# Ergodicity of a system of interacting random walks with asymmetric interaction

## TL;DR

This paper investigates the ergodic behavior of a system of interacting random walks with asymmetric interactions, revealing that ergodicity depends on the strength of interaction and highlighting the unique pile-up phenomenon absent in continuous models.

## Contribution

It introduces a new analysis of ergodicity in inhomogeneous mean field systems of interacting random walks with asymmetric interactions, emphasizing the role of interaction strength.

## Key findings

- Ergodicity occurs only when interaction strength exceeds a certain threshold.
- Piles of particles form due to asymmetric interactions, a phenomenon not seen in continuous-space models.
- The system's behavior is significantly influenced by the interplay of drift, reflection, and interaction asymmetry.

## Abstract

We study N interacting random walks on the positive integers. Each particle has drift {\delta} towards infinity, a reflection at the origin, and a drift towards particles with lower positions. This inhomogeneous mean field system is shown to be ergodic only when the interaction is strong enough. We focus on this latter regime, and point out the effect of piles of particles, a phenomenon absent in models of interacting diffusion in continuous space.