Probabilistic Analysis of Large Complex Geometric Structures

Lehigh University, Bethlehem PA

Investigators

Abstract

Fundamental questions pertaining to large, complex geometric structures often involve sums of spatially dependent terms having short range interactions, but complicated long range dependence. A chief goal is to show that sums of spatially dependent terms behave as though they were sums of independent identically distributed random variables. Thus the goal is to show that such sums satisfy laws of large numbers, including moderate and large deviation principles, that the sums have asymptotically a normal distribution, and that the random point measures defined by these sums satisfy functional central limit theorems. Yukich will employ a spatial dependence structure termed stabilization to establish general thermodynamic limits for functionals in geometric probability and to establish Gaussian limits for such functionals. He will apply the general results to establish limit theory for particular problems in stochastic geometry, ballistic deposition models, random geometric networks and graphs, and spatial statistics. Many real world problems involving the performance analysis of transportation and communication networks, extreme values, signal analysis, and even random packing of spheres can be understood as problems involving functionals of interacting particles, where particles interact locally but have long range dependencies. In many cases the dependencies are weak enough so that sums of such functionals behave as though they were the sum of independent particles and therefore, when the number of particles is large, behave roughly as a sum of independent coin tosses, where each coin has its own probability of coming up heads. This project aims to study the similarities between problems involving complex systems in telecommunications, statistical mechanics, and random networks and problems involving independent coin tosses. In the process Yukich, in collaboration with colleagues from the Lehigh engineering school and Bell labs, hopes to increase our understanding of problems arising in networks, random graphs, combinatorial optimization, and extreme values.