Fluctuations of Matrix Elements of Regular Functions of Gaussian Random Matrices

Abstract

We find the limit of the variance and prove the Central Limit Theorem (CLT) for the matrix elements φ _jk (M), j,k=1,…,n of a regular function φ of the Gaussian matrix M (GOE and GUE) as its size n tends to infinity. We show that unlike the linear eigenvalue statistics Tr φ(M), a traditional object of random matrix theory, whose variance is bounded as n→∞ and the CLT is valid for Tr φ(M)−E{Tr φ(M)}, the variance of φ _jk (M) is O(1/n), and the CLT is valid for $\sqrt{n}(\varphi _{jk}(M)-\mathbf{E}\{\varphi _{jk}(M)\})$ . This shows the role of eigenvectors in the forming of the asymptotic regime of various functions (statistics) of random matrices. Our proof is based on the use of the Fourier transform as a basic characteristic function, unlike the Stieltjes transform and moments, used in majority of works of the field. We also comment on the validity of analogous results for other random matrices.