Three-term asymptotics for discrete Hardy--Rellich constants in high dimension

Sharp Hardy–Rellich constants are spectral thresholds for operators with critical inverse-power potentials. Let $C_{\mathrm{\ell }}(N)$ denote the optimal constant in the $\mathrm{\ell }$th-order discrete Hardy–Rellich inequality on ${\mathbb{Z}}^N$. Recent independent work established the leading high-dimensional behavior $C_{\mathrm{\ell }}(N)\sim 2^{\mathrm{\ell }}{N}^{\mathrm{\ell }}$ for every fixed $\mathrm{\ell }$. We determine the next two orders and prove $$C_{\mathrm{\ell }}(N)=2^{\mathrm{\ell }}{N}^{\mathrm{\ell }}+{\gamma }_{\mathrm{\ell }}{N}^{\mathrm{\ell }-1}+{\delta }_{\mathrm{\ell }}{N}^{\mathrm{\ell }-2}+O_{\mathrm{\ell }}(N^{\mathrm{\ell }-3}),$$ with explicit coefficients ${\gamma }_{\mathrm{\ell }}$ and ${\delta }_{\mathrm{\ell }}$. The central difficulty is a degeneracy that grows with the dimension: after conjugation, the leading operator is scalar on the $2N$ nearest neighbours of the origin. We resolve this cluster using signed-permutation symmetry and an effective operator on four lattice orbits. Weighted torus estimates and Feshbach–Schur reduction justify the finite-dimensional expansion inside the full operator, while a residual bound and Temple’s inequality give the stated remainder. In particular, $C_1(N)=2N-4-20\mathrm{/}(3N)+O(N^{-2})$.