  We study the growth of Laplacian eigenfunctions $ -\Delta \phi_k = \lambda_k
\phi_k$ on compact manifolds $(M,g)$. H\"ormander proved sharp polynomial
bounds on $\| \phi_k\|_{L^{\infty}}$ which are attained on the sphere. On a
`generic' manifold, the behavior seems to be different: both numerics and
Berry's random wave model suggest $\| \phi_k\|_{L^{\infty}} \lesssim
\sqrt{\log{\lambda_k}}$ as the typical behavior. We propose a mechanism,
centered around an $L^1-$analogue of the spectral projector, for explaining the
slow growth in the generic case: for $\phi_{n+1}(x_0)$ to be large, it is
necessary that either (1) several of the first $n$ eigenfunctions were large in
$x_0$ or (2) that $\phi_{n+1}$ is strongly correlated with a suitable linear
combination of the first $n$ eigenfunctions on most of the manifold or (3)
both. An interesting byproduct is quantum entanglement for Laplacian
eigenfunctions: the existence of two distinct points $x,y \in M$ such that the
sequences $(\phi_k(x))_{k=1}^{\infty}$ and $(\phi_k(y))_{k=1}^{\infty}$ do not
behave like independent random variables. The existence of such points is not
to be expected for generic manifolds but common for the classical manifolds and
subtly intertwined with eigenfunction concentration.
