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Nonintersecting Paths, Noncolliding Diffusion Processes and Representation Theory by Katori, Makoto; Tanemura, Hideki is a scholarly article available to read on EtoBox.

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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 Lindstr\"om-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 no

Author

Katori, Makoto; Tanemura, Hideki

Published

2005

Language

EN