On bilinear forms based on the resolvent of large random matrices

TL;DR

This paper investigates the asymptotic behavior of bilinear forms involving the resolvent of large, non-centered random matrices with separable variance profiles, relevant for understanding complex functionals beyond eigenvalues.

Contribution

It provides new theoretical results on the limiting distribution of bilinear forms of resolvents for large non-centered random matrices with separable variance profiles.

Findings

Derived the limiting behavior of $u_n^* Q_n(z) v_n$ as matrix dimensions grow

Established results applicable to non-centered Gram matrices and related functionals

Contributed to the understanding of eigenvalue and resolvent-based statistics in large random matrices

Abstract

Consider a matrix $\Sigma_n$ with random independent entries, each non-centered with a separable variance profile. In this article, we study the limiting behavior of the random bilinear form $u_n^* Q_n(z) v_n$, where $u_n$ and $v_n$ are deterministic vectors, and Q_n(z) is the resolvent associated to $\Sigma_n \Sigma_n^*$ as the dimensions of matrix $\Sigma_n$ go to infinity at the same pace. Such quantities arise in the study of functionals of $\Sigma_n \Sigma_n^*$ which do not only depend on the eigenvalues of $\Sigma_n \Sigma_n^*$, and are pivotal in the study of problems related to non-centered Gram matrices such as central limit theorems, individual entries of the resolvent, and eigenvalue separation.