Vicious Walkers and Hook Young Tableaux

TL;DR

This paper generalizes the vicious walker model by establishing a bijection with hook Young diagrams, and demonstrates that the scaled distribution of walker movements converges to the Tracy--Widom distribution, similar to eigenvalue distributions in random matrix theory.

Contribution

It introduces a novel bijection between non-intersecting walkers and hook Young diagrams, enabling analysis of movement probabilities and revealing Tracy--Widom distribution in the scaling limit.

Findings

Distribution of walker movements converges to Tracy--Widom distribution.

Bijection between path configurations and hook Young diagrams established.

Scaling limit matches the distribution of largest eigenvalues in GUE.

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 walker's movements. Applying the saddle point method, we reveal that the scaling limit gives the Tracy--Widom distribution, which is same with the limit distribution of the largest eigenvalues of the Gaussian unitary ensemble.