Gaussian Multiplicative Chaos for the Sine Process

The sine process is the limit of the radial parts of unitary matrix of growing dimension, scaled in such a way that the average distance between eigenvalues be one. Under this scaling, the ratio of the characteristic polynomial of our matrix converges to a random holomorphic function, a stochastic Euler product. The main result of the talk is the convergence of our stochastic Euler product to the Gaussian multiplicative chaos, the exponential of the Gaussian random field induced by the -1/2 Sobolev seminorm.
The concept of the ​​Gaussian multiplicative chaos goes back to the work on hydrodynamic turbulence of Andrei Nikolaevich Kolmogorov and his school. Starting from the log-normal hypothesis of Kolmogorov-Oboukhov, Mandelbrot, Peyrière and Kahane have given a rigorous construction of the Gaussian multiplicative chaos. In addition to Kahane’s original method, several approaches have recently been proposed to constructing the Gaussian multiplicative chaos, in particular, the method of Shamov, the method of Berestycki, and, in the critical case, the method of Lacoin, each relying, but in different ways, on the Girsanov Theorem. An elementary construction of the exponential of a random field, well-suited to working with non-Gaussian fields, will be given in the talk.
Theorems on convergence of the powers of the absolute value of the characteristic polynomial of a random matrix to Gaussian multiplicative chaos go back to the work of Yan Fyodorov and his collaborators. For a wide class of matrix models, in different régimes, such convergence was established, in particular, by Berestycki, Lambert, Webb and other researchers; related results on the asymptotic of the maximum of the characteristic polynomial are due to Fyodorov-Hiary-Keating, Arguin, Belius and Bourgade, Zeitouni and Paquette, Chhaibi - Najnudel - Nikeghbali, and many other authors.
In the talk, we will directly consider the case of point processes with an infinite number of particles. The key rôle in the proof is played by the analysis of the scaling limit of the Borodin-Okounkov-Geronimo-Case formula, an explicit formula for the remainder term in the Strong Szegő Theorem in the form of Ibragimov.
The classical Kotelnikov Theorem (1933) states that the set of integers is complete and minimal for the Paley-Wiener space. Ghosh proved that realizations of the sine process are almost surely complete in the Paley-Wiener space; for general determinantal point processes, the completeness theorem was proved in joint work with Qiu and Shamov. A corollary of the convergence of the scaled stochastic Euler product to the Gaussian multiplicative chaos is that a realization of the sine process, once a particle is removed, is almost surely complete and minimal in the Paley-Wiener space: in other words, the sine process has excess one.