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# Coupling, Attractiveness and Hydrodynamics for Conservative Particle Systems

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### Abstract

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 $$(\xi_t,\zeta_t)_{t\geq 0}$$ satisfies: (A) if $$\xi_0\leq\zeta_0$$ (coordinate-wise), then for all $$t\geq 0$$, $$\xi_t\leq\zeta_t$$ a.s. In this paper, we consider generalized misanthrope models which are conservative particle systems on $$\Z^d$$ such that, in each transition, $$k$$ particles may jump from a site $$x$$ to another site $$y$$, with $$k\geq 1$$. 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 $$k\le 2$$) 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.

### Most cited references5

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### Scaling Limits of Interacting Particle Systems

(1999)
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### Stochastic Inequalities on Partially Ordered Spaces

(1977)
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### Processus des misanthropes

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### Author and article information

###### Journal
02 March 2009
2009-10-19
10.1214/09-AIHP347
0903.0316