Two CLTs for Sparse Random Matrices

Let $G=G(n,p_n)$ be a homogeneous Erd\"os-R\'enyi graph, and $A$ its adjacency matrix with eigenvalues $\lambda_1(A) \geq \lambda_2(A) \geq ... \geq \lambda_n(A).$ Local laws have been used to show that $lambda_2(A)$ can exhibit fundamentally different behaviors: Tracy-Widom ($p_n \gg n^{-2/3}$), normal ($n^{-7/9} \ll p_n \ll~n^{-2/3}$), and a mix of both ($p_n=cn^{-2/3}$). Additionally, this technique renders the largest eigenvalue $\lambda_1(A),$ separated from the rest of the spectrum for $p_n \gg n^{-1},$ has Gaussian fluctuations when $p_n \geq n^{-1}(\log{n})^{6+c}$ for some $c>0.$ This paper shows this remains true in the range $Bn^{-1}(\log{n})^4 \leq p_n \leq 1-Bn^{-1}(\log{n})^4$ with $B>0$ universal, the tool behind it being a central limit theorem for the eigenvalue statistics of $A$ that is justified via the method of moments.