Recurrence and transience for long-range reversible random walks on a random point process

TL;DR

This paper investigates the recurrence and transience behavior of long-range reversible random walks on random point processes, providing conditions and explicit estimates for different regimes.

Contribution

It establishes almost sure transience and recurrence criteria, and constructs explicit fluxes and resistance estimates for these random walks.

Findings

Proves almost sure transience and recurrence under specific conditions.

Provides effective resistance estimates in recurrent cases.

Constructs finite-energy fluxes in transient regimes.

Abstract

We consider reversible random walks in random environment obtained from symmetric long--range jump rates on a random point process. We prove almost sure transience and recurrence results under suitable assumptions on the point process and the jump rate function. For recurrent models we obtain almost sure estimates on effective resistances in finite boxes. For transient models we construct explicit fluxes with finite energy on the associated electrical network.