• Corpus ID: 252383384

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

@inproceedings{Baumler2022RecurrenceAT,
  title={Recurrence and transience of symmetric random walks with long-range jumps},
  author={Johannes Baumler},
  year={2022}
}

• J. Baumler
• Published 20 September 2022
• Mathematics

. Let X 1 , X 2 , . . . be i.i.d. random variables with values in Z d satisfying P ( X 1 = x ) = P ( X 1 = − x ) = Θ ( k x k − s ) for some s > d . We show that the random walk defined by S n = P nk =1 X k is recurrent for d ∈ { 1 , 2 } and s ≥ 2 d , and transient otherwise. This also shows that for an electric network in dimension d ∈ { 1 , 2 } the condition c { x,y } ≤ C k x − y k − 2 d implies recurrence, whereas c { x,y } ≥ c k x − y k − s for some c > 0 and s < 2 d 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)].