Spectral properties of deterministic matrices multiplied by rotationally invariant random non-Hermitian ensembles

In this paper, we study spectral properties of multiplicative deformations of non-Hermitian random matrices. We consider matrices of the form $\mathbf{A}\mathbf{B}$, where $\mathbf{A}$ is a deterministic $N\times N$ matrix (not necessarily Hermitian) and $\mathbf{B}$ is a rotationally invariant random matrix. We show that, as $N\to \mathrm{\infty }$, the boundary of the complex eigenvalue distribution of $\mathbf{A}\mathbf{B}$ is governed by simple equations involving the ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ transforms of $\mathbf{B}$.