Vicious walkers and hook Young tableaux

2002/09/30 by Kazuhiro Hikami, Takashi Imamura
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Generalization #Hook #Limit (mathematics) #Markov Chains and Monte Carlo Methods #Path (computing) #Point (geometry) #Random Matrices and Applications #Saddle point #Unitary state #cond-mat.stat-mech #math.PR #nlin.SI

paper · pdf · doi:10.1088/0305-4470/36/12/311

published as J. Phys. A: Math. Gen. 36, 3033--3048 (2003) · 23 pages, 5 figures

openalex publication_date 2003/03/13 · arxiv created 2003/03/14 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

We consider a generalization of the vicious walker model. Using a bijection map between the path configuration of the non-intersecting random walkers and the hook Young diagram, we compute the probability concerning the number of movements of the walker. Applying the saddle point method, we reveal that the scaling limit gives the Tracy–Widom distribution, which is the same with the limit distribution of the largest eigenvalues of the Gaussian unitary ensemble.

Citations

• On the distribution of the length of the longest increasing subsequence of random permutations
• Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem
• On the distributions of the lengths of the longest monotone subsequences in random words