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

An optimal result on localization in random displacements models