The Gaussian multiplicative chaos for the sine-process

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 role.