@article{EJP527,
author = {Véronique Gayrard and Gérard Ben Arous},
title = {Elementary potential theory on the hypercube.},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {13},
year = {2008},
keywords = {random walk on hypercubes, lumping.},
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 $0<\alpha_0<1$.},
pages = {no. 59, 1726-1807},
issn = {1083-6489},
doi = {10.1214/EJP.v13-527},
url = {http://ejp.ejpecp.org/article/view/527}}