Quasi-Symmetries of Determinantal Point Processes

Abstract

The main result of this paper is that determinantal point processes on the real line corresponding to projection operators with integrable kernels are quasi-invariant, in the continuous case, under the group of diffeomorphisms with compact support (Theorem 1.4); in the discrete case, under the group of all finite permutations of the phase space (Theorem 1.6). The Radon-Nikodym derivative is computed explicitly and is given by a regularized multiplicative functional. Theorem 1.4 applies, in particular, to the sine-process, as well as to determinantal point processes with the Bessel and the Airy kernels; Theorem 1.6 to the discrete sine-process and the Gamma kernel process. The paper answers a question of Grigori Olshanski.

The argument on regularization of multiplicative functionals has been simplified. Section 4 has become shorter. Subsections 2.9, 2.13, 2.14, 2.15 have been added: in particular, formula (43) simplifies the argument for unbounded functions