Some problems on mean-field spin glasses

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Spin glasses are disordered spin systems invented by theoretical physicists in the early 70's. The aim is to describe the strange magnetic behaviors of certain alloys. Over the past decades, physicists' investigation has generated a series of methodologies and predictions. In the community of mathematics, spin glass models are purely probabilistic objects that present highly complex random structures. The general aim of this research project is to pursue mathematical investigation of physical phenomena including the complexity of the random Hamiltonian, the structure of the Gibbs measure and the thermodynamic limit of the free energy. As well as being important in the field of probability, the tools and results obtained in this research plan are relevant to a variety of scientific branches including computer science, theoretical biology and social networks. More precisely, this proposal is concerned about the Sherrington-Kirkpatrick (SK) model. The PI will investigate the structure and the role of the functional order parameter in the low temperature regime. In addition, the conjectured ultrametricity of the Gibbs measure and the problem of chaos in temperature will be analyzed following the Guerra-Talagrand replica symmetry breaking bound. The PI will as well study the central limit theorem for some physical quantities such as the overlap and magnetization especially at the critical temperature using Stein's method. Another direction is the SK model with heavy tail disorder. As now higher order interactions are allowed, the system has been predicted to exhibit unusual spin glass phenomena. The last part of the project involves the bipartite version of the SK model, which was used to model collective properties of two interactive large groups of individuals motivated by the studies of the social and neural networks, biology and economics. The PI will be concentrated on the computation of the limiting free energy.