Topics in Disordered Systems

Disordered systems are statistical mechanics models in random environments. This text considers the equilibrium properties of a few examples of disordered Ising models from a probability theory and mathematical physics approach, but the material should be of interest to condensed matter physicists, material scientists, applied mathematicians and theoretical computer scientists. The two main types of systems considered are disordered ferromagnets and spin glasses. The emphasis is on questions concerning the nunmber of ground states (at zero temperature) or the nunber of pure Gibbs states (at nonzero temperatures), and these questions are often connected to issues concerning percolation and related models of geometric or combinatorial probability. The conceptual issue of how to approach the thermodynamic (i.e. infinite volume) in systems which may have many complex competing states is considered, along with the more specific problem of whether and in what sense Parisi’s analysis of the meanfield (or “infinite-range”) model is relevant with regard to the low temperature behaviour of short-range spin glasses.