In this talk, we discuss the fluctuation of f(X) as a matrix, where X is a large square random matrix with centered, independent, identically distributed entries and f is an analytic function. In particular, we show that for a generic deterministic matrix A of the same size as X, the trace of f(X)A is approximately Gaussian which decomposes into two independent modes corresponding to tracial and traceless parts of A. We also briefly discuss the proof that mainly relies on Hermitization of X and its resolvents.
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