  The results of a comparative study of electronic spectral properties as well
as phonon Probability Distribution Function (PDF) are reported.
  The study is done using Dynamical Mean Field Theory for the half-filled
spinless and spinful Holstein model and for a single hole in Holstein t-J
model.
  Qualitative changes in phonon PDF signals a polaronic crossover while an
electronic Metal Insulator Transition (MIT) due to pairing into a bipolaronic
state can be seen from a gap in the electronic DOS or by the vanishing of the
quasi particle spectral weight. MIT can be observed only at sufficiently high
density provided the Coulomb repulsion is neglected. In the limit of low
density the carrier show a polaronic behavior trough the development of a
definite polarization shifting the average value of the phonon PDF. At large
densities occupied and empty sites result an a bimodal the phonon PDF.
  At the polaron crossover fluctuation of phonon coordinates tends to be larger
than those at any other coupling. In this regime Born-Oppenheimer approximation
is shown to fail also when phonon frequency is much less then electron
bandwidth. For a single hole in the t-J model a further source of localization
is due to magnetic superexchange energy which tends to localize the spin-defect
spreading due to the presence of the hole.
