Irregular or random growth is a ubiquitous phenomenon in nature, occurring in settings ranging from the growth of bacterial colonies and tumors to the formation of coffee rings and the spread of forest fires. The last twenty years have seen a renaissance in our understanding of mathematical models of these phenomena, particularly in two space-time dimensions where many such models are conjectured (and sometimes proven) to have the same statistical properties. This project supports fundamental research into mathematical models of this type, both in the two dimensional settings which have been the focus of much recent work and in higher dimensions where much less is known. In addition to the proposed research, this project includes outreach components aimed at connecting students at levels ranging from middle school to graduate school to fun mathematics in general and to the world of random growth models in particular.
The PI will study the structure of a class of models at the interface of rigorous statistical mechanics and probability known as random walks in random potentials, both in positive and in zero temperature. This class of models is broad. On the lattice, it includes first passage percolation and directed last passage percolation, directed polymers in random media, as well as random walks in both static and dynamic random environments, with general admissible steps. Continuum models like the continuum directed polymer and the associated stochastic heat equation with multiplicative noise will also be considered. The main goal of this project is to improve our understanding of the infinite volume structure of such models through specific examples and to work toward a unified treatment of that structure in the greatest generality that can be achieved. Concretely, the main goal of the project will be to characterize the existence and uniqueness properties of semi-infinite and bi-infinite volume measures (respectively geodesics). The work will also consider related problems, including investigating how infinite volume measures relate to the regularity properties of the limiting free energy (respectively shape function), developing computational tools in exactly solvable models and on understanding the structure of the environment viewed from the perspective of paths drawn from the infinite volume measure (respectively a geodesic).
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.
