Integrable Probability

The project focuses on studying large time and scale behavior for a variety of probabilistic systems. Many of those were designed to model different natural processes such as bacterial growth, crystal melting, very cold gases at atomic levels, etc. An accurate analysis at large times or scales is typically very difficult, and the project concentrates on models with additional algebraic structure that originate in seemingly unrelated areas of mathematics. This structure helps to discover new phenomena that tend to be universal (i.e., present in a very wide range of systems). As a result, using sophisticated mathematical tools, the principal investigator seeks to find new universal laws that play the role of the famous and familiar bell curve law and that can often be observed through physical and numerical experiments.

The goal of the emerging field of integrable probability is to identify and analyze exactly solvable probabilistic models. The models and results are often easy to describe, yet difficult to find, and they carry essential information about broad universality classes of stochastic processes. The project aims at developing a bridge between deep algebraic and representation theoretic structures on one end, and probabilistic systems on the other end, that would allow to utilize the former in order to discover and study the latter. The framework of Macdonald processes introduced and developed in the last five years has been quite successful on this path, and it continues to grow. Known applications include interacting particle systems, random growing interfaces in (1+1) and (2+1) dimensions, random matrices and log-gases, and directed polymers in random media. The framework is now poised to expand to include the theory of solvable lattice models of statistical mechanics, offering completely new perspective and new results in this well-established domain.