Complexity of Disordered Systems

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是如何转化为整个系统的新行为的。这一建议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，并且极端(网络枢纽、低能量配置、最优轨迹)在其中发挥重要作用。除了对概率论很重要外，本提案中获得的结果也将相关并应用于许多科学分支，因为大多数问题是为了理解物理、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，该提案涉及首次通过渗流(多孔介质中流体流动的一个例子)和平均场自旋玻璃模型(无序磁铁与受阻相互作用的例子)的项目。主要问题与模型的复杂性有关，即存在大量极端及其在空间中的位置。特别是，作者计划研究Sherrington-Kirkpatrick、混合p-自旋和二体模型中的泛函序参数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关于随机势(聚合物模型)中长化学链的增长界面和波动的基本问题。物理学的预测表明，在许多这样的模型中，涨落应该与偏离标准高斯的极限定律(例如，与随机矩阵理论中的特雷西-维多姆分布有关)呈亚线性增长。这一建议在可积模型的范围之外继续和扩展了PI对这些系统的研究。特别是，这项提议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。

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.