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We consider the zero-range process with arbitrary bounded monotone rates onthe complete graph, in the regime where the number of sites diverges while thedensity of particles per site converges. We determine the asymptotics of themixing time from any initial configuration, and establish the cutoffphenomenon. The intuitive picture is that the system separates into a slowlyevolving solid phase and a quickly relaxing liquid phase: as time passes, thesolid phase dissolves into the liquid phase, and the mixing time is essentiallythe time at which the system becomes completely liquid. Our proof uses the pathcoupling technique of Bubley and Dyer, and the analysis of a suitablehydrodynamic limit. To the best of our knowledge, even the order of magnitudeof the mixing time was unknown, except in the special case of constant rates.

Cutoff for the mean-field zero-range process with bounded monotone rates