Probabilistic Analysis of Large Complex Geometric Structures

与大型复杂几何结构有关的基本问题通常涉及具有短程相互作用但复杂长程依赖性的空间依赖项的总和。一个主要的目标是表明，空间依赖项的总和的行为，好像他们是独立的同分布的随机变量的总和。 因此，我们的目标是要表明，这些款项满足法律的大数，包括中等和大偏差原则，该款项有渐近正态分布，并确定这些款项的随机点措施满足功能中心极限定理。 Yukich将采用一种称为稳定的空间依赖结构来建立几何概率中泛函的一般热力学极限，并为这些泛函建立高斯极限。 他将应用一般结果为随机几何，弹道沉积模型，随机几何网络和图形以及空间统计中的特定问题建立极限理论。许多涉及运输和通信网络性能分析，极值，信号分析甚至球体随机填充的真实的世界问题可以理解为涉及相互作用粒子泛函的问题，其中粒子局部相互作用但具有长程依赖性。 在许多情况下，相关性很弱，以至于这些泛函的和表现得好像是独立粒子的和，因此，当粒子数很大时，它们大致表现为独立抛硬币的和，其中每个硬币都有自己的正面朝上的概率。 本课题的目的是研究电信、统计力学、随机网络中的复杂系统问题与独立抛硬币问题之间的相似性。在这个过程中，Yukich与Lehigh工程学院和贝尔实验室的同事合作，希望增加我们对网络，随机图，组合优化和极值中出现的问题的理解。

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.