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

Abstract

Let $X_1,X_2,\dots$ be i.i.d. random variables with values in ${\mathbb{Z}}^d$ satisfying $\mathbb{P}\left(X_1=x\right)=\mathbb{P}\left(X_1=-x\right)=\mathrm{\Theta }\left(\Vert x{\Vert }^{-s}\right)$ 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 $s\ge 2d$, and transient otherwise. This also shows that for an electric network in dimension $d\in \{1,2\}$ the condition $c_{\{x,y\}}\le C\Vert x-y{\Vert }^{-2d}$ implies recurrence, whereas $c_{\{x,y\}}\ge c\Vert x-y{\Vert }^{-s}$ 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)].