Colored Stochastic Vertex Models

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

The goal of the project is large time and scale analysis of complex probabilistic systems. Many of those are designed to model physical or biological processes with numerous components, such as bacterial growth, combustion fronts, very cold gases at the atomic level, and so on. Predicting the behavior of such a system at large times is very challenging, and for mathematical analysis one needs to rely on certain special structure of the model at hand. This structure originates in algebra and representation theory, and it ultimately allows one to access large time and scale characteristics at a very fine resolution. The resulting probabilistic laws tend to be occurring in a substantially wider range of probabilistic systems, which can be observed through numerical and physical experiments. This project is about stochastic systems that can be analyzed by essentially algebraic methods; such systems are often call exactly solvable or integrable. The overarching goal is to develop a theory of stochastic vertex models related to quantum affine groups of arbitrary ranks. On one hand, this class of models is very general, uniting many previously studied integrable probabilistic models (such as ASEP, directed polymers in random environment, and quantum Boson systems), as well as adding many new ones (such as multi-species or colored exclusion processes). On the other hand, the representation theoretic framework available for these models makes it feasible to extend previous analysis far beyond what is currently known and, more importantly, discover new qualitative and quantitative universal phenomena. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.