Random matrix products: Universality and least singular values

TL;DR

This paper proves the universality of local spectral statistics and Gaussian limits for products of independent iid random matrices, extending previous results to multiple matrices and non-Gaussian entries.

Contribution

It generalizes universality and Gaussian limit results from single matrices to products of multiple matrices with non-Gaussian entries, using moment matching and singular value bounds.

Findings

Established local universality of correlation functions for matrix products

Proved Gaussian limits for linear spectral statistics of matrix products

Derived explicit variances for the limiting distributions

Abstract

We establish, under a moment matching hypothesis, the local universality of the correlation functions associated with products of $M$ independent iid random matrices, as $M$ is fixed, and the sizes of the matrices tend to infinity. This generalizes an earlier result of Tao and the third author for the case $M=1$. We also prove Gaussian limits for the centered linear spectral statistics of products of $M$ independent iid random matrices. This is done in two steps. First, we establish the result for product random matrices with Gaussian entries, and then extend to the general case of non-Gaussian entries by another moment matching argument. Prior to our result, Gaussian limits were known only for the case $M=1$. In a similar fashion, we establish Gaussian limits for the centered linear spectral statistics of products of independent truncated random unitary matrices. In both cases, we are able to obtain explicit expressions for the limiting variances. The main difficulty in our study is that the entries of the product matrix are no longer independent. Our key technical lemma is a lower bound on the least singular value of the translated linearization matrix associated with the product of $M$ normalized independent random matrices with independent and identically distributed subgaussian entries. This lemma is of independent interest.