# The Spectrum of Heavy Tailed Random Matrices

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We obtain uniform asymptotics for polynomials orthogonal on a fixed and varying arc of the unit circle with a positive analytic weight function. We also complete the proof of the large s asymptotic expansion for the Fredholm determinant with the kernel sinz/(Ï€ z) on the interval [0,s],...

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Let ( X) be i.i.d. complex random variables such that | X| is in the domain of attraction of an Î±-stable law, with 0 < Î± < 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, we prove that there...

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There are two parts in this paper. In the first part we construct the Markov chain in random environment (MCRE), the skew product Markov chain and pâ€“ $$ \ifmmode\expandafter\vec\else\expandafter\vecabove\fi{\theta } $$ chain from a random transition matrix and a twoâ€“dimensional...

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The circular law asserts that the spectral measure of eigenvalues of rescaled random matrices without symmetry assumption converges to the uniform measure on the unit disk. We prove a local version of this law at any point $$z$$ away from the unit circle. More precisely, if $$ | |z| - 1 | \ge...

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The ubiquitous presence of complexity in nature makes it necessary to seek new mathematical tools which can probe physical systems beyond linear or perturbative approximations. The random matrix theory is one such tool in which the statistical behavior of a system is modeled by an ensemble of...

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- Eigenvectors of some large sample covariance matrix ensembles. Ledoit, Olivier; Péché, Sandrine // Probability Theory & Related Fields;Jul2011, Vol. 151 Issue 1/2, p233
We consider sample covariance matrices $${S_N=\frac{1}{p}\Sigma_N^{1/2}X_NX_N^* \Sigma_N^{1/2}}$$ where X is a N Ã— p real or complex matrix with i.i.d. entries with finite 12th moment and Î£ is a N Ã— N positive definite matrix. In addition we assume that the spectral measure of Î£...