The connection between free probability and random matrices is well known: large random matrices in general position are asymptotically free. In recent papers by Marcus, Spielman and Srivastava they introduced the notion of finite free convolution in connection to expected characteristic polynomials of random matrices. We will describe the theory of cumulants for this convolution developed in joint work with Daniel Perales and give recent results on Berry-Essen theorem, infinite divisibility, Cramer theorem and appearance of "spikes" for this convolution.
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