Determinantal Point Processes Associated with Hilbert Spaces of Holomorphic Functions

• Published in: Communications in mathematical physics Vol. 351; no. 1; pp. 1 - 44
• Main Authors: Bufetov, Alexander I., Qiu, Yanqi
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2017; Springer Nature B.V; Springer Verlag
• Subjects: Classical and Quantum Gravitation; Complex Systems; Computation; Derivatives; Dynamical Systems; Equivalence; Functionals; Kernels; Mathematical and Computational Physics; Mathematical Physics; Mathematics; Palm; Physics; Physics and Astronomy; Quantum Physics; Radon; Relativity Theory; Theoretical; Determinantal point processes; generalized Fock spaces; regularized multiplicative functionals; generalized Bergman spaces; Palm measures; rigidity
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-017-2840-y

We study determinantal point processes on C induced by the reproducing kernels of generalized Fock spaces as well as those on the unit disc D induced by the reproducing kernels of generalized Bergman spaces. In the first case, we show that all reduced Palm measures 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 of different orders are singular. In the second case, we show that all reduced Palm measures, 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.