Quasi-symmetries of determinantal point processes. (English) Zbl 1430.60045
Summary: 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 G. Olshanski [Adv. Math. 226, No. 3, 2305–2350 (2011; Zbl 1218.60004)].
MSC:
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
| 60B10 | Convergence of probability measures |
| 60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |
| 33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |
Keywords:
determinantal point processes; Gibbs property; palm measures; integrable kernels; multiplicative functionalsCitations:
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