Quasi-Symmetries of Determinantal Point Processes

The classical De Finetti Theorem (1937) states that an exchangeable collection of random variables is a mixture of Bernoulli sequences.
The first result of the talk is that determinantal point processes on Z induced by integrable kernels are quasi-invariant under the action of the infinite symmetric group. The Radon-Nikodym derivative is a regularized multiplicative functional on the space of configurations. A key example is the discrete sine-process of Borodin, Okounkov and Olshanski.
The second result is a continuous counterpart of the first: namely, it is proved that determinantal point processes with integrable kernles on R, a class that includes processes arising in random matrix theory such as Dyson's sine-process, or the processes with the Bessel kernel or the Airy kernel studied by Tracy and Widom, are quasi-invariant under the action of the group of diffeomorphisms of the line with compact support.
While no analogues of these results in higher dimensions are known, in joint work with Yanqi Qiu it is shown that for determinantal point processes corresponding to Hilbert spaces of holomorphic functions on the complex plane C or on the unit disk D, the quasi-invariance under the action of the group of diffeomorphisms with compact support also holds.