## The Annals of Probability

### Cutoff for the mean-field zero-range process

#### 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).

#### Article information

Source
Ann. Probab., Volume 47, Number 5 (2019), 3170-3201.

Dates
First available in Project Euclid: 22 October 2019

https://projecteuclid.org/euclid.aop/1571731448

Digital Object Identifier
doi:10.1214/19-AOP1336

Mathematical Reviews number (MathSciNet)
MR4021248

Zentralblatt MATH identifier
07145314

#### Citation

Merle, Mathieu; Salez, Justin. Cutoff for the mean-field zero-range process. Ann. Probab. 47 (2019), no. 5, 3170--3201. doi:10.1214/19-AOP1336. https://projecteuclid.org/euclid.aop/1571731448

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