# Finite-rank perturbations of random band matrices via infinitesimal free probability

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Free probability provides a unifying framework for studying random multi-matrix models in the large $N$ limit. Typically, the purview of these techniques is limited to invariant or mean-field ensembles. Nevertheless, we show that random band matrices fit quite naturally into this framework. Our considerations extend to the infinitesimal level, where finer results can be stated for the $\frac{1}{N}$ correction. As an application, we consider the question of outliers for finite-rank perturbations of our model. In particular, we find outliers at the classical positions from the deformed Wigner ensemble. No background in free probability is assumed.