Analysis of large systems is one of the central problems in theoretical and mathematical physics, and for a long time it
remains one of the main problems of statistical mechanics. In recent decades, this topic has become even more relevant in
connection with the problem of big data processing, the development of quantum information, applied problems in
combinatorics and other areas.
The primary aim of the project is the development of new methods of mathematical physics in the study of the phenomenon
of limit shapes - deterministic structures in systems consisting of a large number of random elements. The methods used in
this work are characterized by a synthesis of traditional approaches to this problem and methods of the modern theory of
integrable systems. To implement this approach it is necessary not only to apply, but also to develop the methods of
integrable systems, both quantum and classical. The stated aim of the project will be achieved by solving specific tasks that
can be grouped into three categories.
- Limit shapes in models of statistical mechanics.
a) Description of the geometry and properties of limit shapes in two-dimensional models of statistical mechanics. One of the
most significant discoveries in this direction was a complete description of the limit shapes for dimer models in regions with
critical boundary conditions (Okunkov et al.). We plan to take the next step and develop a new methodology using
Hamiltonian methods of integrable models. The direction is based on a recent series of works by Reshetikhin et al., where it
was shown that the corresponding Euler-Lagrange equations have an infinite number of integrals. Limit and correlation
functions will be studied for generalized boundary conditions of the domain wall type. The new methodology will be
developed based on the tangent method.
b) The question of the limit shapes in systems without a height function has remained open for many years. The answer is
well known for the Ising model - the Dobrushin-Kotecky-Shlosman drops. Their discovery was the most striking and
significant achievement in the study of the Ising model after Onsager discovered its exact solution. One of the objectives of
the project is to study the formation of such drops in more general models (such as noncritical dimer models, six-vertex
model in the antiferroelectric phase, etc.) in noncritical phases. The solution of this problem requires the creation of new
methods for calculating the surface tension energy and calculating finer asymptotics of partition functions and correlation
functions.
c) The limit shapes of large systems will be investigated by numerical methods that we will develop and apply based on
the Monte Carlo method and its modifications, such as the population annealing algorithm. A parallel version of these
algorithms will be implemented on video cards. These new methods will be applied to study subtle asymptotic phenomena
and scaling in limit shapes, as well as correlation functions. - Asymptotic representation theory and combinatorics.
a) Limit distributions of indecomposable components in tensor products of finite-dimensional representations of Lie
superalgebras and quantum groups at roots of unity in the limit when the number of factors tends to infinity will be obtained
and studied. For Lie superalgebras, this problem was studied for a very special case and we are planning to present a
complete solution to the problem. To solve this problem in the case of quantum groups at interesting roots of unity, a very
subtle analysis of the decomposition of these representations into tilting modules is required. For other Lie algebras, this
problem is still open. In both cases, it has important applications and extends far beyond existing results.
b) Similar asymptotic problems for representations of affine Kac-Moody algebras are completely new. Several results
available in the literature do not provide a meaningful description of the asymptotics even for special cases. Calculation of
these asymptotics is another task of the project. For example, in the case of sl(n), it is directly related to quantum information
theory. The affine version is expected to be just as important.
c) Asymptotic problems of combinatorics, in particular, the statistics of large partitions and their generalizations will be
studied in the context of the asymptotics of the moments of random large distributions. One of the important issues related
to this problem is the asymptotic behavior of the spectrum of the spin Calogero-Moser model and the corresponding density
matrices. - Classical and quantum integrable and superintegrable systems.
a) It is planned to study the relationship between classical superintegrable systems and cryptography problems, and also to
use these results both for constructing new crypto-algorithms and for finding new superintegrable systems.
b) Infinite-dimensional analogs of superintegrable systems will be constructed on moduli spaces associated with simple
finite-dimensional Lie groups, on loop groups and the corresponding Kac-Moody groups. Quantum analogs of these systems
will also be built. This is a completely new approach based on the recent work of a project leader.
c) We will study the limit of systems of Calogero-Moser type and their generalizations in the case when the rank of the Lie
algebra tends to infinity. The solution of this problem is relevant for the above problems of asymptotic representation theory
and combinatorics. In particular, very little is known about such asymptotics for root systems B, C, and D.
Project is supported by the Russian Science Foundation under grant 21-11-00141
English (United Kingdom) 
Russian (Russia)