LEAPS-MPS: Random growth models, spin glasses and stability of complex networks

该项目的重点是从自然系统（例如生态系统和社交网络）产生的三个随机模型，具有复杂的行为，如相变，自组织和混沌。其中包括描述细胞增殖的随机生长模型、铁磁合金的自旋玻璃模型和生态系统的动力学模型。从这些项目中获得的结果不仅在概率论中很重要，而且在生物学，统计物理学和数据科学等其他科学分支中也有广泛的应用。教育部分包括开发新的研究生课程，为当地本科生提供的活动，以及为美国本科生提供的暑期课程。该项目还提供研究生水平的研究培训机会。PI正在研究第一次通过渗流的极限形状和波动，这是一种经典的随机增长模型，通常用于描述多孔介质中的流体流动或细胞生长。此外，PI使用迭代方案研究了Sherrington-Kirkpatrick和其他平均场自旋玻璃模型的基本性质，特别是在低温设置下的基本性质。此外，PI正在使用Kac-Rice公式和随机矩阵理论的方法研究大型动态系统的“稳定平衡”。该奖项反映了NSF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。

This project focuses on three stochastic models that arise from natural systems (for example ecosystems and social networks) with complex behavior, such as phase transition, self-organization and chaos. These include a random growth model describing cell proliferation, a spin glass model for ferromagnetic alloys, and a dynamic model for ecosystems. The results obtained from these projects are not only important in probability theory but also have wide applications in other branches of sciences such as biology, statistical physics and data sciences. The educational component includes the development of a new graduate course, activities for local undergraduate students, and summer programs for undergraduate students in the US. The project also provides research training opportunities at a graduate level.The PI is studying the limit shape and the fluctuations in first-passage percolation, a classic random growth model that is often used to describe fluid flow in porous media or the growth of cells. Moreover, the PI investigates fundamental properties of the Sherrington-Kirkpatrick and other mean-field spin glass models using iterative schemes, especially those under the low-temperature setting. Furthermore, the PI is studying ``stable equilibria’' of large dynamical systems using the Kac-Rice formula and approaches from random matrix theory. The results obtained there will be used to understand the relationship between complexity and stability in large ecosystems.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.