Discrete Schr\"odinger 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 Schr"odinger operators $H_\omega = - \Delta + V_\omega$on$\ell^2 (\ZZ^d)$where$\Delta$ is the discrete Laplacian and $V_{\omega }$ the multiplication by the function $V_\omega (x) = \sum_{k \in \ZZ^d} \omega_k u(x-k)$\mathrm{.}Here$\omega_k$,$k \in \ZZ^d$, are i.i.d. random variables and $u\in {\mathrm{\ell }}^1({\text{ZZ}}^d;\text{RR})$ is a so-called single-site potential. Since $u$ may change sign, certain properties of $H_{\omega }$ depend in a non-monotone way on the random parameters ${\omega }_k$. This requires new methods at certain stages of the localization proof.