  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.
