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

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 $\mathbf{x}, \mathbf{y} \in \mathbb{C}^{\ell}$.