  We prove Lifshitz behavior at the bottom of the spectrum for non--negative
random potentials, i.\,e.\ show that the IDS is exponentially small at low
energies. The theory is developed for the breather potential and generalized to
all non--negative random potentials in a second step. Since our models need not
be ergodic, we need to identify the minimum of the spectrum. We deduce an
initial length scale estimate from the Lifshitz bound for the breather model
and combine it with a recent Wegner estimate to establishes Anderson
localization via multi-scale analysis. Finally, for ergodic models, we
complement the Lifshitz behavior with a lower bound. We provide detailed proofs
accessible to non-experts.
