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

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) = Theta (||x||(-s)) for some s > d. We show that the random walk defined by S-n = Sigma(n)(k=1) X-k is recurrent for d is an element of{1, 2} and s >= 2d, and transient otherwise. This also shows that for an electric network in dimension d is an element of{1, 2} the condition c({x,y}) <= C||x - y||(-2d) implies recurrence, whereas c({x,y}) >= 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 weightdependent random connection model, which was previously studied by Gracar et al. [Electron. J. Probab. 27. 1-31 (2022)].