Next-order asymptotic expansion of the optimal Hardy weight in dimensions d ≥ 3

Establish the conjectured next-order asymptotic expansion of the optimal Hardy weight associated with the simple random walk on \(\mathbb{Z}^d\), \(d\geq 3\), namely \(w(x)=\frac{(d-2)^2}{8d|x|^2}+\frac{Q(\omega^x)}{|x|^4}+\mathcal{O}(|x|^{-6})\), where \(\omega^x=x/|x|\) and \(Q(z)=\frac{(d-2)^2}{16d}\left(\frac{5(d+2)^2}{8}\sum_{j=1}^d z_j^4-(2d+3)\right)\).

Background

For transition weights on $\mathbb{Z}^d$ with $d\geq 3$, the paper proves that the Hardy weight constructed from the square root of the Green function has leading asymptotic behavior $$w_{\mu}(x)=\frac{(d-2)^2\sigma_\mu^2}{8|x|_\mu^2}+\mathcal{O}(|x|^{-4})$$ under a suitable moment condition.

In the special case of the simple random walk, the authors note that the explicit higher-order term in the Green-function expansion appears sufficient to derive a further term in the Hardy-weight expansion. They explicitly label the displayed formula as a conjecture, leaving the proposed $|x|^{-4}$ correction and its angular dependence unresolved.