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We prove Lifshitz behavior at the bottom of the spectrum for non--negativerandom potentials, i.\,e.\ show that the IDS is exponentially small at lowenergies. The theory is developed for the breather potential and generalized toall non--negative random potentials in a second step. Since our models need notbe ergodic, we need to identify the minimum of the spectrum. We deduce aninitial length scale estimate from the Lifshitz bound for the breather modeland combine it with a recent Wegner estimate to establishes Andersonlocalization via multi-scale analysis. Finally, for ergodic models, wecomplement the Lifshitz behavior with a lower bound. We provide detailed proofsaccessible to non-experts.

Lifshitz asymptotics and localization for random breather models