Abstract
Let be i.i.d. random variables with values in satisfying for some . We show that the random walk defined by is recurrent for and , and transient otherwise. This also shows that for an electric network in dimension the condition implies recurrence, whereas for some and 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)].
Acknowledgments
I thank Yuki Tokushige for making me aware of this problem and for many helpful comments on an earlier version of this paper. I thank Markus Heydenreich for making me aware of the applications of Theorem 1.3 to the random-connection model. I thank Noam Berger and Christian Mönch for useful discussions. I thank an anonymous referee for very many helpful remarks and comments. This work is supported by TopMath, the graduate program of the Elite Network of Bavaria and the graduate center of TUM Graduate School.
Citation
Johannes Bäumler. "Recurrence and transience of symmetric random walks with long-range jumps." Electron. J. Probab. 28 1 - 24, 2023. https://doi.org/10.1214/23-EJP998
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