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

TL;DR

This paper establishes a central limit theorem for linear eigenvalue statistics of large random graphs by analyzing fluctuations of a specific function of the adjacency matrix and proving Gaussian behavior in the limit.

Contribution

It introduces a new CLT for eigenvalue statistics of large random graphs using fluctuation analysis and Wick relations, extending previous results.

Findings

Fluctuations normalized by n^{-1/2} satisfy Wick relations.

Proves CLT for trace of the resolvent G(z).

Extends CLT to linear eigenvalue statistics for functions with controlled growth.

Abstract

We consider the adjacency matrix $A$ of a large random graph and study fluctuations of the function $f_n(z,u)=\frac{1}{n}\sum_{k=1}^n\exp\{-uG_{kk}(z)\}$ with $G(z)=(z-iA)^{-1}$. We prove that the moments of fluctuations normalized by $n^{-1/2}$ in the limit $n\to\infty$ satisfy the Wick relations for the Gaussian random variables. This allows us to prove central limit theorem for $\hbox{Tr}G(z)$ and then extend the result on the linear eigenvalue statistics $\hbox{Tr}\phi(A)$ of any function $\phi:\mathbb{R}\to\mathbb{R}$ which increases, together with its first two derivatives, at infinity not faster than an exponential.