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

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)].