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 = 0,1,2ldots }), namely, reduced Palm measures (mathsf {G}_{mathbf {x}}) and (mathsf {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 (dmathsf {G}_{mathbf {x}}/d mathsf {G}_{mathbf {y}}) for (mathbf {x}, mathbf {y} in mathbb {C}^{ell }).