CAREER: Complexity of Disordered Systems

Northwestern University, Evanston IL

Investigators

Abstract

A classic aim of probability theory is to understand how the random interaction of small individuals (particles, internet users, investors) translates to novel behavior of the entire system. The goal of this project is to analyze such complex systems where the interactions have high-dimensional dependence structures and where the extremes (network hubs, low energy configurations, optimal trajectories) play a significant role. As well as being important to probability theory, the results to be obtained in this project will be relevant and applied to many branches of science, as most of the questions were introduced to understand the behavior of various optimization problems in physics, computer science, theoretical biology, and social networks. The project includes summer programs in probability aimed at US undergraduate and graduate students. The project involves problems on mean field spin glass models and on first-passage percolation in the d-dimensional integer lattice. The PI plans to investigate the structure of the Parisi measure in the Sherrington-Kirkpatrick model at low temperature, the phenomena of full replica symmetry breaking, and extensions of the Parisi theory to other disordered complex systems. The PI also plans to develop an intrinsic theory of geodesic rays in first passage percolation and its relation to fluctuations predictions (in two-dimensions given by the Kardar-Parisi-Zhang theory), outside the scope of integrable models. As part of the educational component of the project, the PI will continue his mentorship and training of students and post-doctoral fellows. The project includes probability summer programs (schools and boot-camps) aimed at US undergraduate and graduate students.