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Abstract : |
A lock step walk is a one-dimensional integer lattice walk in discrete time. Suppose that initially there are innitely many walkers on the non-negative 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 that no two walkers can occupy the same site at the same time. Hence we describe the walk as vicious. It is proved that as time tends to innity, a certain limiting conditional distribution of the displacement of the leftmost walker is identical to the limiting distribution of the (scaled) largest eigenvalue of a random GOE matrix (GOE TracyWidom distribution). The proof is based on the bijection between path congurations and semistandard Young tableaux established recently by Guttmann, Owczarek and Viennot. The distribution of semistandard Young tableaux is analyzed using the Hankel determinant expression for the probability obtained from the work of Rains and the author. The asymptotics of the Hankel determinant are then obtained by applying the Deift-Zhou steepest-descent method to the Riemann-Hilbert problem for the related orthogonal polynomials. 1, |
