Quasi-Symmetries of Determinantal Point Processes

TL;DR

This paper demonstrates that certain determinantal point processes are quasi-invariant under specific groups of transformations, providing explicit formulas for the Radon-Nikodym derivatives and extending results to well-known processes like sine, Bessel, and Airy.

Contribution

It establishes quasi-invariance of determinantal point processes with integrable kernels under diffeomorphisms and permutations, with explicit Radon-Nikodym derivatives, answering a question by Olshanski.

Findings

Determinantal point processes are quasi-invariant under diffeomorphisms with compact support.

Explicit Radon-Nikodym derivatives are computed as regularized multiplicative functionals.

Results apply to sine, Bessel, Airy, and Gamma kernel 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.