  Using a standard definition of fractional powers on the universal cover
$\exp:S\to \mathbb{C}^*$ seen as an infinite helicoid embedded in
$\mathbb{R}^3$, we study the statistics of pairs from the countable family
$\{n^\alpha \, : \, n \in \exp^{-1}(\Lambda) \}$ for every complex grid
$\Lambda$ and every real parameter $\alpha \in \, ]0,1[\,$. We prove the
convergence of the empirical pair correlations measures towards a rotation
invariant measure with explicit density. In particular, with the scaling factor
$N\mapsto N^{1-\alpha}$, we prove that there exists an exotic pair correlation
function which exhibits a level repulsion phenomenon. For other scaling
factors, we prove that either the pair correlations are Poissonian or there is
a total loss of mass. In addition, we give an error term for this convergence,
with explicit dependence on parameters of the grid $\Lambda$.
