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

We consider discrete Schr"odinger operators of the form $H=-\mathrm{\Delta }+V$ on ${\mathrm{\ell }}^2({\mathbb{Z}}^d)$, where $\mathrm{\Delta }$ is the discrete Laplacian and $V$ is a bounded potential. Given $\mathrm{\Gamma }\subset {\mathbb{Z}}^d$, the $\mathrm{\Gamma }$-trimming of $H$ is the restriction of $H$ to ${\mathrm{\ell }}^2({\mathbb{Z}}^d\setminus \mathrm{\Gamma })$, denoted by $H_{\mathrm{\Gamma }}$. We investigate the dependence of the ground state energy $E_\Gamma(H)=\inf \sigma (H_\Gamma)$on$\Gamma$\mathrm{.}Weshowthatforrelativelydensepropersubsets$\Gamma$of$\Z^d$wealwayshave$E_\Gamma(H)>E_\emptyset(H)$. We use this lifting of the ground state energy to establish Wegner estimates and localization at the bottom of the spectrum for $\mathrm{\Gamma }$-trimmed Anderson models, i.e., Anderson models with the random potential supported by the set $\mathrm{\Gamma }$