Dynamical and Spatial Asymptotics of Large Disordered Systems

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This project will study the asymptotic behaviors of several stochastic models in probability theory, in terms of long-time dynamics and static spatial limits. These models find wide applications in various disciplines, including condensed matter physics, material science, computer science, and biology, in the study of objects such as quantum particles in disordered media, the growth of bacterial colonies, traffic flow, and the kinetic theory of gases. A focus is to understand universality, the phenomenon where microscopically different probabilistic models produce the same limiting behavior. This project also contains educational components, including curriculum development and supporting K-12 extracurricular math programs.The specific models to be investigated fall into three categories. The first is the Anderson model described by the lattice Schrödinger equation with i.i.d. random potentials. The main objective is to mathematically establish the localization phenomenon, where wave packets do not spread. The principal investigator (PI) plans to carry out comprehensive studies of this model under reduced regularity assumptions. The second theme of this project is the Kardar-Parisi-Zhang (KPZ) universality, which describes the scaling limit of various random growth processes. In the past quarter-century, enormous progress has been made on those with exact-solvable structures. The PI will use geometric and probabilistic methods to study the asymptotics of several such exactly-solvable models, including local environment limits and scaling limits under large deviation, and a limiting random geometry termed the directed landscape. The ultimate goal is to extend KPZ universality beyond exact-solvability. The third topic concerns Gibbs samplers, which are Monte Carlo Markov Chain (MCMC) algorithms used to sample high-dimensional distributions. The focus is on the continuous state space setting, where tools to analyze time evolution are relatively limited. A particular instance is Kac's walk from kinetic theory, whose order of mixing time was only determined in recent years. The PI plans to develop a general framework to understand the mechanism behind the evolution of these Gibbs samplers, and prove predicted cutoffs for them.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.