Non-Asymptotic Random Matrix Theory and Random Graphs

This 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 Principal Investigator 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. A special emphasis will be placed on the study of sparse matrices as these matrices naturally appear in signal reconstruction and big data analysis. Another direction of the 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 project provides research training opportunities for graduate students.

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 Principal Investigator 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. Another area of study will be the geometric properties of such matrices considered as linear operators between certain normed spaces. Such results can find applications in computer science and signal reconstruction where random matrices are widely used for signal encoding and decoding.

Another part of this research will address the problems arising in geometry of random graphs. The Principal Investigator will also concentrate on studying the process of growth of random graphs by analyzing the evolution of corresponding random matrices.

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.