非相交路径、非碰撞扩散过程与表示论
Nonintersecting Paths, Noncolliding Diffusion Processes and Representation Theory
摘要：
一维对称简单随机游动系统，其中没有一个步行者在给定的时间内遇到其他人，被称为恶性步行者模型。它是由迈克尔·费舍尔介绍的，并在他的玻尔兹曼奖章演讲中描述了该模型在各种润湿和熔化现象中的应用。在本报告中，我们解释有趣的联系之间的表示论，概率论，和随机矩阵理论使用这个简单的扩散粒子系统。$N$步行者的每一次恶意步行都由时空平面上的不相交格路径的$N$元组表示。在不相交的格点路径和半标准的杨氏图之间建立了一个简单的双射。基于这种双射和称为Schur函数的对称多项式的一些知识，我们可以给出恶性游动的分拆函数的行列式表达式，它被认为是概率论中的Karlin-McGregor公式（或计数组合学中的Lindström-Gessel-Viennot公式）的特殊情况。利用Schur函数的一个基本性质，我们可以取恶性游动的扩散标度极限，定义一个非碰撞的布朗粒子系统。该扩散过程求解了带有漂移项的随机微分方程，漂移项作为与粒子间距离的倒数成正比的两体排斥力，因此它与Dyson的布朗运动模型一致。换句话说，所得到的非碰撞布朗粒子系统与埃尔米特矩阵值过程的特征值过程在分布上等价。
英文摘要：
The system of one-dimensional symmetric simple random walks, in which none of walkers have met others in a given time period, is called the vicious walker model. It was introduced by Michael Fisher and applications of the model to various wetting and melting phenomena were described in his Boltzmann medal lecture. In the present report, we explain interesting connections among representation theory, probability theory, and random matrix theory using this simple diffusion particle system. Each vicious walk of $N$ walkers is represented by an $N$-tuple of nonintersecting lattice paths on the spatio-temporal plane. There is established a simple bijection between nonintersecting lattice paths and semistandard Young tableaux. Based on this bijection and some knowledge of symmetric polynomials called the Schur functions, we can give a determinantal expression to the partition function of vicious walks, which is regarded as a special case of the Karlin-McGregor formula in the probability theory (or the Lindström-Gessel-Viennot formula in the enumerative combinatorics). Due to a basic property of Schur function, we can take the diffusion scaling limit of the vicious walks and define a noncolliding system of Brownian particles. This diffusion process solves the stochastic differential equations with the drift terms acting as the repulsive two-body forces proportional to the inverse of distances between particles, and thus it is identified with Dyson's Brownian motion model. In other words, the obtained noncolliding system of Brownian particles is equivalent in distribution with the eigenvalue process of a Hermitian matrix-valued process.
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