  Consider the Schr\"odinger operator $-\triangle+\lambda V$ with non-negative
iid random potential $V$ of strength $\lambda>0$. We prove existence and
uniqueness of the associated landscape function on the whole space, and show
that its correlations decay exponentially. As a main ingredient we establish
the (annealed and quenched) exponential decay of the Green function of
$-\triangle+\lambda V$ using Agmon's positivity method, rank-one perturbation
in dimensions $d\ge 3$, and first-passage percolation in dimensions $d=1,2$.
