Last Passage Percolation and Random Matrix

主要研究者：Jinho Baik，Princeton UniversityDMS-0208557本项目是关于组合数学中的最长递增子序列问题，它也可以被解释为概率/统计力学中的最后一次通过渗流问题。我们考虑一个在每个格点上随机分配时间的二维格点上的最大通过时间。基本的兴趣是最大通过时间的概率性质，随着格的大小变得很大。其他等价形式的问题包括纸牌游戏，二维随机增长模型，相互作用粒子系统，排队论，定向聚合物和所谓的“恶意”随机漫步。近年来的研究表明，该领域与随机矩阵理论之间存在着有趣的联系。随机矩阵理论是近50年来数学和物理学中的一个活跃的研究领域。我们想更多地了解这种联系，并进一步研究这些看似不同的领域之间的关系。本课题是在这一方向上的进一步工作，从更大的框架上看，本课题可以看作是研究随机环境下系统规模变大时极大化过程的基本性质的工作。这种过程可以是晶体的生长，纸张燃烧的火锋形状，或者计算机或蜂窝网络中的最快通道。这些问题也是统计学、统计物理学和工程学的主题。本项目的主要目标之一是研究一类系统在模型规模变大时的各种与模型微观结构无关的普适性质。

PI: Jinho Baik, Princeton UniversityDMS-0208557This project is about the longest increasing subsequence problems incombinatorics, which can also be interpreted as last passage percolation problems inprobability/statistical mechanics. We consider a maximal passage time in a 2-dimensional lattice with a random time assigned at each lattice site. The basic interest is the probabilistic properties of the maximal passage time as the size of lattice becomes large. Other equivalent forms of questions include a card game, random growth models in 2-dimension, interacting particle systems, queuing theory, directed polymersand so-called ``vicious'' random walks. Recent progresses show that there areinteresting connections of this field to the random matrix theory. Random matrix theory has been an active field of research in both mathematics and physics for last 50 years. We would like to understand more on this connection and also investigate further relations between these seemingly different fields. This project is a further work in this direction.From a greater framework, the subject of this project can be regarded as a work on the basic properties of a maximization process in a random environment when the size of system becomes large. Such process could be growth of crystal, shape of fire front of paper burning, or fastest passage in a computer or cellular network. These problems are also subjects in statistics, statistical physics, and engineering. One of the main goal of this project is an investigation of various universal properties, which are independent of the microstructure of the models, of a class of systems when the size of model becomes large.