非衝突乱歩系・シューア関数・ランダム行列

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タイトル別名
  • Vicious Walker Model, Schur Function and Random Matrices
  • ヒショウトツ ランポケイ シューア カンスウ ランダム ギョウレツ

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We consider the vicious walker model, which was introduced by Michael Fisher in his Boltzmann medal lecture in 1983 as a mathematical model of wetting and melting phenomena. It is a system of particles performing noncolliding random walk in one dimension. Using nonintersecting property of the paths of vicious walkers and by elementary calculus of deter minants, we show that the Green function of the system is equal to the Schur function, which plays an important role in the representation theory of symmetric group, and its two kinds of determinantal expressions are derived. MacMahon conjecture, Bender-Knuth conjecture and Macdonald equality for the summations of Schur functions are discussed from the viewpoint of vicious walker model. By taking the diffusion scaling limit of the vicious walker model, a system of noncolliding Brownian particles is constructed and its relation to the distribution of eigenvalues of real symmetric random matrices is clarified.

収録刊行物

  • 応用数理

    応用数理 13 (4), 296-307, 2003

    一般社団法人 日本応用数理学会

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