Recurrence of locally perturbed random walks with long jumps

We study the effect of local perturbations on the recurrence of random walks with long jumps. Such walks serve as discrete models for infinite-horizon Lorentz processes, in which a particle can take arbitrarily long steps in specific directions. Motivated by a question of Sinai in the finite-horizon case and its extension by Szász to the infinite-horizon setting, we give recurrence and transience criteria for long-jump walks on Z2 and certain classes of graphs, and we prove that local perturbations in a bounded region do not change the recurrence property. Our proofs combine the Markov chain approach with the electrical network method, making the arguments transparent to a broad audience in probability.