  Eigenstates of local many-body interacting systems that are far from spectral
edges are thought to be ergodic and close to being random states. This is
consistent with the eigenstate thermalization hypothesis and volume-law scaling
of entanglement. We point out that systematic departures from complete
randomness are generically present in mid-spectrum eigenstates, and focus on
the departure of the entanglement entropy from the random-state prediction. We
show that the departure is (partly) due to spatial correlations and due to
orthogonality to the eigenstates at the spectral edge, which imposes structure
on the mid-spectrum eigenstates.
