Random Polymer Measures

This project explores how complex and unpredictable growth patterns occur in nature—for example, how infections spread, how crystals form, or how traffic jams develop. These systems can look chaotic, but they often follow deep mathematical rules. By studying the mathematics behind these random and irregular processes, the project aims to uncover the hidden patterns that govern them. Although the work is rooted in abstract mathematics, it has real-world applications in areas like medicine, environmental science, and engineering. Better understanding how random growth and movement work can help us manage diseases, improve materials, and design smarter technologies. This research not only advances our knowledge of mathematics but also contributes to solving real problems that affect people’s lives.

This project focuses on the mathematical study of random motion in random environments, with particular emphasis on models of random growth. The principal investigator has developed and applied a key analytical tool--Busemann functions--to a broad class of stochastic models, including directed and undirected percolation, random polymers, and random walks in random environments, across discrete, semi-discrete, and continuous frameworks. Notably, this includes work on the Kardar-Parisi-Zhang (KPZ) equation, a central object in the study of stochastic growth. Busemann functions provide deep structural insights: they solve energy-entropy variational formulas, describe infinite-volume Gibbs measures in positive temperature regimes, and characterize geodesic rays in zero-temperature settings. They also yield eternal solutions and stationary distributions in associated random dynamical systems (RDS), playing a central role in understanding their stability properties and in identifying shock locations in inviscid or zero-temperature models. This project aims to extend this framework to a wider class of stochastic Hamilton-Jacobi equations--both viscous and inviscid, with time-dependent or time-independent Hamiltonians--and to resolve several open problems that emerged from prior work. Chief among these is a longstanding question in the theory of lattice percolation: the differentiability of the shape function.

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.

Award Number: 2450951
Principal Investigator: Firas Rassoul-Agha
Funds Obligated: $200,000
State: UT