Quantitative estimates of discrete harmonic measures

Abstract

A theorem of Bourgain states that the harmonic measure for a domain in ${\mathbb{R}}^d$ 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 ${\mathbb{Z}}^d$, $d\ge 2$. By refining the argument, we prove that for all $\b>0$ there exists $ρ(d,\b)<d$ and $N(d,\b)$, such that for any $n>N(d,\b)$, any $x\in {\mathbb{Z}}^d$, and any $A\subset \{1,...,n{\}}^d$ $$ | \{y\in\Z^d\colon ν_{A,x}(y) \geq n^{-\b} \}| \leq n^{ρ(d,\b)}, $$ 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 $\b \to \infty$.

16 pages, 2 figures. Part (B) of the theorem is new