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

TL;DR

This paper studies the spectral boundaries of large deformed matrices composed of a deterministic part and a rotationally invariant random part, revealing simple boundary equations for eigenvalue distributions.

Contribution

It derives explicit boundary equations for eigenvalues of large matrices deformed by rotationally invariant random matrices, extending random matrix theory insights.

Findings

Eigenvalue distributions satisfy simple boundary equations in the large-N limit.

Results apply to various explicit random matrix ensembles.

Numerical simulations support the theoretical boundary equations.

Abstract

One of the great miracles of random matrix theory is that, in the $N \to \infty$ 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.