Spectral Properties of Random Matrices

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This project will advance our knowledge of random matrices, a set of operators that serves as a paradigm for many high-dimensional systems. While the Gaussian distribution emerges universally from probabilistic models exhibiting independence, random matrix statistics appear from correlated systems, so that the scope of their applications have considerably expanded recently. For example, integrable systems, growth models, analytic number theory and quantum mechanics exhibit random matrix statistics. Moreover, random matrix techniques have improved our theoretical understanding of numerical analysis and deep learning networks. Methods from probability (random walks and coupling), analysis (homogenization) and mathematical physics (loop equations) were recently advanced in connection with spectral analysis of random matrices, to prove eigenvalues and eigenvectors universality. The PI will further develop such tools for the following problems: (1) Delocalization for band matrices. The PI aims at understanding universality beyond mean field models. One key question concerns band matrices in dimension 1 up to the ``Anderson'' transition. A recent approach to this problem involves a mean field reduction and the connection with quantum unique ergodicity, and it will be developed further; (2) Universality of extremal statistics. This includes extreme spectral spacings, large values of the characteristic polynomial, fluctuations of individual eigenvalues and more generally universal aspects of log-correlated random fields;(3) Coulomb gases, the Gaussian free field and Gaussian multiplicative chaos. Recent progress gave convergence of the electric field of 2D Coulomb gases to the Gaussian free field. A natural extension concerns convergence of the characteristic polynomial to Gaussian multiplicative chaos;(4) Eigenvectors of non-Hermitian random matrices. Physicists predicted in the 1990s the typical size of overlaps between eigenvectors from the Ginibre ensemble. This was recently upgraded to an explicit limiting distribution of diagonal overlaps, and to correlations of off-diagonal overlaps. Extension of this new integrability to the joint law of all overlaps will be studied.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.