Mathematical Studies of Short-Ranged Spin Glasses

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The work that is supported under this grant addresses the mathematical foundations of the theory of disordered magnets known as spin glasses.Although there exist proposed solutions of idealized (and unrealistic) spin glass models, we are interested in answering fundamental statistical mechanical questions that bear on the behavior of real laboratory spin glasses. These questions, many of which remain controversial despitetwo decades of intensive study, include the nature of ordering in the spin glass phase, the number of ground states (and low-temperature pure phases), the nature of the spin glass metastate (an ensemble of thermodynamic phases), understanding anomalous dynamical behavior such as slow relaxation and aging, and proving the presence or absence of a phase transition. The methods used and concepts introduced should be relevant not only for spin glasses but should also be applicable to other disordered systems, many of which remain poorly understood.Our deep physical and mathematical understanding of ordered systems in the solid and liquid state - for example, crystals, ferromagnets, superconductors, liquid crystals, and many others - has been of fundamental scientific and technological importance throughout the second half of this century. However, there exist many systems, both familiar and unfamiliar in everyday use, in which randomness or disorder playsa key role, and in which our mathematical and physical understanding remains comparatively primitive. One familiar example is ordinary window glass, where the atoms or molecules are "stuck" in random locations (as opposed to a regular crystalline array as would be found, for example, in ice). Spin glasses are disordered magnetic systems which are thought to be prototypes forthis kind of macroscopic "frozen-in" disorder, and they may be more amenable to mathematical analysis than other materials in this class. Nevertheless, little fundamental progress has been made even here. This work is aimed at resolving basic mathematical and physical issues concerningthese materials and at providing a general theoretical approach for a wide variety of disordered systems.