  Within generalized random energy models, we study the effects of energy
discreteness and of entropy extensivity in the low temperature phase. At zero
temperature, discreteness of the energy induces replica symmetry breaking, in
contrast to the continuous case where the ground state is unique. However, when
the ground state energy has an extensive entropy, the distribution of overlaps
P(q) instead tends towards a single delta function in the large volume limit.
Considering now the whole frozen phase, we find that P(q) varies continuously
with temperature, and that state-to-state fluctuations of entropy wash out the
differences between the discrete and continuous energy models.
