We study the mixing time of the unit-rate zero-range process on the complete graph, in the regime where the number @math of sites tends to infinity while the density of particles per site stabilizes to some limit @math . We prove that the worst-case total-variation distance to equilibrium drops abruptly from @math to @math at time @math . More generally, we determine the mixing time from an arbitrary initial configuration. The answer turns out to depend on the largest initial heights in a remarkably explicit way. The intuitive picture is that the system separates into a slowly evolving solid phase and a quickly relaxing liquid phase. As time passes, the solid phase dissolves into the liquid phase, and the mixing time is essentially the time at which the system becomes completely liquid. Our proof combines meta-stability, separation of timescale, fluid limits, propagation of chaos, entropy, and a spectral estimate by Morris (2006).
The zero-range process has a long history. In the classical setting, the particles evolve on an infinite transitive graph like the lattice @math , and the description of the set of stationary laws constitutes by itself an important question. More recently, hydrodynamic limits and complex phenomena such as metastability and condensation have received a considerable attention in both the mathematical and physical communities. The works are too numerous to be all cited, and we refer the interested reader to the comprehensive survey @cite_0 and the references therein.
@cite_0
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We review recent progress on the zero-range process, a model of interacting particles which hop between the sites of a lattice with rates that depend on the occupancy of the departure site. We discuss several applications which have stimulated interest in the model such as shaken granular gases and network dynamics; we also discuss how the model may be used as a coarse-grained description of driven phase-separating systems. A useful property of the zero-range process is that the steady state has a factorized form. We show how this form enables one to analyse in detail condensation transitions, wherein a finite fraction of particles accumulate at a single site. We review condensation transitions in homogeneous and heterogeneous systems and also summarize recent progress in understanding the dynamics of condensation. We then turn to several generalizations which also, under certain specified conditions, share the property of a factorized steady state. These include several species of particles; hop rates which depend on both the departure and the destination sites; continuous masses; parallel discrete-time updating; non-conservation of particles and sites.