##### Abstract

After a brief introduction to the problem of directed polymers in a random
medium, several aspects of the problem are addressed in more detail.

It is possible to show that, in high enough dimension and above a certain temperature,
a phase exists in which the free energy is given by the annealed free energy,.
with probability one. The proof of this statement is extended to obtain upper and
lower bounds on the temperature of the transition between this phase and a low
temperature phase and hence prove the existence of the phase transition.

The mean field solution is reviewed and a method of obtaining high dimension
expansions around it is presented. The method uses the idea of "n -tree approximations"
as a systematic way of including the correlations present in finite dimensions.
Using this approach 1/d expansions are obtained for the free energy,
transition temperature, transverse fluctuations and overlaps. The results are consistent
with the existence of a finite upper critical dimension above which the
behaviour is of a mean field nature.

Directed polymers are then considered on disordered hierarchical lattices. On
these lattices the problem reduces to the study of the stable laws that occur
when one combines random variables in a nonlinear way. Recursion relations for
various properties of the system are obtained, which are studied using numerical
and analytical techniques. It is possible to obtain a perturbative expansion for
the free energy, overlaps and non -integer moments of the partition function.

Finally, a generalisation of the standard directed polymer problem is considered,
in which one allows the walks to contribute positive and negative weights to the
partition sum. The solution of the mean field limit of this problem is obtained
indirectly, by using the relationship between the mean field directed polymer problem,
the random energy model (REM) and the generalised random energy model
(GREM). Numerical simulations are presented which give good confirmation of
the analytical predictions. It is observed that in solving this generalised polymer
problem one has also obtained a formula for the largest Lyapounov exponent of a
product of large, sparse random matrices.