Spin Glasses and Other Models of Disordered Media

概率论的主要目标之一是理解大量个体及其微小的相互作用如何转化为整个系统的新行为。该项目的目标是分析这种复杂的无序系统，其中相互作用具有高维依赖结构，其中极端（低能量配置，最优轨迹）起着重要作用。除了对概率论很重要外，本项目获得的结果将与许多科学分支相关并应用，因为大多数问题都是为了理解物理学、计算机科学、理论生物学和社会网络中各种优化问题的行为而引入的。其中许多项目是与研究生合作的；PI将继续指导和培训学生和博士后研究人员。本课题主要研究平均场自旋玻璃模型和第一通道渗流模型的问题。这些问题包括量化平均场自旋玻璃模型的低温结构的研究，其长期目标是严格建立完整复制对称破破的Parisi描述，研究超越经典整数晶格的第一通道渗透中的极限形状、涨落和测地射线，以及建立一类随机动力系统中稳定和不稳定平衡数的进一步性质。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.