Order of current variance and diffusivity in the rate one totally asymmetric zero range process

TL;DR

This paper establishes the t^{2/3} variance and t^{1/3} diffusivity scaling in a totally asymmetric zero range process, advancing understanding of universality in one-dimensional interacting particle systems.

Contribution

It proves the specific scaling laws for current variance and diffusivity in the zero range process, extending methods to larger state spaces and linking to broader universality conjectures.

Findings

Variance of current is of order t^{2/3}.

Diffusivity scales as t^{1/3}.

Results imply similar variance scaling for tagged particles in exclusion processes.

Abstract

We prove that the variance of the current across a characteristic is of order t^{2/3} in a stationary constant rate totally asymmetric zero range process, and that the diffusivity has order t^{1/3}. This is a step towards proving universality of this scaling behavior in the class of one-dimensional interacting systems with one conserved quantity and concave hydrodynamic flux. The proof proceeds via couplings to show the corresponding moment bounds for a second class particle. We build on the methods developed by Balazs-Seppalainen for asymmetric simple exclusion. However, some modifications were needed to handle the larger state space. Our results translate into t^{2/3}-order of variance of the tagged particle on the characteristics of totally asymmetric simple exclusion.