Sprinkled Decoupling for Hammersley's Process

2025/06/23 by Leandro P. R. Pimentel, Pimentel, Leandro P. R., Roberto Viveros +1
Decision Sciences · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Probability and Risk Models

paper · pdf · doi:10.48550/arxiv.2506.19008

openalex publication_date 2025/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Hilário, 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.

Citations

• A sprinkled decoupling inequality for Gaussian processes and applications
• Cylinders' percolation: decoupling and applications
• Evolution of a passive particle in a one-dimensional diffusive environment
• Coalescence estimates for the corner growth model with exponential weights
• Random walk on random walks: higher dimensions
• Local Behavior of Airy Processes
• Sharpness of the phase transition for continuum percolation in R²
• Random Walk on Random Walks
• On the local fluctuations of last-passage percolation models
• Hammersley’s process with sources and sinks

Related

• Hammersley’s process with sources and sinks
• Sharp threshold for the ballisticity of the random walk on the exclusion process
• Polynomial ballisticity conditions in mixing environments
• Law of large numbers for ballistic random walks in dynamic random environments under lateral decoupling
• Busemann functions and the speed of a second class particle in the rarefaction fan