The greedy walk on an inhomogeneous Poisson process

TL;DR

This paper investigates the behavior of a deterministic greedy walk on an inhomogeneous Poisson process, establishing conditions under which it visits all points and identifying threshold functions for this property.

Contribution

It provides necessary and sufficient conditions for the greedy walk to visit all points on an inhomogeneous Poisson process, including precise threshold functions.

Findings

Conditions for the walk to visit all points are characterized.

Threshold functions for visiting all points are identified.

The results depend on the mean measure of the Poisson process.

Abstract

The greedy walk is a deterministic walk that always moves from its current position to the nearest not yet visited point. In this paper we consider the greedy walk on an inhomogeneous Poisson point process on the real line. Our primary interest is whether the walk visits all points of the point process, and we determine sufficient and necessary conditions on the mean measure of the point process for this to happen. Moreover, we provide precise results on threshold functions for the property of visiting all points.