Higher order fluctuations of extremal eigenvalues of sparse random matrices

We study higher-order fluctuations of extremal eigenvalues of sparse random matrices on the regime Nε q N1/2 where q is the sparsity parameter. In the case N1/9 q N1/6, it was known that eigenvalue rigidity can be recovered by removing asymptotically Gaussian fluctuations. We consider the regime Nε q N1/6 and apply a higher-order random correction to the spectral edge in order to capture sub-leading order fluctuations of extremal eigenvalues. We establish local semicircle law near the edge under corrections and recover the eigenvalue rigidity by removing asymptotically Gaussian fluctuations arising from higher-order random corrections.