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

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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.