Determinantal structure of the overlaps for induced spherical unitary ensemble

TL;DR

This paper investigates the determinantal structure of overlaps in the induced spherical unitary ensemble, demonstrating universality in scaling limits across different regimes, thus confirming universal behavior of overlaps.

Contribution

It establishes the determinantal structure of overlaps and proves universality of their scaling limits in various regimes for the induced spherical unitary ensemble.

Findings

Universality of the overlap in different regimes

Determinantal structure of the $k$-th conditional expectation

Confirmation of universal behavior of overlaps

Abstract

In this note, we study the determinantal structure of the $k$-th conditional expectation of the overlap for induced spherical unitary ensemble. We will show the universality for the scaling limits of the $k$-the conditional expectation of the overlap in the three regimes, strongly non-unitary, weakly non-unitary, and the singular origin regimes. As a consequence, we will confirm the universality for the overlap.