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- 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?
These problems are formulated in very different terms, but their solutions are
similar. Their focus is a remarkable object, a determinantal point process. The first example of a determinantal point process emerged in 1960s in Dyson’s work on
matrices with random elements (as in the last of the problems above). From that time
these processes regularly find new applications.
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
- Orthogonal polynomial ensembles. Random matrices (Fisher, Wishart,
Wigner, Dyson). Radial part of the Haar measure on the unitary group.
- Scaling limit of the circular unitary ensemble. Dyson’s sine process.
- Random Young tableaux. Schur measures. Discrete sine process by
Borodin, Okounkov and Olshansky. Szegő theorems and Borodin—Okounkov
formula.
- Determinantal processes with Bessel and Airy kernels.
- Macchi—Soshnikov—Shirai—Takahashi theorem on existence of
determinantal point process.
- Palm measures. Shirai—Takahashi theorem on Palm measures. Rigidity
of determinantal point processes.
- Quasi-symmetries of determinantal point processes.
- Limit theorems for determinantal point processes. Soshnikov’s central
limit theorem and estimates on the rate of convergence.
- *Gaussian multiplicative chaos for sine process.
 RSS: Forthcoming seminars
 Lecturers
Bufetov Alexander Igorevich
Gorbunov Sergei Milhailovich
Klimenko Alexey Vladimirovich
Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center |
