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Spectrum of Non-Hermitian Heavy Tailed Random Matrices

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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 α-stable law, with 0 < α < 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 ~ n 1/α and a probability measure μ α on \({\mathbb{C}}\) depending only on α such that with probability one, the empirical distribution of the eigenvalues of the rescaled matrix \({(a_n^{-1}X_{jk})_{1\leq j,k\leq n}}\) converges weakly to μ α as n → ∞. 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 μ α . In contrast with the Hermitian case, we find that μ α is not heavy tailed.

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Authors and Affiliations

  1. IMT UMR 5219 CNRS and Université Paul-Sabatier Toulouse III, Toulouse, France

    Charles Bordenave

  2. Dipartimento di Matematica, Università Roma Tre, Rome, Italy

    Pietro Caputo

  3. LAMA UMR 8050 CNRS and Université Paris-Est Marne-la-Vallée, Paris, France

    Djalil Chafaï

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  1. Charles Bordenave
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  2. Pietro Caputo
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  3. Djalil Chafaï
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Correspondence to Djalil Chafaï.

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Communicated by P. Forrester

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Bordenave, C., Caputo, P. & Chafaï, D. Spectrum of Non-Hermitian Heavy Tailed Random Matrices. Commun. Math. Phys. 307, 513–560 (2011). https://doi.org/10.1007/s00220-011-1331-9

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