Last Passage Percolation and Random Matrix

PI: Jinho Baik，普林斯顿大学dms -0208557这个项目是关于组合数学中最长增加子序列问题，也可以解释为概率/统计力学中的最后一段渗透问题。我们考虑二维晶格中每个晶格点随机分配时间的最大通过时间。最基本的兴趣是当晶格变大时，最大通过时间的概率性质。其他类似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“恶性”随机漫步。最近的进展表明，这一领域与随机矩阵理论有着有趣的联系。近50年来，随机矩阵理论一直是数学和物理学研究的一个活跃领域。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。这个项目是在这个方向上的进一步工作。从一个更大的框架来看，这个项目的主题可以被看作是当系统规模变大时，在随机环境中最大化过程的基本性质的工作。这样的过程可以是晶体的生长，纸张燃烧的火锋的形状，或者是计算机或蜂窝网络的最快通道。这些问题也是统计学、统计物理和工程学的主题。该项目的主要目标之一是研究当模型尺寸变大时一类系统的各种通用特性，这些特性与模型的微观结构无关。

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.