Spherical Induced Ensembles with Symplectic Symmetry

TL;DR

This paper studies the complex eigenvalues of the induced spherical Ginibre ensemble with symplectic symmetry, establishing local universality and deriving scaling limits of correlation functions in various regimes.

Contribution

It introduces a differential equation approach for analyzing correlation kernels, advancing understanding of spectral properties in symplectic ensembles.

Findings

Proves local universality of eigenvalue point processes along the real axis.

Derives scaling limits of correlation functions at regular points and spectral singularities.

Develops a differential equation method for asymptotic analysis of Pfaffian point processes.

Abstract

We consider the complex eigenvalues of the induced spherical Ginibre ensemble with symplectic symmetry and establish the local universality of these point processes along the real axis. We derive scaling limits of all correlation functions at regular points both in the strong and weak non-unitary regimes as well as at the origin having spectral singularity. A key ingredient of our proof is a derivation of a differential equation satisfied by the correlation kernels of the associated Pfaffian point processes, thereby allowing us to perform asymptotic analysis.