Using path-integral techniques, we compute exactly the distribution of the maximal height $H_p$ 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 $p\ge 1$. For large $p$, we show that $⟨H_p⟩\sim \sqrt{2p}$ (excursions) whereas $⟨H_p⟩\sim \sqrt{p}$ (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.

• Received 3 July 2008

DOI:https://doi.org/10.1103/PhysRevLett.101.150601