  We consider elliptic differential operators on either the entire Euclidean
space $\mathbb{R}^d$ or on subsets consisting of a cube $\Lambda_L$ of integer
length $L$. For eigenfunctions of the operator, and more general solutions of
elliptic differential quations, we derive several quantitative unique
continuation results. The first result is of local nature and estimates the
vanishing order of a solution. The second is a sampling result and compares the
$L^2$-norm of a solution over a union of equidistributed $\delta$-balls in
space with the $L^2$-norm on the entire space. In the case where the space
$\mathbb{R}^d$ is replaced by a finite cube $\Lambda_L$ we derive similar
estimates. A particular feature of our bound is that they are uniform as long
as the coefficients of the operator are chosen from an appropriate ensemble,
they are quantitative and explicit with respect to the radius $\delta$, they
are $L$-independent and stable under small shifts of the $\delta$-balls. Our
proof applies to second order terms which have slowly varying coefficients on
the relevant length scale. The results can be also interpreted as special cases
of uncertainty relations, observability estimates, or spectral inequalities.
