 00378 E. Bolthausen, K. MuenchBerndl
 Quantitative estimates of discrete harmonic measures
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Sep 27, 00

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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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