Kolmogorov formulated the axiomatic foundations of probability theory within the framework of random variables as measurable maps on probability spaces. These maps may take values in finite-dimensional spaces (random vectors), in spaces of functions on an interval (stochastic processes), and, in exactly the same way, one can consider random measures.

The naturality of this viewpoint is justified by examples. In various problems—from representation theory and mathematical physics to analytic number theory—one encounters random variables that themselves take values in a space of measures.

How does the characteristic polynomial of a random matrix behave as its size tends to infinity?

To what does the exponential of a power series with independent Gaussian coefficients converge on the unit circle?

How does a random polynomial whose zeros are the particles of a two-dimensional Coulomb gas on the circle behave as the number of particles tends to infinity?

How does the Riemann zeta function behave at infinity along the critical line?

The central object in all the questions above is a certain random holomorphic function. In all the limits mentioned, this function oscillates strongly, but nevertheless has a limit. Despite the smoothness of the original function, it is precisely the viewpoint of this function as a random measure that allows the existence of the limit.

The limiting random measure—Gaussian multiplicative chaos—was constructed by Jean-Pierre Kahane, who considered a completely different problem. His construction continues the work of Mandelbrot and Peyrière, who proposed a rigorous interpretation of the log-normal Kolmogorov–Obukhov hypothesis that had appeared in the theory of homogeneous isotropic turbulence.

The properties of this measure are both interesting and unusual. For example, it is "concentrated" on a set whose Hausdorff dimension is almost surely non-integer; the logarithm of the moment of the measure of a small ball does not depend linearly on the order of the moment— phenomenon, which is called the multifractal spectrum and reflects the complex local behavior. More surprisingly, this measure is uncorrelated on disjoint subsets, despite the existence of an uniqueness theorem for the pre-limit object in the examples above.

Our goal will be, starting from the basics, to understand the recent developments, among which is the emergence of Gaussian multiplicative chaos in random matrix models.

Program

1. Kolmogorov's theory of homogeneous isotropic turbulence. Landau's criticism. The Kolmogorov–Obukhov log-normal hypothesis.
2. Random fields and random measures.
3. Gaussian multiplicative chaos — Kahane's construction.
4. Gaussian multiplicative chaos — Shamov's construction.
5. Gaussian multiplicative chaos — Berestycki's construction.
6. Critical Gaussian multiplicative chaos — Lacoin's construction.
7. Gaussian multiplicative chaos in random matrix models.

Lecturers
Bufetov Alexander Igorevich
Gorbunov Sergei Milhailovich

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. 075-15-2025-303).

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