The spectrum of heavy-tailed random matrices
| dc.creator | Arous, Gerard Ben | |
| dc.creator | Guionnet, Alice | |
| dc.date | 2007-07-14 | |
| dc.date.accessioned | 2026-07-07T08:18:27Z | |
| dc.date.available | 2026-07-07T08:18:27Z | |
| dc.description | Let $X_N$ be an $N\ts N$ random symmetric matrix with independent equidistributed entries. If the law $P$ of the entries has a finite second moment, it was shown by Wigner \cite{wigner} that the empirical distribution of the eigenvalues of $X_N$, once renormalized by $\sqrt{N}$, 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 $N^{\frac{1}α}$, the corresponding spectral distribution converges in expectation towards a law $μ_α$ which only depends on $α$. We characterize $μ_α$ 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. | |
| dc.identifier | https://arxiv.org/abs/0707.2159 | |
| dc.identifier | http://arxiv.org/abs/0707.2159 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/134436 | |
| dc.subject | Probability | |
| dc.subject | Mathematical Physics | |
| dc.subject | 15A52; 60E07 | |
| dc.title | The spectrum of heavy-tailed random matrices | |
| dc.type | text |