Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Monotonicity and condensation in homogeneous stochastic particle systems

Thomas Rafferty, Paul Chleboun, and Stefan Grosskinsky

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We study stochastic particle systems that conserve the particle density and exhibit a condensation transition due to particle interactions. We restrict our analysis to spatially homogeneous systems on fixed finite lattices with stationary product measures, which includes previously studied zero-range or misanthrope processes. All known examples of such condensing processes are non-monotone, i.e. the dynamics do not preserve a partial ordering of the state space and the canonical measures (with a fixed number of particles) are not monotonically ordered. For our main result we prove that condensing homogeneous particle systems with finite critical density are necessarily non-monotone. On fixed finite lattices condensation can occur even when the critical density is infinite, in this case we give an example of a condensing process that numerical evidence suggests is monotone, and give a partial proof of its monotonicity.


Nous étudions un système de particules aléatoires qui conserve la densité et fait apparaître un phénomène de condensation en raison des interactions entre particules. Nous restreignons notre analyse au cas des systèmes homogènes en espace sur un réseau fini avec mesure stationnaire produit, ce qui inclut les cas étudiés précédemment du processus « zero-range » et du processus misanthrope. Tous les exemples connus qui montrent un phénomène de condensation sont non-monotones, c’est-à-dire que la dynamique ne préserve aucun ordre partiel sur l’espace d’états et les mesures canoniques (avec un nombre fixe de particules) ne sont pas ordonnées. Notre résultat principal montre que tout système de particules homogène avec densité critique finie qui montre un phénomène de condensation est nécessairement non monotone. Sur un réseau fini, la condensation peut apparaître même quand la densité critique est finie, dans ce cas nous donnons un exemple d’un processus avec condensation dont une étude numérique montre la monotonie, et nous donnons une preuve partielle de cette monotonie.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 2 (2018), 790-818.

Received: 8 May 2015
Revised: 6 February 2017
Accepted: 19 February 2017
First available in Project Euclid: 25 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C22: Interacting particle systems [See also 60K35] 82C26: Dynamic and nonequilibrium phase transitions (general)

Stochastic particle systems Condensation Monotonicity Stationary product measures


Rafferty, Thomas; Chleboun, Paul; Grosskinsky, Stefan. Monotonicity and condensation in homogeneous stochastic particle systems. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 2, 790--818. doi:10.1214/17-AIHP821.

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