On the Marchenko-Pastur and Circular Laws for some Classes of Random Matrices with Dependent Entries

By Radosław Adamczak. 2011

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In the first part of the article we prove limit theorems of Marchenko-Pastur type for the average spectral distribution of random matrices with dependent entries satisfying a weak law of large numbers, uniform bounds on moments and a martingale like condition investigated previously by Goetze and Tikhomirov. Examples include log-concave unconditional distributions on the space of matrices. In the second part we specialize to random matrices with independent isotropic unconditional log-concave rows for which (using the Tao-Vu replacement principle) we prove the circular law.

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Around the circular law (2011, background): “its covariance matrix is identity) then we expect that EμX tends to the circular law, see [1] for i.”