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Random vicious walks and random matrices
Author(s) -
Baik Jinho
Publication year - 2000
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/1097-0312(200011)53:11<1385::aid-cpa3>3.0.co;2-t
Subject(s) - mathematics , bijection , random walk , combinatorics , young tableau , integer lattice , random matrix , distribution (mathematics) , eigenvalues and eigenvectors , mathematical analysis , statistics , physics , quantum mechanics , half integer
A lock step walk is a one‐dimensional integer lattice walk in discrete time. Suppose that initially there are infinitely many walkers on the nonnegative 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 infinity, 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 Tracy‐Widom distribution). The proof is based on the bijection between path configurations 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. © 2000 John Wiley & Sons, Inc.