Interacting Particle Systems: Exact Solutions, Scaling Limits, and Relations to Classic Integrable Systems

Universality is a fundamental phenomenon studied in physics; universal behavior reveals fundamental properties of the physical models and the processes they describe. Mathematical models for physical phenomena are analyzed by studying their scaling limits, that is, their behavior over long timespan and broad spatial scope. The broad arsenal of mathematical tools used to understand these models includes computations of exact formulas for observables, non-linear stochastic and deterministic partial differential equations, and many others. This project aims to study the universality of one-dimensional random growing interfaces, which can model a wide range of physical phenomena, for example the spread of fire and the growth of bacterial colonies or liquid crystals. Recent developments in the field provide new mathematical methods to solve problems that were previously out of reach. KPZ (Kardar-Parisi-Zhang) universality implies that the limiting fluctuations of many mathematical models describing one-dimensional random growing interfaces are essentially the same. Such growing interfaces can be described mathematically in many ways, including free energies of directed random polymers, interacting particle systems, stochastic Burgers and Hamilton-Jacobi-Bellman equations, and stochastically perturbed reaction-diffusion equations. These models are expected to converge to the same scaling limit, a Markov process called the KPZ fixed point. Such models are typically studied by computing exact formulas for their observables and analyzing the scaling limits of these formulas. This project aims to study exact solutions of interacting particle systems on various domains. In particular, one of the goals of the project is to explain surprising relationships between random and classic integrable systems, such as the Kadomtsev–Petviashvili (KP) and Korteweg-de Vries (KdV) equations and the Toda lattice. 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.