Fluctuations of functions of sparse Erdős-Rényi graphs

Let $A$ be the (rescaled) adjacency matrix of the Erdős-Rényi graphs $\mathcal{G}(\mathcal{N},\mathcal{p})$. For $N^{-1+\tau }⩽p⩽N^{-\tau }$, we study the fluctuation of $f(A{)}_{ii}$ on the global and mesoscopic spectral scales. We show that the distribution of $f(A{)}_{ii}$ is asymptotically the sum of two independent Gaussian random variables on different scales, where a phase transition occurs on the spectral scale $p$.