  We prove that, for a density of disorder $\rho$ small enough, a certain class
of discrete random Schr\"odinger operators on $\Z^d$ with diluted potentials
exhibits a Lifschitz behaviour from the bottom of the spectrum up to energies
at a distance of the order $\rho^\alpha$ from the bottom of the spectrum, with
$\alpha>2(d+1)/d$. This leads to localization for the energies in this zone for
these low density models. The same results hold for operators on the
continuous, and in particular, with Bernoulli or Poisson random potential.
