Gibbsian line ensembles, vertex models, and geometric aspects of stochastic growth

This project studies stochastic growth models, which are mathematical idealizations of the evolution of 1-dimensional interfaces between two media, such as the boundary of a cell colony, the frontier of a forest fire, or the boundary between the wet and dry portions of a piece of paper after water is dripped on it. In the past decade mathematical structures called line ensembles have played a prominent role in the understanding of stochastic growth models. Line ensembles also show up in modeling seemingly unrelated areas, such as magnetization, traffic, and protein synthesis in cells. This project will expand knowledge of line ensembles and related techniques, which will lead to the solution of previously intractable problems. The project will also incorporate organization of conferences and seminars, mentoring of graduate students, outreach to middle school students, and expository writing.The asymmetric simple exclusion process (ASEP), last passage percolation and polymer models, the stochastic six vertex (S6V) model, the parabolic Airy line ensemble, and the directed landscape are a few central models and objects in the Kardar-Parisi-Zhang (KPZ) universality class, and they share a common feature of bearing connections to line ensembles. This project will exploit and in some cases develop these connections to explore a number of directions: (i) the limiting statistics of models including the S6V model, ASEP, higher spin colored vertex models, and their degenerations, (ii) the upper and lower tail behavior of KPZ models and the relationship of the line ensembles with random geometry, and (iii) questions in more classical areas of probability such as mixing times (of polymer models) and interfaces in the Ising model.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.