  We prove fluctuation bounds for the particle current in totally asymmetric
zero range processes in one dimension with nondecreasing, concave jump rates
whose slope decays exponentially. Fluctuations in the characteristic directions
have order of magnitude $t^{1/3}$. This is in agreement with the expectation
that these systems lie in the same KPZ universality class as the asymmetric
simple exclusion process. The result is via a robust argument formulated for a
broad class of deposition-type processes. Besides this class of zero range
processes, hypotheses of this argument have also been verified in the authors'
earlier papers for the asymmetric simple exclusion and the constant rate zero
range processes, and are currently under development for a bricklayers process
with exponentially increasing jump rates.
