Absolute continuity and singularity of Palm measures of the Ginibre point process

TL;DR

This paper establishes a clear dichotomy for the Ginibre point process and its reduced Palm measures, showing when they are absolutely continuous or singular, and provides explicit formulas for their Radon-Nikodym derivatives.

Contribution

It proves a precise criterion for absolute continuity and singularity among Palm measures of the Ginibre process and derives explicit Radon-Nikodym densities.

Findings

Reduced Palm measures are mutually absolutely continuous if and only if they have the same order.

Reduced Palm measures are singular if they differ in order.

Explicit Radon-Nikodym density formulas are provided.

Abstract

We prove a dichotomy between absolute continuity and singularity of the Ginibre point process $\mathsf{G}$ and its reduced Palm measures $\{\mathsf{G}_{\mathbf{x}}, \mathbf{x} \in \mathbb{C}^{\ell}, \ell = 0,1,2\dots\}$, namely, reduced Palm measures $\G_{\mathbf{x}}$ and $\G_{\mathbf{y}}$ for $\mathbf{x} \in \mathbb{C}^{\ell}$ and $\mathbf{y} \in \mathbb{C}^{n}$ are mutually absolutely continuous if and only if $\ell = n$; they are singular each other if and only if $\ell \not= n$. Furthermore, we give an explicit expression of the Radon-Nikodym density $d\G_{\mathbf{x}}/d \G_{\mathbf{y}}$ for $\mathbf{x}, \mathbf{y} \in \mathbb{C}^{\ell}$.