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