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On the spectral radius of the ratio of Girko matrices

Djalil Chafaï, David García-Zelada, Yuan Yuan Xu

TL;DR

The paper establishes a universal high-dimensional behavior for the spectral radius of the ratio of two independent Girko matrices, showing that after scaling by $\sqrt{n}$ the fluctuations converge to a heavy-tailed law governed by the minimum of an independent Gamma sequence. The spherical (Ginibre) case is analyzed in depth via inversion symmetry, Girko Hermitization, and a replacement principle that leverages local laws and a Green function comparison to reduce to the infinite Ginibre ensemble, with Kostlan’s observations yielding explicit limiting distributions. The core contributions include a rigorous replacement framework, a precise description of the spherical ensemble as a determinantal Coulomb gas with a rotationally invariant equilibrium measure, and a demonstration that edge and bulk statistics are universal for this ratio model. These results illuminate why the spectral radius of the ratio exhibits simpler universality than that of a single Girko matrix and connect random matrix theory with determinantal point processes and potential-theoretic techniques, offering a blueprint for analyzing similar ratio-based non-normal matrix models.

Abstract

Girko matrices have independent and identically distributed entries of mean zero and unit variance. In this note, we consider the random matrix model formed by the ratio of two independent Girko matrices, its entries are dependent and heavy-tailed. Our main message is that divided by the square root of the dimension, the spectral radius of the ratio converges in distribution, when the dimension tends to infinity, to a universal heavy-tailed distribution. We provide a mathematical proof of this high-dimensional phenomenon, under a fourth moment matching with a Gaussian case known as the complex Ginibre ensemble. In this Gaussian case, the model is known as the spherical ensemble, and its spectrum is a determinantal planar Coulomb gas. Its image by the inverse stereographic projection is a rotationally invariant gas on the two-sphere. A crucial observation is the invariance in law of the model under inversion, related to its spherical symmetry, and that makes, in a sense, edge and bulk equivalent. Our approach involves Girko Hermitization, local law estimates for Wigner matrices, lower bound estimates on the smallest singular value, and convergence of kernels of determinantal point processes. The universality of the high-dimensional fluctuation of the spectral radius of the ratio of Girko matrices turns out to be remarkably more accessible mathematically than for a single Girko matrix!

On the spectral radius of the ratio of Girko matrices

TL;DR

The paper establishes a universal high-dimensional behavior for the spectral radius of the ratio of two independent Girko matrices, showing that after scaling by the fluctuations converge to a heavy-tailed law governed by the minimum of an independent Gamma sequence. The spherical (Ginibre) case is analyzed in depth via inversion symmetry, Girko Hermitization, and a replacement principle that leverages local laws and a Green function comparison to reduce to the infinite Ginibre ensemble, with Kostlan’s observations yielding explicit limiting distributions. The core contributions include a rigorous replacement framework, a precise description of the spherical ensemble as a determinantal Coulomb gas with a rotationally invariant equilibrium measure, and a demonstration that edge and bulk statistics are universal for this ratio model. These results illuminate why the spectral radius of the ratio exhibits simpler universality than that of a single Girko matrix and connect random matrix theory with determinantal point processes and potential-theoretic techniques, offering a blueprint for analyzing similar ratio-based non-normal matrix models.

Abstract

Girko matrices have independent and identically distributed entries of mean zero and unit variance. In this note, we consider the random matrix model formed by the ratio of two independent Girko matrices, its entries are dependent and heavy-tailed. Our main message is that divided by the square root of the dimension, the spectral radius of the ratio converges in distribution, when the dimension tends to infinity, to a universal heavy-tailed distribution. We provide a mathematical proof of this high-dimensional phenomenon, under a fourth moment matching with a Gaussian case known as the complex Ginibre ensemble. In this Gaussian case, the model is known as the spherical ensemble, and its spectrum is a determinantal planar Coulomb gas. Its image by the inverse stereographic projection is a rotationally invariant gas on the two-sphere. A crucial observation is the invariance in law of the model under inversion, related to its spherical symmetry, and that makes, in a sense, edge and bulk equivalent. Our approach involves Girko Hermitization, local law estimates for Wigner matrices, lower bound estimates on the smallest singular value, and convergence of kernels of determinantal point processes. The universality of the high-dimensional fluctuation of the spectral radius of the ratio of Girko matrices turns out to be remarkably more accessible mathematically than for a single Girko matrix!
Paper Structure (18 sections, 10 theorems, 112 equations, 3 figures)

This paper contains 18 sections, 10 theorems, 112 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Comments and open problems
  4. Model.
  5. First order universality.
  6. Inner radius of Girko matrices.
  7. Moments assumptions.
  8. Characteristic polynomials
  9. Ratio of other models
  10. Conventions and notation
  11. The spherical ensemble
  12. Coulomb gas and determinantal structure
  13. Kostlan observation and spectral radius
  14. Equilibrium measure
  15. Proof of Theorem \ref{['th:spherical:girko:gininf']} and Theorem \ref{['th:spherical:girko:rho']}
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $M=AB^{-1}$ where $A$ and $B$ are independent $n\times n$ Girko matrices, not necessarily of the same law, both satisfying it:condensity--it:condmom. Let $\mathrm{Gin}_\infty$ be the infinite Ginibre determinantal point process on $\mathbb{C}$, with kernel $K(z,w)=\sqrt{\gamma(z)\gamma(w)}\mathr

Figures (3)

  • Figure 1: Sample of the spherical ensemble and its planar stereographic projection.
  • Figure 2: The point $T(P)\in\mathbb{C}$ is the image of $P\in\mathbb{S}^2\setminus\{N\}$ by the stereographic projection or transform $T$ with respect of the north pole $N$, see \ref{['eq:T']}. See also zbMATH07645442. The image of $N$ is $\infty$. For gases, the gap around $N$ on $\mathbb{S}^2$ is in correspondence with the gap around $\infty$ on $\mathbb{C}\cup\{\infty\}$.
  • Figure 3: Histograms illustrating the universality of the high dimensional behavior, for three choices of entries law : symmetric complex Bernoulli or Rademacher, isotropic complex Gaussian, and symmetric complex discrete heavy tailed (same tail as Zipf or Pareto, $\alpha=4$). For the Bernoulli and Zipf cases, the matrices are also conditioned to be nonsingular by using rejection sampling.

Theorems & Definitions (20)

  • Example 1.1: Moment matching with the standard complex Gaussian distribution
  • Theorem 1.2: From ratio of Girko matrices to infinite Ginibre
  • Theorem 1.3: Spectral radius
  • Theorem 1.4: Replacement principle for local eigenvalue statistics near a point
  • Remark 1.5: Real case
  • proof : Proof of Theorem \ref{['th:spherical:girko:gininf']}
  • proof : Proof of Theorem \ref{['th:spherical:girko:rho']}
  • Theorem 4.1: Comparison theorem
  • Lemma 4.2
  • Lemma 4.3
  • ...and 10 more