Limiting Shape of First-Passage Percolation

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Physicists and other scientists use methods from probability to model the dynamics of large collections of interacting components. Two-dimensional models describe, for example, the growth of bacterial colonies in a petri dish or the spreading of a fluid in a random medium. To better understand such systems, much attention has been paid to how, for example in a bacterial colony, the boundary of the region occupied by the colony changes over time. Kardar, Parisi, and Zhang (KPZ) proposed that a stochastic partial differential equation could be used to describe the evolution of these boundaries. This project will study a large class of systems that are believed to be governed by this equation, with the goal of verifying that the time evolution is well-described by this probabilistic model. The project also provides research training opportunities for graduate students.The KPZ universality class is a collection of models of random growth in the plane that all exhibit the same asymptotic behavior. Over the last two decades there has been great progress understanding exactly solvable (algebraic) models like exponential last-passage percolation and the longest increasing subsequence in a uniformly random permutation. This project will study first-passage percolation. This is a class of models which are believed to be in the KPZ universality class and have the same asymptotic properties as the models mentioned above. But without the algebraic tools of the exactly solvable models, they have proven very difficult to analyze. This project aims to understand basic properties of first-passage percolation such as the relative rate of growth of the process in different directions.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.