  In the spectrum of many-body quantum systems, the low-energy eigenstates were
the traditional focus of research. The interest in the statistical properties
of the full eigenspectrum has grown more recently, in particular in the context
of non-equilibrium questions. Wave functions of interacting lattice quantum
systems can be characterized either by local observables, or by global
properties such as the participation ratio (PR) in a many-body basis or the
entanglement between various partitions. We present a study of the PR and of
the entanglement entropy (EE) between two roughly equal spatial partitions of
the system, in all the eigenfunctions of local Hamiltonians. Motivated by the
similarity of the PR and EE - both are generically larger in the bulk and
smaller near the edges of the spectrum - we quantitatively analyze the
correlation between them. We elucidate the effect of (proximity to)
integrability, showing how low-entanglement and low-PR states appear also in
the middle of the spectrum as one approaches integrable points. We also
determine the precise scaling behavior of the eigenstate-to-eigenstate
fluctuations of the PR and EE with respect to system size, and characterize the
statistical distribution of these quantities near the middle of the spectrum.
