Higher order fluctuations of extremal eigenvalues of sparse random matrices

We consider extremal eigenvalues of sparse random matrices, a class of random matrices including the adjacency matrices of Erd\H{o}s-R\'{e}nyi graphs $\mathcal{G}(N,p)$. Recently, it was shown that the leading order fluctuations of extremal eigenvalues are given by a single random variable associated with the total degree of the graph (Ann. Probab., 48(2):916-962, 2020; Probab. Theory Related Fields, 180:985-1056, 2021). We construct a sequence of random correction terms to capture higher (sub-leading) order fluctuations of extremal eigenvalues in the regime $N^{\epsilon} < pN < N^{1/3-\epsilon}$. Using these random correction terms, we prove a local law up to a shifted edge and recover the rigidity of extremal eigenvalues under some corrections for $pN>N^{\epsilon}$.