  We study the mixing time of the unit-rate zero-range process on the complete
graph, in the regime where the number $n$ of sites tends to infinity while the
density of particles per site stabilizes to some limit $\rho>0$. We prove that
the worst-case total-variation distance to equilibrium drops abruptly from $1$
to $0$ at time $n\left(\rho+\frac{1}{2}\rho^2\right)$. 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).
