# Quantitative estimates of discrete harmonic measures

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 $\b \to \infty$.

## 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 $\b \to \infty$.

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