Non-Asymptotic Approach in Random Matrix Theory

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The proposed research is intended to provide new connections between two areas of mathematics, probability and functional analysis. One of the main objects of investigation is a random matrix, a large rectangular array of random data. The PI strives to understand the properties of such arrays which hold with high probability and the dependence of those properties on the nature of random entries and the structure of the matrix. This study will have potential applications beyond the realm of pure mathematics, as random matrices are used in statistics, computer algorithms, and wireless communication. The PI plans to put a special emphasis on the study of sparse matrices as these matrices naturally appear in signal reconstruction and big data analysis. Another direction of the proposed research is the study of random graphs, which are random networks of nodes connected by roads (edges). Besides representing real transportation networks, graphs can be used to model interaction of atoms in a material, internet communities, etc.The main direction of this research is the non-asymptotic theory of random matrices, a new and rapidly developing area of research analyzing spectral characteristics of a random matrix of a large but fixed size and striving to obtain bounds valid with high probability. The PI intends to study singular values, eigenvalues, and eigenvectors of different ensembles of random matrices of a large size. The results obtained in this direction would have important applications within the random matrix theory in proving limit laws for the spectral characteristics of random matrices. They would be also useful in computer science, as the singular values control the rate of convergence of many numerical algorithms. Another part of the proposed research will address the problems arising in geometry of random graphs This direction is strongly related to random matrix theory as well. The PI will study delocalization and nodal domains of eigenvectors a random graph. The information about them would be valuable in mathematical analysis of congestion in transportation networks.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.