## Electronic Journal of Probability

### Mixing and relaxation time for random walk on wreath product graphs

#### Abstract

Suppose that $G$ and $H$ are finite, connected graphs, $G$ regular, $X$ is a lazy random walk on $G$ and $Z$ is a reversible ergodic Markov chain on $H$. The generalized lamplighter chain $X^{\diamond}$ associated with $X$ and $Z$ is the random walk on the wreath product $H \wr G$, the graph whose vertices consist of pairs $(f,x)$ where $f=(f_v)_{v\in V(G)}$ is a labeling of the vertices of $G$ by elements of $H$ and $x$ is a vertex in $G$. In each step, $^{\diamond}$* moves from a configuration $(f,x)$ by updating x to y using the transition rule of $X$ and then independently updating both $f_x$ and $f_y$ according to the transition probabilities on $H$; $f_z$ for $z$ different of $x,y$ remains unchanged. We estimate the mixing time of $X^{\diamond}$ in terms of the parameters of $H$ and $G$. Further, we show that the relaxation time of $X^{\diamond}$ is the same order as the maximal expected hitting time of $G$ plus $|G|$ times the relaxation time of the chain on $H$.

#### Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 71, 23 pp.

Dates
Accepted: 30 July 2013
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465064296

Digital Object Identifier
doi:10.1214/EJP.v18-2321

Mathematical Reviews number (MathSciNet)
MR3091717

Zentralblatt MATH identifier
06247240

Rights

#### Citation

Komjáthy, Júlia; Peres, Yuval. Mixing and relaxation time for random walk on wreath product graphs. Electron. J. Probab. 18 (2013), paper no. 71, 23 pp. doi:10.1214/EJP.v18-2321. https://projecteuclid.org/euclid.ejp/1465064296

#### References

• Aldous, David; Diaconis, Persi. Shuffling cards and stopping times. Amer. Math. Monthly 93 (1986), no. 5, 333-348.
• D. Aldous and J. A. Fill, Reversible Markov chains and random walks on graphs, University of California, Berkeley, 2002.
• Barlow, Martin T.; Ding, Jian; Nachmias, Asaf; Peres, Yuval. The evolution of the cover time. Combin. Probab. Comput. 20 (2011), no. 3, 331-345.
• Brummelhuis, M. J. A. M.; Hilhorst, H. J. Covering of a finite lattice by a random walk. Phys. A 176 (1991), no. 3, 387-408.
• Dembo, Amir; Peres, Yuval; Rosen, Jay; Zeitouni, Ofer. Cover times for Brownian motion and random walks in two dimensions. Ann. of Math. (2) 160 (2004), no. 2, 433-464.
• Dembo, Amir; Peres, Yuval; Rosen, Jay; Zeitouni, Ofer. Late points for random walks in two dimensions. Ann. Probab. 34 (2006), no. 1, 219-263.
• Diaconis, Persi; Fill, James Allen. Strong stationary times via a new form of duality. Ann. Probab. 18 (1990), no. 4, 1483-1522.
• Ding, Jian; Lee, James R.; Peres, Yuval. Cover times, blanket times, and majorizing measures. Ann. of Math. (2) 175 (2012), no. 3, 1409-1471.
• Feige, Uriel. A tight lower bound on the cover time for random walks on graphs. Random Structures Algorithms 6 (1995), no. 4, 433-438.
• Häggström, Olle; Jonasson, Johan. Rates of convergence for lamplighter processes. Stochastic Process. Appl. 67 (1997), no. 2, 227-249.
• Hunter, Jeffrey J. Variances of first passage times in a Markov chain with applications to mixing times. Linear Algebra Appl. 429 (2008), no. 5-6, 1135-1162.
• J. Komjáthy, J. Miller, and Y. Peres, Uniform mixing time for random walk on lamplighter graphs, Ann. Inst. Henri Poincaré Probab. Stat. (2013).
• 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
• N. Levy, Mixing time for lamplighter graphs, Master's thesis, University of California, Berkeley, 2006.
• L. Lovász and P. Winkler, Efficient stopping rules for markov chains, Proceedings of the twenty-seventh annual ACM symposium on Theory of computing (New York, NY, USA), STOC '95, ACM, 1995, pp. 76-82.
• Miller, Jason; Peres, Yuval. Uniformity of the uncovered set of random walk and cutoff for lamplighter chains. Ann. Probab. 40 (2012), no. 2, 535-577.
• Peres, Yuval; Revelle, David. Mixing times for random walks on finite lamplighter groups. Electron. J. Probab. 9 (2004), no. 26, 825-845.
• Y. Peres and P. Sousi, Mixing times are hitting times of large sets, http://arxiv.org/abs/1108.0133.
• Winkler, Peter; Zuckerman, David. Multiple cover time. Random Structures Algorithms 9 (1996), no. 4, 403-411.