FRG: Collaborative Research: Integrable Probability

Much of modern probability research seeks to understand the behavior of large and complex random systems (for instance, growth in disordered media, cracking, turbulent fluids, or traffic flow) with an aim towards developing theories with predictive and statistical value. While one can try to directly model such systems on computers, their size and complexity often render such attempts fruitless. Instead, one can look for models of such systems that are complex enough to display all of the phenomena under study, yet simple enough to admit exact mathematical computation to probe that behavior. Integrable probability is the theory behind discovering and subsequently analyzing such models. This project seeks to unify the area and various recent breakthroughs and in so doing discover a host of new types of integrable probability systems, new tools for their analysis, and new large-scale universal phenomena.

Integrable probability is an area of research at the interface between probability, mathematical physics, and statistical physics on the one hand, and representation theory and integrable systems on the other. Integrable probabilistic systems are characterized by two properties: It is possible to write down concise and exact formulas for expectations of a variety of interesting observables of the systems; and asymptotics of the systems, observables, and formulas provide access to exact descriptions of new phenomena and universality classes (containing more than just integrable examples). The discovery and analysis of integrable probabilistic systems hinges upon underlying algebraic structure. These integrable probabilistic systems can be viewed as projections of powerful objects whose origins lie in representation theory and integrable systems. There is a rich history of major breakthroughs in the study of integrable probabilistic systems, including the six-vertex model, Ising model, and more recently certain models in the KPZ universality class. The basic mechanisms at the heart of many of these existing results are Schur / Macdonald processes (built off the structure of symmetric polynomials) and quantum integrable systems (built off solutions to the Yang-Baxter equation and the Bethe ansatz). Each mechanism has produced breakthrough results, such as the recent resolution of the 25-year-old physics conjecture that the KPZ stochastic partial differential equation is in the KPZ universality class. Until recently, these two routes to integrable probability have existed relatively separately. The goal of the proposed project is to create a unified theory of integrable probability, combining and generalizing the methods of Schur / Macdonald processes and quantum integrable systems and, complementarily, extracting new analyzable models and uncovering new probabilistic or physical phenomena.