The mean-field Zero-Range process with unbounded monotone rates: mixing time, cutoff, and Poincar\'e constant

TL;DR

This paper analyzes the mixing time and cutoff phenomena of the mean-field Zero-Range process with unbounded monotone rates, providing bounds on the Poincaré constant and extending results to arbitrary geometries.

Contribution

It establishes the mixing time, cutoff, and Poincaré constant bounds for the mean-field Zero-Range process with unbounded rates, extending to general geometries.

Findings

Determines the mixing time and cutoff for the process.

Proves the Poincaré constant is bounded away from zero and infinity.

Extends results to arbitrary geometries using comparison arguments.

Abstract

We consider the mean-field Zero-Range process in the regime where the potential function $r$ is increasing to infinity at sublinear speed, and the density of particles is bounded. We determine the mixing time of the system, and establish cutoff. We also prove that the Poincar\'e constant is bounded away from zero and infinity. This mean-field estimate extends to arbitrary geometries via a comparison argument. Our proof uses the path-coupling method of Bubley and Dyer and stochastic calculus.