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Consider the classical Gaussian unitary ensemble of size $N$ and the realWishart ensemble $W_N(n,I)$. In the limits as $N \to \infty$ and $N/n \to\gamma > 0$, the expected number of eigenvalues that exit the upper bulk edgeis less than one, 0.031 and 0.170 respectively, the latter number beingindependent of $\gamma$. These statements are consequences of quantitativebounds on tail sums of eigenvalues outside the bulk which are established herefor applications in high dimensional covariance matrix estimation.

Tail sums of Wishart and GUE eigenvalues beyond the bulk edge