Determinantal point processes and Gaussian multiplicative chaos (Lecture 1)

To almost every realization of the sine-process one naturally assigns a random entire function, the analogue of the Euler product for the sine, the scaling limit of ratios of characteristic polynomials of a random matrix. The main result of the talk is that the square of the absolute value of our random entire function converges to the Gaussian multiplicative chaos. As a corollary, one obtains that almost every realization with one particle removed is a complete and minimal set for the Paley-Wiener space, whereas if two particles are removed, then the resulting set is a zero set for the Paley-Wiener space. Quasi-invariance of the sine-process under compactly supported diffeomorphisms of the line plays a key rôle.
In joint work with Qiu, the Patterson-Sullivan construction is used to interpolate Bergman functions from a realization of the determinantal point process with the Bergman kernel, in other words, by the Peres-Virág theorem, the zero set of a random series with independent complex Gaussian entries. The invariance of the zero set under the isometries of the Lobachevsky plane plays a key rôle. Conditional measures of the determinantal point process with the Bergman kernel are found explicitly (cf. arXiv:2112.15557, Dec. 2021).