A theorem of Bourgain states that the harmonic measure for a domain in $\R^d$ is supported on a set of Hausdorff dimension strictly less than $d$ \cite{Bourgain}. We apply Bourgain's method to the discrete case, i.e., to the distribution of the first entrance point of a random walk into a subset of $\Z ^d$, $d\geq 2$. By refining the argument, we prove that for all $\b>0$ there exists $\rho (d,\b)<d$ and $N(d,\b)$, such that for any $n>N(d,\b)$, any $x \in \Z^d$, and any $A\subset {1,..., n}^d$ $$ | {y\in\Z^d\colon \nu_{A,x}(y) \geq n^{-\b} }| \leq n^{\rho(d,\b)}, $$ where $\nu_{A,x} (y)$ denotes the probability that $y$ is the first entrance point of the simple random walk starting at $x$ into $A$. Furthermore, $\rho$ must converge to $d$ as $\b \to \infty$.

Bolthausen, E; Münch-Berndl, K (2001). *Quantitative estimates of discrete harmonic measures.* Israel Journal of Mathematics, 124(1):125-141.

## Abstract

A theorem of Bourgain states that the harmonic measure for a domain in $\R^d$ is supported on a set of Hausdorff dimension strictly less than $d$ \cite{Bourgain}. We apply Bourgain's method to the discrete case, i.e., to the distribution of the first entrance point of a random walk into a subset of $\Z ^d$, $d\geq 2$. By refining the argument, we prove that for all $\b>0$ there exists $\rho (d,\b)<d$ and $N(d,\b)$, such that for any $n>N(d,\b)$, any $x \in \Z^d$, and any $A\subset {1,..., n}^d$ $$ | {y\in\Z^d\colon \nu_{A,x}(y) \geq n^{-\b} }| \leq n^{\rho(d,\b)}, $$ where $\nu_{A,x} (y)$ denotes the probability that $y$ is the first entrance point of the simple random walk starting at $x$ into $A$. Furthermore, $\rho$ must converge to $d$ as $\b \to \infty$.

## Citations

## Altmetrics

## Downloads

## Additional indexing

Item Type: | Journal Article, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | 2001 |

Deposited On: | 27 Apr 2010 07:00 |

Last Modified: | 05 Apr 2016 13:25 |

Publisher: | Hebrew University Magnes Press |

ISSN: | 0021-2172 |

Additional Information: | The original publication is available at www.springerlink.com |

Publisher DOI: | https://doi.org/10.1007/BF02772611 |

## Download

TrendTerms displays relevant terms of the abstract of this publication and related documents on a map. The terms and their relations were extracted from ZORA using word statistics. Their timelines are taken from ZORA as well. The bubble size of a term is proportional to the number of documents where the term occurs. Red, orange, yellow and green colors are used for terms that occur in the current document; red indicates high interlinkedness of a term with other terms, orange, yellow and green decreasing interlinkedness. Blue is used for terms that have a relation with the terms in this document, but occur in other documents.

You can navigate and zoom the map. Mouse-hovering a term displays its timeline, clicking it yields the associated documents.