Let (X(jk))(j,k >= 1) be i.i.d complex random variables such that vertical bar X(jk)vertical bar is in the domain of attraction of an alpha-stable law, with 0 < alpha < 2. Our main result is a heavy tailed counterpart of Girko's circular law. Namely, under some additional smoothness assumptions on the law of X(jk), we prove that there exist a deterministic sequence a(n) similar to n(1/alpha) and a probability measure mu(alpha) on C depending only on alpha such that with probability one, the empirical distribution of the eigenvalues of the rescaled matrix (a(n)(-1) X(jk))1 <=(j),(k <= n) converges weakly to mu(alpha) as n -> infinity. Our approach combines Aldous & Steele's objective method with Girko's Hermitization using logarithmic potentials. The underlying limiting object is defined on a bipartized version of Aldous' Poisson Weighted Infinite Tree. Recursive relations on the tree provide some properties of mu(alpha). In contrast with the Hermitian case, we find that mu(alpha) is not heavy tailed.

Bordenave, C., Caputo, P., Chafai, D. (2011). Spectrum of Non-Hermitian Heavy Tailed Random Matrices. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 307(2), 513-560 [10.1007/s00220-011-1331-9].

Spectrum of Non-Hermitian Heavy Tailed Random Matrices

CAPUTO, PIETRO;
2011-01-01

Abstract

Let (X(jk))(j,k >= 1) be i.i.d complex random variables such that vertical bar X(jk)vertical bar is in the domain of attraction of an alpha-stable law, with 0 < alpha < 2. Our main result is a heavy tailed counterpart of Girko's circular law. Namely, under some additional smoothness assumptions on the law of X(jk), we prove that there exist a deterministic sequence a(n) similar to n(1/alpha) and a probability measure mu(alpha) on C depending only on alpha such that with probability one, the empirical distribution of the eigenvalues of the rescaled matrix (a(n)(-1) X(jk))1 <=(j),(k <= n) converges weakly to mu(alpha) as n -> infinity. Our approach combines Aldous & Steele's objective method with Girko's Hermitization using logarithmic potentials. The underlying limiting object is defined on a bipartized version of Aldous' Poisson Weighted Infinite Tree. Recursive relations on the tree provide some properties of mu(alpha). In contrast with the Hermitian case, we find that mu(alpha) is not heavy tailed.
2011
Bordenave, C., Caputo, P., Chafai, D. (2011). Spectrum of Non-Hermitian Heavy Tailed Random Matrices. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 307(2), 513-560 [10.1007/s00220-011-1331-9].
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/121321
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 33
  • ???jsp.display-item.citation.isi??? 31
social impact