Random Polymer Measures

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This project is focused on the interface of probability theory and statistical mechanics. The activity aims at investigating the evolution of systems with complex interactions, such as particles moving in a disordered environment, cars navigating their way through traffic, the surface of a growing crystal, or the boundary of an infected tissue. Complexity is captured by the randomness in the model, both in the environment in which the particles interact or the crystal grows, and in the interaction or growth process itself. The aim of the project is to develop the mathematical laws that govern such systems. To have a simple example in mind, one can think of how the fraction of heads in a large number of tosses of a coin will converge to the probability of getting heads in one toss. Besides its impact on probability theory and mathematics in general, the project will have a direct impact on the understanding of many physical systems involving motion in random or disordered media. Understanding complex interactions has wide implications for science and engineering and thereby for society. The project provides research training opportunities for graduate students and postdoctoral associates.This research is on the subject of random motion in random media. In his recent work, the PI has focused on the study of random polymer models, both in positive and zero temperature. The overarching theme was establishing energy-entropy duality, producing solutions to these variational formulas in terms of Busemann functions, then using these solutions to describe the large-volume systems. The PI already constructed the Busemann functions in a variety of models and used them to analyze infinite geodesics (or Gibbs measures, in the positive temperature case), establish one-force one-solution principles, and study the stability and instability of the related dynamical systems. In this project, the PI will work on extending these results to a quite general setting that will cover zero and positive temperature, directed and undirected, discrete and continuous, as well as higher-dimensional models. The PI will also develop methods to use Busemann functions to study the regularity of the limiting shape function, prove localization, study the size of the polymer and shape fluctuations, and, for random walk in random environment, describe the structure of the Martin boundary.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.