Lower current large deviations for zero-range processes on a ring

We study lower large deviations for the current of totally asymmetric
zero-range processes on a ring with concave current-density relation. We use an
approach by Jensen and Varadhan which has previously been applied to exclusion
processes, to realize current fluctuations by travelling wave density profiles
corresponding to non-entropic weak solutions of the hyperbolic scaling limit of
the process. We further establish a dynamic transition, where large deviations
of the current below a certain value are no longer typically attained by
non-entropic weak solutions, but by condensed profiles, where a non-zero
fraction of all the particles accumulates on a single fixed lattice site. This
leads to a general characterization of the rate function, which is illustrated
by providing detailed results for four generic examples of jump rates,
including constant rates, decreasing rates, unbounded sublinear rates and
asymptotically linear rates. Our results on the dynamic transition are
supported by numerical simulations using a cloning algorithm.