Products of Independent Gaussian Random Matrices

TL;DR

This thesis provides exact formulas for eigen- and singular value correlations in products of independent Gaussian matrices, analyzing their asymptotic behaviors as matrix size or number of factors grow large.

Contribution

It derives exact correlation functions for eigen- and singular values in Gaussian matrix products, enabling detailed asymptotic analysis.

Findings

Exact correlation functions for eigen- and singular values derived

Asymptotic limits for densities and correlations established

Results applicable for arbitrary matrix dimensions and number of factors

Abstract

This thesis reviews recent progress on products of random matrices from the perspective of exactly solved Gaussian random matrix models. We derive exact formulae for the correlation functions for the eigen- and singular values at arbitrary matrix dimension and for an arbitrary number of factors. These exact results are used to study asymptotic limits for the macroscopic densities and the microscopic correlations as either the matrix dimension or the number of factors tends to infinity.