Random Systems from Symmetric Functions and Vertex Models

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

Modern probability theory is an area of pure mathematics instrumental in the growing understanding of natural and social reality. The project studies complex random systems motivated by real-world questions, including the structure of ice and other condensed matter, magnetism, quantum systems, polymers, thermodynamics, and traffic models. The PI aims to discover and analyze systems living in discrete space that capture the essential large-scale and long-time physical phenomena, yet are "integrable", that is, simple enough for exact mathematical treatment. The analysis of integrable random systems is powered by explicit formulas and various symmetries coming from algebraic structures. Of special interest are time-dependent irreversible systems and models with impurities displaying a wide range of conjecturally universal behavior. The project provides research opportunities for graduate students as well as training activities through organization of summer school on integrable probability. The project revolves around new and known integrable stochastic systems whose structure is accessible through the Yang-Baxter equation or symmetric functions. The scope of the studied models includes interacting particle systems on the one-dimensional lattice with local or global interaction (asymmetric simple exclusion process and discrete analogs of the Dyson Brownian motion, respectively), random tilings, and other statistical mechanical systems. The goals are three-fold: (1) discover new models with integrable structure (such as q- and Macdonald analogs of Dyson Brownian motion); (2) find previously overlooked symmetries in well-known models (such as time-reversal symmetries in particle systems with arbitrary initial data); and (3) probe new asymptotic phenomena and establish their universality (such as inhomogeneous deformations of pure Gibbs states and the Gaussian Free Field). The goals will be achieved by finding and analyzing exact formulas for moments and correlations (with the help of symmetric functions and vertex models), and also by developing distributional symmetries using Markov maps, or by introducing many inhomogeneous parameters into the system. 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.