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We prove a sharp    N    transition for the infinitesimal distribution of a periodically banded GUE matrix. For bandwidths    b  N   =  Ω    N    , we further prove that our model is infinitesimally free from the matrix units and the normalized all‐1’s matrix. Our results allow us to extend previous work of Shlyakhtenko on finite‐rank perturbations of Wigner matrices in the infinitesimal framework. For finite‐rank perturbations of our model, we find outliers at the classical positions from the deformed Wigner ensemble. © 2020 Wiley Periodicals LLC

Finite‐Rank Perturbations of Random Band Matrices via Infinitesimal Free Probability