A Note on Mixed Matrix Moments for the Complex Ginibre Ensemble

TL;DR

This paper explores mixed matrix moments in the complex Ginibre ensemble, relating them to overlap functions, leading to new asymptotic problems and partial results, while highlighting open questions for future research.

Contribution

It establishes a connection between mixed matrix moments and overlap functions, introducing new asymptotic problems and providing partial solutions in the context of the complex Ginibre ensemble.

Findings

Derived new asymptotic problems for overlap functions.

Obtained partial results on mixed matrix moments.

Highlighted open problems for future investigation.

Abstract

We consider the mixed matrix moments for the complex Ginibre ensemble. These are well-known. We consider the relation to the expected overlap functions of Chalker and Mehlig. This leads to new asymptotic problems for the overlap. We obtain some results, but we also state some remaining open problems.