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This thesis includes analysis of disordered spin ensembles corresponding toExact Cover, a multi-access channel problem, and composite models combiningsparse and dense interactions. The satisfiability problem in Exact Cover isaddressed using a statistical analysis of a simple branch and bound algorithm.The algorithm can be formulated in the large system limit as a branchingprocess, for which critical properties can be analysed. Far from the criticalpoint a set of differential equations may be used to model the process, andthese are solved by numerical integration and exact bounding methods. Themulti-access channel problem is formulated as an equilibrium statisticalphysics problem for the case of bit transmission on a channel with powercontrol and synchronisation. A sparse code division multiple access method isconsidered and the optimal detection properties are examined in typical case byuse of the replica method, and compared to detection performance achieved byiterative decoding methods. These codes are found to have phenomena closelyresembling the well-understood dense codes. The composite model is introducedas an abstraction of canonical sparse and dense disordered spin models. Themodel includes couplings due to both dense and sparse topologiessimultaneously. Through an exact replica analysis at high temperature, andvariational approaches at low temperature, several phenomena uncharacteristicof either sparse or dense models are demonstrated. An extension of thecomposite interaction structure to a code division multiple access method ispresented. The new type of codes are shown to outperform sparse and dense codesin some regimes both in optimal performance, and in performance achieved byiterative detection methods in finite systems.

Typical case behaviour of spin systems in random graph and composite ensembles