  We consider a divergence-form elliptic difference operator on the lattice
$\mathbb{Z}^d$, with a coefficient matrix that is an i.i.d. perturbation of the
identity matrix. Recently, Bourgain introduced novel techniques from harmonic
analysis to prove the convergence of the Feshbach-Schur perturbation series
related to the averaged Green's function of this model. Our main contribution
is a refinement of Bourgain's approach which improves the key decay rate from
$-2d+\epsilon$ to $-3d+\epsilon$. (The optimal decay rate is conjectured to be
$-3d$.) As an application, we derive estimates on higher derivatives of the
averaged Green's function which go beyond the second derivatives considered by
Delmotte-Deuschel and related works.
