We investigate the large-distance asymptotics of optimal Hardy weights onZd\mathbb Z^d,d≥3d\geq 3, via the super solution construction. For the free discrete Laplacian, the Hardy weight asymptotic is the familiar(d−2)24|x|−2\frac {(d-2)^2}{4}|x|^{-2}as|x|→∞|x|\to \infty. We prove that the inverse-square behavior of the optimal Hardy weight is robust for general elliptic coefficients onZd\mathbb Z^d: (1) averages over large sectors have inverse-square scaling, (2) for ergodic coefficients, there is a pointwise inverse-square upper bound on moments, and (3) for i.i.d. coefficients, there is a matching inverse-square lower bound on moments. The results imply|x|−4|x|^{-4}-scaling for Rellich weights onZd\mathbb Z^d. Analogous results are also new in the continuum setting. The proofs leverage Green’s function estimates rooted in homogenization theory.