  The zero-range process is a stochastic interacting particle system that
exhibits a condensation transition under certain conditions on the dynamics. It
has recently been found that a small perturbation of a generic class of jump
rates leads to a drastic change of the phase diagram and prevents condensation
in an extended parameter range. We complement this study with rigorous results
on a finite critical density and quenched free energy in the thermodynamic
limit, as well as quantitative heuristic results for small and large noise
which are supported by detailed simulation data. While our new results support
the initial findings, they also shed new light on the actual (limited)
relevance in large finite systems, which we discuss via fundamental diagrams
obtained from exact numerics for finite systems.
