Mean field asymptotic for stochastic processes on graphs

Stanford University, Stanford CA

Investigators

Abstract

Over the past three decades, physicists have developed sophisticated non-rigorous techniques for accurately predicting the asymptotic behavior of large complex(random) systems. Mathematicians are making significant progress in developing the corresponding rigorous theories and proving 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. A particular success story from a physics point of view are mean field(disordered) models, where two levels of randomness are often present, one of whom frozen (quenched), in the form of dependence graph or weights for the resulting stochastic process, whose distribution is exchangeable when averaged over the disorder. Specific directions we plan to pursue are (1) Rigorous study of large systems of discrete variables that are strongly interacting according to a mean field model determined by an ensemble of (randomly chosen) graphs, (2) 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,(3) properties of stochastic dynamics for spin systems out of equilibrium, such as aging and the finite size scaling associated with their phase transitions. This project is expected to yield new mathematical ideas and techniques that resolve challenging problems in probability theory, impact its interface with statistical physics, and provide insight to the analysis of typical behavior of large constraint satisfaction networks, thus advancing progress at the core of the NSF cyber-enabled discovery and innovation initiative. Long-term effects on graduate and post-graduate training of students in discrete probability and information theory are also expected.