  Recently, Bauke and Mertens conjectured that the local statistics of energies
in random spin systems with discrete spin space should in most circumstances be
the same as in the random energy model. This was proven in a large class of
models for energies that do not grow too fast with the system size. Considering
the example of the generalized random energy model, we show that the conjecture
breaks down for energies proportional to the volume of the system, and describe
the far more complex behavior that then sets in.
