Zero-range process in random environment

Christophe Bahadoran (1) , Thomas S. Mountford (2) , K. Ravishankar (3) , Ellen Saada (4, 5)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal
2 EPFL - Institut de Mathématiques
3 NYU–ECNU Institute of Mathematical Sciences at NYU Shanghai
4 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
5 CNRS - Centre National de la Recherche Scientifique

We survey our recent articles dealing with one dimensional attractive zero range processes moving under site disorder. We suppose that the underlying random walks are biased to the right and so hyperbolic scaling is expected. Under the conditions of our model the process admits a maximal invariant measure. The initial focus of the project was to find conditions on the initial law to entail convergence in distribution to this maximal distribution, when it has a finite density. Somewhat surprisingly, necessary and sufficient conditions were found. In this part hydrody-namic results were employed chiefly as a tool to show distributional convergence but subsequently we developed a theory for hydrodynamic limits treating profiles possessing densities that did not admit corresponding equilibria. Finally we derived strong local equilibrium results.

Mots clés: Asymmetric attractive zero-range process; site disorder; phase transition; condensation; hydrodynamic limit; strong local equilibrium; large-time convergence

Domaines: Probabilités [math.PR]

Christophe Bahadoran, Thomas S. Mountford, K. Ravishankar, Ellen Saada. Zero-range process in random environment. Cédric Bernardin, François Golse, Patrícia Gonçalves, Valeria Ricci, Ana Jacinta Soares (Editors). From Particle Systems to Partial Differential Equations; International Conference, Particle Systems and PDEs VI, VII and VIII, 2017–2019, Springer Nature Switzerland, pp.51-77, 2021, Springer Proceedings in Mathematics & Statistics, Volume 352, 978-3-030-69783-9. ⟨hal-02914876⟩