Cutoff phenomenon for asymmetric zero range process with monotone rates

TL;DR

This paper studies the mixing times of asymmetric zero range processes with monotone rates, demonstrating cutoff phenomena under certain asymmetry conditions and linking mixing to hydrodynamic limits.

Contribution

It establishes cutoff behavior for asymmetric zero range processes with monotone rates, including new results for convex and strongly asymmetric concave flux cases.

Findings

Cutoff occurs in totally asymmetric case with convex flux.

Cutoff also occurs with strong asymmetry and concave flux.

Mixing time aligns with the system reaching macroscopic equilibrium.

Abstract

We investigate the mixing time of the asymmetric Zero Range process on the segment with a non-decreasing rate. We show that the cutoff holds in the totally asymmetric case with a convex flux, and also with a concave flux if the asymmetry is strong enough. We show that the mixing occurs when the macroscopic system reaches equilibrium. A key ingredient of the proof, of independent interest, is the hydrodynamic limit for irregular initial data.