Hardy and Rellich inequality on lattices

In this paper, we study the asymptotic behaviour of sharp constant in discrete Hardy and Rellich inequality on lattice $\mathbb{Z}^d$ as $d \rightarrow \infty$. process, proved some Hardy-type inequalities for operators $\Delta^m$ $\nabla(\Delta^m)$ non-negative integers $m$ a $d$ dimensional torus. It turns out that grows $d^2$ respectively $ d