Random walk on a population of random walkers

TL;DR

This paper studies the stochastic process of an excitation moving among a population of random walkers on various substrates, providing analytical results and numerical validation for long-time, low-density regimes.

Contribution

It introduces a mapping technique to analyze the excitation's dynamics among random walkers, extending understanding across different substrate topologies.

Findings

Analytical expressions for excitation dynamics at long times.

Validation of theoretical results with numerical simulations.

Insights into how substrate topology affects excitation movement.

Abstract

We consider a population of $N$ labeled random walkers moving on a substrate, and an excitation jumping among the walkers upon contact. The label $\mathcal{X}(t)$ of the walker carrying the excitation at time $t$ can be viewed as a stochastic process, where the transition probabilities are a stochastic process themselves. Upon mapping onto two simpler processes, the quantities characterizing $\mathcal{X}(t)$ can be calculated in the limit of long times and low walkers density. The results are compared with numerical simulations. Several different topologies for the substrate underlying diffusion are considered.