Ergodicity of a system of interacting random walks with asymmetric interaction

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We study N interacting random walks on the positive integers. Each particle has drift δ 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.