Statistical mechanical analysis of (1 + ∞) dimensional disordered systems

Valuable insight into the theory of disordered systems and spin-glasses has been offered by two classes of exactly solvable models: one-dimensional models and mean-field (infinite-range) ones, which, each carry their own specific techniques and restrictions. Both classes of models are now considered as 'exactly solvable' in the sense that in the thermodynamic limit the partition sum can been carried out analytically and the average over the disorder can be performed using methods which are well understood. In this thesis I study equilibrium properties of spin systems with a combination of one-dimensional short- and infinite-range interactions. I find that such systems, under either synchronous or asynchronous spin dynamics, and even in the absence of disorder, lead to phase diagrams with first-order transitions and regions with a multiple number of locally stable states. I then proceed to the study of recurrent neural network models with (1+∞)-dimensional interactions, and find that the competing short- and long-range forces lead to highly complex phase diagrams and that unlike infinite-range (Hopfield-type) models these phase diagrams depend crucially on the number of patterns stored, even away from saturation. To solve the statics of such models for the case of synchronous dynamics I first make a detour to solve the synchronous counterpart of the one-dimensional random-field Ising model, where I prove rigorously that the physics of the two random-field models (synchronous vs. sequential) becomes asymptotically the same, leading to an extensive ground state entropy and an infinite hierarchy of discontinuous transitions close to zero temperature. Finally, I propose and solve the statics of a spin model for the prediction of secondary structure in random hetero-polymers (which are considered as the natural first step to the study of real proteins). The model lies in the class of (1+∞)-dimensional disordered systems as a consequence of having steric- and hydrogen-bonds interactions (one-dimensional short-range) as well as interactions due to monomer pairs' polarity-type effects (infinite-range). The model's phase diagram shows a second-order transition between 'folded' and 'unfolded' states as well as regions where 'folding' is found to depend on initial conditions. (author)