Exact distribution of the maximal height of p vicious walkers.
Using path-integral techniques, we compute exactly the distribution of the maximal height Hp of p nonintersecting Brownian walkers over a unit time interval in one dimension, both for excursions p watermelons with a wall, and bridges p watermelons without a wall, for all integer p>or=1. For large p, we show that <Hp> approximately square root 2p (excursions) whereas <Hp> approximately square root p (bridges). Our exact results prove that previous numerical experiments only measured the preasymptotic behaviors and not the correct asymptotic ones. In addition, our method establishes a physical connection between vicious walkers and random matrix theory.
Schehr
Grégory
G
Laboratoire de Physique Théorique (UMR du CNRS 8627), Université de Paris-Sud, 91405 Orsay Cedex, France.
Majumdar
Satya N
SN
Comtet
Alain
A
Randon-Furling
Julien
J