Gobron, Thierry; Saada, Ellen Couplings, attractiveness and hydrodynamics for conservative particle systems. (English) Zbl 1252.60093 Ann. Inst. Henri Poincaré, Probab. Stat. 46, No. 4, 1132-1177 (2010). Authors’ 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 \(\mathbb{Z}^d\) 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 and 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 \(k\leq 2\)) 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. Reviewer: Mihai Gradinaru (Rennes) Cited in 1 ReviewCited in 12 Documents MSC: 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C22 Interacting particle systems in time-dependent statistical mechanics Keywords:conservative particle systems; attractiveness; couplings; discrepancies; macroscopic stability; hydrodynamic limit; misanthrope process; discrete Hammersley-Aldous-Diaconis process; stick process; solid-on-solid interface dynamics; two-species exclusion model PDFBibTeX XMLCite \textit{T. Gobron} and \textit{E. Saada}, Ann. Inst. Henri Poincaré, Probab. Stat. 46, No. 4, 1132--1177 (2010; Zbl 1252.60093) Full Text: DOI arXiv EuDML References: [1] E. D. Andjel. Invariant measures for the zero range process. Ann. Probab. 10 (1982) 525-547. · Zbl 0492.60096 · doi:10.1214/aop/1176993765 [2] C. Bahadoran, H. Guiol, K. Ravishankar and E. Saada. A constructive approach to Euler hydrodynamics for attractive processes. 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