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Mixing Times for Random Walks on Finite Lamplighter Groups

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Mixing Times for Random Walks on Finite Lamplighter Groups
2.	Creator	Author's name, affiliation, country	Yuval Peres; University of California
2.	Creator	Author's name, affiliation, country	David Revelle; University of California
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	random walks; lamplighter group; mixing time; cover time
3.	Subject	Subject classification	60J10; 60B15
4.	Description	Abstract	Given a finite graph $G$, a vertex of the lamplighter graph $G^\diamondsuit=\mathbb {Z}_2 \wr G$ consists of a zero-one labeling of the vertices of $G$, and a marked vertex of $G$. For transitive $G$ we show that, up to constants, the relaxation time for simple random walk in $G^\diamondsuit$ is the maximal hitting time for simple random walk in $G$, while the mixing time in total variation on $G^\diamondsuit$ is the expected cover time on $G$. The mixing time in the uniform metric on $G^\diamondsuit$ admits a sharp threshold, and equals $|G|$ multiplied by the relaxation time on $G$, up to a factor of $\log |G|$. For $\mathbb {Z}_2 \wr \mathbb {Z}_n^2$, the lamplighter group over the discrete two dimensional torus, the relaxation time is of order $n^2 \log n$, the total variation mixing time is of order $n^2 \log^2 n$, and the uniform mixing time is of order $n^4$. For $\mathbb {Z}_2 \wr \mathbb {Z}_n^d$ when $d\geq 3$, the relaxation time is of order $n^d$, the total variation mixing time is of order $n^d \log n$, and the uniform mixing time is of order $n^{d+2}$. In particular, these three quantities are of different orders of magnitude.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	NSF
7.	Date	(YYYY-MM-DD)	2004-11-17
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/198
10.	Identifier	Digital Object Identifier	10.1214/EJP.v9-198
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 9
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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