On the almost-circular symplectic induced Ginibre ensemble

TL;DR

This paper analyzes the asymptotic behavior of the symplectic induced Ginibre process in a nearly circular regime, revealing new correlation kernels and scaling limits both near and away from the real axis.

Contribution

It introduces new correlation kernels for the symplectic induced Ginibre ensemble and characterizes their scaling limits in the almost-circular regime, connecting to known ensembles.

Findings

New Pfaffian correlation kernel interpolates between known ensembles.

Scaling limits differ near and away from the real axis.

Precise asymptotics for gap probabilities in the regime.

Abstract

We consider the symplectic induced Ginibre process, which is a Pfaffian point process on the plane. Let $N$ be the number of points. We focus on the almost-circular regime where most of the points lie in a thin annulus $\mathcal{S}_{N}$ of width $O(\frac{1}{N})$ as $N \to \infty$. Our main results are the scaling limits of all correlation functions near the real axis, and also away from the real axis. Near the real axis, the limiting correlation functions are Pfaffians with a new correlation kernel, which interpolates the limiting kernels in the bulk of the symplectic Ginibre ensemble and of the anti-symmetric Gaussian Hermitian ensemble of odd size. Away from the real axis, the limiting correlation functions are determinants, and the kernel is the same as the one appearing in the bulk limit of almost-Hermitian random matrices. Furthermore, we obtain precise large $N$ asymptotics for the probability that no points lie outside $\mathcal{S}_{N}$, as well as of several other "semi-large" gap probabilities.