Let \(S_ 0,S_ 1,..\). be a simple (nearest neighbor) symmetric random walk on \({\mathbb{Z}}^ d\) and \[ \tau(B) = \inf \{n\geq 0:S_ n\in B\},\quad B\in {\mathbb{Z}}^ d, \]
\[ H_ B(x,y) = \begin{cases} P_ x(\tau(B)<\infty \text{ and } S_{\tau(B)} = y) &\text{if \(d=2\)} \\ P_ x(S_{\tau(B)} = y| \tau(B)<\infty) &\text{if \(d\geq 3\).} \end{cases} \] For a connected set B of vertices in \({\mathbb{Z}}^ d\) which contains the origin, we denote its cardinality by \(| B|\) and set \(r(B)=\max \{| x|:x\in B\}.\)
The author proves that there exist constants C(d), depending on d only, such that, for all \(y\in B,\) \[ \lim_{| x| \to \infty}H_ B(x,y) \leq \begin{cases} C(2)r(B)^{-1/2}& \text{ if \(d=2,\)} \\ C(d)| B|^{1-2/d}& \text{ if \(d\geq 3\).} \end{cases} \]