The Anderson Transition in Random Matrix Theory

This project will investigate open problems at the intersection of probability theory and physics and develop new techniques for the study of the Anderson transition, a concept in condensed matter physics describing a sharp change from conducting to insulating behavior in certain materials as the density of impurities is increased. This transition plays a pivotal role in the physics of semiconductors and quantum materials. While the Anderson transition has been studied for decades, many foundational theoretical questions remain unresolved. Further, recent advances have revealed a tight connection between these questions and current research in many-body quantum systems. The project aims to provide a rigorous mathematical framework for the Anderson transition and its applications to contemporary quantum physics. The project also incorporates training of graduate students, research opportunities for undergraduate students, and the creation of new expository materials. A variety of mathematical models of disordered systems will be studied using tools from random matrix theory. The project will consider random matrices with heavy-tailed and sparse entries, as well as random operators on tree-like structures. The research aims to rigorously establish the existence of Anderson transitions in such models. It will also explore closely related phenomena, including multifractal eigenstates and anomalous quantum dynamics, which are believed to occur alongside Anderson transitions. The planned approaches will leverage recent insights in random matrix theory related to the resolvent formalism and local spectral laws. 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.