Normal approximation, the Gaussian multiplicative chaos, and excess one for the sine-process

The Soshnikov Central Limit Theorem states that scaled additive statistics of the sine-process converge to the normal law. The first main result of this talk gives a detailed comparison between the law of an additive, sufficiently Sobolev regular, statistic under the sine-process and the normal law. The comparison for low frequencies is obtained by taking the scaling limit in the Borodin-Okounkov-Geronimo-Case formula. The exponential decay for the high frequencies is obtained, under an additional assumption of holomorphicity in a horizontal strip, with the use of an analogue of the Johansson change of variable formula; quasi-invariance of the sine-process under compactly supported diffeomorphisms plays a key rôle in the proof. The corollaries of the normal approximation theorem include the convergence of the random entire function, the infinite product with zeros at the particles, to Gaussian multiplicative chaos. A complementary estimate to the Ghosh completeness theorem follows in turn: indeed, Ghosh proved that reproducing sine-kernels along almost every configuration of the sine-process form a complete set; it is proved in the talk that if one particle is removed, then the set is still complete; whereas if two particles are removed from the configuration, then the resulting set is the zero set for the Paley-Wiener space. The talk extends the results of the preprint https://arxiv.org/abs/1912.13454