Kyoto Journal of Mathematics

Cover times for sequences of reversible Markov chains on random graphs

Yoshihiro Abe

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We provide conditions that classify sequences of random graphs into two types in terms of cover times. One type (type 1) is the class of random graphs on which the cover times are of the order of the maximal hitting times scaled by the logarithm of the size of vertex sets. The other type (type 2) is the class of random graphs on which the cover times are of the order of the maximal hitting times. The conditions are described by some parameters determined by random graphs: the volumes, the diameters with respect to the resistance metric, and the coverings or packings by balls in the resistance metric. We apply the conditions to and classify a number of examples, such as supercritical Galton–Watson trees, the incipient infinite cluster of a critical Galton–Watson tree, and the Sierpinski gasket graph.

Article information

Source
Kyoto J. Math., Volume 54, Number 3 (2014), 555-576.

Dates
First available in Project Euclid: 14 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1408020878

Digital Object Identifier
doi:10.1215/21562261-2693442

Mathematical Reviews number (MathSciNet)
MR3263552

Zentralblatt MATH identifier
1338.60181

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 05C80: Random graphs [See also 60B20]

Citation

Abe, Yoshihiro. Cover times for sequences of reversible Markov chains on random graphs. Kyoto J. Math. 54 (2014), no. 3, 555--576. doi:10.1215/21562261-2693442. https://projecteuclid.org/euclid.kjm/1408020878


Export citation

References

  • [1] Y. Abe, Cover times for sequences of reversible Markov chains on random graphs (extended version), preprint, arXiv:1206.0398v3 [math.PR].
  • [2] D. J. Aldous, Random walk covering of some special trees, J. Math. Anal. Appl. 157 (1991), 271–283.
  • [3] K. B. Athreya and P. E. Ney, Branching Processes, reprint of the 1972 original, Dover, Mineola, N.Y., 2004.
  • [4] M. T. Barlow, “Diffusions on Fractals” in Lectures in Probability Theory and Statistics (Saint-Flour, France, 1995), Lecture Notes in Math. 1690, Springer, Berlin, 1998.
  • [5] M. T. Barlow, J. Ding, A. Nachmias, and Y. Peres, The evolution of the cover time, Combin. Probab. Comput. 20 (2011), 331–345.
  • [6] I. Benjamini and G. Kozma, A resistance bound via an isoperimetric inequality, Combinatorica 25 (2005), 645–650.
  • [7] I. Benjamini and E. Mossel, On the mixing time of a simple random walk on the super critical percolation cluster, Probab. Theory Related Fields 125 (2003), 408–420.
  • [8] A. K. Chandra, P. Raghavan, W. L. Ruzzo, R. Smolensky, and P. Tiwari, The electrical resistance of a graph captures its commute and cover times, Comput. Complexity 6 (1996/1997), 312–340.
  • [9] C. Cooper and A. Frieze, The cover time of the giant component of a random graph, Random Structures Algorithms 32 (2008), 401–439.
  • [10] D. A. Croydon, Random walk on the range of random walk, J. Stat. Phys. 136 (2009), 349–372.
  • [11] D. A. Croydon, B. M. Hambly, and T. Kumagai, Convergence of mixing times for sequences of random walks on finite graphs, Electron. J. Probab. 17 (2012), no. 3.
  • [12] D. A. Croydon and T. Kumagai, Random walks on Galton–Watson trees with infinite variance offspring distribution conditioned to survive, Electron. J. Probab. 13 (2008), 1419–1441.
  • [13] J. Ding, J. R. Lee, and Y. Peres, Cover times, blanket times, and majorizing measures, Ann. of Math. (2) 175 (2012), 1409–1471.
  • [14] J. Jonasson, On the cover time for random walks on random graphs, Combin. Probab. Comput. 7 (1998), 265–279.
  • [15] J. Jonasson and O. Schramm, On the cover time of planar graphs, Electron. Commun. Probab. 5 (2000), 85–90.
  • [16] H. Kesten, Subdiffusive behavior of random walk on a random cluster, Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), 425–487.
  • [17] M. Ledoux and M. Talagrand, Probability in Banach Spaces, Ergeb. Math. Grenzgeb. (3) 23, Springer, Berlin, 1991.
  • [18] D. A. Levin, Y. Peres, and E. L. Wilmer, Markov Chains and Mixing Times, Amer. Math. Soc., Providence, 2009.
  • [19] P. Matthews, Covering problems for Brownian motion on spheres, Ann. Probab. 16 (1988), 189–199.
  • [20] A. G. Pakes, Some new limit theorems for the critical branching process allowing immigration, Stochastic Processes Appl. 3 (1975), 175–185.
  • [21] G. Pete, A note on percolation on $\mathbb{Z}^{d}$: Isoperimetric profile via exponential cluster repulsion, Electron. Commun. Probab. 13 (2008), 377–392.
  • [22] S. Sheffield, Gaussian free fields for mathematicians, Probab. Theory Related Fields 139 (2007), 521–541.
  • [23] D. Shiraishi, Exact value of the resistance exponent for four dimensional random walk trace, Probab. Theory Related Fields 153 (2012), 191–232.
  • [24] R. S. Strichartz, Differential Equations on Fractals: A Tutorial, Princeton Univ. Press, Princeton, 2006.
  • [25] M. Talagrand, The Generic Chaining: Upper and Lower Bounds of Stochastic Processes, Springer Monogr. Math., Springer, Berlin, 2005.