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

Let W-n be a Wigner matrix of dimension nxnnxn with eigenvalues lambda(1,n )>= & ctdot; >= lambda(n,n) and gg be an analytic function on [-2-epsilon,2+epsilon] with polynomial growth. It is known that Tr[g(W-n)]-E[Tr[g(W-n)]] converges in distribution to a normal random variable with mean 00 and a finite variance depending on gg. On the other hand, it is also known that n(2/3)(lambda(1,n)-2) 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[g(W-n)]-E[Tr[g(W-n)]] is asymptotically independent of the point process at the edge of the spectrum. Hence, one gets that n(2/3)(lambda(1),n-2) and Tr[g(W-n)]-E[Tr[g(W-n)]] 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.