Mathematics > Probability
[Submitted on 20 Sep 2022 (v1), last revised 1 Dec 2022 (this version, v2)]
Title:Recurrence and transience of symmetric random walks with long-range jumps
Authors:Johannes Bäumler
Abstract: Let $X_1, X_2, \ldots$ be i.i.d. random variables with values in $\mathbb{Z}^d$ satisfying $\mathbb{P} \left(X_1=x\right) = \mathbb{P} \left(X_1=-x\right) = \Theta \left(\|x\|^{-s}\right)$ 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)].
Comments: 24 pages, 4 figures. Version 2 contains new results about the random connection model
Subjects: Probability (math.PR)
Cite as: arXiv:2209.09901 [math.PR]
(or arXiv:2209.09901v2 [math.PR] for this version)
From: Johannes Bäumler [view email]
[v1] Tue, 20 Sep 2022 17:59:40 UTC (20 KB)
[v2] Thu, 1 Dec 2022 09:52:59 UTC (27 KB)
math.PR