Topics in Small Value Theory of Probability

该研究项目考虑了小值理论研究中各种技术的发展：正值随机变量取值较小的典型行为和罕见事件。典型行为涉及随着指标集的大小而对一族非负随机变量的最小值的期望。这一方向的主题包括由生成树索引的最小长度生成树，高斯向量绝对值的最小值的比较不等式，由排列索引的随机赋值类型问题，在这个方向上的主题包括小球概率、下尾行为和水平穿越概率。作者在过去几年中开发的工具的非常成功的应用将扩展到各种等周型高斯不等式、张量随机场、高斯混沌、布朗涡模型、永久过程、和随机多项式的正性指数。主要目的是在系统研究各种方法和不同应用的基础上，建立和扩展一般的小值理论。概率论中的两个基本现象是典型的行为，如期望值、大数定律和中心极限定理，以及罕见的事件，如极大或极小的值。这一建议旨在通过发展相关物理和生物随机模型的新技术来加深我们对正随机量小值现象的理解。这项研究对本科生和研究生的教育和研究都有帮助。许多未解决的问题和研究结果将作为学生的课程项目。这项研究将提高我们对极端随机事件的理解，并为研究我们的随机环境提供基本工具。

This research project considers the development of various techniques in the study of small value theory: Both typical behaviors and rare events of the type that positive random quantities take smaller values.The typical behavior deals with the expectation of the minimum over a family of non-negative random variables as the size of the index set grows larger.Topics in this direction includeminimum length spanning tree indexed by spanning trees, comparison inequalities for minimum of the absolute value of Gaussian vectors, random assignment type problems indexed by permutations, and the first passage percolation indexed by paths.The rare events of small value type deal with decay probabilities of positive random quantities taking smaller values than typical ones.Topics in this direction include small ball probabilities, lower tail behaviors and level crossing probabilities.The very successful applications of tools developed by the proposer over the last few years will be expanded to a detailed study of various isoperimetric type Gaussian inequalities, tensored random fields, Gaussian chaos, Brownian eddy models, permanental processes, and positivity exponents for random polynomials.The major objective is to build and extend a general small value theory based on systematic study of various methods and diverse applications.Two fundamental phenomena in probability theory are typical behaviors such as expected values, laws of large numbers and central limit theorems, and rare events such as extremely big or small values.This proposal aims to deepen our understanding of small value phenomena for positive random quantities by developing new techniques in the study of relevant physical and biological random models.This research benefits both undergraduate and graduate education and research. Many open problems and results from the proposed study will be used as students course projects.This research should improve our understanding of extremal random events and provide basic tools for the study of our random environment.