Random Matrices And Determinantal Point Process

In mathematics,a determinantal point process is a stochastic point process,the probability distribution of which is characterized as a determinant of some founction.Such processes arise as important tools in random matrix theory,combinatorics,number theory and quantum physics.In addition,it provides a special and effective probability model of repulsion.In this paper,we will first study the basic properties of the determinantal point process and discuss some general methods for obtaining finite determinantal point processes.Secondly,we will specifically survey two models,random domino tilings of the Aztec diamond and a one-dimensional local random growth model,the corner growth model.We will also discuss in detail the relationship between them.Finally,we will learn the asymptotic properties of the Aztec diamond model and the random permutations model.Besides,we will discuss their relationship with the Tracy-Widom distribution and the Airy kernel point process.