Integrable Probability

Introduction

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.

Research Focus

The proposed research is a study of a variety of probabilistic models. Specific examples include:

1. The single- and multi-species asymmetric simple exclusion process
2. A six vertex model
3. Random walks on Hecke, Temperley-Lieb, and Brauer algebras
4. Random tilings models
5. Random representations

The suggested methodology consists of a range of probabilistic, algebraic, analytic, and combinatorial techniques.

Research Questions

The project involves two circles of questions.

Random Walks on Algebras

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
• Developing a general theory of random walks on algebras

Asymptotic Representation Theory

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
• Studying random tilings in random environments

Unifying Idea

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.