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Limiting behavior for the distance of a random walk


 
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1. Title Title of document Limiting behavior for the distance of a random walk
 
2. Creator Author's name, affiliation, country Nathanael Berestycki; University of Cambridge
 
2. Creator Author's name, affiliation, country Rick Durrett; Cornell University
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) random walk, phase transition, adjacent transpositions, random regular graphs, riffle shuffles
 
3. Subject Subject classification 60C05, 60G50, 60J10
 
4. Description Abstract In this paper we study some aspects of the behavior of random walks on large but finite graphs before they have reached their equilibrium distribution. This investigation is motivated by a result we proved recently for the random transposition random walk: the distance from the starting point of the walk has a phase transition from a linear regime to a sublinear regime at time $n/2$. Here, we study the examples of random 3-regular graphs, random adjacent transpositions, and riffle shuffles. In the case of a random 3-regular graph, there is a phase transition where the speed changes from 1/3 to 0 at time $3log_2 n$. A similar result is proved for riffle shuffles, where the speed changes from 1 to 0 at time $log_2 n$. Both these changes occur when a distance equal to the average diameter of the graph is reached. However in the case of random adjacent transpositions, the behavior is more complex. We find that there is no phase transition, even though the distance has different scalings in three different regimes.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Both authors were partially supported by a joint NSF-NIGMS grant DMS-0201037.
 
7. Date (YYYY-MM-DD) 2008-03-10
 
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/490
 
10. Identifier Digital Object Identifier 10.1214/EJP.v13-490
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 13
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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