{"id":62,"date":"2010-06-12T09:04:46","date_gmt":"2010-06-12T07:04:46","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=62"},"modified":"2012-08-10T06:45:38","modified_gmt":"2012-08-10T04:45:38","slug":"spectrum-of-non-hermitian-heavy-tailed-random-matrices","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2010\/06\/12\/spectrum-of-non-hermitian-heavy-tailed-random-matrices\/","title":{"rendered":"Spectrum of non-Hermitian heavy tailed random matrices"},"content":{"rendered":"

\"Spectrum<\/a><\/p>\n

I have recently uploaded the paper arXiv:1006.1713 [math.PR]<\/a> entitled Spectrum of non-Hermitian heavy tailed random matrices<\/em>, written in collaboration with Charles Bordenave<\/a> and Pietro Caputo<\/a>.<\/p>\n

We provide a rigorous analysis of the phenomenon behind the pictures above. Let \\( {(X_{jk})_{j,k\\geq1}} \\) be i.i.d. complex random variables with cumulative distribution function \\( {F} \\). Our main result is a heavy tailed counterpart of Girko's circular law when \\( {F} \\) has an infinite second moment. Roughly, if \\( {F} \\) is in the attraction domain of an \\( {\\alpha} \\)-stable law, with \\( {0<\\alpha<2} \\), then we prove that there exist a deterministic sequence \\( {a_n\\sim n^{1\/\\alpha}} \\) and a probability measure \\( {\\mu_\\alpha} \\) on \\( {\\mathbb{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\\leq j,k\\leq n}} \\) converges weakly to \\( {\\mu_\\alpha} \\) as \\( {n\\rightarrow\\infty} \\). 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 and Hermitized cases, we find that \\( {\\mu_\\alpha} \\) is not heavy tailed.<\/p>\n

Let us first recall that the eigenvalues<\/em> of an \\( {n\\times n} \\) complex matrix \\( {M} \\) are the roots in \\( {\\mathbb{C}} \\) of its characteristic polynomial. We label them \\( {\\lambda_1(M),\\ldots,\\lambda_n(M)} \\) with non growing modules and growing phases. We also denote by \\( {s_1(M)\\geq\\cdots\\geq s_n(M)} \\) the singular values<\/em> of \\( {M} \\), defined for every \\( {1\\leq k\\leq n} \\) by \\( {s_k(M):=\\lambda_k(\\sqrt{MM^*})} \\). We define the empirical spectral measure and the empirical singular values measure as<\/p>\n

\\[ \\mu_M = \\frac 1 n \\sum_{k=1} ^n \\delta_{\\lambda_k (M)} \\quad \\text{and } \\quad \\nu_M = \\frac 1 n \\sum_{k=1} ^n \\delta_{s_k (M)}. \\]<\/p>\n

Let us define the \\( {n\\times n} \\) random matrix \\( {X = (X_{ij}) _{ 1 \\leq i, j \\leq n}} \\). Following Dozier and Silverstein<\/a>, if \\( {F} \\) has finite positive variance \\( {\\sigma^2} \\), then for every \\( {z\\in\\mathbb{C}} \\), there exists a probability measure \\( {\\mathcal{Q}_{\\sigma,z}} \\) on \\( {[0,\\infty)} \\) depending only on \\( {\\sigma} \\) and \\( {z} \\), with explicit Cauchy-Stieltjes transform, such that a.s. (almost surely) <\/a><\/p>\n

\\[ \\nu_{\\frac{1}{\\sqrt{n}}X-zI} \\underset{n\\rightarrow\\infty}{\\rightsquigarrow} \\mathcal{Q}_{\\sigma,z} \\ \\ \\ \\ \\ (1) \\]<\/p>\n

where \\( {\\rightsquigarrow} \\) denotes the weak convergence of probability measures. The proof of (1)<\/a> is based on a classical approach for Hermitian random matrices with bounded second moment: truncation, centralization, recursion on the resolvent, and cubic equation for the limiting Cauchy-Stieltjes transform (fixed point characterization). In the special case \\( {z=0} \\), the statement (1)<\/a> reduces to the quartercircular law theorem (square version of the Marchenko-Pastur theorem (see e.g. Marchenko and Pastur<\/a>, Wachter<\/a>, Yin<\/a>) and the probability measure \\( {Q_{\\sigma,0}} \\) is the quartercircular law with Lebesgue density <\/a><\/p>\n

\\[ x\\mapsto \\frac{1}{\\pi\\sigma^2}\\sqrt{4\\sigma^2-x^2}\\mathbf{1}_{[0,2\\sigma]}(x). \\ \\ \\ \\ \\ (2) \\]<\/p>\n

Girko's<\/a> famous circular law theorem states under the same assumptions that a.s. <\/a><\/p>\n

\\[ \\mu_{\\frac{1}{\\sqrt{n}}X} \\underset{n\\rightarrow\\infty}{\\rightsquigarrow} \\mathcal{U}_\\sigma \\ \\ \\ \\ \\ (3) \\]<\/p>\n

where \\( {\\mathcal{U}_\\sigma} \\) is the uniform law on the disc \\( {\\{z\\in\\mathbb{C};|z|\\leq\\sigma\\}} \\). This statement was established through a long sequence of partial results Mehta<\/a>, Girko<\/a>, Silverstein<\/a>, Edelman<\/a>, Girko<\/a>, Bai<\/a>, Girko<\/a>, Bai and Silverstein<\/a>, Pan and Zhou<\/a>, Götze and Tikhomirov<\/a>, Tao and Vu<\/a>, the general case (3)<\/a> being finally obtained by Tao and Vu<\/a>, by using Girko's Hermitization with logarithmic potentials and uniform integrability, the convergence (1)<\/a>, and polynomial bounds on the extremal singular values. The idea of using directly logarithmic potentials was already present in the work of Goldsheid and Khoruzhenko<\/a> for non-Hermitian random tridiagonal matrices.<\/p>\n

The aim of our work is to investigate what happens when \\( {F} \\) does not have a finite second moment. We shall consider the following hypothesis:<\/p>\n