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

TL;DR

This work addresses the problem of characterizing the limiting spectral boundaries of sums $oldsymbol{A}+oldsymbol{B}$ where $oldsymbol{A}$ is deterministic and $oldsymbol{B}$ is rotationally invariant but non-Hermitian. It develops a unifying framework built on the $oldsymbol{ ilde{oldsymbol{R}}}_1$ and $oldsymbol{ ilde{oldsymbol{R}}}_2$ transforms and their associated multivalued functions to derive two explicit boundary equations, expressed in terms of $oldsymbol{ au}(oldsymbol{A})$, $oldsymbol{ au}(oldsymbol{A}oldsymbol{A}^*)$, $h_{oldsymbol{A}}$, and $h_{oldsymbol{B}}$. The paper verifies the theory across several ensembles (complex Ginibre, elliptic, Haar unitary, Wishart, and two-ring invariant) and demonstrates universal edge behavior in bi-invariant cases, with concrete, testable boundary formulas for each model. Numerical simulations illustrate the predicted spectral boundaries and the geometry of edges (including lemniscates, ellipses, and inner holes), providing practical tools for stability analysis and edge-detection in non-Hermitian random deformations.

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.

Figures (5)

• Figure 1: Empirical eigenvalues from single realizations of non-Hermitian deformations of $\vb{D}$. Left: $\vb{D}+\vb{U}$ with $\vb{U}$ a random unitary matrix ($N=1000$); the lemniscate \ref{['eqlemniscate']} gives the theoretical spectral boundary. Center: $\vb{D}+\sigma\vb{G}$ with $\sigma=1.1$ and $\vb{G}$ a complex Ginibre matrix ($N=1000$); the outer boundary is given by \ref{['outer']} with $r_{+,\vb{B}}=\sigma$. Right: $\vb{D}+\sigma\vb{U}$ with $\sigma=1.1$ and $\vb{U}$ a random unitary matrix ($N=1000$); besides the outer boundary \ref{['outer']}, an inner spectral edge appears at radius $\sqrt{\sigma^2-1}$.
• Figure 2: Complex eigenvalues of a matrix of size $N = 500$ given by $\vb{W}_q + \sigma \vb{G}$, where $\vb{W}_q$ is a complex Wishart matrix with parameter $q = 1/4$, $\sigma = 0.3$, and $\vb{G}$ is a complex Ginibre matrix. The theoretical spectral boundary is also shown using the equation \ref{['theo']}.
• Figure 3: The right panel shows the eigenvalues from a single realization of the matrix $\vb{M}$ with $q = 0.5$ and $N = 1000$. We use the procedure described in Proposition \ref{['propsamoussa']} to determine the spectral boundary. The left panel shows the curve $\mathcal{C}$ computed numerically, and $\varphi(\mathcal{C})$ is drawn on the right as the predicted spectral boundary.
• Figure 4: Empirical eigenvalues of $\vb{U} + \sigma \vb{E}$, where $\vb{U}$ is a random unitary matrix, $\sigma = 0.8$, and $\vb{E}$ is a complex elliptic matrix with parameter $\tau = 0.8$. The boundaries are given by Proposition \ref{['soron']}.
• Figure 5: Complex eigenvalues of the matrix $\vb{M}$ defined in Eq.\ref{['defM']} ($N=500$) for several parameter pairs $(r_1,r_2)$. The theoretical spectral boundaries predicted by Conjecture \ref{['conjecture']} are also shown.

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Definition 3: Complex Ginibre ensemble
• Definition 4: Elliptic Ginibre ensemble
• Definition 5: Haar unitary ensemble
• Definition 6: Complex Wishart ensemble
• Definition 7: Two--ring invariant ensemble
• Proposition 8
• proof
• Proposition 9