Condensation of the Invariant Measures of the Supercritical Zero Range Processes

Abstract

For \(\alpha \ge 1\), let \(g:{\mathbb {N}}\rightarrow {\mathbb {R}}_+\) be given by \(g(0)=0\), \(g(1)=1\), \(g(k)=(k/k-1)^\alpha \), \(k\ge 2\). Consider the homogeneous zero range process on a discrete set in which a particle jumps from a site x, occupied by k particles, to site y with rate \(g(k)p(y-x)\) for some fixed probability \(p:{\mathbb {Z}}\rightarrow [0,1]\). Armendáriz and Loulakis (Probab Theory Relat Fields 145:175–188, 2009, https://doi.org/10.1007/s00440-008-0165-7) proved a strong form of the equivalence of ensembles for the invariant measure of the supercritical zero range process with \(\alpha >2\). We generalize their result to all \(\alpha \ge 1\).