  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. Here we give necessary conditions for
this hypothesis to be true, which we show to hold in wide classes of examples:
short range spin glasses and mean field spin glasses of the SK type. We also
show that, under certain conditions, the conjecture holds even if energy levels
that grow moderately with the volume of the system are considered.
