Mean Field Asymptotic for Stochastic Processes on Graphs

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The common thread in this proposal is the focus on mean field disordered models originally developed within statistical physics. These are usually characterized by two levels of randomness, one of whom frozen (quenched), in the form of dependence graph or weights for the resulting stochastic process, whose distribution is exchangeable (hence being a mean field model). In connection with such models we focus on the following directions. The rigorous study of probabilistic models for large systems of discrete variables that are strongly interacting according to a random graph structure. The aim is to develop a better mathematical understanding of their phase transitions and the existence of multiple Gibbs measures. A related effort has to do with the development of novel connections between the asymptotic behavior of the spectrum of structured models of random matrices and properties of the corresponding random graphs and graph embeddings.Understanding the scope and reasons behind some unexpected asymptotic behavior of stochastic dynamics for spin systems out of equilibrium. A prominent example is the aging phenomenon where memory accumulates in the system with time and its relation to a yet to be understood specific form of breakup of the fluctuation-dissipation theorem (which relates the time derivative of the correlation function at equilibrium to the effect of small perturbations in Markovian dynamics). Over the past three decades, physicists have developed sophisticated non-rigorous techniques for accurately predicting the asymptotic behavior of large complex random systems. Their studies were motivated by the desire to understand the collective behavior of various states of the matter, and the phase transitions connecting them. For instance, the objective of understanding water at the critical temperature at which it turns into vapor, or magnets at the critical temperature at which they lose spontaneous magnetization, has motivated the development of a sophisticated unified theory of critical phenomena.Over time, it has become apparent that the models and intuitions developed by physicists rest on deep mathematical principles, whose reach is much broader than simply physical systems. More recently, mathematicians are making significant progress in developing the corresponding rigorous theories and proving some of these predictions. Probability theory is at the forefront of this convergence, starting with the theory of large deviations and continuing with the emerging vibrant activity in the study of stochastic dynamics of interacting particles, large random matrices, Gibbs measures and planar objects with conformal symmetries.This project focuses on developing this line of research, in particular to develop the mathematics of large random systems without (finite-dimensional) geometric structures. These are called by physicists `mean field systems' and enjoy a remarkable degree of universality.