Ratio of Girko Matrices

• The ratio of Girko matrices is defined as M = A B⁻¹, where A and B are independent Girko matrices with i.i.d. complex entries, exhibiting heavy-tailed properties.
• It features universal spectral characteristics, determinantal point processes, and a spherical ensemble structure that connects to Coulomb gas behavior.
• Analysis leverages Hermitization, local laws, and fourth moment matching to rigorously establish spectral radius limits and edge-to-bulk symmetry.

The ratio of Girko matrices refers to random matrix models constructed from two independent Girko matrices—each having independent, identically distributed complex entries with mean zero and unit variance—by forming the matrix quotient $M = A B^{-1}$. This object occupies a central role in non-Hermitian random matrix theory, especially regarding universal spectral properties and their connections to determinantal point processes, spherical ensembles, and high-dimensional universality phenomena (Chafaï et al., 21 Oct 2025).

1. Definition and Construction

A Girko matrix $A \in \mathbb{C}^{n \times n}$ consists of i.i.d. entries $a_{ij}$ with $\mathbb{E} a_{ij} = 0$, $\mathbb{E} |a_{ij}|^2 = 1$. The ratio $M = A B^{-1}$, with $A$ and $B$ independent Girko matrices, produces entries that are highly dependent and heavy-tailed. In the Gaussian case (entries standard complex normal), the ratio model is equivalent to the spherical ensemble, whose joint eigenvalue density is determinantal and expresses a Coulomb gas structure in the complex plane.

2. Spectral Radius and High-Dimensional Fluctuations

The core result for the spectral radius $ρ_{\max}(M)$ is that, when rescaled by $1/\sqrt{n}$, its limiting distribution is universal and heavy-tailed. Explicitly,

$$\frac{\rho_{\max}(M)}{\sqrt{n}}\xrightarrow[n \to \infty]{d} \frac{1}{\sqrt{\min_{k\ge1}\gamma_k}}$$

where the $\gamma_k$ are independent gamma random variables $\gamma_k \sim \mathrm{Gamma}(k,1)$. The tail probability decays as $1/x^2$, indicating a strong presence of heavy-tailed behavior. This universality holds provided the fourth moments of the entries match those of the standard complex normal, a condition verified by a Green function comparison or Ornstein–Uhlenbeck interpolation in the proof.

Correspondingly, the minimum spectral radius satisfies

$$\sqrt{n}\,\rho_{\min}(M)\xrightarrow[n \to \infty]{d} \sqrt{\min_{k\ge1} \gamma_k}$$

highlighting the edge-to-bulk symmetry in the spectral statistics of the ratio ensemble.

3. Spherical Ensemble, Coulomb Gas, and Stereographic Symmetry

For $A, B$ Ginibre matrices, $M$'s eigenvalues follow a spherical ensemble: the joint density is

$$\prod_{i<j}|z_i-z_j|^2 \prod_{i=1}^n (1+|z_i|^2)^{-(n+1)}$$

This is interpreted as a determinantal planar Coulomb gas with potential $Q(|z|) = \ln(1 + |z|^2)$. Under stereographic projection $T^{-1}(z)$, the distribution lifts to a rotationally invariant gas on the two-sphere $S^2$, leading to invariance under $z \mapsto 1/z$. This symmetry equates edge and bulk behaviors: the largest modulus eigenvalue statistics are determined by the smallest gaps near the origin, and vice versa.

4. Hermitization and Proof Strategy

Analysis of the ratio leverages Girko Hermitization, which relates the eigenvalue problem for the non-Hermitian quotient to the log-determinant of the Hermitized matrix

$(A-zB)(A-zB)^*$

The empirical spectral distribution $\mu_M$ is generated via

$$\int f(z)\,d\mu_M(z) = \frac{1}{4\pi}\int \Delta f(z)\,\log\det[(A-zB)(A-zB)^*]\,d^2z$$

Rigorous proof utilizes local laws for resolvents, control of the smallest singular value (to avoid spurious small denominators), and kernel convergence for determinantal point processes. Universality is established by a fourth moment matching theorem: the gap probabilities and kernel statistics of the non-Gaussian Girko ratio model are asymptotically identical to those in the Ginibre (Gaussian) case.

5. Universality and Accessibility

The limiting distribution for the spectral radius of the ratio model is universally characterized by independent gamma variables, regardless of the fine details of the entry distributions (beyond the fourth moment match). Remarkably, this universality is easier to establish mathematically for the ratio model than for the single Girko matrix spectral radius, due to the spherical symmetry and kernel determinantal structure of the spherical ensemble.

Invariance under inversion ($z \mapsto 1/z$) arising from the stereographic setting makes the analysis of edge and bulk statistics equivalent—a property not present in the single-matrix case.

6. Mathematical Formulations

Key formulas summarizing the theory:

• Empirical spectral measure and radius:

$$\mu_M = \frac{1}{n}\sum_{\lambda\in\operatorname{spec}(M)}\delta_\lambda,\quad \rho_{\max}(M) = \max_{\lambda\in\operatorname{spec}(M)}|\lambda|$$

• Spherical ensemble density:

$$\prod_{i<j}|z_i-z_j|^2 \prod_{i=1}^n (1+|z_i|^2)^{-(n+1)}$$

• Stereographic projection:

$$T(x)=\frac{x_1+ix_2}{1-x_3},\qquad T^{-1}(z)=\frac{(2\Re z,2\Im z,|z|^2-1)}{1+|z|^2}$$

7. Implications and Applications

The universality of the scaled spectral radius for the ratio of Girko matrices advances understanding of spectral edges in non-Hermitian ensembles, relevant for systems modeled by matrix quotients, spherical ensembles, and random linear systems with noise. The heavy-tailed nature of the spectral edge can inform limits of stability and transitions in physical and engineering applications, and motivates further investigation of singular value distributions, kernel universality, and connections with Coulomb gases and integrable systems.

8. Open Problems and Perspectives

One notable observation is that, despite the heavier dependencies and tail behavior in the ratio ensemble, the global fluctuation analysis is mathematically more tractable than for single Girko matrices, due to the structure inherited from spherical symmetry and determinantal point processes. Future directions include extensions to more general products/ratios, analysis of finer singular value statistics, and exploration of multi-matrix universality classes under less restrictive moment matching.

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The ratio of Girko matrices, when analyzed under high dimensionality and fourth moment regularity, yields universal and explicitly characterizable spectral edge distributions exhibiting deep symmetries and connections to spherical ensembles, determinantal point processes, and random matrix universality (Chafaï et al., 21 Oct 2025).