Electronic Communications in Probability

Mixing time for the random walk on the range of the random walk on tori

Jiří Černý and Artem Sapozhnikov

Full-text: Open access

Abstract

Consider the subgraph of the discrete $d$-dimensional torus of size length $N$, $d\geq 3$, induced by the range of the simple random walk on the torus run until the time $uN^d$. We prove that for all $d\geq 3$ and $u>0$, the mixing time for the random walk on this subgraph is of order $N^2$ with probability at least $1 - Ce^{-(\log N)^2}$.

Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 26, 10 pp.

Dates
Received: 15 December 2015
Accepted: 4 March 2016
First available in Project Euclid: 10 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1457617917

Digital Object Identifier
doi:10.1214/16-ECP4750

Mathematical Reviews number (MathSciNet)
MR3485395

Zentralblatt MATH identifier
1338.60240

Subjects
Primary: 60K37: Processes in random environments 58J35: Heat and other parabolic equation methods

Keywords
Random walk mixing time isoperimetric inequality random interlacements coupling

Rights
Creative Commons Attribution 4.0 International License.

Citation

Černý, Jiří; Sapozhnikov, Artem. Mixing time for the random walk on the range of the random walk on tori. Electron. Commun. Probab. 21 (2016), paper no. 26, 10 pp. doi:10.1214/16-ECP4750. https://projecteuclid.org/euclid.ecp/1457617917


Export citation

References

  • [1] Martin T. Barlow and Edwin A. Perkins, Symmetric Markov chains in $\mathbb Z^d$: How fast can they move?, Probab. Theory Related Fields 82 (1989), no. 1, 95–108.
  • [2] Jiří Černý and Serguei Popov, On the internal distance in the interlacement set, Electron. J. Probab. 17 (2012), no. 29, 25.
  • [3] Jiří Černý and Augusto Teixeira, Random walks on torus and random interlacements: Macroscopic coupling and phase transition, to appear in Ann. Appl. Probab. arXiv:1411.7795
  • [4] Alexander Drewitz, Balázs Ráth, and Artëm Sapozhnikov, On chemical distances and shape theorems in percolation models with long-range correlations, J. Math. Phys. 55 (2014), no. 8, 083307, 30.
  • [5] B. Morris and Yuval Peres, Evolving sets, mixing and heat kernel bounds, Probab. Theory Related Fields 133 (2005), no. 2, 245–266.
  • [6] Serguei Popov and Augusto Teixeira, Soft local times and decoupling of random interlacements, J. Eur. Math. Soc. 17 (2015), no. 10, 2545–2593.
  • [7] Eviatar Procaccia, Ron Rosenthal, and Artem Sapozhnikov, Quenched invariance principle for simple random walk on clusters in correlated percolation models, to appear in Probab. Theory Related Fields, arXiv:1310.4764
  • [8] Eviatar B. Procaccia and Eric Shellef, On the range of a random walk in a torus and random interlacements, Ann. Probab. 42 (2014), no. 4, 1590–1634.
  • [9] Balázs Ráth and Artëm Sapozhnikov, The effect of small quenched noise on connectivity properties of random interlacements, Electron. J. Probab. 18 (2013), no. 4, 20.
  • [10] Artem Sapozhnikov, Random walks on infinite percolation clusters in models with long-range correlations, to appear in Ann. Probab., arXiv:1410.0605
  • [11] Alain-Sol Sznitman, Vacant set of random interlacements and percolation, Ann. of Math. (2) 171 (2010), no. 3, 2039–2087.
  • [12] Alain-Sol Sznitman, Decoupling inequalities and interlacement percolation on $G\times \mathbb Z$, Invent. Math. 187 (2012), no. 3, 645–706.
  • [13] Augusto Teixeira, On the size of a finite vacant cluster of random interlacements with small intensity, Probab. Theory Related Fields 150 (2011), no. 3–4, 529–574.
  • [14] Augusto Teixeira and David Windisch, On the fragmentation of a torus by random walk, Comm. Pure Appl. Math. 64 (2011), no. 12, 1599–1646.