Periodic Reporting for period 1 - IProbability (Integrable Probability)
Reporting period: 2022-06-01 to 2024-11-30
This project is devoted to integrable probability. The key feature of the field is the prominent role of methods and ideas from other parts of mathematics (such as representation theory, combinatorics, integrable systems, and others) which are applied to stochastic models. This philosophy often leads to very precise limit theorems which seem to be inaccessible by more standard probabilistic techniques.
The proposed research is a study of a variety of probabilistic models. Specific examples include the single- and multi-species asymmetric simple exclusion process, a six vertex model, random walks on Hecke, Temperley-Lieb, and Brauer algebras, random tilings models, and random representations. The suggested methodology consists of a range of probabilistic, algebraic, analytic, and combinatorial techniques.
The project involves two circles of questions. The first one focuses on random walks on algebras and their applications to interacting particle systems. The specific objectives include studying the Kardar-Parisi-Zhang type fluctuations for the multi-species asymmetric simple exclusion process, computing limit shapes and fluctuations around them for a general six vertex model, introducing and studying integrable three-dimensional analogues of a six vertex model, and developing a general theory of random walks on algebras.
The second one focuses on asymptotic representation theory. This area deals with the probabilistic description of representations of big groups. Such questions turn out to be related to a plethora of other probabilistic models, in particular, to models of statistical mechanics. The goals of this part include bringing this interplay to a new level, developing asymptotic representation theory of quantum groups, and studying random tilings in random environment.
The unifying idea behind these questions is a systematic use of precise relations for the study of asymptotic behavior of stochastic models which are out of reach of any other techniques.
The proposed research is a study of a variety of probabilistic models. Specific examples include the single- and multi-species asymmetric simple exclusion process, a six vertex model, random walks on Hecke, Temperley-Lieb, and Brauer algebras, random tilings models, and random representations. The suggested methodology consists of a range of probabilistic, algebraic, analytic, and combinatorial techniques.
The project involves two circles of questions. The first one focuses on random walks on algebras and their applications to interacting particle systems. The specific objectives include studying the Kardar-Parisi-Zhang type fluctuations for the multi-species asymmetric simple exclusion process, computing limit shapes and fluctuations around them for a general six vertex model, introducing and studying integrable three-dimensional analogues of a six vertex model, and developing a general theory of random walks on algebras.
The second one focuses on asymptotic representation theory. This area deals with the probabilistic description of representations of big groups. Such questions turn out to be related to a plethora of other probabilistic models, in particular, to models of statistical mechanics. The goals of this part include bringing this interplay to a new level, developing asymptotic representation theory of quantum groups, and studying random tilings in random environment.
The unifying idea behind these questions is a systematic use of precise relations for the study of asymptotic behavior of stochastic models which are out of reach of any other techniques.
A significant progress was achieved for a few of research goals indicated in the proposal. Several new methods were developed and now are applied to the problems ( In particular, Problems 1, 2, 9,10, 15,16,19) of the proposal, with some papers already published and close to be ready to the publication. On the other hand, some of the more ambitious goals (especially Problem 8) of the proposal are still widely open, despite a lot of work in their direction as well -- the hope is, that there would be more progress in their direction in the remaining years of the project. It is also interesting to note that the problems indicated in the proposal are of wide interest, with many groups throughout the world working on them, which in particular resulted that Problems 5, 6, 7 were alrady largely resolved in works not related with this project.
In more detail, one paper is currently published (related to Problems 1 and 3), and several preprints are publicly available on Arxiv and submitted to the journals.
In more detail, one paper is currently published (related to Problems 1 and 3), and several preprints are publicly available on Arxiv and submitted to the journals.
One of the main results beyond the state of the arts so far is the analysis of the domino tilings in random environment. Despite the fact that the model is well-known and well-studied in the last thirty years, this type of results seems to be completely new and beyond the previously existing methods.
Another important result beyond the state of the arts is the solution of Problems 15 and 16, dealing with the interplay of random matrices, free probability, and extreme characters of infinite-dimensional unitary group.
Another important result beyond the state of the arts is the solution of Problems 15 and 16, dealing with the interplay of random matrices, free probability, and extreme characters of infinite-dimensional unitary group.