Integrable Probability and Random Matrices: 2D Structures, Limit Theorems

The goal of this research is to achieve a better understanding of the limit behavior of a class of stochastic systems related to statistical mechanics, random matrix theory and representation theory. Examples of such systems include random stepped surfaces in three-dimensional space which, in particular, model melting crystals; square-ice model which is a mathematical two-dimensional approximation for a thin layer of ice; and interacting particle systems used for modeling (e.g., a one-lane highway, the growth of plankton in the ocean). Our aim is to extract macroscopic properties of very large systems starting from their microscopic definitions, with main accents on the appearance of the universal random fields and distributions.

There are two main distinctions of the systems we study. First is that we concentrate on and explore 2d structures which generalize many classical 1d probabilistic models such as eigenvalues of a random matrix. The two-dimensional extensions that we consider give new and often more natural interpretations of earlier one-dimensional results and also pave the way to prove new interesting asymptotic results about well-known one-dimensional models. Second, most models in this research enjoy a rich algebraic structure, which usually means that expectations of many observables can be computed in a concise manner. The techniques include usage of symmetric functions of representation-theoretic origin, eigenfunctions of difference operators, orthogonal polynomials, etc. This exact solvability provides tools for delicate asymptotic analysis and gives access to the properties of the universal objects appearing in the limit, such as Tracy-Widom distributions, the GUE-eigenvalues distribution, the GUE-corners process (its two-dimensional extension), and the Gaussian Free Field. The results obtained for the exactly solvable models are generally believed to extend to a large variety of similar stochastic systems.