Recurrence and transience of symmetric random walks with long-range jumps

TL;DR

This paper characterizes when symmetric random walks with long-range jumps are recurrent or transient based on the decay rate of jump probabilities, providing new electric network proofs and applications to percolation clusters.

Contribution

It offers a new electric network-based proof for recurrence and transience criteria of long-range random walks, and applies these results to specific percolation models.

Findings

Random walk recurrence for $d ext{ in } extstylerace{1,2}$ when $s ext{ and } d$ satisfy certain conditions.

Transience of the walk when the decay rate $s$ is below a threshold related to the dimension.

Recurrence of walks on long-range percolation clusters and weight-dependent random connection models.

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)].