# The random walk on the random connection model

## TL;DR

This paper investigates the behavior of random walks on a continuum long-range percolation model, showing recurrence or transience depending on the intensity, and relates findings to classical discrete models.

## Contribution

It extends the understanding of random walk behavior to continuum models with long-range connections, aligning properties with classical discrete percolation results.

## Key findings

- Recurrence for all Poisson intensities
- Transience for sufficiently large intensities
- Addresses a conjecture related to the model

## Abstract

We study the behavior of the random walk in a continuum independent long-range percolation model, in which two given vertices $x$ and $y$ are connected with probability that asymptotically behaves like $|x-y|^{-\alpha}$ with $\alpha>d$, where $d$ denotes the dimension of the underlying Euclidean space. More precisely, focus is on the random connection model in which the vertex set is given by the realization of a homogeneous Poisson point process. We show that this random graph exhibits the same properties as classical discrete long-range percolation models studied in [3] with regard to recurrence and transience of the random walk. The recurrence results are valid for every intensity of the Poisson point process while the transience results hold for large enough intensity. Moreover, we address a question which is related to a conjecture in [16] for this graph.