  We investigate the effects of finite size corrections on the overlap
probabilities in the Generalized Random Energy Model (GREM) in two situations
where replica symmetry is broken in the thermodynamic limit. Our calculations
do not use replicas, but shed some light on what the replica method should give
for finite size corrections. In the gradual freezing situation, which is known
to exhibit full replica symmetry breaking, we show that the finite size
corrections lead to a modification of the simple relations between the sample
averages of the overlaps $ Y_k $ between $ k $ configurations predicted by
replica theory. This can be interpreted as fluctuations in the replica block
size with a \emph{negative} variance. The mechanism is similar to the one we
found recently in the random energy model [1]. We also consider a simultaneous
freezing situation, which is known to exhibit one step replica symmetry
breaking. We show that finite size corrections lead to full replica symmetry
breaking and give a more complete derivation of the results presented in [2]
for the directed polymer on a tree.
