Asymptotic spectral independence of Wigner ensembles

TL;DR

This paper proves that eigenvalue clusters in Wigner ensembles become asymptotically independent when separated by large distances, under certain averaging conditions, using advanced probabilistic techniques.

Contribution

It establishes asymptotic spectral independence of eigenvalue clusters in Wigner matrices, extending understanding of eigenvalue correlations at macroscopic scales.

Findings

Eigenvalue clusters become independent at large distances

Correlation functions converge to those of independent processes

Results rely on advanced probabilistic machinery

Abstract

We consider the joint distribution of eigenvalue clusters of the Wigner ensemble separated by macroscopic distances (i.e., on the same scale as the difference between the edges of the semicircle law). We prove that under an averaging condition, the correlation function governing any finite collection of clusters converges to that of independence point processes. The proof relies heavily on the machinery developed by Erdos, Ramirez, Schlein, and Yau.