Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Hydrodynamic limit for a system of independent, sub-ballistic random walks in a common random environment

Milton Jara and Jonathon Peterson

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Abstract

We consider a system of independent random walks in a common random environment. Previously, a hydrodynamic limit for the system of RWRE was proved under the assumption that the random walks were transient with positive speed (Electron. J. Probab. 15 (2010) 1024–1040). In this paper we instead consider the case where the random walks are transient but with a sublinear speed of the order $n^{\kappa}$ for some $\kappa\in(0,1)$ and prove a quenched hydrodynamic limit for the system of random walks with time scaled by $n^{1/\kappa}$ and space scaled by $n$. The most interesting feature of the hydrodynamic limit is that the influence of the environment does not average out under the hydrodynamic scaling; that is, the asymptotic particle density depends on the specific environment chosen. The hydrodynamic limit for the system of RWRE is obtained by first proving a hydrodynamic limit for a system of independent particles in a directed trap environment.

Résumé

Nous considérons un système de particules indépendantes évoluant dans un milieu aléatoire commun. Auparavant, la limite hydrodynamique de ce système de particules en milieu aléatoire a été obtenue quand les particules sont transientes avec une vitesse positive (Electron. J. Probab. 15 (2010) 1024–1040). Dans cet article nous considérons le cas où les particules sont transientes mais ont une vitesse sous-linéaire d’ordre $n^{\kappa}$ pour $\kappa\in(0,1)$ et nous montrons l’existence d’une limite hydrodynamique du système de particules avec une échelle du temps $n^{1/\kappa}$ et une échelle spatiale $n$. La propriété la plus intéressante de cette limite hydrodynamique est que le milieu n’est pas moyenné par la limite ; c’est-à-dire, la densité asymptotique des particules dépend de la réalisation du milieu choisi. La limite hydrodynamique du système de particules est déduite à partir de la limite hydrodynamique d’un système de particules indépendantes dans un milieu aléatoire dirigé.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 1747-1792.

Dates
Received: 14 January 2016
Revised: 9 May 2016
Accepted: 1 June 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1511773725

Digital Object Identifier
doi:10.1214/16-AIHP770

Mathematical Reviews number (MathSciNet)
MR3729634

Zentralblatt MATH identifier
1382.60121

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K37: Processes in random environments

Keywords
Hydrodynamic limit Random walk in random environment Directed traps

Citation

Jara, Milton; Peterson, Jonathon. Hydrodynamic limit for a system of independent, sub-ballistic random walks in a common random environment. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 1747--1792. doi:10.1214/16-AIHP770. https://projecteuclid.org/euclid.aihp/1511773725


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