  Shannon-R\'enyi entropies and associated participation spectra quantify how
much a many-body wave-function is localized in a given configuration basis.
Using these tools, we present an analysis of the ground-state wave functions of
various quantum spin systems in one and two dimensions. General ideas and a
review of the current status of this field are first given, with a particular
emphasis on universal subleading terms characterizing different quantum phases
of matter, and associated transitions. We highlight the connection with the
related entanglement entropies and spectra when this is possible.
  In a second part, new results are presented for the participation spectra of
interacting spin models, mostly based on quantum Monte Carlo simulations, but
also using perturbation theory in some cases. For full antiferromagnetic
systems, participation spectra are analyzed in terms of ferromagnetic domain
walls which experience a repulsive pairwise interaction. This confinement
potential is either linear for long-range N\'eel order, or logarithmic for
quasi-long-range order. The case of subsystems is also analyzed in great detail
for a 2d dimerized Heisenberg model undergoing a quantum phase transition
between a gapped paramagnet and a N\'eel phase. Participation spectra of line
shaped (1d) sub-systems are quantitatively compared with finite temperature
participation spectra of ansatz effective boundary (1d) entanglement
Hamiltonians. While short-range models describe almost perfectly the gapped
side, the N\'eel regime is best compared using long-range effective
Hamiltonians. Spectral comparisons performed using Kullback-Leibler
divergences, a tool potentially useful for entanglement spectra, provide a
quantitative way to identify both the best boundary entanglement Hamiltonian
and temperature.
