Particle detection among Random Walks as a non-reversible Random Interlacements Process

TL;DR

This paper models particle detection in a lattice using non-reversible random interlacements, extending the framework to include non-reversible Markov chains and analyzing detection probabilities in percolation regimes.

Contribution

It introduces a novel extension of random interlacements to non-reversible Markov chains and applies this to particle detection problems in percolation models.

Findings

Detection probability depends on particle density u

Target escape is impossible for large u

Extension of random interlacements to non-reversible walks

Abstract

Consider the random set composed of particles initially distributed on Zd, d >= 2, according to a Poisson point process of intensity u > 0 and moving as independent simple symmetric random walks, the trap particles. We are interested in the detection by these particles of a target particle, initially at the origin and able to move with finite mean speed. The escape strategy for the target particle is to stay inside the infinite cluster of empty sites, assuming u is in the subcritical site percolation regime of particle occupation. By translating the problem to the framework of percolation of Random Interlacements we also prove that for u large enough the target doesn't escape. In doing this we extend the random interlacements formalism in order to allow non reversible random walks. As far a we know this is the first example of Random Interlacements for non-reversible Markov chains.