Determinantal point processes associated with Hilbert spaces of holomorphic functions

TL;DR

This paper investigates determinantal point processes linked to Hilbert spaces of holomorphic functions, revealing their measure equivalences, singularities, and invariance properties on complex planes and unit discs.

Contribution

It provides explicit formulas for Radon-Nikodym derivatives and demonstrates measure equivalence and singularity results for these processes, advancing understanding of their structure and invariance.

Findings

All reduced Palm measures of the same order are equivalent in Fock spaces.

Reduced Palm measures of different orders are singular in Fock spaces.

All reduced Palm measures are equivalent in Bergman spaces.

Abstract

We study determinantal point processes on $\mathbb{C}$ induced by the reproducing kernels of generalized Fock spaces as well as those on the unit disc $\mathbb{D}$ induced by the reproducing kernels of generalized Bergman spaces. In the first case, we show that all reduced Palm measures $\textit{of the same order}$ are equivalent. The Radon-Nikodym derivatives are computed explicitly using regularized multiplicative functionals. We also show that these determinantal point processes are rigid in the sense of Ghosh and Peres, hence reduced Palm measures $\textit{of different orders}$ are singular. In the second case, we show that all reduced Palm measures, $\textit{of all orders}$, are equivalent. The Radon-Nikodym derivatives are computed using regularized multiplicative functionals associated with certain Blaschke products. The quasi-invariance of these determinantal point processes under the group of diffeomorphisms with compact supports follows as a corollary.