  We study Derrida's generalized random energy model in the presence of uniform
external field. We compute the fluctuations of the ground state and of the
partition function in the thermodynamic limit for all admissible values of
parameters. We find that the fluctuations are described by a hierarchical
structure which is obtained by a certain coarse-graining of the initial
hierarchical structure of the GREM with external field. We provide an explicit
formula for the free energy of the model. We also derive some large deviation
results providing an expression for the free energy in a class of models with
Gaussian Hamiltonians and external field. Finally, we prove that the
coarse-grained parts of the system emerging in the thermodynamic limit tend to
have a certain optimal magnetization, as prescribed by strength of external
field and by parameters of the GREM.
