Complexity of Disordered Systems

无序系统的复杂性

• 批准号: 1407554
• 负责人: Antonio Auffinger
• 金额: $ 14.41万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2015-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1407554&HistoricalAwards=false
• 关键词: Complexity; Disordered; Systems

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.

概率论的经典目标之一是理解小个体（互联网用户、粒子、投资者）以看似随机的方式相互作用如何转化为整个系统的新颖行为。该提案的目标是分析此类系统，其中相互作用具有高维依赖结构，并且极端情况（网络中心、低能量配置、最佳轨迹）发挥重要作用。除了对概率论很重要之外，该提案中获得的结果也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、和社交网络。更具体地说，该提案涉及第一通道渗滤（多孔介质中流体流动的示例）和平均场自旋玻璃模型（相互作用受阻的无序磁体的示例）的项目。 主要问题与模型的复杂性有关，即大量极端情况的存在及其在空间中的位置。特别是，提议者计划研究 Sherrington-Kirkpatrick、混合 p-自旋和二部模型中函数序参数所起的作用及其与相应哈密顿量极值的数量和位置的关系。该提案进一步解决了随机势（聚合物模型）中长化学链不断增长的界面和波动的基本问题。物理学预测表明，在许多此类模型中，波动应按照偏离标准高斯分布的限制定律（例如，与随机矩阵理论中的 Tracy-Widom 分布相关）呈亚线性缩放。该提案继续并扩展了 PI 对这些系统的研究，超出了可积模型的范围。特别是，该提案旨在研究标度指数的普遍行为、极限形状的性质和测地线的几何形状。