Complexity of Disordered Systems

概率论的一个经典目标是理解小个体（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行为。本提案的目标是分析这样的系统，其中相互作用具有高维依赖结构，其中极端（网络集线器，低能量配置，最优轨迹）发挥重要作用。除了对概率论很重要外，本提案中获得的结果将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、理论生物学和社会网络中各种优化问题的行为而引入的。更具体地说，该提案涉及第一通道渗透（多孔介质中流体流动的一个例子）和平均场自旋玻璃模型（相互作用受挫的无序磁铁的例子）的项目。主要问题与模型的复杂性有关，也就是说，大量极端情况的存在及其在空间中的位置。特别地，作者计划研究功能序参数在Sherrington-Kirkpatrick、混合p-自旋和二部模型中所起的作用及其与相应哈密顿量的极值数目和位置的关系。该提案进一步解决了在随机电位（聚合物模型）中生长界面和长化学链波动的基本问题。来自物理学的预测表明，在许多这样的模型中，波动应该按照偏离标准高斯分布的限制定律（例如，这与随机矩阵理论中的Tracy-Widom分布有关）按次线性缩放。这一建议继续并扩展了可积模型范围外这些系统的PI研究。特别是，本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。

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.