Scaling Limit of Vicious Walkers, Schur Function, and Gaussian Random Matrix Ensemble

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We consider the diffusion scaling limit of the vicious walkers and derive the time-dependent spatial-distribution function of walkers. The dependence on initial configurations of walkers is generally described by using the symmetric polynomials called the Schur functions. In the special case in the scaling limit that all walkers are started from the origin, the probability density is simplified and it shows that the positions of walkers on the real axis at time one is identically distributed with the eigenvalues of random matrices in the Gaussian orthogonal ensemble. Since the diffusion scaling limit makes the vicious walkers converge to the nonintersecting Brownian motions in distribution, the present study will provide a new method to analyze intersection problems of Brownian motions in one-dimension.