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 $\ell \ne n$. Furthermore, we give an explicit expression of the Radon-Nikodym density $d\G_{\mathbf{x}}/d \G_{\mathbf{y}}$ for $\mathbf{x},\mathbf{y}\in {\mathbb{C}}^{\ell }$.

47 pages, version (ii)

References in corpus (1)

• Rigidity and Tolerance in Gaussian zeroes and Ginibre eigenvalues: quantitative estimates