Deformed single ring theorems

Abstract: Given a sequence of deterministic matrices $A = A_N$ and a sequence of deterministic nonnegative matrices $Σ=Σ_N$ such that $A\to a$ and $Σ\to σ$ in $\ast$-distribution for some operators $a$ and $σ$ in a finite von Neumann algebra $\mathcal{A}$. Let $U =U_N$ and $V=V_N$ be independent Haar-distributed unitary matrices. We use free probability techniques to prove that, under mild assumptions, the empirical eigenvalue distribution of $UΣV^*+A$ converges to the Brown measure of $T+a$, where $T\in\mathcal{A}$ is an $R$-diagonal operator freely independent from $a$ and $\vert T\vert$ has the same distribution as $σ$. The assumptions can be removed if $A$ is Hermitian or unitary. By putting $A= 0$, our result removes a regularity assumption in the single ring theorem by Guionnet, Krishnapur and Zeitouni. We also prove a local convergence on optimal scale, extending the local single ring theorem of Bao, Erdős and Schnelli.

• The paper establishes conditional convergence of eigenvalues for deformed non-Hermitian random matrices to the Brown measure.
• It introduces a boundedness condition via the Cauchy transform to remove strict regularity assumptions from classical single ring theorems.
• The study refines local eigenvalue density approximations using subordination techniques, aligning theoretical predictions with empirical observations.

Analysis of "Deformed Single Ring Theorems" (2210.11147)

Introduction

The paper "Deformed Single Ring Theorems" by Ching-Wei Ho and Ping Zhong investigates the spectral distribution of non-Hermitian random matrices modified by certain deterministic matrices using free probability techniques. This work extends existing single ring theorems by providing new conditions under which empirical eigenvalue distributions converge to specific operator limits in von Neumann algebras. The authors refine and generalize prior results by removing certain regularity assumptions and introducing a "deformed" single ring theorem. They also establish a local version of this theorem, which examines eigenvalue distribution at finer scales.

Non-Hermitian Random Matrices and Free Probability

Non-Hermitian random matrices are pivotal in contemporary random matrix theory due to their applications across physics and data science. The single ring theorem, studied by Feinberg and Zee, describes the limiting distribution of eigenvalues of matrices of the form $U\Sigma V^*$, where $U$ and $V$ are Haar-distributed unitary matrices and $\Sigma$ is a deterministic diagonal matrix. The key result is that these matrices' eigenvalues are confined to an annular region, or "single ring".

The authors explore the form $U\Sigma V^* + A$, where $A$ is an additional deterministic matrix, demonstrating that existing theorems can be supplemented by free probability, specifically using free convolution and the Brown measure. They address the empirical eigenvalue distribution of these deformed matrices, under boundaries where $A$ may lack certain hermitian properties.

Deformed Single Ring Theorem

The paper's principal contribution is the derivation of a conditional convergence theorem for the eigenvalues of matrices of the form $U\Sigma V^* + A$ to the Brown measure of the corresponding $R$-diagonal operator sum. Under certain assumptions, such as the invertibility of $\Sigma$ with a bounded condition number, the authors demonstrate that the eigenvalue distribution of these matrices converges to the Brown measure, offering insights into the spectral nature of these deformations.

One of the key innovations of the work is substituting the regularity assumption previously necessary for the baseline single ring theorem with a condition that stems from the Cauchy transform's boundedness for the singular value distributions. This approach broadens the theorem’s applicability, extending to settings where $A$ lacks Hermitian or unitary form.

Local Single Ring Results

Ho and Zhong further extend their results by examining the local behavior of eigenvalues in the bulk spectrum, providing a deformed local single ring theorem. They refine eigenvalue density approximations at optimal scales, aligning with the theoretical density as the matrix size grows. This local theorem employs techniques involving subordination functions and approximate subordination from free probability theory to achieve precise controls over eigenvalue fluctuations.

Implications and Future Directions

The implications of this research are twofold: it enhances the theoretical understanding of spectral distributions in more general settings and offers potential applications in physics and quantitative finance, where non-Hermitian models are prevalent.

The paper sets the stage for future examination of non-Hermitian models under broader conditions or incorporating additional matrix types, such as orthogonal matrices, into the framework. Furthermore, exploring numerical simulations to verify theoretical results could present another avenue for future research, along with potential expansions into understanding the interplay between deterministic matrix perturbations and resulting eigenvalue distributions.

Conclusion

Ching-Wei Ho and Ping Zhong significantly advance the study of non-Hermitian random matrices through "Deformed Single Ring Theorems". Their work provides essential theoretical extensions to the spectral distribution understanding, marrying free probability with empirical eigenvalue convergence in non-traditional matrix constructs. This analysis underlines the dynamic progression of random matrix theory's frontier, aligning classical results with contemporary mathematical frameworks.