  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 pre-asymptotic behaviors and not the correct asymptotic ones. In
addition, our method establishes a physical connection between vicious walkers
and random matrix theory.
