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北京概率论联合讨论班(9月27日)
2004-9-27 [学术报告]
报告题目 Vicious Walks, Schur Functions and Random Matrix Theory
报 告 人
Professor Makoto KATORI (Chuo University, Tokyo, Japan）
时 间 2004.9.27 5:00-6:00pm
地 点 中国科学院数学与系统科学研究院思源楼712
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 1984.In the present talk, I will try to explain interesting connections among representation theory, random matrix theory and probability 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. We show that 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 Karlin-McGregor formula in probability theory. Due to a 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 idetified with Dyson 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.
备 注 下次报告时间：10月11日
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发表于 2004-9-26 10:14
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