Random vicious walks and random matrices

TL;DR

This paper establishes a connection between a one-dimensional random walk model with non-colliding walkers and the distribution of the largest eigenvalue of a GOE random matrix, using combinatorial and analytical techniques.

Contribution

It proves a new link between non-colliding random walks and random matrix theory via Young tableaux and Hankel determinants.

Findings

Displacement distribution matches GOE Tracy-Widom distribution in large time limit.

Bijection between path configurations and semistandard Young tableaux is utilized.

Asymptotics derived using Riemann-Hilbert problem and steepest-descent method.

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.