Spectral and Hierarchical Properties of Random Matrices

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The spectral statistics of random matrices describe key properties of the energy levels (eigenstates) of many disordered physical systems. The scope of their applications has expanded recently. For example, integrable growth models, quantum mechanics, and analytic number theory exhibit random matrix eigenvalue distributions. Determinants of random matrices also serve as tools for the analysis of many high-dimensional random functions. Applications include criteria for the possibility of optimization of loss functions in deep learning. More recently, a new connection has emerged, a hierarchical structure behind the eigenvalues of random matrices and L-functions, their extremes being characterized by analogy with branching processes. This project will extend and combine recent methods developed for these random matrix spectral and hierarchical universalities to enlarge their scope, and apply these techniques to random landscapes. The project provides research training opportunities for graduate students.The PI will first work on universal connections between random matrices, branching processes, L-functions and gaussian multiplicative chaos, by studying the Fyodorov-Keating conjectures up to tightness of the maximum, their universality in the class of Wigner matrices and beta ensembles, and the emergence of the 2d Gaussian multiplicative chaos measures from the characteristic polynomial of random normal matrices. The PI will extend the Fisher-Hartwig asymptotic formulas to settings allowing any temperature. The PI will also study the topological complexity of random landscapes and the universality of their critical exponents, as the signal and noise vary.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.