Article Dans Une Revue Communications in Mathematical Physics Année : 2011

Spectrum of non-Hermitian heavy tailed random matrices

Résumé

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.

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Dates et versions

hal-00490516 , version 1 (08-06-2010)
hal-00490516 , version 2 (10-06-2010)
hal-00490516 , version 3 (03-05-2011)
hal-00490516 , version 4 (23-08-2011)
hal-00490516 , version 5 (13-10-2011)

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Charles Bordenave, Pietro Caputo, Djalil Chafai. Spectrum of non-Hermitian heavy tailed random matrices. Communications in Mathematical Physics, 2011, 307 (2), pp.513-560. ⟨10.1007/s00220-011-1331-9⟩. ⟨hal-00490516v5⟩
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