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Chapitre D'ouvrage Année : 2014

Lecture notes on the circular law

Résumé

The circular law theorem states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension n tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices. In these expository notes, we present a proof in a Gaussian case, due to Mehta and Silverstein, based on a formula by Ginibre, and a proof of the universal case by revisiting the approach of Tao and Vu, based on the Hermitization of Girko, the logarithmic potential, and the control of the small singular values. We also discuss some related models and open problems.
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Dates et versions

hal-01015134 , version 1 (25-06-2014)

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  • HAL Id : hal-01015134 , version 1

Citer

Charles Bordenave, Djalil Chafaï. Lecture notes on the circular law. Van H. Vu. Modern Aspects of Random Matrix Theory, American Mathematical Society, pp.1-35, 2014, Proceedings of Symposia in Applied Mathematics (PSAPM volume 72), 978-0-8218-9471-2. ⟨hal-01015134⟩
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