[Paper Review] Limiting Spectral Distributions of Sums of Products of Non-Hermitian Random Matrices

Holger Kösters, А. Н. Тихомиров|arXiv (Cornell University)|Jun 14, 2015

Random Matrices and Applications41 references4 citations

TL;DR

This paper establishes the limiting spectral distribution of sums of products of non-Hermitian i.i.d. random matrices and their inverses, showing universality and stability under a free probability convolution. It proves that the eigenvalue distribution of the sum of m independent copies of such products converges to the same limiting distribution as a single product after proper rescaling, extending prior results on Ginibre matrices and establishing a stable limit law via free multiplicative convolution and S-transforms.

ABSTRACT

For fixed $l,m \ge 1$, let $\mathbf{X}_n^{(0)},\mathbf{X}_n^{(1)},\dots,\mathbf{X}_n^{(l)}$ be independent random $n imes n$ matrices with independent entries, let $\mathbf{F}_n^{(0)} := \mathbf{X}_n^{(0)} (\mathbf{X}_n^{(1)})^{-1} \cdots (\mathbf{X}_n^{(l)})^{-1}$, and let $\mathbf{F}_n^{(1)},\dots,\mathbf{F}_n^{(m)}$ be independent random matrices of the same form as $\mathbf{F}_n^{(0)}$. We investigate the limiting spectral distributions of the matrices $\mathbf{F}_n^{(0)}$ and $\mathbf{F}_n^{(1)} + \dots + \mathbf{F}_n^{(m)}$ as $n o \infty$. Our main result shows that the sum $\mathbf{F}_n^{(1)} + \dots + \mathbf{F}_n^{(m)}$ has the same limiting eigenvalue distribution as $\mathbf{F}_n^{(0)}$ after appropriate rescaling. This extends recent findings by Tikhomirov and Timushev (2014). To obtain our results, we apply the general framework recently introduced in Götze, Kösters and Tikhomirov (2014) to sums of products of independent random matrices and their inverses. We establish the universality of the limiting singular value and eigenvalue distributions, and we provide a closer description of the limiting distributions in terms of free probability theory.

Motivation & Objective

• To determine the limiting spectral distribution of sums of products of independent non-Hermitian random matrices and their inverses as the matrix size n → ∞.
• To establish universality of the limiting eigenvalue distribution across different entry distributions satisfying moment conditions.
• To show that the limiting distribution of such matrix sums is stable under a free multiplicative convolution operation.
• To characterize the limiting distribution explicitly using free probability tools, particularly the S-transform and the R-transform.
• To extend previous results on Ginibre matrices to more general products involving inverses and sums of i.i.d. matrices.

Proposed method

• Applies the general framework from Götze, Kösters, and Tikhomirov (2014) to analyze sums of products of independent random matrices and their inverses.
• Uses the concept of free multiplicative convolution ⊠ and additive convolution ⊞ to describe the limiting eigenvalue distribution.
• Employs the S-transform to characterize the limiting distribution of products of random matrices, particularly for the matrix Fₙ⁽⁰⁾ = Xₙ⁽⁰⁾(Xₙ⁽¹⁾)⁻¹⋯(Xₙ⁽ˡ⁾)⁻¹.
• Applies the Helffer–Sjöstrand functional calculus to control the spectral measure and derive convergence in probability.
• Uses inequalities for singular values (e.g., Horn's inequality) to verify the necessary moment and stability conditions (C0–C2).
• Leverages bounds on small singular values of non-Hermitian random matrices (e.g., from [22, 23]) to ensure invertibility and stability of the inverse matrices with high probability as n → ∞.

Experimental results

Research questions

• RQ1 Does the limiting spectral distribution of a sum of m independent copies of a product of non-Hermitian random matrices with i.i.d. entries coincide with that of a single such product after appropriate rescaling?
• RQ2 Is the limiting eigenvalue distribution of such matrix products universal across different entry distributions satisfying only finite second moments?
• RQ3 Can the limiting distribution be described explicitly using free probability theory, particularly via the S-transform and free multiplicative convolution?
• RQ4 Does the limiting distribution of these matrix products exhibit stability under free convolution, analogous to classical stable laws?
• RQ5 What is the precise relationship between the limiting spectral distribution of Fₙ⁽⁰⁾ and the rescaled sum Fₙ⁽¹⁾ + ⋯ + Fₙ⁽ᵐ⁾ in the large n limit?

Key findings

• The limiting spectral distribution of the sum m⁻⁽ˡ⁺¹⁾ᐟ²(Fₙ⁽¹⁾ + ⋯ + Fₙ⁽ᵐ⁾) coincides with that of Fₙ⁽⁰⁾ after proper rescaling, establishing a form of stability under free convolution.
• The limiting eigenvalue distribution of Fₙ⁽⁰⁾ is given by the free multiplicative convolution H(Q⁻¹(γ₁⁻¹ ⊠ γ₁^⊞ˡ¹ ⊠ ⋯ ⊠ γ₁^⊞ˡᵏ)), where H denotes the Herglotz transform and Q⁻¹ the R-transform inverse.
• The limiting distribution is universal: it depends only on the number of matrices and the exponents in the product, not on the specific distribution of the matrix entries, provided they satisfy the moment conditions (1.2)–(1.4).
• The S-transform of the limiting distribution of Fₙ⁽⁰⁾ is shown to be σₛ(2/(l+1)), confirming its stability under the free multiplicative convolution with m copies.
• The convergence of the empirical eigenvalue distribution μₙ to the non-random limit μ holds weakly in probability as n → ∞.
• The results extend previous findings by Tikhomirov and Timushev (2014) to include matrix products involving inverses and sums of independent copies, with rigorous justification via singular value bounds and free probability.