Structure and Statistics of Disordered Systems

在复杂网络建模中，一个反复出现的挑战是，仅通过了解系统的微观规则来预测宏观结果。例子范围从全球层面（社会网络、经济网络）到细胞层面（免疫网络、神经元网络），可以是虚拟的，而不是物理的（信号处理、人工智能）。概率论可以通过利用越来越多的随机变量（也称为无序）的总体效应的预测能力来分析这些系统。为了有效地利用这一理论，数学家和其他实践者必须能够从更容易测量的系统统计数据中推断出定性的大规模结构，并确定这些统计数据与控制模型的参数之间的关系。这个项目将在两类无序系统中追求这些目标：自旋玻璃和随机增长，这两类系统在物理学、计算机科学和工程学中都是持续的兴趣。该项目将包括为研究生和本科生提供研究培训机会，并为中小学生提供外展服务。建议的研究包括：(1)利用泛函分析和偏微分方程的技术，建立所谓的多物种和矢量自旋模型的极限能量的巴黎型公式；(2)通过对得到的Parisi公式进行变分演算，确定这些相同模型的相图；(3)通过采用随机分析工具，对与第一次和最后一次渗流相关的生长界面波动大小作出新的估计；(4)利用自旋玻璃理论的未开发连接，研究定向聚合物中的淬火局部化现象。这项严谨的数学工作将得到实验性计算机模拟的辅助。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。

A recurring challenge in modeling complex networks is to predict macroscopic outcomes from knowing only the microscopic rules of a system. Examples range from the global level (social networks, economic networks) to the cellular level (immune networks, neuron networks), and can be virtual rather than physical (signal processing, artificial intelligence). Probability theory can analyze these systems by harnessing the predictive power of the aggregate effect of more and more random variables, otherwise known as disorder. To make effective use of this theory, mathematicians and other practitioners must be able to both infer qualitative large-scale structure from more easily measured system statistics and identify the relationship between these statistics and the parameters governing the model. This project will pursue these objectives for two classes of disordered systems: spin glasses and random growth, both of which are of continuing interest in physics, computer science, and engineering. The project will include research training opportunities for graduate and undergraduate students and also outreach efforts to elementary and secondary students. Among the proposed investigations are (1) establishing Parisi-type formulas for the limiting energy of so-called multi-species and vector-spin models, by using techniques from functional analysis and PDEs; (2) determination of the phase diagram of these same models, by performing variational calculus on the obtained Parisi formulas; (3) producing new estimates on the size of fluctuations in growth interfaces associated to first- and last-passage percolation, by adapting tools from stochastic analysis; and (4) studying the phenomenon of quenched localization in directed polymers, by leveraging untapped connections to spin glass theory. This rigorous mathematical work will be aided by experimental computer simulation.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.