Random Matrices with Application to Quantum Computing and Econometrics

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This project develops new statistical and probabilistic methods and extends some existing methods with novel applications in statistics and econometrics. The core topic of these subjects is random matrix theory. The efficiency of the proposed methodologies will be demonstrated via simulations and applications to real data sets. The outcomes of the project will enhance the applicability of methods for high-dimensional data settings, and the research results will be applied in statistics and econometrics. The project will promote teaching, training, and learning activities at the University of Minnesota. The main educational goals are (a) to train PhD students at the University of Minnesota; and (b) to promote collaboration among experts and students. The research results will be disseminated through conference presentations and publications. The investigator plans to develop new methodologies to investigate properties of a few types of random matrices. They include Haar orthogonal/unitary matrices, sample correlation matrices, Macdonald measures and circular orthogonal ensembles. The investigator will then apply them to answer statistics and econometrics problems. The project consists of the following main themes: (1) the investigator plans to study the approximation of Haar-invariant orthogonal/unitary matrices by independent normals. The solutions are known when the approximation errors are measured by some well-known distances, including the total variation distance. Besides their relevance in theoretical research, they are also applied to data storage. The investigator plans to study the same approximation question under the Wasserstein distance; (2) the investigator plans to explore the Macdonald measure and show the eigenvalues are asymptotically a Gaussian Free Field. A function of these eigenvalues is shown to converge to a Gaussian multiplicative chaos. Nowadays active investigations such as the construction of the Liouville measure in 2d-Liouville quantum gravity or thick points of the Gaussian Free Field have appeared in many branches. Inspired by this understanding, the investigator will study the Macdonald measure; (3) the investigator plans to study the largest entries of sample correlation matrices for dependent data. The same problem for independent data is well-understood. Although the dependent case is more applicable, the technical steps to analyze them are more intensive. The investigator proposes new methods to solve this question; and (4) by using a new method on asymptotic independence between sums and maxima of dependent data, which was established recently by the investigator and his co-authors, it is planned to study problems dealing with high-dimensional panel data arising in econometrics.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.