Structured Random Matrices for Quantum Chaos and Manifold Learning

The study of large random matrices has been driven by an ever-increasing range of applications, beginning a century ago with the work of Wishart in statistics, followed by their application to nuclear physics by Wigner in the 1950s; more recently, they have informed sweeping conjectures in number theory, models of ecosystems and neural networks, and heuristics for machine learning, financial mathematics, and quantum information theory. The success of random matrix theory is due to (1) their mathematical structure allowing for explicit computations, and (2) the universality phenomenon, that random matrix distributions appear in diverse settings with no apparent connection to matrices. Despite significant recent advances, a rigorous mathematical understanding of the broad universality phenomenon is lacking. This project investigates classes of structured random matrices, in order to close the gap between computable models and real-world systems. The types of structure considered are motivated by two application domains: (1) Quantum chaos theory, which conjectures that random matrix statistics govern the behavior of a broad class of physical quantum systems; (2) Manifold learning, where random matrices help understand when it is possible to extract a meaningful signal from noisy measurements, e.g., in medical imaging. The project also enriches the experience of graduate and undergraduate students through research opportunities and course development. For quantum chaos theory, the project investigates the underlying mechanisms of universality by developing robust arguments for the main signatures of quantum chaos, that is, eigenvalue repulsion and eigenvector delocalization, for structured random Markov matrices associated with expander graphs and manifolds with curvature conditions. For manifold learning theory, the project studies inner product kernel matrices formed from noisy data with manifold structure, providing theoretical guarantees for the performance of spectral-based unsupervised manifold learning algorithms. 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.