Correlations between eigenvalues of large random matrices with independent entries

TL;DR

This paper derives correlation functions for eigenvalues of large Hermitian random matrices with independent entries, using diagrammatic and RG methods, and discusses universality across different ensemble classes.

Contribution

It introduces a combined diagrammatic and RG approach to compute eigenvalue correlations for matrices with independent entries, extending understanding of universality.

Findings

Derived general form for one, two, and three-point Green functions

Re-derived Green functions using RG approach, showing invariance under certain ensemble choices

Compared correlation functions across different ensemble classes to discuss universality

Abstract

We derive the connected correlation functions for eigenvalues of large Hermitian random matrices with independently distributed elements using both a diagrammatic and a renormalization group (RG) inspired approach. With the diagrammatic method we obtain a general form for the one, two and three-point connected Green function for this class of ensembles when matrix elements are identically distributed, and then discuss the derivation of higher order functions by the same approach. Using the RG approach we re-derive the one and two-point Green functions and show they are unchanged by choosing certain ensembles with non-identically distributed elements. Throughout, we compare the Green functions we obtain to those from the class of ensembles with unitary invariant distributions and discuss universality in both ensemble classes.