Thermodynamic limit for the invariant measures in supercritical zero range processes

• Published in: Probability theory and related fields Vol. 145; no. 1-2; pp. 175 - 188
• Main Authors: Armendáriz, Inés, Loulakis, Michail
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.09.2009; Springer; Springer Nature B.V
• Subjects: Applied mathematics; Computational mathematics; Condensation; Density; Economics; Exact sciences and technology; Finance; General topics; Insurance; Invariants; Limit theorems; Management; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Nonparametric inference; Operations Research/Decision Theory; Phase transitions; Probability; Probability and statistics; Probability Theory and Stochastic Processes; Probability theory on algebraic and topological structures; Quantitative Finance; Sciences and techniques of general use; Statistics; Statistics for Business; Studies; Theoretical; Thermodynamics; Topological manifolds; Equivalence of ensembles; 60F10; 82C22; Zero range processes; Subexponential distributions; Large deviations; 60K35; Condensation; Density estimation; Fluctuations; Probability theory; Order statistic; Critical density; Joint distribution; Convergence; Limit theorem; Zero range processes 60K35; Invariant measure; Particle system; Thermodynamic limit; Law of large numbers; Marginal distribution; Supercritical process
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-008-0165-7

We prove a strong form of the equivalence of ensembles for the invariant measures of zero range processes conditioned to a supercritical density of particles. It is known that in this case there is a single site that accomodates a macroscopically large number of the particles in the system. We show that in the thermodynamic limit the rest of the sites have joint distribution equal to the grand canonical measure at the critical density. This improves the result of Großkinsky, Schütz and Spohn, where convergence is obtained for the finite dimensional marginals. We obtain as corollaries limit theorems for the order statistics of the components and for the fluctuations of the bulk.