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Order of current variance and diffusivity in the rate one totally asymmetric zero range process. (English) Zbl 1151.82381
Summary: 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 M. Balázs and T. Seppäläinen [Order of current variance and diffusivity in the asymmetric simple exclusion process, preprint, arXiv:math/0608400] 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.

##### MSC:
 82C22 Interacting particle systems in time-dependent statistical mechanics 82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics 60J05 Discrete-time Markov processes on general state spaces 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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