On the Correlation Functions of the Characteristic Polynomials of Random Matrices with Independent Entries: Interpolation Between Complex and Real Cases

TL;DR

This paper explores how the correlation functions of characteristic polynomials in random matrices with independent entries vary between complex and real cases, depending on the entries' second moment, revealing a continuum of behaviors.

Contribution

It introduces an interpolation framework that links the correlation functions of complex and real random matrices based on their second moment, extending understanding of their asymptotic behavior.

Findings

Correlation functions resemble those of the Complex Ginibre Ensemble scaled by moments.

Behavior depends continuously on the second moment of the entries.

The results unify complex and real matrix cases under a common framework.

Abstract

The paper is concerned with the correlation functions of the characteristic polynomials of random matrices with independent complex entries. We investigate how the asymptotic behavior of the correlation functions depends on the second moment of the common probability law of the matrix entries, a sort of ``reality measure'' of the entries. It is shown that the correlation functions behave like that for the Complex Ginibre Ensemble up to a factor depending only on the second moment and the fourth absolute moment of the common probability law of the matrix entries.