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.
"On the Marchenko-Pastur and Circular Laws for some Classes of Random Matrices with Dependent Entries." Electron. J. Probab. 16 1065 - 1095, 2011. https://doi.org/10.1214/EJP.v16-899