Sprinkled Decoupling for Hammersley's Process

TL;DR

This paper establishes a sprinkled decoupling inequality for the stationary Hammersley's process and applies it to analyze a fugitive detection problem and the ballistic behavior of a random walk in a dynamic environment.

Contribution

It introduces a new decoupling inequality for Hammersley's process and uses it to solve problems related to particle evasion and random walk behavior.

Findings

A fugitive can evade particles if their jump range is large.

A random walk in the environment is ballistic under certain conditions.

The decoupling inequality enables analysis of complex particle interactions.

Abstract

In this article we prove a sprinkled decoupling inequality for the stationary Hammersley's interacting particle process. Inspired by the work of Baldasso and Texeira (2018), and Hil\'ario, Kious and Texeira (2020), we apply this inequality to study two distinct problems on the top of this particle process. First, we analyze a detection problem, demonstrating that a fugitive can evade particles, provided that their jump range is sufficiently large. Second, we show that a random walk in a dynamic random environment exhibits ballistic behavior with respect to the characteristic speed of the particle system, under a weak assumption on the probability of being away of this critical speed.