Independence of linear spectral statistics and the point process at the edge of Wigner matrices

Let Formula: see text be a Wigner matrix of dimension Formula: see text with eigenvalues Formula: see text and Formula: see text be an analytic function on Formula: see text with polynomial growth. It is known that Tr[Formula: see text)]-ETr[[Formula: see text)]] converges in distribution to a normal random variable with mean Formula: see text and a finite variance depending on Formula: see text. On the other hand, it is also known that Formula: see text converges in distribution to the GOE Tracy widom law. In this paper we prove that whenever the entries of the Wigner matrix are sub-Gaussian, Tr[Formula: see text)]-ETr[[Formula: see text)]] is asymptotically independent of the point process at the edge of the spectrum. Hence, one gets that Formula: see text and Tr[Formula: see text)]-ETr[[Formula: see text)]] are asymptotically independent. The main ingredient of the proof is based on a recent paper by Banerjee A new combinatorial approach for tracy–widom law of wigner matrices, preprint (2022), arXiv:2201.00300 . The result of this paper can be viewed as a first step to find the joint distribution of eigenvalues in the bulk and the edge.