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We investigate the fluctuations around the mean of the Stieltjes transform ofthe empirical spectral distribution of any selfadjoint noncommutativepolynomial in a Wigner matrix and a deterministic diagonal matrix. We obtainthe convergence in distribution to a centred complex Gaussian process whosecovariance is expressed in terms of operator-valued subordination functions.

Fluctuations of the Stieltjes transform of the empirical spectral  distribution of selfadjoint polynomials in Wigner and deterministic diagonal  matrices