Rigorous results on Spin Glasses

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The main questions in this proposal originate from the models that were introduced by the physicists about forty years ago with the goal of understanding the unusual magnetic behaviour of certain metal alloys, called spin glasses. The study of these models produced many new ideas that found unexpected applications beyond their original motivation. One of the most interesting applications was to the area of optimization problems. To give one example, let us consider a large group of people where any two are either friends or enemies, and let us divide them into two groups attempting to keep friends together and separate the enemies. This can not always be done perfectly, because if two of your friends are enemies then either you lose one of your friends or they will stay enemies inside your group; these are called frustrated triples. This problem of dividing the group in an optimal way is known to be very difficult to solve in general (imagine a situation with numerous frustrated triples), but there are many interesting questions one can ask about what happens in a typical situation. For example, one can ask how optimal solutions typically look like, or how other almost optimal solutions are related to the optimal ones. The physicists were very successful in applying the ideas developed in the study of spin glasses to various optimization problems arising in mathematics (traveling salesman problem, graph partitioning), computer science (random satisfiability of Boolean formulas) and biology (modelling brain activity), and their main contributions can be roughly divided into two categories. On the one hand, they have developed a deep and sophisticated theoretical understanding of what happens in a typical situation in these optimization problems (where a typical situation is modelled by choosing the parameters of the problem randomly). On the other hand, having this picture in mind allowed them to come up with efficient algorithms for solving some of these problems in practice. The main focus of this proposal is on the first category, namely, theoretical picture in one family of models -- the so called diluted spin glass models, which includes a modification of the above problem of splitting a group of people into two groups in the case when each person interacts with only a small number of other people. The main goal of the proposal is to find a rigorous mathematical explanation for the picture proposed by the physicists in these models. In particular, the proposal attempts to prove the famous Mézard--Parisi formula for the free energy, as well as describe the structure of the Gibbs measure in these models (where the main open problem is called reproducibility hypothesis). Just like the work of the physicists was based on the ideas developed in the context of the models of spin glasses, this proposal builds upon a significant progress achieved in recent years in the rigorous mathematical theory behind the original spin glass models.