Thermodynamic Limit for the Invariant Measures in Supercritical Zero Range Processes

TL;DR

This paper establishes a strong form of the equivalence of ensembles for zero range processes at supercritical density, showing the bulk distribution converges to the grand canonical measure at critical density, with implications for order statistics and fluctuations.

Contribution

It proves a stronger form of ensemble equivalence for zero range processes, extending previous finite-dimensional results to the full distribution in the thermodynamic limit.

Findings

Bulk sites follow the grand canonical measure at critical density

Single site hosts a macroscopically large number of particles

Results include limit theorems for order statistics and fluctuations

Abstract

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 Gro\ss kinsky, Sch\"{u}tz 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.