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Discrete Schrödinger operators with random alloy-type potential

We review recent results on localization for discrete alloy-type models based on the multiscale analysis and the fractional moment method, respectively. The discrete alloy-type model is a family of Schrödinger operators $H_ω= - Δ+ V_ω$ on $\ell^2 (\ZZ^d)$ where $Δ$ is the discrete Laplacian and $V_ω$ the multiplication by the function $V_ω(x) = \sum_{k \in \ZZ^d} ω_k u(x-k)$. Here $ω_k$, $k \in \ZZ^d$, are i.i.d. random variables and $u \in \ell^1 (\ZZ^d ; \RR)$ is a so-called single-site potential. Since $u$ may change sign, certain properties of $H_ω$ depend in a non-monotone way on the random parameters $ω_k$. This requires new methods at certain stages of the localization proof.