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September 2019 Cutoff for the mean-field zero-range process
Mathieu Merle, Justin Salez
Ann. Probab. 47(5): 3170-3201 (September 2019). DOI: 10.1214/19-AOP1336

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 $n(\rho +\frac{1}{2}\rho^{2})$. 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 metastability, separation of timescales, fluid limits, propagation of chaos, entropy and a spectral estimate by Morris (Ann. Probab. 34 (2006) 1645–1664).

Citation

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Mathieu Merle. Justin Salez. "Cutoff for the mean-field zero-range process." Ann. Probab. 47 (5) 3170 - 3201, September 2019. https://doi.org/10.1214/19-AOP1336

Information

Received: 1 April 2018; Published: September 2019
First available in Project Euclid: 22 October 2019

zbMATH: 07145314
MathSciNet: MR4021248
Digital Object Identifier: 10.1214/19-AOP1336

Subjects:
Primary: 37A25, 60J27, 82C22

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 5 • September 2019
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