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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 theconvergence of the empirical pair correlations measures towards a rotationinvariant measure with explicit density. In particular, with the scaling factor$N\mapsto N^{1-\alpha}$, we prove that there exists an exotic pair correlationfunction which exhibits a level repulsion phenomenon. For other scalingfactors, we prove that either the pair correlations are Poissonian or there isa total loss of mass. In addition, we give an error term for this convergence,with explicit dependence on parameters of the grid $\Lambda$.

Effective pair correlations of fractional powers of complex grid points