Freeman Dyson has calculated the pair correlation function of the spectrum of a random infinite hermitian matrix in 1962. The function he found, by taking determinants, defines further correlation functions of a random countable subset of the real line — the sine process. This process is of significant interest due to its connection with areas such as representation theory, mathematical physics, and analytic number theory.

For example, in the 1980s, Jimbo, Miwa, Mori, and Sato discovered its connection with integrable systems. In 1973, Montgomery conjectured that it describes the statistical behavior of the zeros of the Riemann zeta function. In the 2000s, Borodin and Olshanski found that this process arises in the representation theory of the infinite unitary group. In 2015, Ghosh showed that any entire function growing no faster than exponentially can be reconstructed from its values on the points of almost any subset drawn from the sine process.

During the course we will discuss the "characteristic polynomial" of the sine process — a random entire function vanishing exactly at the points of the random subset. We will discuss its distribution at infinity and explain how this distribution is connected to an important 20th-century problem: the Toeplitz determinants and orthogonal polynomials asymptotics. We will also discuss "physical" generalizations of the sine process and their connection with integrable systems.

No knowledge of the last semester's material is required.

Program

1. Toeplitz and Hankel determinants, their scaling limits.
2. Multiplicative functionals of determinantal point processes.
3. Gaussian multiplicative chaos for the sine process.
4. Circular beta-ensemble, its matrix representation.
5. Scaling limit of the circular beta-ensembles. The Killip-Stoiciu construction of the sine-beta process via a system of stochastic differential equations.
6. Evolution of zeros of a holomorphic function under the action of the heat equation. Calogero-Sutherland model. Its integrability, Lax representation, and connection to the Jack polynomials.
7. Deformation of the Riemann xi-function by the heat equation. The De Brujin constant, its positivity.

Lecturers
Bufetov Alexander Igorevich
Gorbunov Sergei Milhailovich

Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center