2022/10/18 by Diaconu, Simona
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2210.09625
Let G=G(n,pn) be a homogeneous Erdös-Rényi graph, and A its adjacency matrix with eigenvalues λ1(A) ≥ λ2(A) ≥ ... ≥ λn(A). Local laws have been used to show that lambda2(A) can exhibit fundamentally different behaviors: Tracy-Widom (pn ≫ n-2/3), normal (n-7/9 ≪ pn ≪~n-2/3), and a mix of both (pn=cn-2/3). Additionally, this technique renders the largest eigenvalue λ1(A), separated from the rest of the spectrum for pn ≫ n-1, has Gaussian fluctuations when pn ≥ n-1(logn)6+c for some c>0. This paper shows this remains true in the range Bn-1(logn)4 ≤ pn ≤ 1-Bn-1(logn)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.