Random vicious walks and random matrices

Abstract

Lock step walker model is a one-dimensional integer lattice walker model in discrete time. Suppose that initially there are infinitely many walkers on the non-negative even integer sites. At each tick of time, each 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. It is proved that in the large time limit, a certain conditional probability of the displacement of the leftmost walker is identical to the limiting distribution of the properly scaled largest eigenvalue of a random GOE matrix (GOE Tracy-Widom distribution). The proof is based on the bijection between path configurations and semistandard Young tableaux established recently by Guttmann, Owczarek and Viennot. Statistics of semistandard Young tableaux is analyzed using the Hankel determinant expression for the probability from the work of Rains and the author. The asymptotics of the Hankel determinant is obtained by applying the Deift-Zhou steepest-descent method to the Riemann-Hilbert problem for the related orthogonal polynomials.