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

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}$,namely,reducedPalmmeasures$\G{$\mathbf{x}$}$and$\G_{$\mathbf{y}$}$for$$\mathbf{x}$ \in $\mathbb{C}$^{\ell}$and$$\mathbf{y}$ \in $\mathbb{C}$^{n}$aremutuallyabsolutelycontinuousifandonlyif$\ell = n$; they are singular each other if and only if $\mathrm{\ell }≠n$. Furthermore, we give an explicit expression of the Radon-Nikodym density $d{\text{G}}_{\mathbf{x}}\mathrm{/}d{\text{G}}_{\mathbf{y}}$ for $$\mathbf{x}$, $\mathbf{y}$ \in $\mathbb{C}$^{\ell}$.