Exact distribution of the maximal height of p vicious walkers

Details Exact distribution of the maximal height of p vicious walkers Exact distribution of the maximal height of p vicious walkers

2008-07-03

arxiv.org/abs/0807.0522v2

Phys. Rev. Lett. 101, 150601 (2008)

Physics

Condensed Matter

Statistical Mechanics

4 pages, 2 figures; minor modifications, added references; published version

Scientific paper

10.1103/PhysRevLett.101.150601

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.

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