  We study the criticality and subcriticality of powers $(-\Delta)^\alpha$ with
$\alpha>0$ of the discrete Laplacian $-\Delta$ acting on $\ell^2(\mathbb{N})$.
We prove that these positive powers of the Laplacian are critical if and only
if $\alpha \ge 3/2$. We complement our analysis with Hardy type inequalities
for $(-\Delta)^\alpha$ in the subcritical regimes $\alpha \in (0,3/2)$. As an
illustration of the critical case, we describe the negative eigenvalues
emerging by coupling the discrete bilaplacian with an arbitrarily small
localized potential.
