Complexity of Disordered Systems

University Of Chicago, Chicago IL

Investigators

Abstract

One of the classic goals of probability theory is to understand how the interaction of small individuals (internet users, particles, investors) in seemingly random ways translates to novel behavior of the entire system. The goal of this proposal is to analyze such 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 obtained in this proposal 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. More specifically, the proposal involves projects on first-passage percolation (an example of fluid flow in a porous medium) and on mean field spin glass models (example of disordered magnets with frustrated interactions). The major questions are tied to the complexity of the models, that is, the presence of a large number of extremes and their location in space. In particular the proposer plans to investigate the role played by the functional order parameters in the Sherrington-Kirkpatrick, mixed p-spin and bipartite models and its relation with the number and location of extremes of the corresponding Hamiltonians. The proposal further addresses fundamental questions on growing interfaces and fluctuations of long chemical chains in a random potential (polymer models). Predictions from physics indicate that, in many of these models, fluctuations should scale sub-linearly with limiting laws that deviate from the standard Gaussian (for instance, which relate to the Tracy-Widom distribution from random matrix theory). This proposal continues and expands the research of the PI on these systems outside the scope of integrable models. In particular, this proposal aims to investigate the universal behavior of scaling exponents, the nature of the limit shape and the geometry of geodesics.