Cutoff for the mean-field zero-range process

TL;DR

This paper analyzes the mixing time of the zero-range process on a complete graph, revealing an abrupt cutoff at a specific time related to particle density, and describes the system's phase separation and dissolution dynamics.

Contribution

It provides an explicit characterization of the cutoff phenomenon for the zero-range process, including dependence on initial configurations and a detailed analysis of phase separation.

Findings

The mixing time exhibits an abrupt cutoff at time n(ρ + 0.5ρ^2).

The system separates into solid and liquid phases during evolution.

The mixing time depends explicitly on the largest initial heights.

Abstract

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).