Extremal Graph Theory and Bootstrap Percolation

极值图论和 Bootstrap 渗滤

• 批准号: 0603769
• 负责人: Jozsef Balog
• 金额: --
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-15 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603769&HistoricalAwards=false
• 关键词: Extremal; Graph; Theory; Bootstrap; Percolation

Abstract for award of Balog DMS-0302804The proposed research areas are extremal graph theory and bootstrap percolation. They are not far from each other, as many probabilistic tools are used in the first one, and many combinatorial ideas are needed in the second one. The need of computer science and demands from applications where discrete models play more and more important roles, increase the importance of extremal graph theory and suggests an algorithmic point of view. For about forty years now, percolation theory has been an active area of research at the interface of probability theory, combinatorics and physics. Interest in various aspects of standard percolation remains high, including estimates of critical probabilities. Lately more and more variants of the standard percolation models have been studied, in particular, the family of processes known as bootstrap percolation. Recent applications arise from different aspects, for example from spatio-temporal dynamical systems. Computer experiments performed by physicists have suggested interesting non-trivial large-scale behavior, and many deep mathematical results have been proved about a number of models.The proposer is aiming to study the percolation process at the critical probability.The work of the proposer is an extension of Turan's Theorem into several directions. One direction is to describe graph families which do not contain certain induced subgraphs. The other is to study Turan type of questions on hypergraphs, in particular on triple systems, and to develop general tools like regularity and stability theorems.Bootstrap percolation, a member of the family of random cellular automata, is a process on graphs, where each site is open or closed with a certain probability, and these states are changing with time.Studying bootstrap percolation, the main aim of the proposer is to describe the phase transition, estimate the critical probability, and the size of the window around the critical probability. The plan is to prove that the transitions are sharp, and to investigate different models, whose understanding would be helpful in the applications.

Balog DMS-0302804 获奖摘要提议的研究领域是极值图论和自举渗透。它们相距并不远，因为第一个中使用了许多概率工具，而第二个中则需要许多组合思想。计算机科学的需求以及离散模型发挥越来越重要作用的应用的需求，增加了极值图论的重要性，并提出了算法的观点。大约四十年来，渗流理论一直是概率论、组合学和物理学交叉领域的一个活跃研究领域。人们对标准渗透的各个方面仍然很感兴趣，包括临界概率的估计。最近，越来越多的标准渗滤模型变体被研究，特别是被称为引导渗滤的过程家族。最近的应用来自不同的方面，例如时空动力系统。 物理学家进行的计算机实验提出了有趣的非平凡的大规模行为，并且许多深刻的数学结果已被证明关于许多模型。提议者的目的是研究临界概率下的渗滤过程。提议者的工作是图兰定理在几个方向上的延伸。一个方向是描述不包含某些导出子图的图族。另一个是研究关于超图的图兰类型问题，特别是关于三重系统的问题，并开发诸如正则性和稳定性定理之类的通用工具。Bootstrap 渗流是随机元胞自动机家族的一员，是图上的过程，其中每个站点以一定的概率打开或关闭，并且这些状态随着时间而变化。研究bootstrap percolation，提议者的主要目的是描述相变，估计临界概率，以及窗口的大小围绕临界概率。该计划是为了证明过渡是尖锐的，并研究不同的模型，其理解将有助于应用。