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])]-E[Tr[[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])]-E[Tr[[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])]-E[Tr[[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.