Determinantal structure and bulk universality of conditional overlaps in the complex Ginibre ensemble

TL;DR

This paper explores the determinantal structure of conditional overlaps in the complex Ginibre ensemble, revealing their universality in the bulk scaling limit and connecting eigenvector overlaps with eigenvalue correlations.

Contribution

It demonstrates the emergence of a determinantal structure for conditional overlaps and introduces the universal bulk scaling limit away from the origin.

Findings

Determinantal structure for overlaps is established at finite matrix size.

Bulk scaling limit of overlaps is universal and matches the origin limit.

Connection between eigenvector overlaps and eigenvalue correlation functions is clarified.

Abstract

In these proceedings we summarise how the determinantal structure for the conditional overlaps among left and right eigenvectors emerges in the complex Ginibre ensemble at finite matrix size. An emphasis is put on the underlying structure of orthogonal polynomials in the complex plane and its analogy to the determinantal structure of $k$-point complex eigenvalue correlation functions. The off-diagonal overlap is shown to follow from the diagonal overlap conditioned on $k\geq2$ complex eigenvalues. As a new result we present the local bulk scaling limit of the conditional overlaps away from the origin. It is shown to agree with the limit at the origin and is thus universal within this ensemble.