### 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 language | English (US) |
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Pages (from-to) | 715-751 |

Number of pages | 37 |

Journal | Communications in Mathematical Physics |

Volume | 278 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2008 |

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### ASJC Scopus subject areas

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

### Cite this

*Communications in Mathematical Physics*,

*278*(3), 715-751. https://doi.org/10.1007/s00220-007-0389-x

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

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 278, no. 3, pp. 715-751. https://doi.org/10.1007/s00220-007-0389-x

}

TY - JOUR

T1 - The spectrum of heavy tailed random matrices

AU - Arous, Gérard Ben

AU - Guionnet, Alice

PY - 2008/3

Y1 - 2008/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=39149116408&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=39149116408&partnerID=8YFLogxK

U2 - 10.1007/s00220-007-0389-x

DO - 10.1007/s00220-007-0389-x

M3 - Article

VL - 278

SP - 715

EP - 751

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -