Random vicious walks and random matrices

A lock step walk is a one-dimensional integer lattice walk in discrete time . Suppose that initially there are infinitely many walkers on the nonnegati ve even integer sites. At each moment of time, every walker moves either to its left or to its right with equal probability. The only constraint is th at no two walkers can occupy the same site at the same time. Hence we descr ibe the walk as vicious. It is proved that as time tends to infinity, a cer tain limiting conditional distribution of the displacement of the leftmost walker is identical to the limiting distribution of the (scaled) largest ei genvalue of a random GOE matrix (GOE Tracy-Widom distribution). The proof i s based on the bijection between path configurations and semistandard Young tableaux established recently by Guttmann, Owczarek, and Viennot. The dist ribution of semistandard Young tableaux is analyzed using the Hankel determ inant expression for the probability obtained from the work of Rains and th e author. The asymptotics of the Hankel determinant are then obtained by ap plying the Deift-Zhou steepest-descent method to the Riemann-Hilbert proble m for the related orthogonal polynomials. (C) 2000 John Wiley & Sons, Inc.