Higher order fluctuations of extremal eigenvalues of sparse random matrices

Jaehun Lee

Published 2021-08-26Version 1

We study higher-order fluctuations of extremal eigenvalues of sparse random matrices on the regime $N^{\epsilon}\ll q \ll N^{1/2}$ where $q$ is the sparsity parameter. In the case $N^{1/9}\ll q\ll N^{1/6}$, it was known that eigenvalue rigidity can be recovered by removing asymptotically Gaussian fluctuations. We consider the regime $N^{\epsilon} \ll q\ll N^{1/6}$ 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.