It is known that the normalized average of an i.i.d. sequence of free random variables with _nite variance converge to the semicircle law not only in distribution, but also on the level of densities. This superconvergence phenomenon, observed _rst by Bercovici and Voiculescu, was subsequently extended to arbitrary limit laws for free additive convolution. In a joint work with H. Bercovici and JC Wang, using subordination techniques, we show that the same phenomenon occurs for the multiplicative versions of free convolution on the positive line and on the unit circle. In the second part, I will discuss briey another joint work with CW Ho on the Brown measures of free Brownian motions with certain nontrivial initial conditions, extending a recent work of Driver-Hall-Kemp, where the same subordination functions appeared sur-prisingly. The famous circular law (due to Girko, Bai, Tao, Vu and many others) states that n_n square random matrices with independent and i.i.d. entries that have mean zero and variance 1/n convergences to the uniform distribution on the unit disk as the size n tends to in_nity. Our result for circular Brownian motion provides a density formula for the candidate of limit of those random matrices perturbed by any deterministic Hermitian matrices that converge to some limit.

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