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Hitting probabilities of random walks on $${\mathbb{Z}}^ d$$. (English) Zbl 0626.60067
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}$
Reviewer: Mufa Chen

##### MSC:
 60G50 Sums of independent random variables; random walks 60K35 Interacting random processes; statistical mechanics type models; percolation theory
##### Keywords:
hitting probability; diffusion limited aggregation
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