On the support of harmonic measure for the random walk (Q1366938)

Consider simple random walk on the \(d\)-dimensional cube \(Q^d(n)=\{0,\ldots,n-1\}^d\), starting at \(v\in Q^d(n)\), and denote by \(\mu_{v,A}\) the harmonic measure of \(A\subset Q^d(n)\), that is, \(\mu_{v,A}(S)\) is the probability that the walker hits \(A\) first in \(S\subset A\). The author proves that, for any \(\varepsilon>0\) and sufficiently large \(n\), there is, for any \(v\in Q^d(n)\) and \(A\subset Q^d(n)\), a subset \(S(v,A)\) of \(A\) with \(\# S(v,A)\leq \varepsilon n^d\) such that \(\mu_{v,A}(S(v,A))\geq 1-\varepsilon\). This is an extension of a two-dimensional result by Øksendal.