Spectral boundaries of deterministic matrices deformed by rotationally invariant random non-Hermitian ensembles

One of the great miracles of random matrix theory is that, in the $N\to \mathrm{\infty }$ limit, many otherwise intractable matrix problems with horrendously complicated finite-$N$ expressions admit remarkably simple and elegant asymptotic solutions. In this paper, we illustrate this phenomenon in the context of spectral boundaries (or spectral edges) for deformed random matrices. Specifically, 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. In the large-$N$ limit, we show that the complex eigenvalue distribution of $\mathbf{A}+\mathbf{B}$ satisfies remarkably simple boundary equations that depend on the ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ transforms of $\mathbf{B}$. We illustrate our results on several explicit random matrix ensembles and support them with numerical simulations.