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

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 in Balázs and Seppäläinen (Order of current variance and diffusivity in the asymmetric simple exclusion process, 2006) for 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.