  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.
