  We prove spectral and dynamical localization for the multi-dimensional random
displacement model near the bottom of its spectrum by showing that the approach
through multiscale analysis is applicable. In particular, we show that a
previously known Lifshitz tail bound can be extended to our setting and prove a
new Wegner estimate. A key tool is given by a quantitative form of a property
of a related single-site Neumann problem which can be described as "bubbles
tend to the corners".
