  This thesis includes analysis of disordered spin ensembles corresponding to
Exact Cover, a multi-access channel problem, and composite models combining
sparse and dense interactions. The satisfiability problem in Exact Cover is
addressed using a statistical analysis of a simple branch and bound algorithm.
The algorithm can be formulated in the large system limit as a branching
process, for which critical properties can be analysed. Far from the critical
point a set of differential equations may be used to model the process, and
these are solved by numerical integration and exact bounding methods. The
multi-access channel problem is formulated as an equilibrium statistical
physics problem for the case of bit transmission on a channel with power
control and synchronisation. A sparse code division multiple access method is
considered and the optimal detection properties are examined in typical case by
use of the replica method, and compared to detection performance achieved by
iterative decoding methods. These codes are found to have phenomena closely
resembling the well-understood dense codes. The composite model is introduced
as an abstraction of canonical sparse and dense disordered spin models. The
model includes couplings due to both dense and sparse topologies
simultaneously. Through an exact replica analysis at high temperature, and
variational approaches at low temperature, several phenomena uncharacteristic
of either sparse or dense models are demonstrated. An extension of the
composite interaction structure to a code division multiple access method is
presented. The new type of codes are shown to outperform sparse and dense codes
in some regimes both in optimal performance, and in performance achieved by
iterative detection methods in finite systems.
