Invariant Ensembles of Random Matrices: New Techniques, New Horizons

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The contemporary subject of Random Matrix Theory has emerged as a viable model framework for the analysis of complex random systems comprised of many highly correlated components. Such systems are ubiquitous in mathematics and nature, ranging from complex number-theoretic functions, to atomic nuclei, to public transit systems. This project will develop new techniques for the spectral analysis of a broad class of random matrices whose distribution is immune to the action of a continuous group of symmetries. The novel approach which will be developed has the potential to dramatically expand our current understanding of this important paradigm.The goal of this project is to bring new methods from algebraic combinatorics and geometry (Weingarten calculus, Hurwitz theory) to the study of invariant ensembles of random matrices, developing a new approach to the spectral theory of these ensembles which is very general and robust. Specifically, the goal of the project is to develop an analogue of Wigner's moment method which specifically targets invariant ensembles. This approach will be used to analyze spectral statistics of invariant ensembles of random matrices, in both the single-matrix and more general multi-matrix cases. The moment method developed is not limited to random matrices with an absolutely continuous law; in particular, it is applicable to discrete mixtures of orbital measures. This leads to important new connections between random matrix theory and statistical mechanics.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.