We consider random elliptic equations in dimension d≥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≥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.
