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We consider a random, uniformly elliptic coefficient field $a$ on the lattice ${\mathbb{Z}}^d$. The distribution $\langle ·\rangle$ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green’s function $G(t,x,y)$ satisfy optimal annealed estimates which are $L^2$ and $L^1$, respectively, in probability, i.e., they obtained bounds on $\langle |{\nabla }_{x}G(t,x,y){{|}^2\rangle }^{1/2}$ and $\langle \left|{\nabla }_x{\nabla }_{y}G(t,x,y)\right|\rangle$. In particular, the elliptic Green’s function $G(x,y)$ satisfies optimal annealed bounds. In their recent work, the authors...