Probability and Mean Field Models for Spin Glasses

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The unexpected behavior of mean-field models for spin glasses was discovered in the seventies by G. Parisi. These models are simple mathematical objects. It is remarkable that they exhibit such a subtle behavior, and that their rigorous study is so challenging. The PI believes that elucidating these models will bring the discovery of an entire new direction in probability theory. He has been trying for 10 years to build new methods to study them. One of the fundamental objectives (the proof of the Parisi formula giving the free energy of the Sherrington-Kirkpatrick model) has been attained, but this success cannot hide the fact that overall our understanding remains rather limited. In particular, two central questions, (known as the Ultrametricity conjecture and the Chaos problem) seem as hard as ever. They are possibly related to questions of analysis. The PI plans to work on these very hard questions, but also will strive to make incremental progress on less daunting issues, in particular the study of new mathematically canonical mean field Hamiltonians that exhibit replica symmetric equations of a new type. Mathematics and Physics have influenced each other in fundamental ways since at least Galileo, and this mutual influence is stronger than ever. In the seventies, Physicists discovered that certain alloys (called spin glasses) responded in an unconventional way to magnetic simulation. They introduced new models to explain these behaviors. These models are simple and natural mathematical objects. Yet for a long time there existed no mathematical method to study them. The early work of the physicists uses heuristic methods that need not be completely reliable. The present project it a step in the long range program of eliminating this discrepancy, by developing new tools from mathematics, and in particular probability theory, to completely understand these fundamental models. The long rang outcome of this program could be a new impetus on probability theory, one of the most important branch of mathematics for applications.