论文标题
与远程跳跃的对称随机步行的复发和瞬态
Recurrence and transience of symmetric random walks with long-range jumps
论文作者
论文摘要
令$ x_1，x_2，\ ldots $ be i.i.d.随机变量带有$ \ Mathbb {z}^d $满足$ \ mathbb {p} \ left（x_1 = x \ right）= \ Mathbb {p} \ left（x_1 = -x = -x \ orir）我们表明，由$ s_n = \ sum_ {k = 1}^{n} x_k $定义的随机步行是$ d \ in \ {1,2 \} $和$ s \ geq 2d $的$ d \ n} x_k $，否则。这还表明，对于尺寸的电网络$ d \ in \ {1,2 \} $，条件$ c _ {\ {x，y \}} \ leq c \ | x-y \ | x-y \ |^{ - 2d} $暗示重新恢复，而$ c _ { \ | x-y \ |^{ - s} $对于某些$ c> 0 $，$ s <2d $表示瞬变。这一事实以前已经知道，但是我们提供了仅使用电网的新证明。我们还使用这些结果来显示某些远程渗滤簇上随机步行的复发。特别是，我们显示了几种二维权重依赖性随机连接模型的复发，这是Gracar等人先前研究的。 [电子。 J. Probab。 27。1-31（2022）]。
Let $X_1, X_2, \ldots$ be i.i.d. random variables with values in $\mathbb{Z}^d$ satisfying $\mathbb{P} \left(X_1=x\right) = \mathbb{P} \left(X_1=-x\right) = Θ\left(\|x\|^{-s}\right)$ for some $s>d$. We show that the random walk defined by $S_n = \sum_{k=1}^{n} X_k$ is recurrent for $d\in \{1,2\}$ and $s \geq 2d$, and transient otherwise. This also shows that for an electric network in dimension $d\in \{1,2\}$ the condition $c_{\{x,y\}} \leq C \|x-y\|^{-2d}$ implies recurrence, whereas $c_{\{x,y\}} \geq c \|x-y\|^{-s}$ for some $c>0$ and $s<2d$ implies transience. This fact was already previously known, but we give a new proof of it that uses only electric networks. We also use these results to show the recurrence of random walks on certain long-range percolation clusters. In particular, we show recurrence for several cases of the two-dimensional weight-dependent random connection model, which was previously studied by Gracar et al. [Electron. J. Probab. 27. 1-31 (2022)].
