Geometry of Physical Models Governed by Diffusion
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Over the past decades probability theory has had tremendous success in elucidating scaling limits and large-scale phenomena. However, systems that have no apparent scaling limit are abundant in both the physical and life sciences, and there is practical need to better understand such systems. This research project aims to advance understanding of such systems by studying models augmented with additional symmetries that share the local behavior of the classical disordered systems under study. Graduate students and postdoctoral associates will receive training through involvement in the research.The topics under study in this project divide into three subareas: (1) stationary diffusion limited aggregation (DLA) and Hastings-Levitov models, (2) DLA in a wedge, and (3) chemical distance in random interlacements. These models are of interest for statistical mechanics and mathematical physics. They share the features of being generated by diffusion or the harmonic measure, and of attempting to grasp the nature of a physical phenomenon that is not amenable more classical models in the field, such as DLA or Bernoulli percolation. The methods to be employed borrow ideas and tools from various mathematical disciplines, including complex analysis, harmonic analysis, differential equations, and ergodic theory.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.