Random Levy Matrices Revisited

TL;DR

This paper compares eigenvalue densities of Levy-distributed Wigner matrices and free random Levy matrices, showing their spectral stability, similar tail behaviors, and how their addition leads to a maximal entropy ensemble, supported by analytical and numerical results.

Contribution

It provides a detailed comparison of eigenvalue spectra between Wigner-Levy and free random Levy matrices, including analytical corrections and the relation via matrix addition.

Findings

Eigenvalue densities match analytical predictions for N=100.

Spectra show weak dependence on matrix size N.

Addition of Wigner-Levy matrices yields free Levy spectra, illustrating a matrix central limit theorem.

Abstract

We compare eigenvalue densities of Wigner random matrices whose elements are independent identically distributed (iid) random numbers with a Levy distribution and maximally random matrices with a rotationally invariant measure exhibiting a power law spectrum given by stable laws of free random variables. We compute the eigenvalue density of Wigner-Levy (WL) matrices using (and correcting) the method by Bouchaud and Cizeau (BC), and of free random Levy (FRL) rotationally invariant matrices by adapting results of free probability calculus. We compare the two types of eigenvalue spectra. Both ensembles are spectrally stable with respect to the matrix addition. The discussed ensemble of FRL matrices is maximally random in the sense that it maximizes Shannon's entropy. We find a perfect agreement between the numerically sampled spectra and the analytical results already for matrices of dimension N=100. The numerical spectra show very weak dependence on the matrix size N as can be noticed by comparing spectra for N=400. After a pertinent rescaling spectra of Wigner-Levy matrices and of symmetric FRL matrices have the same tail behavior. As we discuss towards the end of the paper the correlations of large eigenvalues in the two ensembles are however different. We illustrate the relation between the two types of stability and show that the addition of many randomly rotated Wigner-Levy matrices leads by a matrix central limit theorem to FRL spectra, providing an explicit realization of the maximal randomness principle.