This project is about the longest increasing subsequence problems in combinatorics, which can also be interpreted as last passage percolation problems in probability/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 polymers and so-called ``vicious'' random walks. Recent progresses show that there are interesting 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.
