Probabilistic Analysis of Large Complex Geometric Structures

在随机几何和应用概率中出现的许多问题，以及在网络、空间统计学和统计力学中出现的问题，可以用大型随机几何结构的行为来理解，其中“大”意味着随机性涉及越来越多的随机变量。“几何”是指问题在很大程度上依赖于底层空间的几何形状。涉及这些复杂结构的问题涉及理解具有短程相互作用但复杂的长程相关性的空间相关项的总和的行为。离散随机几何中感兴趣的问题涉及I.I.D.的凸壳泛函。样本，渐近量化误差，极大值点的极限行为，以及欧氏空间中广义镶嵌的极限行为。涉及空间数据的感兴趣的问题包括嵌入高维欧氏空间的非线性数据云的维度估计、熵估计、表面积分和体积积分的估计，以及建立数据传输的最小代价网络和能量标度定律。在每一种情况下，人们都试图量化在这些问题中出现的泛函的平均行为。一个主要目标是证明空间相关项的和就像它们是独立的同分布随机变量的和一样。证明了这些和满足大数定律，它们具有渐近正态分布，由这些和定义的随机点测度满足泛函中心极限定理，也就是说，证明了它们的标度行为是用布朗单来理解的。本项目旨在解决工业界和学术界研究人员感兴趣的几何概率问题。例如：(1)给定一个未知的物体或身体(如人体内的梗塞或地下储油层)，我们如何有效地利用对该物体的随机探头来找到其表面积和体积的可靠估计值？(2)考虑到大量的空间数据，我们如何仅使用数据的点间距离来确定数据的固有属性，包括其固有维度？(3)考虑到万维网这样的网络，人们如何才能找到最有效地通过它传输和发送信息的方法，从而最大限度地减少成本和旅行时间？同样，给定一个通信网络，如何最佳地放置发射机以最大化覆盖范围？(Iv)给定任何复杂的网络，包括航空公司和其他交通网络，如何有效地将车辆送到最大限度地覆盖？这个项目的目标是开发理论工具来解决这些和相关的问题，并开发在工业中使用的高效算法。

Many questions arising in stochastic geometry and applied probability, as well as questions arising in networks, spatial statistics, and statistical mechanics, may be understood in terms of the behavior of large random geometric structures, where `large' means that the randomness involves a growing number of random variables. `Geometric' means that the problems depend heavily on the geometry of the underlying space. Problems involving these complex structures involve understanding the behavior of sums of spatially dependent terms having short range interactions, but complicated long range dependence. Problems of interest in discrete stochastic geometry involve functionals of convex hulls of i.i.d. samples, asymptotic quantization error, the limit behavior of maximal points, and the limit behavior of generalized tessellations in Euclidean space. Problems of interest involving spatial data include dimension estimation of non-linear data clouds embedded in a high dimensional Euclidean space, estimation of entropy, estimation of surface and volume integrals, as well as establishing minimal cost networks for data transmission and energy scaling laws. In each case, one seeks to quantify the `mean' or average behavior of functionals arising in these problems. A chief goal is to show that sums of spatially dependent terms behave as though they were sums of independent identically distributed random variables. One thus wants to show that such sums satisfy laws of large numbers, that they have asymptotically a normal distribution, and that the random point measures defined by these sums satisfy functional central limit theorems, that is to say show their scaling behavior is understood in terms of Brownian sheets.This project aims to solve problems in geometric probability which are of interest to researchers in both industry and academia. Examples include the following: (i) given an unknown object or body (such as an infarction in the human body or an underground deposit of oil) how can we use effectively use random probes of the object to find reliable estimators of its surface area and volume? (ii) given a huge amount of spatial data, how do we use only the interpoint distances of the data to determine intrinsic properties of the data, including its intrinsic dimension? (iii) given a network such as the world wide web, how does one best find ways to efficiently transmit and route information through it, minimizing cost and travel time? Similarly, given a communication network, how does one optimally place transmitters to maximize coverage?(iv) given any complex network, including airline and other transportation networks, how does one efficiently route vehicles to maximizerevenue? The goal of this project is to develop theoretical tools to solve these and related problems and to develop efficient algorithms of use in industry.