# Graph distances of continuum long-range percolation

## TL;DR

This paper investigates a continuum long-range percolation model on finite regions of Euclidean space, analyzing how the graph distances change and identifying phase transitions at specific parameter values.

## Contribution

It demonstrates phase transitions in continuum long-range percolation at critical parameters, extending classical lattice results to a Poisson point process setting.

## Key findings

- Phase transitions at s=d and s=2d in the model
- Analysis techniques based on Poisson point process properties
- Analogies with classical lattice long-range percolation

## Abstract

We consider a version of continuum long-range percolation on finite boxes of $\mathbb{R}^d$ in which the vertex set is given by the points of a Poisson point process and each pair of two vertices at distance $r$ is connected with probability proportional to $r^{-s}$ for a certain constant $s$. We explore the graph-theoretical distance in this model. The aim of this paper is to show that this random graph model undergoes phase transitions at values $s=d$ and $s=2d$ in analogy to classical long-range percolation on $\mathbb{Z}^d$, by using techniques which are based on an analysis of the underlying Poisson point process.