Random matrices whose entries come from a stationary Gaussian process are studied. It is shown that the limiting spectral distribution is determined by the absolutely continuous component of the spectral measure of the stationary process, a phenomenon resembling that in the situation where the entries of the matrix are i.i.d. On the other hand, the discrete component contributes to the limiting behaviour of the eigenvalues in a completely different way.

The random matrix results obtained are used to understand when a free convolution of two measures is absolutely continuous with respect to the Lebesgue measure. It is shown that if the support of a probability measure is contained in the positive half line, and is bounded away from zero, then its free multiplicative convolution with the semicircle law is absolutely continuous. For the proof, a result concerning the Hadamard product of a deterministic matrix and a scaled Wigner matrix is needed.

This talk is based on joint works with Rajat Subhra Hazra and Deepayan Sarkar.