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

TL;DR

This paper proves that in large random graphs with increasing edges per vertex, the fluctuations of linear eigenvalue statistics follow a Gaussian distribution with a specific variance, extending the understanding of spectral behavior in such graphs.

Contribution

It establishes a central limit theorem for eigenvalue fluctuations in large random graphs within the diluted regime, linking their variance to that of the Wigner ensemble.

Findings

Fluctuations converge to a Gaussian distribution.

Variance matches the non-Gaussian part of Wigner ensemble.

Applicable to a wide class of test functions.

Abstract

We study the linear eigenvalue statistics of large random graphs in the regimes when the mean number of edges for each vertex tends to infinity. We prove that for a rather wide class of test functions the fluctuations of linear eigenvalue statistics converges in distribution to a Gaussian random variable with zero mean and variance which coincides with "non gaussian" part of the Wigner ensemble variance.