  Consider the classical Gaussian unitary ensemble of size $N$ and the real
Wishart 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 edge
is less than one, 0.031 and 0.170 respectively, the latter number being
independent of $\gamma$. 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.
