短课程 An introduction to Kardar?Parisi?Zhang (KPZ) theory
报告人 Zhipeng Liu, University of Kansas
报告时间 6月 4日与5日 19:00-21:30， 6月 6日与7日 15:00-17:30
普适性概念与现象，如高斯普适类，广泛出现在复杂结构与随机系统中，在概率，数学物理与统计力学领域中扮演着核心角色。完全不同于涨落指数为1/2的高斯普适类， 涨落指数为1/3的KPZ普适类则刻画了随机增长界面， 一定的随机PDE, 随机矩阵，交互粒子系统与随机环境下聚合物等众多模型的长时间与大尺度统计行为。本课程内容如下：
In the past twenty years, much interest has been focused on a university class, the so-called KPZ universality class, named after Kardar, Parisi and Zhang, who, in 1986, introduced an equation (the KPZ equation) for the growth of random surfaces. It is believed this universality class includes many models in mathematical physics which have the same scaling rates for the time correlations, spatial correlations and the height fluctuations. The limiting fluctuations are also conjectured to be universal and only depend on the initial data.
In the talks, we will focus on the most developed model in the KPZ universality class, the totally asymmetric simple exclusion process (TASEP). We will discuss how this model was first solved by using the Robinson-Schensted-Knuth correspondence, a technique in combinatorics, and the orthogonal polynomials. The limiting one point distributions are given by the Tracy-Widom distributions, which were first introduced in the random matrix theory. Then we will consider how to derive the limiting spatial processes, which are called the Airy processes, for some classic initial conditions.
In the second part of the talks, we will consider TASEP on a periodic domain. We will show how to obtain the transition probability of periodic TASEP by using the so-called coordinate Bethe ansatz, which was developed in physics and introduced to TASEP by Schutz. Starting from the transition probability formula, we will discuss how to analyze the limiting one point distribution in the so-called relaxation time scale, which is the crossover regime between the Gaussian and KPZ universality. Finally we will discuss some recent results about the limiting space-time fluctuation fields of periodic TASEP in the relaxation time scale.