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Elementary potential theory on the hypercube.


 
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1. Title Title of document Elementary potential theory on the hypercube.
 
2. Creator Author's name, affiliation, country Véronique Gayrard; CNRS
 
2. Creator Author's name, affiliation, country Gérard Ben Arous; Courant Institute for mathematical Sciences, NYU
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) random walk on hypercubes, lumping.
 
3. Subject Subject classification 82C44, 60K35
 
4. Description Abstract This work addresses potential theoretic questions for the standard nearest neighbor random walk on the hypercube $\{-1,+1\}^N$. For a large class of subsets $A\subset\{-1,+1\}^N$ we give precise estimates for the harmonic measure of $A$, the mean hitting time of $A$, and the Laplace transform of this hitting time. In particular, we give precise sufficient conditions for the harmonic measure to be asymptotically uniform, and for the hitting time to be asymptotically exponentially distributed, as $N\rightarrow\infty$. Our approach relies on a $d$-dimensional extension of the Ehrenfest urn scheme called lumping and covers the case where $d$ is allowed to diverge with $N$ as long as $d\leq \alpha_0\frac{N}{\log N}$ for some constant $0<\alpha_0<1$.
 
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7. Date (YYYY-MM-DD) 2008-10-04
 
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/527
 
10. Identifier Digital Object Identifier 10.1214/EJP.v13-527
 
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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