Bounded-hop percolation

TL;DR

This paper studies the minimum number of hops needed in a two-type continuum percolation model relevant to wireless networks, providing asymptotic probabilities and distributional limits as network density increases.

Contribution

It introduces a novel analysis of hop distribution in a two-type continuum percolation model, linking Euclidean and chemical distances in the supercritical regime.

Findings

Derived the distributional limit of rescaled minimum hops as intensity grows.

Explicit expression for the probability of connection within a given number of hops.

Established the relationship between Euclidean and chemical distances in the model.

Abstract

Motivated by an application in wireless telecommunication networks, we consider a two-type continuum-percolation problem involving a homogeneous Poisson point process of users and a stationary and ergodic point process of base stations. Starting from a randomly chosen point of the Poisson point process, we investigate distribution of the minimum number of hops that are needed to reach some point of the second point process. In the supercritical regime of continuum percolation, we use the close relationship between Euclidean and chemical distance to identify the distributional limit of the rescaled minimum number of hops that are needed to connect a typical Poisson point to a point of the second point process as its intensity tends to infinity. In particular, we obtain an explicit expression for the asymptotic probability that a typical Poisson point connects to a point of the second point process in a given number of hops.