Greedy Clearing of Persistent Poissonian Dust

TL;DR

This paper analyzes a particle's deterministic movement among Poisson points with marks, showing that even a small density of double marks guarantees the eventual removal of all points.

Contribution

It introduces a model of particle motion on marked Poisson points and proves that any positive density of double marks ensures complete removal of points.

Findings

Any positive density of double marks leads to total removal of points.

The particle's deterministic jumps effectively clear the entire process.

The model demonstrates the impact of mark density on clearing efficiency.

Abstract

Given a Poisson point process on R, assign either one or two marks to each point of this process, independently of the others. We study the motion of a particle that jumps deterministically from its current location to the nearest point of the Poisson point process which still contains at least one mark, and removes one mark per each visit. A point of the Poisson point process which is left with no marks is removed from the system. We prove that the presence of any positive density of double marks leads to the eventual removal of every Poissonian point.