# A 3D Ginibre point field

## TL;DR

This paper introduces a novel three-dimensional random point field based on quaternion determinants, providing explicit formulas and asymptotic analysis of the kernel and orthogonal polynomials in the bulk and at the center.

## Contribution

It develops a new 3D Ginibre point field using quaternion determinants, with explicit formulas and asymptotic behavior analysis.

## Key findings

- Explicit formulas for polynomials and kernels
- Asymptotic behavior in bulk and at the center
- Extension of Ginibre ensemble to 3D using quaternions

## Abstract

We introduce a three-dimensional random point field using the concept of the quaternion determinant. Orthogonal polynomials on the space of pure quaternions are defined, and used to construct a kernel function similar to the Ginibre kernel. We find explicit formulas for the polynomials and the kernel, and calculate their asymptotics in the bulk and at the center of coordinates.