• Several particles randomly and independently wander on the set of integer points on a line. What is the probability that their trajectories do not intersect for the first moments of time? What is the conditional distribution of their positions assuming the trajectories do not intersect?
• Consider a finite graph. A subtree of this graph that contains all its vertices is called its spanning tree. How many spanning trees does the graph have? How many of them contain a given edge? A given set of edges?
• Consider a power series, whose coefficients are independent standard complex Gaussian random variables. What is a probability that the sum of this series does not vanish inside the circle of radius $1/2$?
• All elements of a complex matrix are independent and Gaussian. What is the distribution of its minimal singular value?

The theory of determinantal point processes is an actively developing area of mathematics. We will start from the basics, but quite soon arrive at some open problems. We will stress the relation of this field with the classical analysis.

Tentative syllabus

1. Orthogonal polynomial ensembles. Random matrices (Fisher, Wishart, Wigner, Dyson). Radial part of the Haar measure on the unitary group.
2. Scaling limit of the circular unitary ensemble. Dyson’s sine process.
3. Random Young tableaux. Schur measures. Discrete sine process by Borodin, Okounkov and Olshansky. Szegő theorems and Borodin—Okounkov formula.
4. Determinantal processes with Bessel and Airy kernels.
5. Macchi—Soshnikov—Shirai—Takahashi theorem on existence of determinantal point process.
6. Palm measures. Shirai—Takahashi theorem on Palm measures. Rigidity of determinantal point processes.
7. Quasi-symmetries of determinantal point processes.
8. Limit theorems for determinantal point processes. Soshnikov’s central limit theorem and estimates on the rate of convergence.
9. *Gaussian multiplicative chaos for sine process.

Lecturers
Bufetov Alexander Igorevich
Gorbunov Sergei Milhailovich
Klimenko Alexey Vladimirovich

Financial support
The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no.  075-15-2022-265).

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