math.PR

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# Title: Spectrum of non-Hermitian heavy tailed random matrices

Abstract: Let (X_{jk})_{j,k>=1} be i.i.d. complex random variables such that |X_{jk}| 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 exists a deterministic sequence a_n ~ 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 tends to 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.
 Comments: Expanded version of a paper published in Communications in Mathematical Physics 307, 513-560 (2011) Subjects: Probability (math.PR); Spectral Theory (math.SP) Journal reference: Communications in Mathematical Physics 307, 2 (2011) 513-560 DOI: 10.1007/s00220-011-1331-9 Cite as: arXiv:1006.1713 [math.PR] (or arXiv:1006.1713v5 [math.PR] for this version)

## Submission history

From: Djalil Chafai [view email]
[v1] Wed, 9 Jun 2010 06:36:01 GMT (47kb)
[v2] Fri, 11 Jun 2010 12:46:38 GMT (48kb)
[v3] Tue, 3 May 2011 13:01:19 GMT (49kb)
[v4] Wed, 24 Aug 2011 06:17:33 GMT (50kb)
[v5] Fri, 14 Oct 2011 09:23:17 GMT (50kb)
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