## The Annals of Probability

### Quasi-symmetries of determinantal point processes

Alexander I. Bufetov

#### Abstract

The main result of this paper is that determinantal point processes on $\mathbb{R}$ 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.

#### Article information

Source
Ann. Probab., Volume 46, Number 2 (2018), 956-1003.

Dates
Revised: May 2017
First available in Project Euclid: 9 March 2018

https://projecteuclid.org/euclid.aop/1520586273

Digital Object Identifier
doi:10.1214/17-AOP1198

Mathematical Reviews number (MathSciNet)
MR3773378

Zentralblatt MATH identifier
06864077

#### Citation

Bufetov, Alexander I. Quasi-symmetries of determinantal point processes. Ann. Probab. 46 (2018), no. 2, 956--1003. doi:10.1214/17-AOP1198. https://projecteuclid.org/euclid.aop/1520586273

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