Free random Lévy and Wigner-Lévy matrices

Zdzisław Burda, Jerzy Jurkiewicz, MacIej A. Nowak, G. Papp, Ismail Zahed

Research output: Contribution to journalArticle

19 Citations (Scopus)

Abstract

We compare eigenvalue densities of Wigner random matrices whose elements are independent identically distributed random numbers with a Lévy 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-Lévy matrices using (and correcting) the method by Bouchaud and Cizeau, and of free random Lévy (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-Lévy 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-Lévy matrices leads by a matrix central limit theorem to FRL spectra, providing an explicit realization of the maximal randomness principle.

Original languageEnglish
Article number051126
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume75
Issue number5
DOIs
Publication statusPublished - May 30 2007

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matrices
Random Matrices
eigenvalues
Ensemble
Eigenvalue
Free Probability
Weak Dependence
Tail Behavior
Stable Laws
Shannon Entropy
random numbers
Random number
Largest Eigenvalue
Rescaling
random variables
Invariant Measure
calculus
Central limit theorem
Identically distributed
Randomness

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Condensed Matter Physics
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Free random Lévy and Wigner-Lévy matrices. / Burda, Zdzisław; Jurkiewicz, Jerzy; Nowak, MacIej A.; Papp, G.; Zahed, Ismail.

In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol. 75, No. 5, 051126, 30.05.2007.

Research output: Contribution to journalArticle

Burda, Zdzisław ; Jurkiewicz, Jerzy ; Nowak, MacIej A. ; Papp, G. ; Zahed, Ismail. / Free random Lévy and Wigner-Lévy matrices. In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics. 2007 ; Vol. 75, No. 5.
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