  We study the statistics of pairs from the sequence
$(n^\alpha)_{n\in\mathbb{N}^*}$, for every parameter $\alpha \in \, ]0,1[$. We
prove the convergence of the empirical pair correlation measures towards a
measure with an explicit density. In particular, when using 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.
