  In this thesis, we consider several Random Energy Models. This includes
Derrida's Random Energy Model (REM) and Generalized Random Energy Model (GREM)
and a nonhierarchical version (BK-GREM) by Bolthausen and Kistler. The limiting
free energy in all these models along with Word GREM, a model proposed by us,
turn out to be a cute consequence of large deviation principle (LDP). This LDP
argument allows us to consider non-Gaussian driving distributions as well as
external field. We could also consider random trees as the underlying tree
structure in GREM. In all these models, as expected, limiting free energy is
not 'universal' unlike the SK model. However it is 'rate specific'.
Consideration of non-Gaussian driving distribution as well as different driving
distributions for the different levels of the underlying trees in GREM leads to
interesting phenomena. For example in REM, if the Hamiltonian is Binomial with
parameter $N$ and $p$ then the existence of phase transition depends on the
parameter $p$. More precisely, phase transition takes place only when
$p>{1/2}$. For another example, consider a 2 level GREM with exponential
driving distribution at the first level and Gaussian in the second with equal
weights at both the levels. Then even if the limiting ratio for the second
level particles, $p_2$ is 0.00001 (very small), the system reduces to a
Gaussian REM. On the other hand, if we consider a 2 level GREM with Gaussian
driving distribution at the first level and exponential in the second, the
system will never reduce to a Gaussian REM. In either case, the system will
never reduce to that of an exponential REM. etc.
