The spectrum of heavy tailed random matrices

Gérard Ben Arous, Alice Guionnet

Research output: Contribution to journalArticle

Abstract

Let X N be an N → N random symmetric matrix with independent equidistributed entries. If the law P of the entries has a finite second moment, it was shown by Wigner [14] that the empirical distribution of the eigenvalues of X N , once renormalized by √N , converges almost surely and in expectation to the so-called semicircular distribution as N goes to infinity. In this paper we study the same question when P is in the domain of attraction of an α-stable law. We prove that if we renormalize the eigenvalues by a constant a N of order N1α, the corresponding spectral distribution converges in expectation towards a law μ which only depends on α. We characterize μα and study some of its properties; it is a heavy-tailed probability measure which is absolutely continuous with respect to Lebesgue measure except possibly on a compact set of capacity zero.

Original languageEnglish (US)
Pages (from-to)715-751
Number of pages37
JournalCommunications in Mathematical Physics
Volume278
Issue number3
DOIs
StatePublished - Mar 2008

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Random Matrices
Eigenvalue
Converge
entry
Stable Laws
Spectral Distribution
Empirical Distribution
Domain of Attraction
eigenvalues
Lebesgue Measure
Absolutely Continuous
Symmetric matrix
Compact Set
Probability Measure
Infinity
Moment
infinity
attraction
Zero
moments

ASJC Scopus subject areas

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

Cite this

The spectrum of heavy tailed random matrices. / Arous, Gérard Ben; Guionnet, Alice.

In: Communications in Mathematical Physics, Vol. 278, No. 3, 03.2008, p. 715-751.

Research output: Contribution to journalArticle

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