Critical and Subcritical Growth Models

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The projects supported by this award involve studying mathematical models for random growth. Some examples include bacterial or tumor spread, and fluid flow through porous media. The research questions center on geometric aspects of optimal growth paths, as well as the overall size and speed of growth. The proposed work has connections to other areas of mathematics and physics, like the structure of disordered magnets, and satisfaction problems from computer science. The projects call for work by undergraduate and graduate students, as well as postdoctoral researchers, and provide research training opportunities for graduate students.This project contains questions in probability theory and mathematical physics, and centers on percolation-type growth models including first-passage percolation (FPP) and Bernoulli percolation. These are models that were introduced in the 1950's, but despite decades of effort by researchers, many of their fundamental properties remain elusive. The proposed projects include determination of fractal properties and scaling limits of box-crossing paths in Bernoulli percolation, the effect of random noise on passage-time asymptotics in critical FPP, and the geometry and topological structure of the growing set in sub-critical (usual) FPP. It is expected that results obtained in these studies will affect work on epidemic models, disordered spin systems, and polymer models.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.