Random matrices and branching processes

This project will advance our knowledge of random matrices, which arise from many correlated systems in physics, data science, pure and applied mathematics. While the Gaussian universality class is intimately related to the notion of independence, the random matrix universality class was proposed by Wigner to model stable energy levels of heavy nuclei. The models he introduced have since been understood to serve as paradigms for deep phenomena such as energy level repulsion, eigenstate thermalization, hierarchical self-assembly, disorder in high dimension, and the distribution of the zeros of the Riemann zeta function. This project will provide deeper probabilistic understanding for longstanding problems. The awardee will also engage in a variety of organizational, educational and outreach activities, including conference organization, course development, talks for a range of audiences, and mentoring of students. A new line of research has appeared recently in random matrix theory, establishing a connection with the active part of probability theory which studies logarithmically correlated fields, such as branching Brownian motion and the two-dimensional Gaussian free field. Processes in this class are also intimately related to the Gaussian multiplicative chaos random measures. Some of these connections were made rigorous thanks to the discovery of a hierarchical structure behind random matrices and L-functions. The PI will work on dynamical and branching techniques to discover new statistics, for the following problems regarding random spectra and other complex systems. (1) Branching in integrable random matrix theory and number theory, first in relation to the Fyodorov-Hiary-Keating conjectures, then with studies on a hierarchy behind the multiplicative Fourier transform. (2) Universality of extreme statistics in random matrix theory, with logarithmically correlations for classes of random matrix models. (3) Fisher-Hartwig asymptotics, with the development of probabilistic techniques for the analysis of Toeplitz determinants with singularities and generalizations, and the connection with Gaussian multiplicative chaos random measures. (4) Quantitative universality, with the analysis of the spectral form factor of random matrices, the main observable used in physics to identify the random matrix universality class. 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.