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We consider discrete Schr\"odinger operators of the form $H=-\Delta +V$ on$\ell^2(\Z^d)$, where $\Delta$ is the discrete Laplacian and $V$ is a boundedpotential. Given $\Gamma \subset \Z^d$, the $\Gamma$-trimming of $H$ is therestriction of $H$ to $\ell^2(\Z^d\setminus\Gamma)$, denoted by $H_\Gamma$. Weinvestigate the dependence of the ground state energy $E_\Gamma(H)=\inf \sigma(H_\Gamma)$ on $\Gamma$. We show that for relatively dense proper subsets$\Gamma$ of $\Z^d$ we always have $E_\Gamma(H)>E_\emptyset(H)$. We use thislifting of the ground state energy to establish Wegner estimates andlocalization at the bottom of the spectrum for $\Gamma$-trimmed Andersonmodels, i.e., Anderson models with the random potential supported by the set$\Gamma$

Ground state energy of trimmed discrete Schr\"odinger operators and localization for trimmed Anderson models