Tail sums of Wishart and Gaussian eigenvalues beyond the bulk edge

Consider the classical Gaussian unitary ensemble of size N and the real white Wishart ensemble with N variables and n degrees of freedom. In the limits of large N and n, with positive ratio γ in the Wishart case, the expected number of eigenvalues that exit the upper bulk edge is less than one, approaching 0.031 and 0.170 respectively, the latter number being independent of γ. These statements are consequences of quantitative bounds on tail sums of eigenvalues outside the bulk which are established here for applications in high dimensional covariance matrix estimation.