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.
Categories: math.PR math-ph math.DS math.MP