Spin Glasses and Other Models of Disordered Media

点击翻译按钮获取中文摘要

One of the main goals of probability theory is to understand how a large number individuals and their small interactions translate to novel behavior of an entire system. The goal of this project is to analyze such complex disordered systems where the interactions have high-dimensional dependence structures and where the extremes (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. Many of these projects are in collaboration with graduate students; the PI will continue mentoring and training of students and post-doctoral researchers. The project concerns problems on mean field spin glass models and on first-passage percolation models. Those problems include a study to quantify the low-temperature structure of mean field spin glass models, with the long range goal of rigorously establish the Parisi description of full replica symmetry breaking, to study limit shapes, fluctuations and geodesics rays in first passage percolation beyond the classical case of the integer lattice and to establish further properties on the number of stable and unstable equilibria in a class of random dynamical systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.