Central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs

Mariya Shcherbina, Brunello Tirozzi
2010 Journal of Mathematical Physics  
We consider the adjacency matrix A of a large random graph and study fluctuations of the function f_n(z,u)=1/n∑_k=1^n{-uG_kk(z)} with G(z)=(z-iA)^-1. We prove that the moments of fluctuations normalized by n^-1/2 in the limit n→∞ satisfy the Wick relations for the Gaussian random variables. This allows us to prove central limit theorem for TrG(z) and then extend the result on the linear eigenvalue statistics Trϕ(A) of any function ϕ:R→R which increases, together with its first two derivatives, at infinity not faster than an exponential.
doi:10.1063/1.3299297 fatcat:hetbyqbxaza6nodtez6uuzkbce