Free random Lévy matrices

Zdzislaw Burda, Romuald A. Janik, Jerzy Jurkiewicz, Maciej A. Nowak, G. Papp, Ismail Zahed

Research output: Contribution to journalArticle

33 Citations (Scopus)

Abstract

Using the theory of free random variables and the Coulomb gas analogy, we construct stable random matrix ensembles that are random matrix generalizations of the classical one-dimensional stable Levy distributions. We show that the resolvents for the corresponding matrices obey transcendental equations in the large size limit. We solve these equations in a number of cases, and show that the eigenvalue distributions exhibit Levy tails. For the analytically known Lévy measures we explicitly construct the density of states using the method of orthogonal polynomials. We show that the Levy tail distributions are characterized by a different novel form of microscopic universality.

Original languageEnglish
Article number021106
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume65
Issue number2
DOIs
Publication statusPublished - Feb 2002

Fingerprint

Random Matrices
Tail
Lévy Distribution
Coulomb Gas
Eigenvalue Distribution
Stable Distribution
Transcendental
Density of States
matrices
Resolvent
Orthogonal Polynomials
Universality
Analogy
Ensemble
Random variable
random variables
polynomials
eigenvalues
gases
Generalization

ASJC Scopus subject areas

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

Cite this

Free random Lévy matrices. / Burda, Zdzislaw; Janik, Romuald A.; Jurkiewicz, Jerzy; Nowak, Maciej A.; Papp, G.; Zahed, Ismail.

In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol. 65, No. 2, 021106, 02.2002.

Research output: Contribution to journalArticle

Burda, Zdzislaw ; Janik, Romuald A. ; Jurkiewicz, Jerzy ; Nowak, Maciej A. ; Papp, G. ; Zahed, Ismail. / Free random Lévy matrices. In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics. 2002 ; Vol. 65, No. 2.
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