  We introduce a simple zero-range process with constant rates and one fast
rate for a particular occupation number, which diverges with the system size.
Surprisingly, this minor modification induces a condensation transition in the
thermodynamic limit, where the structure of the condensed phase depends on the
scaling of the fast rate. We study this transition and its dependence on system
parameters in detail on a rigorous level using size-biased sampling. This
approach generalizes to any particle system with stationary product measures,
and the techniques used in this paper provide a foundation for a more
systematic understanding of condensing models with a non-trivial condensed
phase.
