Inhomogeneous long-range percolation in the strong decay regime: recurrence in one dimension

TL;DR

This paper establishes a criterion for recurrence in one-dimensional inhomogeneous long-range percolation graphs, emphasizing the role of scarce long-edges, and complements existing results on recurrence and transience.

Contribution

It provides a new sufficient condition for recurrence in one-dimensional long-range percolation models based on edge distribution.

Findings

Recurrence criterion linked to scarcity of long-edges

Complements previous recurrence and transience results

Advances understanding of spatial random graphs in one dimension

Abstract

We provide a sufficient criterion for the recurrence of spatial random graphs on the real line based on the scarceness of long-edges. In particular, this complements earlier recurrence results obtained by Gracar et al. (Electron. J. Probab. 27 (2022)) and a transience criterion derived by the author (Probab. Theory Related Fields 189, no. 3-4 (2024)).