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Random walk with long-range constraints


 
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1. Title Title of document Random walk with long-range constraints
 
2. Creator Author's name, affiliation, country Yinon Spinka; Tel-Aviv University; Israel
 
2. Creator Author's name, affiliation, country Ron Peled; Tel Aviv Unviersity; Israel
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Random walk, random graph homomorphism, phase transition, Lipschitz function
 
3. Subject Subject classification 82B41, 60C05, 82B20, 82B26, 60D05, 05A16, 60K35
 
4. Description Abstract We consider a model of a random height function with long-range constraints on a discrete segment. This model was suggested by Benjamini, Yadin and Yehudayoff and is a generalization of simple random walk. The random function is uniformly sampled from all graph homomorphisms from the graph $P_{n,d}$ to the integers $\mathbb{Z}$, where the graph $P_{n,d}$ is the discrete segment $\{0,1,\ldots, n\}$ with edges between vertices of different parity whose distance is at most $2d+1$. Such a graph homomorphism can be viewed as a height function whose values change by exactly one along edges of the graph $P_{n,d}$. We also consider a similarly defined model on the discrete torus.

Benjamini, Yadin and Yehudayoff conjectured that this model undergoes a phase transition from a delocalized to a localized phase when $d$ grows beyond a threshold $c\log n$. We establish this conjecture with the precise threshold $\log_2 n$. Our results provide information on the typical range and variance of the height function for every given pair of $n$ and $d$, including the critical case when $d-\log_2 n$ tends to a constant.

In addition, we identify the local limit of the model, when $d$ is constant and $n$ tends to infinity, as an explicitly defined Markov chain.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Israeli Science Foundation and Marie Curie
 
7. Date (YYYY-MM-DD) 2014-06-23
 
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/3060
 
10. Identifier Digital Object Identifier 10.1214/EJP.v19-3060
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 19
 
12. Language English=en en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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