Localization for random Schr\"odinger operators with low density potentials

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.