Disordered systems are a very active area of research in statistical mechanics, both in theoretical and in mathematical physics. Physically they are motivated by material, such as alloys, that have an irregular spatial structure. E.g., in spin glasses, one has ferromagnetic atoms (such as iron) embedded in a matrix of a conducting metal, (such as gold or copper). The magnetic properties of such material are dominated by the interaction of the magnetic moments of the ferromagnetic atoms. But the interaction is modulated by the conductor and oscillates in sign with the distance between these atom. A reasonable way of modelling this is through random interactions between Ising spins. The resulting model turns out to be quite hard to analyse, mainly because of the competing tendencies of the interactions. There has been a lot of progress in the mathematical understanding of such systems over the last decade.
For the impatient, here is a short introduction to just to the mean field theory of spin glasses intended to bring the reader quickly towards the Parisi solution and Guerra’s bounds.
A lot of interesting stuff is missing, of course.
A good source on material on mathematical statistical mechanics are also the Proceedings of the 2005 Les Houches Summer School on Mathematical Statistical Mechanics, that appeared this year at Elsevier. They contain in depth lecture notes on many topics of current interest.