  We study estimates of the Green's function in $\mathbb{R}^d$ with $d \ge 2$,
for the linear second order elliptic equation in divergence form with variable
uniformly elliptic coefficients. In the case $d \ge 3$, we obtain estimates on
the Green's function, its gradient, and the second mixed derivatives which
scale optimally in space, in terms of the "minimal radius" $r_*$ introduced in
[Gloria, Neukamm, and Otto: A regularity theory for random elliptic operators;
ArXiv e-prints (2014)]. As an application, our result implies optimal
stochastic Gaussian bounds in the realm of homogenization of equations with
random coefficient fields with finite range of dependence. In two dimensions,
where in general the Green's function does not exist, we construct its gradient
and show the corresponding estimates on the gradient and mixed second
derivatives. Since we do not use any scalar methods in the argument, the result
holds in the case of uniformly elliptic systems as well.
