Cutoff for the mean-field zero-range process

DOI10.1214/19-AOP1336zbMATH Open1448.60159arXiv1804.04608OpenAlexW2981349303MaRDI QIDQ2280554

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Publication date: 18 December 2019

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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 $ho>0$. We prove that the worst-case total-variation distance to equilibrium drops abruptly from $1$ to $0$ at time $nleft(ho+frac12h{o}^{2}ight)$. 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).

Full work available at URL: https://arxiv.org/abs/1804.04608

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