  We consider random elliptic equations in dimension $d\geq 3$ at small
ellipticity contrast. We derive the large-distance asymptotic expansion of the
annealed Green's function up to order $4$ in $d=3$ and up to order $d+2$ for
$d\geq 4$. We also derive asymptotic expansions of its derivatives. The
obtained precision lies far beyond what is established in prior results in
stochastic homogenization theory. Our proof builds on a recent breakthrough in
perturbative stochastic homogenization by Bourgain in a refined version shown
by Kim and the second author, and on Fourier-analytic techniques of Uchiyama.
