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

Abstract

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