[Paper Review] Euclidean random matrices and their applications in physics

Arthur Goetschy, S. E. Skipetrov|arXiv (Cornell University)|Mar 12, 2013

Random Matrices and ApplicationsMathematics7 references19 citations

TL;DR

This paper presents a comprehensive theoretical framework for Euclidean random matrices (ERMs), focusing on eigenvalue density in both Hermitian and non-Hermitian cases using diagrammatic, free probability, and field-theoretic methods. It establishes analytical tools to describe eigenvalue statistics in spatially correlated random matrices, with key results on the semicircle and circular laws, and applies them to problems in condensed matter physics, optics, and quantum chaos.

ABSTRACT

We review the state of the art of the theory of Euclidean random matrices, focusing on the density of their eigenvalues. Both Hermitian and non-Hermitian matrices are considered and links with simpler, standard random matrix ensembles are established. We discuss applications of Euclidean random matrices to contemporary problems in condensed matter physics, optics, and quantum chaos.

Motivation & Objective

• To develop a unified analytical framework for the eigenvalue density of large Euclidean random matrices (ERMs), which are defined by spatially correlated random matrix elements.
• To extend standard random matrix theory to ERMs by addressing the nontrivial, spatially correlated statistics of matrix elements that lack a known joint probability distribution.
• To apply the developed formalism to physical systems such as vibrational modes in disordered solids, electron glass dynamics, and wave propagation in random media.
• To establish connections between ERM theory and established ensembles like Gaussian and Wishart matrices through free probability and diagrammatic techniques.
• To provide analytical predictions for eigenvalue statistics that can be validated against numerical diagonalization and used in physical applications.

Proposed method

• Employing a field-theoretic representation of ERMs as $ A = HTH^ angle $, where $ H $ has i.i.d. entries, enabling the use of diagrammatic techniques.
• Using the resolvent and $ \mathcal{R} $-transform formalism to relate eigenvalue density to the Stieltjes transform of the spectral measure.
• Applying the Dyson gas picture to model eigenvalue statistics via a one-body potential and mean-field approximation.
• Implementing a diagrammatic expansion to derive self-consistent equations for the resolvent and eigenvalue density in the high- and low-density limits.
• Utilizing free probability theory to generalize results from standard ensembles (Gaussian, Wishart) to ERMs through the $ \mathcal{R} $-transform and $ \mathcal{S} $-transform.
• Extending the formalism to non-Hermitian ERMs via Hermitization and the use of left/right eigenvector biorthogonal bases, with applications to the random Green’s matrix.

Experimental results

Research questions

• RQ1 How can the eigenvalue density of Hermitian Euclidean random matrices be analytically derived despite the lack of a known joint probability distribution for their elements?
• RQ2 What is the role of spatial correlations in the eigenvalue statistics of ERMs, and how do they differ from standard random matrix ensembles?
• RQ3 How can the diagrammatic and free probability approaches be systematically applied to derive the eigenvalue density in both high- and low-density limits of ERMs?
• RQ4 What is the analytical description of the eigenvalue density for non-Hermitian ERMs, and how does it compare to the circular law and other known results?
• RQ5 In what physical systems do ERMs provide a natural description, and how can the analytical framework be used to predict observable phenomena such as the boson peak or Anderson localization?

Key findings

• The eigenvalue density of Hermitian ERMs converges to the semicircle law in the high-density limit, with corrections arising from spatial correlations.
• For non-Hermitian ERMs, the eigenvalue density follows the circular law in the bulk, with edge effects and non-universal behavior near the spectral boundary.
• The diagrammatic approach yields self-consistent equations for the resolvent, which can be solved analytically in the $ N \to \infty $ limit, enabling prediction of the eigenvalue density.
• The field representation $ A = HTH^ angle $ allows the use of free probability tools, enabling the derivation of the $ \mathcal{R} $-transform for ERMs from known results on $ T $ and $ H $.
• The two-point correlation function of eigenvalues is derived via the irreducible vertex $ U(z,z') $, with a closed-form expression in terms of the resolvent and matrix $ T $.
• The theory successfully predicts the boson peak in vibrational spectra of disordered solids and collective emission in atomic ensembles, with results matching numerical simulations.