An optimal result on localization in random displacements models

Abstract

Abstract We study random displacements models with a long-range particle-media interaction potential 𝔲 ⁢ ( r , θ ) = 𝔣 ⁢ ( θ ) ⁢ r - A {\mathfrak{u}(r,\theta)=\mathfrak{f}(\theta)r^{-A}} in polar coordinates, with a smooth function 𝔣 {\mathfrak{f}} which can be sign-indefinite. Spectral and dynamical localization, with an asymptotically exponential decay of eigenfunction correlators, is proved under the optimal condition A > d {A>d} .