Two CLTs for Sparse Random Matrices

Abstract

Let $G=G(n,p_n)$ be a homogeneous Erdös-Rényi graph, and $A$ its adjacency matrix with eigenvalues $\hat{I}{»}_1(A)\ge \hat{I}{»}_2(A)\ge ...\ge \hat{I}{»}_n(A).$ Local laws have been used to show that $lambd{a}_2(A)$ 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 ($p_n=c{n}^{-2/3}$). Additionally, this technique renders the largest eigenvalue $\hat{I}{»}_1(A),$ separated from the rest of the spectrum for $p_n\gg n^{-1},$ has Gaussian fluctuations when $p_n\ge n^{-1}(\log n{)}^{6+c}$ for some $c>0.$ This paper shows this remains true in the range $B{n}^{-1}(\log n{)}^4\le p_n\le 1-B{n}^{-1}(\log n{)}^4$ 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.