Open Access
2008 Elementary potential theory on the hypercube.
Véronique Gayrard, Gérard Ben Arous
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Electron. J. Probab. 13: 1726-1807 (2008). DOI: 10.1214/EJP.v13-527


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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Véronique Gayrard. Gérard Ben Arous. "Elementary potential theory on the hypercube.." Electron. J. Probab. 13 1726 - 1807, 2008.


Accepted: 4 October 2008; Published: 2008
First available in Project Euclid: 1 June 2016

zbMATH: 1187.82105
MathSciNet: MR2448129
Digital Object Identifier: 10.1214/EJP.v13-527

Primary: 82C44
Secondary: 60K35

Keywords: lumping , random walk on hypercubes

Vol.13 • 2008
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