. https://projecteuclid.org/euclid.ecp/1465315627


Export citation

References

  • Aizenman, M.; Holley, R. Rapid convergence to equilibrium of stochastic Ising models in the Dobrushin Shlosman regime. Percolation theory and ergodic theory of infinite particle systems (Minneapolis, Minn., 1984–1985), 1–11, IMA Vol. Math. Appl., 8, Springer, New York, 1987.
  • D. Aldous and J. Fill. Reversible Markov Chains and Random Walks on Graphs. In preparation, available at http://www.stat.berkeley.edu/~aldous/RWG/book.html.
  • Benjamini, Itai. Instability of the Liouville property for quasi-isometric graphs and manifolds of polynomial volume growth. J. Theoret. Probab. 4 (1991), no. 3, 631–637.
  • I. Benjamini, G. Kozma, and N. C. Wormald. The mixing time of the giant component of a random graph. Preprint, available at http://arxiv.org/abs/math/0610459.
  • Diaconis, P.; Saloff-Coste, L. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 (1996), no. 3, 695–750.
  • Fountoulakis, N.; Reed, B. A. Faster mixing and small bottlenecks. Probab. Theory Related Fields 137 (2007), no. 3-4, 475–486.
  • Fountoulakis, N.; Reed, B. A. The evolution of the mixing rate of a simple random walk on the giant component of a random graph. Random Structures Algorithms 33 (2008), no. 1, 68–86.
  • Goel, Sharad; Montenegro, Ravi; Tetali, Prasad. Mixing time bounds via the spectral profile. Electron. J. Probab. 11 (2006), no. 1, 1–26 (electronic).
  • Kozma, Gady. On the precision of the spectral profile. ALEA Lat. Am. J. Probab. Math. Stat. 3 (2007), 321–329.
  • 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
  • Lovász, László; Kannan, Ravi. Faster mixing via average conductance. Annual ACM Symposium on Theory of Computing (Atlanta, GA, 1999), 282–287, ACM, New York, 1999.
  • Morris, B.; Peres, Yuval. Evolving sets, mixing and heat kernel bounds. Probab. Theory Related Fields 133 (2005), no. 2, 245–266.
  • Nachmias, Asaf; Peres, Yuval. Critical random graphs: diameter and mixing time. Ann. Probab. 36 (2008), no. 4, 1267–1286.
  • Y. Peres and P. Sousi. Mixing times are hitting times of large sets. Preprint, available at http://arxiv.org/abs/1108.0133.