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

Abstract

We consider the adjacency matrix $A$ of a large random graph and study fluctuations of the function $f_n(z,u)=\frac{1}{n}{\sum }_{k=1}^n\exp \{-u{G}_{kk}(z)\}$ 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 $TrG(z)$ and then extend the result on the linear eigenvalue statistics $Tr\varphi (A)$ of any function $\varphi :\mathbb{R}\to \mathbb{R}$ which increases, together with its first two derivatives, at infinity not faster than an exponential.

22 pages

References in corpus (3)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• Random incidence matrices: moments of the spectral density
• Strong eigenfunction correlations near the Anderson localization transition

Cited by in corpus (4)

• Central limit theorems for linear statistics of heavy tailed random matrices
• Central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs. Diluted regime
• On the correlation functions of the characteristic polynomials of the sparse hermitian random matrices
• Global eigenvalue fluctuations of random biregular bipartite graphs