We analyze the spectral distribution of symmetric random matrices with correlated entries. While we assume that the diagonals of these random matrices are stochastically independent, the elements of the diagonals are taken to be correlated. Depending on the strength of correlation, the limiting spectral distribution is either the famous semicircle distribution, the distribution derived for Toeplitz matrices by Bryc, Dembo and Jiang (2006), or the free convolution of the two distributions.
"A phase transition for the limiting spectral density of random matrices." Electron. J. Probab. 18 1 - 17, 2013. https://doi.org/10.1214/EJP.v18-2118