Couplings, attractiveness and hydrodynamics for conservative particle systems

Gobron, Thierry; Saada, Ellen

Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 4, pp. 1132-1177.

L'attractivité est un outil fondamental d'étude des systèmes à une infinité de particules en interaction; la construction du couplage de base est la méthode habituelle pour démontrer cette propriété (par exemple pour l'exclusion simple). Le processus couplé markovien (ξt, ζt)t≥0 obtenu vérifie: (A) si ξ0≤ζ0 (coordonnée par coordonnée), alors pour tout t≥0, ξt≤ζt p.s. Nous considérons dans cet article des systèmes de particules conservatifs sur ℤd qui généralisent le processus des misanthropes en ce que, à chaque transition, k particules peuvent sauter d'un site x vers un autre site y, avec k≥1. Ces modèles incluent l'exclusion simple, où k=1, mais, au-delà de cette valeur, le couplage de base n'est plus valide et il faut une autre construction. Nous obtenons des conditions nécessaires et suffisantes pour l'attractivité sur les taux de transition; nous construisons un processus couplé markovien qui à la fois satisfait (A), et fait décroitre les discrépances entre ses deux marginales. Nous déterminons les probabilités invariantes et invariantes par translation extrémales sous des conditions générales d'irréductibilité. Nous appliquons nos résultats à des exemples incluant un modèle d'exclusion asymétrique à deux espèces avec conservation de la charge (où k≤2) issu d'une dynamique d'interfaces ‘solid-on-solid', et un modèle de batons (où k n'est pas borné) en correspondance avec un processus de Hammersley-Aldous-Diaconis discret généralisé. Nous obtenons la limite hydrodynamique de ces deux modèles unidimensionnels.

Attractiveness is a fundamental tool to study interacting particle systems and the basic coupling construction is a usual route to prove this property, as for instance in simple exclusion. The derived markovian coupled process (ξt, ζt)t≥0 satisfies: (A) if ξ0≤ζ0 (coordinate-wise), then for all t≥0, ξt≤ζt a.s. In this paper, we consider generalized misanthrope models which are conservative particle systems on ℤd such that, in each transition, k particles may jump from a site x to another site y, with k≥1. These models include simple exclusion for which k=1, but, beyond that value, the basic coupling construction is not possible and a more refined one is required. We give necessary and sufficient conditions on the rates to insure attractiveness; we construct a markovian coupled process which both satisfies (A) and makes discrepancies between its two marginals non-increasing. We determine the extremal invariant and translation invariant probability measures under general irreducibility conditions. We apply our results to examples including a two-species asymmetric exclusion process with charge conservation (for which k≤2) which arises from a solid-on-solid interface dynamics, and a stick process (for which k is unbounded) in correspondence with a generalized discrete Hammersley-Aldous-Diaconis model. We derive the hydrodynamic limit of these two one-dimensional models.

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