Probabilistic Analysis of Random Geometric Structures

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0203720Yukich This proposal focuses on the rigorous probabilistic analysis of random structures in Euclidean space. Building on previous research, we focus on the stochastic analysis of (i) random packing, random growing interfaces, and loose sphere packing, all in two or more dimensions, and (ii) functionals of Boolean models and random Euclidean point sets in computational geometry and combinatorial optimization. The aim in (i) is to establish thermodynamic limits, Gaussian fluctuations, and functional central limit theorems for functionals associated with packing models, interface models, as well as for random Euclidean point sets in general. All of these models exhibit finite range interactions and long-range dependence. Part (ii) focuses on the probabilistic analysis of functionals of random point sets in Euclidean space. The proposed work in (ii) will further our understanding of solutions to Euclidean optimization problems, random graphs in computational geometry, and the statistical analysis of random data. One of the central problems in physical, chemical, and biological processes involves understanding the growth of random interfaces. Such interfaces arise naturally in understanding chemical deposition, the morphology of vascular networks, growth patterns in fracture analysis, and the porosity of media. Dynamic random interfaces, ones which change with time, are useful in understanding and modeling crystal growth as well as understanding patterns in nature which change with time. This includes diverse phenomena such as the spread of disease in a population, the growth patterns in a forest fire, as well as the fracture analysis of metals such as those found in aircraft. Much of our understanding of random interfaces is only experimental. This proposal seeks to put our understanding of random interfaces on a solid theoretical foundation. Understanding the behavior of random interfaces currently forms a central part of the principal investigator's collaboration with scientists from the Lehigh University engineering school as well as with scientists from Lucent Technologies.