On correlation function of high moments of Wigner random matrices

TL;DR

This paper investigates the correlation function of high moments of Wigner matrices, revealing its universal form in the large matrix limit and connecting it to properties of the GUE ensemble.

Contribution

It provides a new analysis of the correlation function for high moments of Wigner matrices, showing universality and linking it to the GUE's Inverse Participation Ratio analog.

Findings

Correlation function limit is independent of matrix element distribution.

The analysis extends to moments proportional to n^{2/3}.

The proof combines Sinai-Soshnikov methods with GUE properties.

Abstract

We consider the Wigner ensemble of Hermitian n-dimensional random matrices and study the correlation function K(s',s") of their moments in the limit when the numbers s', s" of the moments are proportional to n to the power 2/3. We show that the limiting expression of K does not depend on the moments of the random matrix elements. The proof is based on a combination of the arguments by Ya. Sinai and A. Soshnikov with the detailed study of a moment analog of the Inverse Participation Ratio of the Gaussian Unitary Invariant Ensemble (GUE).