CAREER: Complexity of Disordered Systems

概率论的一个经典目标是理解小个体（粒子、互联网用户、投资者）的随机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样的复杂系统，其中的相互作用具有高维依赖结构，极端（网络枢纽，低能量配置，最佳轨迹）发挥着重要作用。除了对概率论很重要之外，这个项目中获得的结果将与许多科学分支相关并应用于许多科学分支，因为大多数问题都是为了理解物理学，计算机科学，理论生物学和社交网络中各种优化问题的行为而引入的。该项目包括针对美国本科生和研究生的概率暑期课程。该项目涉及平均场自旋玻璃模型和d维整数晶格中的首次通过渗流问题。PI计划研究低温下Sherrington-Kirkpatrick模型中Parisi测度的结构，完全复制对称性破缺的现象，以及Parisi理论对其他无序复杂系统的扩展。PI还计划在可积模型的范围之外，在第一次通过渗透中开发测地线射线的内在理论及其与波动预测的关系（在Kardar-Parisi-Zhang理论给出的二维中）。作为该项目教育部分的一部分，PI将继续指导和培训学生和博士后研究员。该项目包括针对美国本科生和研究生的概率暑期课程（学校和训练营）。

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.