### 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 ⊂ {-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 → ∞. 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 ≤?α_{0}N/log N for some constant 0 < α_{0} < 1.

Original language | English (US) |
---|---|

Article number | 59 |

Pages (from-to) | 1726-1807 |

Number of pages | 82 |

Journal | Electronic Journal of Probability |

Volume | 13 |

State | Published - Oct 4 2008 |

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### Keywords

- Lumping
- Random walk on hypercubes

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Electronic Journal of Probability*,

*13*, 1726-1807. [59].

**Elementary potential theory on the hypercube.** / Arous, Gérard Ben; Gayrard, Véronique.

Research output: Contribution to journal › Article

*Electronic Journal of Probability*, vol. 13, 59, pp. 1726-1807.

}

TY - JOUR

T1 - Elementary potential theory on the hypercube

AU - Arous, Gérard Ben

AU - Gayrard, Véronique

PY - 2008/10/4

Y1 - 2008/10/4

N2 - 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 ⊂ {-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 → ∞. 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 ≤?α0N/log N for some constant 0 < α0 < 1.

AB - 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 ⊂ {-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 → ∞. 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 ≤?α0N/log N for some constant 0 < α0 < 1.

KW - Lumping

KW - Random walk on hypercubes

UR - http://www.scopus.com/inward/record.url?scp=54249167187&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=54249167187&partnerID=8YFLogxK

M3 - Article

VL - 13

SP - 1726

EP - 1807

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

M1 - 59

ER -