[Paper Review] Asymptotic Spectra of Matrix-Valued Functions of Independent Random Matrices and Free Probability

Friedrich Götze, Holger Kösters+1 more|arXiv (Cornell University)|Aug 7, 2014

Random Matrices and Applications30 references4 citations

TL;DR

This paper establishes a general framework for proving universality of singular value and eigenvalue distributions of matrix-valued functions of independent random matrices using free probability and the Lindeberg principle. It derives a system of equations involving the S-transform and Stieltjes transforms to identify the limiting eigenvalue distribution from the singular value distribution, with explicit formulas for the limiting density in terms of the S-transform and auxiliary functions.

ABSTRACT

We investigate the universality of singular value and eigenvalue distributions of matrix valued functions of independent random matrices and apply these general results in several examples. In particular we determine the limit distribution and prove universality under general conditions for singular value and eigenvalue distributions of products of independent matrices from spherical ensembles.

Motivation & Objective

• To establish universality of singular value and eigenvalue distributions for matrix-valued functions of independent random matrices under general conditions.
• To develop a general method to identify the limiting eigenvalue distribution from the limiting singular value distribution using free probability tools.
• To prove that the limiting eigenvalue distribution can be derived from the S-transform of the singular value distribution via a system of equations involving Stieltjes transforms.
• To extend the applicability of free probability techniques to non-Hermitian and non-Gaussian random matrix models with independent entries.
• To provide a systematic approach for computing the limiting spectral density in complex matrix models, particularly products of random matrices from spherical ensembles.

Proposed method

• Applies the Lindeberg principle to replace i.i.d. matrix entries with Gaussian entries, proving universality under a Lindeberg-type condition and rank/smoothness constraints.
• Uses Girko’s Hermitization principle to relate eigenvalue distribution of a matrix F to the singular value distributions of shifted matrices F − αI for α ∈ ℂ.
• Derives a system of equations (1.1) involving the Stieltjes transform g(z,α) and an auxiliary function w(z,α), using the S-transform S(z) of the singular value distribution of F.
• Employs asymptotic freeness of the matrices in (1.2) and the calculus of R- and S-transforms to derive the key system of equations for g(z,α).
• Establishes analytic continuation and continuity of the Stieltjes transform g(z,α) in z and α, enabling the limit z → 0 to be taken.
• Identifies the limiting eigenvalue density f(u,v) via logarithmic potential theory, expressing it as f(u,v) = (1/(2π|α|²)) · (u ∂ψ/∂u + v ∂ψ/∂v), where ψ(α) = −w(0,α)g(0,α).

Experimental results

Research questions

• RQ1 Under what general conditions is the singular value distribution of a matrix-valued function of independent random matrices universal across different entry distributions?
• RQ2 How can the limiting eigenvalue distribution of such a matrix be derived from its limiting singular value distribution using free probability tools?
• RQ3 What is the precise relationship between the S-transform of the singular value distribution and the Stieltjes transform of the eigenvalue distribution?
• RQ4 Can the limiting eigenvalue density be explicitly computed from the S-transform and auxiliary functions in the limit z → 0?
• RQ5 To what extent does the method apply to products of random matrices from spherical ensembles, and what are the resulting limiting spectral laws?

Key findings

• The limiting eigenvalue distribution of a matrix F = F(X⁽¹⁾, ..., X⁽ᵐ⁾) is determined by the S-transform of its limiting singular value distribution and the asymptotic freeness of the associated block matrices.
• A system of equations (1.1) involving g(z,α) and w(z,α) is derived to characterize the Stieltjes transform of the singular value distribution of F − αI, enabling the computation of the eigenvalue distribution.
• The function ψ(α) = −w(0,α)g(0,α) is shown to be closely related to the logarithmic potential of the limiting eigenvalue distribution, allowing the density f(u,v) to be expressed in terms of its partial derivatives.
• The limiting eigenvalue density is explicitly given by f(u,v) = (1/(2π|α|²)) · (u ∂ψ/∂u + v ∂ψ/∂v), where α = u + iv.
• The method is applied to products of independent matrices from spherical ensembles, proving universality and computing the limiting spectral distribution in the Gaussian case using free probability.
• Analyticity and continuity of the Stieltjes transform g(z,α) are established in a neighborhood of the imaginary axis, enabling the rigorous derivation of the limit z → 0 and the identification of the eigenvalue density.