Paper detail

On bilinear forms based on the resolvent of large random matrices

Consider a matrix $Σ_n$ 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 $Σ_n Σ_n^*$ as the dimensions of matrix $Σ_n$ go to infinity at the same pace. Such quantities arise in the study of functionals of $Σ_n Σ_n^*$ which do not only depend on the eigenvalues of $Σ_n Σ_n^*$, 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.