Probabilistic Models with Boundary: Symmetries and Asymptotics

A major goal of modern probability is to understand the macroscopic behavior of large random systems. This project studies a class of random growth models taking place in different geometric settings and will develop new tools effective for these structures; the aim is to understand the behavior of these systems and the impact of the underlying geometry on this behavior. These systems, for example, might be used to model the growth of cancer along a wall or a cylinder. The extensive algebraic structure underlying integrable or exactly solvable models without boundary has been successfully used to study a variety of probabilistic questions for these models. Many of these models are expected to exhibit universality, meaning that the behavior studied should occur in a wide variety of other models. However, once non-trivial boundary conditions are imposed, our understanding is incomplete. The proposal aims to develop a better understanding of the algebraic structures involved once boundary conditions are imposed and to use this structure to attack probabilistic problems. In particular, the work aims to find new hidden symmetries for these models and to establish asymptotic results via new exact formulas for models with boundary. Undergraduate students will participate in the research, continuing the awardee's record of student mentorship, and the work will be disseminated at seminars and conferences. 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.