How can a clairvoyant particle escape the exclusion process?

TL;DR

This paper investigates a pursuit problem where a target with predictive capabilities can evade detection in a dynamic particle system, demonstrating conditions under which indefinite evasion is possible.

Contribution

It introduces a novel renormalisation scheme for dependent percolation models and applies it to prove the target can escape detection with positive probability for large enough movement range.

Findings

Target can avoid detection forever for large R and certain densities

Develops a general renormalisation scheme for dependent percolation models

Establishes space-time decoupling for the exclusion process

Abstract

We study a detection problem in the following setting: On the one-dimensional integer lattice, at time zero, place nodes on each site independently with probability $\rho \in [0,1)$ and let them evolve as a simple symmetric exclusion process. At time zero, place a target at the origin. The target moves only at integer times, and can move to any site that is within distance $R$ from its current position. Assume also that the target can predict the future movement of all nodes. We prove that, for $R$ large enough (depending on the value of $\rho$) it is possible for the target to avoid detection forever with positive probability. The proof of this result uses two ingredients of independent interest. First we establish a renormalisation scheme that can be used to prove percolation for dependent oriented models under a certain decoupling condition. This result is general and does not rely on the specifities of the model. As an application, we prove our main theorem for different dynamics, such as independent random walks and independent renewal chains. We also proof existence of oriented percolation for random interlacements and for its vacant set for large dimensions . The second step of the proof is a space-time decoupling for the exclusion process.