This work addresses potential theoretic questions for the standard nearest neighbor random walk on the hypercube ${-1,+1}^N$. For a large class of subsets $Asubset{-1,+1}^N$ we give precise estimates for the harmonic measure of $A$, the mean hitting time of $A$, and the Laplace transform of this hitting time. In particular, we give precise sufficient conditions for the harmonic measure to be asymptotically uniform, and for the hitting time to be asymptotically exponentially distributed, as $Nightarrowinfty$. Our approach relies on a $d$-dimensional extension of the Ehrenfest urn scheme called lumping and covers the case where $d$ is allowed to diverge with $N$ as long as $dleq lpha_0rac{N}{log N}$ for some constant $0展开▼
机译：这项工作解决了关于超立方体$ {-1，+ 1 } ^ N $上标准最近邻居随机游走的潜在理论问题。对于一大类子集$ A subset {-1，+ 1 } ^ N $，我们给出了谐波量度$ A $，平均命中时间$ A $以及该值的Laplace变换的精确估计打时间。特别是，我们给出了精确的充分条件，使得调和量度渐近一致，并且命中时间渐近地呈指数分布，如$ N rightarrow infty $。我们的方法依赖于Ehrenfest骨灰盒计划的$ d $维扩展，称为集总，并涵盖了只要$ d leq alpha_0 frac {N} { log N} $表示常数$ 0 < alpha_0 <1 $。
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