Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Ergodicity of a system of interacting random walks with asymmetric interaction

Luisa Andreis, Amine Asselah, and Paolo Dai Pra

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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.


Nous étudions $N$ marches aléatoires interagissantes sur les entiers naturels. Chaque particule a une dérive $\delta$ vers l’infini, une réflexion à l’origine, ainsi qu’une dérive vers les particules de positions plus petites. Nous montrons que ce système de champ moyen inhomogène est ergodique lorsque l’interaction est assez forte. Nous nous concentrons sur ce dernier régime, et mettons en lumière le rôle des empilements de particules sur un même site, phénomène absent dans les modèles continus.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 1 (2019), 590-606.

Received: 23 February 2017
Revised: 8 February 2018
Accepted: 23 February 2018
First available in Project Euclid: 18 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C44: Dynamics of disordered systems (random Ising systems, etc.)

Interacting particle systems Mean-field interaction Non-reversibility


Andreis, Luisa; Asselah, Amine; Dai Pra, Paolo. Ergodicity of a system of interacting random walks with asymmetric interaction. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 1, 590--606. doi:10.1214/18-AIHP893.

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